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Course Lectures
Abstracts of Participants
Fellows Project Reports
WHOI9103
199 Sumer Study Prgr
in
Gephysical Fluid Dycs
Stellar Fluid Dynamics
by
Rick Salmon, Director,
Edited by
Barbara Ewing DeRemer
Woods Hole Oceanographic Institution
Woods Hole, Massachusetts 02543
Octber 1990
Tecca Report
Funding was provided by the National Science Foundation
through Grant No. OCE 8901012.
Reproduction in whole or in part is permitted for any purpose ofthe United States
Government. This report should be cited as Woods Hole Oceanog. Inst. Tech. Rept.,
WHOI9103.
Approved for public release; distribution unlimited.
Apprved fo Ditrbution:
~ .At;t7 John W. ar~
Aste Dir fo Education
PREFACE
The 1990 program on "Stellar Fluid Dynamics" marked our deepest penetration into
astrophysical fluid dynamics since 1963. Introductory lectus by Ed Spiegel and Jean
Paul Zahn, with a supplement on solar MH by Steve Chidrss, paved the way for more
specialized lectures on solar oscilauons (Balfort), radiauvelydrven stellar winds
(Owocki), and neutron stas (Arons). Norm Lebovitz gave us a beauuful synthesis of the
theory of poly tropes, and Leon Golub chalenged our theoreucal impulses with the latest x
ray images of the solar corona. There was considerable focus on stellar convecuon (Zahn,
Stein, Ghosal), and on flows with strong magneuc fields.
As usual, the lecture subjects ranged considerably beyond the special topic of the
summer, with GFD fiing its traduonal role as a clearghouse for new ideas among the
many fields concerned with rotaung, diferenuallyheated fluids. Some of these topics
(symmetr groups, wavelets, negauve energy modes) seem about to burst upon the fluid
mechanics scene, while others, such as the flow on a theedimensional sphere in four
diensional space, may yet be a few years away.
This was a year in which many familiar faces were absent or tardy, and some new ones
appeared. Former fellows Andrew Gilbert and Andrew Woods joined the staff, and
newcomer Phil Morrson, who gave us added breadth in the dicuon of theoreucal plasma
physics, seemed to enjoy discussing everythng with everybody. Our nie fellows (from
the USA, Canada, England, Germany, and Irland) came to us frm diverse backgrunds
in astronomy, mathemaucs, physics, and fluid dynamics; all seemed to thrive in the
interdiscipliar atmosphere of Walsh Cottage.
1990 was also the year that computers came to Walsh Cottage  with a vengeance.
Thans to a generous gift from the Mellon Foundation, we were able to buy or borrow two
Sun workstations, a laser priter, and two personal computers. After a bumpy star that
nearly overwhelmed the dictor, we were rescued by the computer expertse and generous
assistance of Glenn Fled and Steve Meacham.
Once again, we gratefully acknowledge the support of the National Science Foundation
and the Offce of Naval Research, and the capable assistace of Jake Peiron and his staff
in the Educauon Office of the Woos Hole Oceanogrphic Instituuon. Special than go to
Barbara EwingDeRemer, our administrtive assistat and editor, who kept things running
smoothly in the cottage.
Rick Salmon, 1990 diector
i
1990 GFD Participants
The Fellows
Neil Balmfort UK University of Cambridge
George Bell USA Woods Hole Oceanographic Institution
ColmCile Caulfield Eir University of Cambridge
Brian Chaboyer Canada Yale University
Richard Kerswell UK Massachusetts Institute ofT echnology
Stefan Linz Germany Universitat des Saarlaudes
Nathan Platt USA Brown University
Wendy Welch USA University of Washington
Eric Won USA Columbia University
The Staff and Visitors
James L. Anderson Stevens Institute of Technology
University of California, Berkeley
Centro de Investigaci��n Cientlica y de Educaci��n Superior
New York University
Massachusetts Institute of Technology
University of Oxford
Columbia University
University of Cambridge
r.
~!,i
E
l'!
Jonathan Arons
Antoine Bad
Stephen Childress
Glenn Fled
Andrew Fowler
Sandip Ghosal
Andrew Gilbert
Dan Givoli Technion
Jackson Herrg
Rainer Hollerbach
Harvard University
National Center for Atmspheric Research
University of California, San Diego
Michigan Technical University
Stanford University
Leon Golub
Glenn Iedey
Joseph B. Keller
ii
Norman Lebovitz
Albert Libchaber
Wilem Mals
Philip J. Morrson
Stan Owocki
Michael Proctor
David Rose
Robert Rosner
Rick Salmon
Engelbert Schuckig
Andrew Soward
Edward Spiegel
Robert Stein
Melvin Stern
Olivier Thual
George Veronis
John Weiss
Nigel Weiss
John A. Whitehead
Andrew Woods
JeanPaul Zahn
University of Chicago
University of Chicago
Massachusetts Institute of Technology
University of Texas
University of Delaware
University of Cambridge
New York University
University of Chicago
University of California, San Diego
New York University
The University, Newcastle upon Tyne
Columbia University
Michigan State University
Florida State University
National Center for Atmospheric Research
Yale University
Aware Inc.
University of Cambridge
Woods Hole Oceanographic Institution
University of California, San Diego
Observatoire MidiPyr��n��es, Toulouse
iii
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CONTENS PAGE
I WAIH CoAGE DIAy........................................ 6
II PHOT: OUR PRNCIPAL LECT,
JeanPaul Zahn & Edward spiegel................... 11
III LECTS
Stellar Fluid Dynamics I: E. spiegel. . . . . . . . . . . . . . . . . . . . . .12
Stellar Fluid Dynamics II: E. Spiegel. . . . . . . . . . . . . . . . . . . . . 22
Stellar Fluid Dynamics III: E. Spiegel. . . . . . . . . . . . . . . . . . . . 30
Stellar Fluid Dynamics IV: E. Spiegel. . . . . . . . . . . . . . . . . . . . . 43
Stellar Fluid Dynamics V: Rotating Stars, J. P. Zahn. . . . . . .52
Stellar Fluid Dynamics VI: Rotating Stars, J.P. Z ahn. . . . . . 7 0
Stellar Fluid Dynamics VII: Rotating Stars, J. P. Z ahn. . . . . 82
Stellar Fluid Dynamics VI I I : Rotating Stars, J.P. Z ahn. . . . 88
Stellar Fluid Dynamics IX: Rotating Stars, J. P. Z ahn. . . . . 100 "i~.
¡,
"Stellar Fluid Dynamics X: E. Spiegel. . . . . . . . . . . . . . . . . . . . . i 12 T
IV SPECIAL LECTS BY STEPH CHILDRESS:
An Introduction to Solar MHD (Summary)............ 125
V PHOT: Observing volcanic eruptions on the porch
of Walsh cottage.................................. 130
i
VI ABTRCTS OF SEMNAR
Relativistic FluctuationDissipation Theorems , Radiative
Hydrodynamics and Galaxy FormationJa es Anderson 131
Solar Acoustic OscillationsN.J. Balmforth 133
Inviscid Models Associated with Vortex ReconnectionStephen Childress 136
Rossby Wave Radiation from Strong EddiesGlenn R. Flierl 139
Convection and ChaosAndrew C. Fowler 141
On Thermonuclear ConvectionSandip Ghosal 142
On Fast Dyamo Action in Steady Chaotic Flow
A.D.Gilbert, S. Childress,and U. Frisch 144
Arificial Boundary Conditions for Wave ,Problem
in Unbounded DomainsDan Givoli 145
Very High Resolution Solar Xray ImagingLeon Golub 146
Coherent Structures and statistical Theory of TurbulenceJackson Herring 151
A modalo¿.4 Dyamo in the Limit of Asymptotically
Small ViscosityRainer Hollerbach 154
2
VI ABTRCTS OF SEMNAR (continued)
Dyamics of Loalized Structures with Galilean InvarianceChristian Elphick, G.R. Ierley,Oded Regev and E.A. Spiegel 156
Fast Reaction, Slow Diffusion, cue Shortening and
Harmonic MapslJoseph B. Keller 157
Polytropes and Their PerturbationsNorman R. Lebovitz 158
Interface Dyamics: Playing with Symetries
A. Libchaber 159
Some Aspects of Convection in Binary Fluid MixturesStefan J. Linz 160
Hydromagnetic Instability Due to Elliptical Flow
Dav id W. Hughes andWillem V.R. Malkus 162
The Free Energ "Principle", Negative Energy Modes
and StabilityP. J. Morrison 164
Radiati vely Driven Stellar WindsStan Owocki 166
Chaos and Noise in Dyamical System with Slow
Invariant SubspacesMichael R. E. Proctor 167
Calculating Transient Coronal LopsDavid W. Rose 168
Blocking a Barotropic Shear FlowMelvin Stern 169
3
VI ABTRCTS OF SEMINAR (continued)
A Fluid Mechanicist' s Introduction to Lie Symetry
Groups and Partial Differential EquationsRick Salmon ' 170
Vortices in Tight Emrace
E. L. Schucking andE. A. Spiegel 177
0( w Galactic DyamosA. M. Soward 185
The Catastrophe Structure of Thermohaline ConvectionOlivier Thual 186
Simulations of Solar ConvectionBob stein and Ake Nordlund 187
Instability of Flow with TemperatureDependent Viscosity:
A Model of Magma Dyamics
J . A. Whï tehead andKarl R. Helfrich 188
A Long Laoratory Salt FingerGeorge Veronis 189
Applications of Compactly Supported Wavelets to the
Numerical Solution of Partial Differential EquationsJohn Weiss 191
Traveling Waves and Oscillations in Compressible
KagnetoconvectionNigel Weiss 195
Kodeling Kesogranules and Exploders On the SunNigel Weiss 197
4
VI ABTRCTS OF SEMNAR (continued)
Fluid Mechanics and Meltinq
Andrew W. Woods 200
Convective Penetration in Stellar Interiors
JeanPaul Zahn 203
VII FELLWS PROJCT REPORTS
The Structure and stability of Rapidly
Rotatinq PolytropesNeil J. Balmforth 212
The Nonlinear Evolution of a Perturbed
Axisymetric EddyGeorge I. Bell 232
The Rise and Fall of Buoyant PlumesC. P. Caulfield 250
Transport of a Chemical in Stellar Radiative ZonesBr ian Chaboyer 267
Maqnetic Flux Tues and ConvectionRichard Kerswell 277
Naturally Driven Dispersion in Tilted Porous LayersStefan J. Linz 299
Beavior of a Fifth Order System of ODE's
with I.ntermittencyNathan Platt 320
Difussion in Poiseuille FlowWendell T. Welch 336
Classification of Similarity Solutions of
the TwoDimensional Convection EquationsEric C. Won 355
5
¡;
I Walsh Cottage Diary
June 18 Monday Arvals. Opening ceremonies.
June 19 Tuesday 10:00 Ed Spiegel
Stellar Fluid Dynacs I
June 20 Wednesday 10:00 Ed Spiegel
Stellar Fluid Dynamics II
14:00 Steve Childrss
Intoduction to Solar MHD (Part 1)
June 21 Thursday 10:00 Ed Spiegel
Stellar Fluid Dynamics ILL
14:00 Steve Childrss
Introduction to Solar MHD (Part 2)
June 22 Friday 10:00 Ed Spiegel
Stellar Fluid Dynamics IV
June 25 Monday 10:00 JeanPaul Za
Stellar Fluid Dynaics V
June 26 Tuesday 10:00 JeanPaul Za
Stellar Fluid Dynamics VI
13:30 Jonathan Arons
Radio Pulsars
June 27 Wedesday 10:00 JeanPaul Za
Stellar Fluid Dynaics VII
13:30 Jonathan Arons
Radiation Gas Dynamics of Accretion
onto Neutron Stars
17:30 GFD 11, Fisheries 17
June 28 Thurday 10:00 JeanPaul Za
Stellar Fluid Dynamics VIII
14:30 Andrew Fowler
Convection and chaos
6
June 29 Friday 10:00 JeanPaul Za
Stellar Fluid Dynamics IX
July 2 Monday 10:00 Ed Spiegel
Stellar Fluid Dynaics X
July 3 Tuesday 10:00 Neil Balfort
Solar Oscillations
17:30 GFD 8, New Alchemy 22
July 4 Wedesday Holiday
July 5 Thursday 10:00 Stan Owocki
Radiativelydriven Stellar Wind (Part 1)
July 6 Friday 10:00 Stan Owocki
Radiativelydriven Stellar Wind (Part 2)
July 9 Monday 10:00 Norman Lebovitz
Poly tropes and their Pernubations
July 10 Tuesday 10:00 Andrw Gilber
On Fast Dynamo Action in Stead Chatic Flow
July 11 Wednesday 10:00 Joe Keller
Fast Reaction and Slow Difion
13:30 Rick Salon
Application of Lie Symmetry Groups
to Partal Diferental Equtions (Pan 1)
July 12 Thursday 10:00 David Rose
Calculating Transient Coronal Loops
17:30 GFD 7, Marne Policy 9
July 13 Friday 10:00 Rick Salon
Application of Lie Symmetry Groups
to Paral Diferental Equtions (Pan 2)
July 16 Monday 10:00 Melvin Stern
Blocking a Barotropic Shear Flow
July 17 Tuesday 10:00 Andrew Woos
Fluid Mechanics and Melting
7
July 18 Wedesday 10:00 Jack Whtehead
Instabilty of Flow in a Slot
with Temperaturedependent Viscosity
17:30 GFD 2, Geology and Geophysics 15
July 19 Thursday 10:00 Nigel Weiss
Travellng Waves and Oscilations in Compressible
Mag netoconvection
July 20 Friday 10:00 Stefan Linz
Convection in Binary Fluid Mixtures
July 23 Monday 10:00 Leon Golub
Highresolution Observations of the Solar
Magnetic Field
July 24 Tuesday 10:00 Jack Herrg
Coherent Structues and Statistical
Turbulence Theory
July 25 Wedesday 10:00 Nigel Weiss
Modellng Mesogranules and Exploders
on the Sun
July 26 Thursday 10:00 JeanPaul Za
Penetrative Convection in Stars
17:50 GFD 5, Ocean Engineerig 14
July 27 Friday 10:00 Jim Anderson
, Fluctutiondissipation in Radiative Transport
July 30 Monday 10:00 Phil Morrson
The Free Energy Principle, Negative Energy Modes,
and Stabilty (Part 1)
14:00 Dan Givoli
Artifcial Boundary Conditions for Wave Problems
in Unbounded Domains
July 31 Tuesday 10:00 Rainer Hollerbach
An (l2 Dynamo in the Limit of
Asymptotically Small Viscosity
August 1 Wedesday 10:00 Andrw Soward
a(i Galactic Dynamos
August 2 Thurday 10:00 Michael Proctor
The Effect of Noise on Dynamical Systems
with Slow Invariant Subs paces
8
17:30 GFD 35, Administration 11
August 3 Friday 10:00 Phil Morrson
The Free Energy Principle, Negative Energy Modes,
and Stabilty (Part 2)
August 6 Monday 10:00 Albert Libchaber
Interface Dynaics
August 7 Tuesday 10:00 Steve Childress
Inviscid Problems Related to Vortex Reconnection
August 8 Wedesday 10:00 Engelbert Sçhuckig
Two Vortices in Tight Embrace
August 9 Thursday 10:00 Sandip Ghosal
On Thermonuclear Convection
17:50 GFD 19, Facilties 15
August 10 Friday 10:00 Olivier Thual
The Catastrophe Structue of
Thermhaline Convection
August 13 Monday 10:00 Glenn Ierley
Dynamics of Localized Structures with
Galiean I nvariance
August 14 Tuesday 10:00 John Weiss
Application of Compactly Supported Wavelets
to Partial Diferental Equations
August 15 Wednesday 10:00 Glenn Fled
Rossby Wave Radiationjrom Strong Vortices
August 16 Thursday 10:00 George Veronis
A Long Laboratory Salt Finger
17:30 GFD 20, Biology 11
August 17 Friday 10:00 Robert Stein
Simulatiions of Solar Convection
August 20 Monday 10:00 Colmcile Caulfeld
The Rise and Fall of Buoyant Plumes
11:15 Wendy Welch
Difion in Poiseuille Flow
13:30 Bryan Chaboyer
Transport in Stellar Radiative Zones
9
August 21 Tuesday 10:00 George Bell
The Stabilty of Axisymmetric Vortices
11:15 Stefan Unz
Convection in Tilted Porous Laers
13:30 Nathan Platt
Behaior of a Fifhorder System of Ordinary
Diferential Equations with Intermittency
15:00 Fellows 39, Staff 23
August 22 Wednesday 10:00 Neil Balmfort
The Structure and Stability of Rapidly Rotating
Poly tropes
11:15 Richard Kerswell
Magnetic Flux Tubes and Convection
13:30 Eric Won
Symmetries and Similarity Solutions of the
TwoDimensional Convection Equations
August 24 Friday FarewelL.
10
OUR PRINCIPAL LECTRS,
Jean Paul Zahn and Edward spiegel
11
STELLAR FLUID DYNAMICS
E.A. Spiegel
Astronomy Department
Columbia Uiuversity
New York, NY 10027
Lecture I
1 Astrophysical Fluid Dynamics
Astrophysics is a branch of astronomy, so when I say "astronomers" in these lectures, I
include astrophysicists. In fact, astronomy is mostly astrophysics so I think that this is
a good characterization. On the forefront of astronomy, the job is to isolate the relevant
physics in different kinds of celestial objects with a view to modeling. It is only after
some agreement about the nature of the objects has been reached that careful studies are
warranted. In good astrophysics, physical arguments, often based on rough estimates, are
the key to success.
As more phenomena in astronomical objects are recognized t�� be fluid dynamcal,
the need for intuitive understanding of such processes grows. In this school, we have
practiced the development of understanding fluid processes by careful analysis of simple
models. Often, over the decades, practitioners have felt that such studies may not be
relevant to their disciplines, but more and more, the language of G.F.D. inquiry is heard
in discussions by these same practitioners. The reason that these words have acquired
increased meanng and comprehensibility comes from their careful elucidation in simple
pilot studies. That, at any rate, is the credo of Walsh College.
Astronomers have studied what they cal cosmical gas (or aero) dynamcs for years.
That is astrophysics in which the physics is fluid dynamcs. What we shal discuss here is
to be caled Astrophysical Fluid Dynamics to emphasize that the aim is, like that of GFD,
to extract relevant models that are suffciently simple to be analyzed in detai by whatever
means are needed. The results of such analysis are to be used to inform astrophysical
studies but, as the name implies, our subject here is fluid dynamics and it is an end in
itself. No apologies need be made for that.
Modem developments in the mathematical sciences are also afecting the progress in
astronomy. In a$tromathematic$, the object is to paralel the work in astrophysics by using
general mathematical ideas to isolate mathematical processes that elucidate the behavior
of a cosmic object. For example, if we suspect that a gala��tic pattern is engendered by
an instabilty, we need not decide exactly which instabilty it is in order to begin to write
down an equation to describe such a pattern. If we can isolate a suitable equation for the
purose by qualtative mathematical arguments, we can later go back and argue about
which among possible physical processes is responsible for the instabilty. A.F.D. is a
good source of examples of this approach.
12
In the lectures sketched here, the concentration is on stellar fluid dynamics. That
fits in nicely with the Astro in A.F.D. There wil be no attempt to provide a systematic
course on the background astrophysics, though some introductory material wil be given
as the need seems to arise. The main aim is to introduce some fluid dynamical problems
that seem peculiarly stellar. Afterwards, the lectures of Jean~Paul Zahn about rotating
stars wil restore sweet reason.
2 Large Scale Structures
Our vision of the the universe has gone through many revisions over the past few mil
lenia. In the decades of the forties and fifties it was generaly assumed that the universe
is homogenous and isotropic. That people were uncomfortable with this idealization is
attested by the name they gave it: the Cosmological Principle. Today, though wider pos
sibilties are considered and the situation seems more confused, there is real improvement
as the outlook has become more Coperiucan. We are not going to discuss cosmology here,
but it seems worth saying just a few words about some of the salient features to place the
stellar situation in context. .
It now seems reasonably certain that the visible matter in the universe constitutes only
a few percent of the total mass. This visible mass is concentrated in galaxes that appear
to be arranged in a hierarchical distribution. A simple way to think about the distribution
of galaxes is to imagine that they lie on a fractal set, as has long been conjectured (see
the books of Peebles or of Mandelbrot). Such a set has lacunae, voids where there are no
galaxes, and the galaxian distribution is rather fiamentar. The dimension of the fractal
is a matter of debate (see Thieberger, et al., in "The Ubiquity of Chaos", AAAS 1990, S.
Krasner, ed.). What matters is that the galaxes appear to be markers in ��. cosmic flow
about which our only knowledge is that locally (in space and time) it is an expansion.
What happens elsewhere in spacetime is not known except by speculations that form a
sort of mathematical theology of some char. We do not know the nature of the invisible
matter.
The galaxes themselves consist of stars, gas and dust in differing mitures. They have
diverse morphologies that are matched to these melanges. Our galaXy, the Milky way, has
a mass of approximately 1012 Me of visible matter where Me is the sun's mass:: 2 x 1033
gm. Much of this is in a disk that is suspected of being embedded in an invisible halo of
ten or a hundred times this mass. We are not going to worry about the cosmic flow nor
about the circulations in galaxes. Our attention is to be focussed on the fluid dynamcs
within stars. Before we get to that, let us review briefly the fluid dynamcs itself.
.
'.
3 Fluid Dynamics
The work usualy discussed in this course has to do with GFD, and that is only a special
case of AFD. So although many in the audience know fluid dynamcs, we stil ought to
13
set down a simple version of it in a somewhat general way. It wil not be necessary,
however, to derive the Boussinesq approximation. Stil, we ought to mention a question
that does occupy astrophysicists wanting to think about cosmic fluid dynamics. Is a simple
continuum a vald description of the astrophysical plasmas with which they generally have
to deal? There is a lot of discusion of this point and it in vol ves comparing mean free paths
to the scales of motion. In fact, even when those comparisons d�� not seem to support the
use of the fluid picture, it has o��t~n been used with the defence that collective interaction
resulting from long range em and gravitational forces makes for fluid behavior. Let us
skip all that dreary stuff and simply adopt the fluid model to describe motions in stars.
Another point that must be mentioned in the interests of respectablity is that, in
most astrophysical circumstances, length scales are so large that the Reynolds number is
generally astronomical. Hence turbulent flows are the rule in this subject. In the face
of that remark, s'ensible people would change subjects, but we are not among them and
have been here for thirty years and more trying to work around this dilemma that natural
scientists face in thinking about fluid motions. But this problem may make it clear why
we are sometimes schematic in AFD.
So consider this simple equation for the momentum balance of a fluid:
'du =F
dt
where the velocity field u depends on x and t, d/dt = at + u. V is the material derivative
and F is a body force per unit mass. For the simplest case, we take F to be conservative.
Then we have a special case of inviscid fluid dynamcs with F =  VV.
Consider some simple models for V:
(1) V = V(x,t)  specified potential only;
(2) V = V(x, t) + h(p)  includes local coupling of the fluid with itself (h is a point
function of p);
(3) V  V(x, to) + h(p) + Hlpj  with both local and nonlocal couplings of the density
field (H is a functi()nal of p).
For a conservative force, as assumed, Kelvin's Circulation Theorem gives the perma
nence of irrotationalty. Hence, if the flow is initialy irrotational, there exists a function
ø such that u = V ø~ It follows that
1
øt + (V ø)2 + V = O.2 '
This is Bernoul's Theorem in fluid mechanics and the HamiltonJ acob equation in clas
sical mechancs.
We shal also assume the kinematic condition that mass is conserved:
Pt + v. (pu) = O.
For an irrotational, barotropic fluid we have H = O. Then, on letting p  R2 and
"p = Rei';, we can wIite the equations of motion and continuity concisely as1 V2Ri"pt= 2V2"p + (V + h(p) + 2R )"p.
14
Other interesting cases exist, such as the choice H¡pJ =  V2 RI2R, which leads to the
nonlinear Schrödinger equation
i1/t = _~V21/ + ¡h(I1/12) + V(x, t)J 1/.
2
4 Expanding Coordinate System
Most people in this audience know all about transforiing to rotating coordinates, so
instead of going back over this, let us look at another transformation that is common
in astrophysics  the transformation to expanding coordinates. This is a device that
has been much used in cosmology and also in the study of fluid dynaiics in pulsating
stars. Flow in an expanding system is a pleasant introduction to AFD since this choice
of coordinates leads to fictitious forces in complete analogy to those of rotating fluids.
Inspired by Hubble's law, we set u = H(t) x withdp ,
dt = pV . u
and obtain, for a homogeneous system,
p
 = 3H.
p
Hence, H = De(ln(pol p)t), where De = dldt. We introduce the scale factor R(t) defined
by R3(t) = pi Po where Po is the density of a fiducial epoch. Set x = Ry and let v = Ry,
where the dot means time derivative. Then it is possible to show that v = x  Hx
(where H = Ill R) and v R( t) = const, which is the analogue of the conservation of
anguar momentum. That is, R(t),increases with time, causing IPI = ¡mvl to decrease.
Qualtatively, the image is that the wavelength conjugate to P increases. The analogous
effect for photons is called the cosmological red shift.
The possibiltes for R(t) (in the simplest Newtonian models) are shown in figure 1
for various energy densities. Against this background, we want to write the equations of
motion. However, we shal follow the order of the lectures as they were given and defer
those equations to Lecture II since the questions at this point caused a nonnegligible
deflection in the direction of Lecture 1.
5 The Sun
"The sun is round like a bal or an orange," is what first graders were taught fifty years'
ago. Later, it was necessary to make this statement more quantitiative, but we need not
go into that. Let us consider a sphericaly symmetric hydrostatic sun. Then an estimate
of the central temperature can be derived:
dp
 = pg
dr
15
R continual
expansion
Figure 1: Scale factor as a function of time.
where 9 = GM/r2 and, for a perfect gas, with 'R as gas constant and ¡. as mean molecular
weight, we have
p'RTp=.
¡.
Hence in order of magnitude, we have p '" GMp/ R, where R is the star's radius, and so
p'RT '" GMp =* 'RT '" GM¡.,¡. R R
up to factors of order unity. For the sun this gives T '" 107 K at depth.
Then from the measured rate of eiission of luminous energy per unit time  the lu
minosity  we can determine the thermal time of the star, the socaled KelvinHelmholtz
time scale. From the balances already given, we see that the pGtential energy has the
magnitude GM2 / R and that this is comparable to the thermal energy in the star. Hence
the thermal time scale may be estimated as
GM2TKH= RL .
For the sun, L0 ~ 4 x 1033 ergs/sec, so TKH= 30 millon years. Since the time scale for
adjusting any hydrostatic imbalance is short compared to TKH the acoustic travel time
across a solar radius is about an hour  the assumption of hydrostatic balance is well'
justified.
Simple considerations explain the global (dare we' say structural?) stabilty of the sun.
The heat source is provided by nuclear reactions in the core where hydrogen fuses into
16
helium. For particles to have suffcient energy to overcome Coulomb repulsion so that the
nuclei can interact strongly, the temperature must be high enough. If the temperature
increases, the energy production by nuclear fusion correspondingly increases and the star
expands and cools. A temperature is reached at wruch the energy loss matches that
produced in the core. However, at the small scales, there may be instabilties that do not
profoundly modify the mean structure.
6 Radiation
In hot stars, we must allow for the effects of the pressure of the radiation, P..ad = laT4,
where a is a constant. The radiation pressure is comparable to the gas pressure when
aT4 '" 'RpT. This occurs for T3 / p '" 'R/ a. On the other hand, we have already seen that
'RT '" GM/ Rand p '" M/ R3. We combine these estimates and find that the mass at
which radiation pressure and gas pressure are of the same order is M '" 40Me. Above
this mass, radiation pressure dominates, but we never see any stars with masses much in
excess of about 60Me. Some think that trus is a result of vibrational instabilties related
to radiation pressure.
Radiation coming up from within a star may be compared to a fluid ilowing through
a porous medium. The radiation produces a force of levitation on the stellar material
and, in the hot stars, this force may compete with the gravitational force. The situation
resembles that in a fluidized bed. There a fluid flows through a porous medium made of
particles that are not attached to each other. When the drag per particle equals the mean
weight of the particles the bed of particles expands and turns into a fluid. In the stellar
case, the porous medium is already a fluid (a gas or plasma) but there is also a critical
case where the levitating force of the out flowing radiation compensates the weight of the
particles.
The bolometric luminosity of a star, L, is its total rate of emission of light. When
this luminosity is in excess of a cerain critical value, the Eddington luminosity, LE, the
radiative force per unt mass of stellar material exceeds the gravitational force in the
outer layers. It is generaly presumed that the material in a star' that found itself in
this situation would be blown away. This does not happen when a bed of particles is
first fluidized since the drag force per particle diminishes when the density of paricles
decreass. In the stellar case, normal stars with masses in excess of about 60Me exceed
the Eddington limit.
7 Surface Properties
From the hydrostatic picture we can derive a number of observable properties of stars.
Because of the complexities of the microphysics, these derivations require numerical in
tegrations. But we can at least see what is involved physicaly. Let us leave out rotation
and magnetic fields for this purpose.
i 7
Knowing the central temperature of the star, we can estimate that the outward radia
tive flux, in the absence of convection, is "" KT / R where K is the radiative conductivity
and R is the radius. Here we are supposing that the surface temperature is much less
than T, the temperature in the deep interior. If we assume simple dependences of K on
the state variables, such as power laws, we can derive the dependence of the luminosity
on the bulk properties such as mass and cheiical composition.
Notice that these derivations require no statements about the source of energy. If the
source were turned off, the qualitative aspects we have just discussed would hardly be
changed. However, if we want the static state to last for much more than a thermal time,
we do need an energy source, and that is provided by thermonuclear reactions. These
tend to be energy sensitive, so there is some fine tuiung in the central conditions as the
hydrostatics comes into accord with the nuclear reaction rates.
Another delicate problem in the hydrostatic stellar structure theory is the determina
tion of surface conditions. This theory tells us the luminosity and the radius of a star,
given its chemical composition and a few plausible simplifying conditions. If the star
radiated like a blackbody of radius R and temperature Te (e for effective) then
L = 471 R2 O'T4 .e
The actual sunace temperature is not' far from this. Indeed the operational defiiution
of temperature is ambiguous in the surface layers, which are clearly out of thermal equi
librium, so Te could provide one definition of surface temperature. By the way, we have
just passed into the subject called stellar atmospheres, as distinct from stellar structure
theory, which we have been discussing so far, however loosely.
The meshing of these two main topics of stellar physics is like the interaction of
oceanography and meteorology and the key observation that must be explained by this
joining of forces is the socaled HR diagram. This is a plot of log L vs. some observed
spectral property of stars that measures Te. We shal not have time to go into this
spectral lore, even though it is the backbone of classical astrophysics. When I was a
student, we often were handed spectra on exams and asked to read off (in effect) the
surface temperature. We shal bypass this process so, in figure 2, the abscissa is Te
straightaway, with Te decreasing to the left in accordance with astophysical tradition.
The ful calculations show that, for a given chemical composition, assumed unform
throughout the star, and with no rotation, the equilibria form a one parameter famly of
solutions, with the mass as parameter. The locus of this family depends on the values of
parameters such as the chemical composition, but for reasonable choices, one finds that
a large majority of stars lie on a curve like that shown. Of course, the real observations
show some revealng additional details (many of which are understood in terms of stellar
evolution and the development of chemical inhomogeneities), but the main point is that
newly born stars do fal on this main sequence, as it is caled. The maxmum mass is ,at
the upper left and it decreases downward along the sequence. At the upper end, L goes
like a large power of M, 5 or so. The development and verification of al this is the stuf
of stellar evolution theory, which we shal not go into here.
18
logL .
Eddington Cutoff
50 solar masses
o
short lifetime
HERTZSPRUNGRUSSELL
DIAGRA
Sun
'
low mass
slow lOtauon
long lifetime
600
~ Kumar
cutoff
105 18 200
. tff
Figure 2: Schematic HR
We have shown only a section of the theoretical main sequence in the figure, which
corresponds to what is actually oberved. As we have suggested, the upper cutoff is
probably connected to radiative processes, such as the Eddington limit. In any case, for
large masses, the lifetime on the main sequence is smal, as is easily estimated. The energy
available from nuclear reactions is some slight fraction of the rest energy Mc2. Divide
this by L and you get a time which is millons of years at the top of the main sequence.
This is very short compared to the similar estimate for the iun, which runs to tens of
billons of year.
Another feature of the observations that is not realy understood is the socaled
luminosity function. This is the number of stars at each luminosity and it increases
with decreasing L. This is a direct consequence of the mass spectrum of newly formed
stars. For smal enough masses, the central temperature is so low that there are effectively
no nuclear reactions. That Kumar limit is somewhere below O.lM0. Those unproductive
stars include the socaled brown dwarfs. Their numbers remain a mystery that the space
telescope was suppposed to dispeL.
8 Convection
In computing the main sequence shown in figure 2, there are aleady a lot of diffculties
to be surmounted. . Not only do we need to deal with the microphysics of ionization, opac
ity, energy generation and radiative transfer, we have to do something about convection.
When we write th~ hydrostatic equation, since the stars are typicaly composed of perfect
19
gas, tlus is coupled to temperature and to an energy transport equation. So we need to
compute the radial temperature gradient. Once we have this, we can check for convective
instabilty. A static model is locally susceptible to convection when the radial gradi
ent of specific entropy (essentially what meteorologists call the potential temperature) is
negative. That is, for constant molecular weight,
dS = Cp (dT +.J).
dr T dr Cp
Here we are assuming that we are in the outer layers so that the gravitational attraction
is constant. When this gradient is negative, we confront convective instabilty.
In stars, when the the gradient is convectively unstable, the resulting Rayleigh number
is typically astronomical, because of the great length scales involved. In fact, it is very
hard to obtain the static solutions for stars before convection is allowed for, unless you
compute them yourself, since astronomers never publish them. It is therefore not a simple
matter to find out what the conventional Rayleigh numbers in stars really are. In any
case, stars are so nonBoussinesq that the meaning of the Rayleigh number has to be
thought about a bit as well.
When astronomers do encounter unstable entropy gradients, they frequently replace
them by zero gradients, the neutrally stable value, and compute the model that way. This
use of the notion of convective equilibrium goes back to the last century and it has not
greatly wavered. The problem in putting the whole region of convection onto a simple
adiabat is that some stellar models, such as that for the sun, are sensitive to which adiabat
is chosen. So algorithms have been devised for the purpose. Whether you regard tlus as
science or voodoo depends on your background and goals. As you can readily imagine,
the main delicacy hinges on the treatment of the boundar layers. Let us leave this
astrophysical skeleton in its closet.
What is found theoreticaly is that strong convective instabilty occurs in the outer
layers of stars in the lower half of the main sequence. For example, it is believed that
the outer one third (in radius) of the sun is vigorously convective. That convection
occurs largely because of the high opacity (low thermal conductivity) of parialy ionized
hydrogen and to some extent because of its high specific heat.
In the upper half of the main sequence, the hydrogen is fuly ionized, so convection
is not strong at al, though almost al stars do have some sunace convective instability.
On the other hand, these hot stars have interior nuclear reactions which are temperature
sensitive. This promotes strong temperature gradents and convective cores are the rule in
the upper main sequence. We shal not have time to get much involved in thermonuclear .
convection. Our interest will be confied to the fluid dynmics of the outer layers of stars.
Observations suggest that there is strong fluid dynamcal activity in the envelopes
of both hot ("early") and cool ("late") stars. The cool stars have vigorously convective
envelopes. But in the very hot stars, which tend to rotate rapidly (R02  g), the profies
of spectral lines show the evidence of line broadening (through Doppler effects) by motions
with speeds that may even be supersonic. All sorts of other clues point to vigorous activity
20
in those hot stellar envelopes, and we wil offer some suggestions about the causes in
lecture 3. In the intermediate case of the 50called Astars, the fluid dynamical activity is
relatively low. This permts certain peculiarities, as astronomers call them, to manifest
themselves.
Notes submitted by N. Platt and R. Kerswell.
21
Lecture II
1 Expanding Flows
Our opening example of AFD is the study of a fluid in expansion. The simplest case is
that of a uniform, unbounded medium such as cosmologists study. Even for cosmology,
this is an oversimplification. The universe is fied with radiation, which is observed to be
extremely isotropic with an almost perfect black body radiation spectrum at T '" 3 K.
According to standard big bang cosmology, this radiation consists of photons emitted from
the early unverse as orginaly predicted by Gamow's students Alpher and Herman in the
1940's. We shal leave out the effects of this radiation, which were pronounced in the early
days of whatever cosmic event we are going through just now. But we note in passing
that the great degree of isotropy of this radiation attests t~ the great homogeneity of our
universe out to large distances. That is the basis for thinkng that a unform Eulerian flow
makes a good modeL. However, the seemingly fractal ditribution of the visible galaxes
points to a chaotic Lagrangian flow.
We return to the description of expandig coordinates begu in Lecture I and consider
a fluid with velocity u = H(t)x. From conservation of mas we. have, for homogeneous
density,
ôp
 = pV. u8t (1)
which gives
p
 = V. u = 3H.
p
(2)
Thus,
d (¡_1/3
H =  In .!dt Po
and we defie R( t) = (pI Po)1/3, which is caled the scale factor. Two objects initialy
separated by a distance ro at time t = to wi have a separtion at time t of r = R(t)ro,
where we have set R(to) = i.
Now consider conservation of momentum in an isentropic selgravitating fluid:
(3)
8eu+ u. Vu = VV, (4)
where V is the gravitational potential. There is no ter corrsponding to pressure or
entropy ~adients because the fluid is homogeneous. Substitution of u = H(t)x yields
(H + H2)x = VV: (5)
On takng the divergence of the above equation and using Poisson's equation we get
3(H + H2) = 41rGp, (6)
22
which also gives an equation for R( t),
 4 2R = rrGpo/ R .
3 (7)
To obtain the peculiar momentum of a particle, such as a galaxy, we introduce the
coordinate transformation
x = R(t)i:. (8 )
We differentiate this and solve for
Ri: = x  H x, (9 )
which is the velocity relative to the expanding background. Multiply this by Rand
differentiate to obtain d ( 2.) (, R)
dt R i: . = R X  R x . . (i 0)
From equation (7), we see that, if the gravity of the unform background was the only
force acting on the particle, then the right hand side of equation (10) would be zero. Then
we would get R25c = constant and so the pecular velocity (for a particle of constant mass)
decreases like Rl. Thus, with respect to an expanding backgrund, the momentum (P)
will decrease,
P = Po. (ll)R
This result clarfies the wayan expanding gas cools by analogyg with the wayan expanding
gas loses' anguar momentum. Thus, R2i: is like the anguar momentum of a paricle and
the coordinate velocity i: is analogous to an anguar velocity.
The Boltzman distribution is j(P) ,. eE/IcT where E is the energy, given by E =
(P2 + m2)1/2 (the speed of light has been set equal to one), k = Boltzma's constant
and T = temperature. Thus, for a nonrelativistic paricle, E :: m2 + p2/(2m) and so for
the momentum distribution to be time independant (recal equation (ll)), we must have
T '" R2. For relativistic particles (in particular photons), m':': P,hence E '" P. Thus,
for f( P) to be time independant we must have T '" Rl. We see that relativistic and
nonrelativistic paricles in the unverse cool at dierent rates. Therein lies the means to
understand why the background radiation is at only 3K. On the other hand, there is no
explanation here for the large pecular motions in the gas of galaxes (a few hundreds of
km/sec).
For more general motions, we want to g�� into coordinates sugested by this simple flow.
Starting from the usual Euler equation
pDtu= VppVV, (12)
where Dt = 8i + u. V, we may transform into the exandig coordinates given by (8).
We set u = i, whence Ru = u  Hx. Again, the quantity u is a coordiate velocity,
analogous to an angular velocity, so the physical velocity is Ru.
23
We may also transform p, p andV, but we shal not take time for this here. The main
point is that the left hand side of the Euler equation, in expanding coordinates, becomes
Dt + 2Hu + Ili + V (~H2i2) (13 )
where Dt = at + u . V and V = Rn. We have three fictitioUs forces in this transform
of the inertial term, in the sense that a fictitious force is a real force that cannot be
felt by an inertial observer. There is the cosmic drag, Hu, the expanding analogue of
the Coriolis force. However, it is antiparallel to the pecular motion, and represents a
dissipative term. The other two terms are alo famar looking to students of rotating
fluids: there is a term like a centrifugal potential and an analogue of the Euler force which
comes when the rigidly rotating frame you go into has a time dependent rotation rate.
This kind of dynamcs has not been much ~plored as yet, but it arses in stars as well
as cosmology. For example, in a radialy pulsating star, we have a periodic R(t) and can
ask what happens to the criterion for the onset of convection (Poyet and 5., Astron. J.,
1979).
2 Stars
2.1 Stellar Evolution
There are several stages in the evolution of a star. For most of its lifetime, a star burns
hydrogen in its core, converting it into helium. This permts a simple estimate of stellar
lifetimes. For example, the Sun has a total mas' of 2 x '1033 g, 10% of which is in a
core hot enough for burnng. The amount of energy reeased per gr of hydrogen is
approximately 1% of its restmass energy, mB~' Thus the hydrogenburng li��etim~ of
the Sun is
2 x 1033 x 0.1 x 0.01 mBc2 1010T =  yr.
L0 (14)
where L0 = 4 x 1033 erg/sec.
After the hydrogen in the core is exhausted, the core slowly contracts until the tem
perature increases to the point where the helum begins to bur. The star wi go through
successive burng and contracting stages, each time burng heavier elements. The fial
stage reaced depends on the total mass of the star. The highest mass star are able to
convert sicon into iron. As no exothermc nuclear reaction involving iron is possible, no
further central nuclear reactions occur. Continued contration heats the core until iron
breaks down endothermically. The temperatur drops, and so does the pressure. The star
collapses supersonicaly until the core density becomes large. The envelope rebounds and
the star becomes a supernova, according to one version of the story. There is a lot of fluid
dynamcs in this but most of it is in the mids of the Crays.
24
core
niulent envelope
stale
stle photoshpere
T=60K
100 Ia tlck
corona, powers
the solar wind
6
T = 10 K
Figure 1: Schematic diagram of the Sun
2.2 The Solar Atmosphere
In the outer layers of the Sun, hydrogen is parialy ionized, hence there are free electrons
which can be captured by a neutral hydrogen atom. As the binding energy of the second
electron to, H is, only 0.7 eV, a passing photon can be easily absorbed in a freefree
tranition. This makes for very high opacities. So raation is an ineffcient mean of heat
tranfer in the solar atmosphere and convection must occur.
However, within 1000 km of the sunace (the place where the depth measured in photon
mean free paths is of order unity), the sun's atmosphere is convectively stable. This place
of transition roughly defies the photosphere wher most of the sunght we see originates.
From there outward, the temperature incrases and we go into the hot (2 x 106 K) corona
that envelops the sun. The corona is not hydrstatic; it is exanding to feed the solar
wind. A schematic diagram of the Sun is shown in figue 1, which suggests that a 1 l layer
model (as in the ocean) might be useful for analyzig the solar atmosphere.
2.3 Equations of a Simple Stellar Atmosphere
Let us charactereize an atmosphere as the portion of a star where the gravitational ac
celeration (g) can be safely considered to be' constant. In a caresian coordinate system,
where z is in the vertical direction, the equation of hydrostatic balance is
dp
 = gp
dz
where p = pressure and p = density. Assumng an ideal ga, we have, ~
(15)
Ie"p= =pT
m (16)
25
where ñi = mean molecular mass, usually written ¡.mH, where mg is the mass of unt
atomic weight. We sometimes write the gas constant as n = klmg, which permts us to
leave the mean molecular weight, ¡., in evidence. For the purposes of this discussion we
shall assume constant ¡., though it may in reality depend on the state of the material or
its age.
Thus, we have two equations in three unknowns (p, p and T). In the simplest case,
we assume T = constant and obtain
p =
zig (17)p.e
p  p.ezig (18)
where
H nT (19)
¡.g
Thus, in an undisturbed atmosphere that consists of an ideal gas at constant temperature,
the pressure and density decay exponentially with height.
To investigate the timevaring pressure and density fields, let us consider each field
as having a static component and a much smaler timedependent component:
P  Po(z) + p'(x,t)
p  Po(z)+p'(x,t)
(20)
(21)
where Po and Po are given by the static solutions shown above in equations (17) and (18).
These perturbations are assumed to evolve isentropicaly and sice S = ev1n(pl P"), we
have Dp zDp
Dt = C Dt' (22)
where CZ == ¡Pol Po = (¡ kTo) 1m, ¡ = Cp/ c" and we are stil assumng the ideal gas law. Sub
stituting these pertubations into the momentum and continuity equations and neglecting
second order terms yields:
Ô1 8p'
Po ût =  8z '
lJ 8p'
Po ût =  8y ,
lJ 8p'"
Po at =  8z  P 9 (23) ,
and .
8p' J lJPou) 8(pov) 8(Po1J) l 0 (24)
at + 1 8z + 8y + 8z 
where u, v and 11 are the velocities in the z, y and z directions resectively. From equation
(22) we alo have 8p' Po (8P' apo)
 Pogw =¡ +w .ût Po at 8z
Now we can eliminate u, v and 11 from equations (23), (24) and (25) to yield the following
equations in p' and p' oIiy:
(25)
82p' 2' 8p'
V p =gût2 8z (26)
26
r!.
i _:)
I .' H
Figure 2: Dispersion relation for Lagrangian perturbation with T = constant
and
~22 ¡pI  7::iJ = ( 1) (;: + gpl) .
Let us look for solutions of the form
(~ ) = ( = ) . ex (i(leg. X  wt + lez)  z/2HI
where p and p are amplitudes, Ita is a twodimensional horizontal wave number, x is
a twodimensional position vector, w a frequency aid IC is a vertical wave number. We
have choosen the ex;ponential varation in z to consrve wave energ. Substitution into
equations (26) and (27) yields the following dispersion relation (Lamb 1924):
w4 _ ((it~ + 1t2)C2 + w:1 w2 + c21c~W~ = 0 (29)
(27)
(28)
where wi == 7g/(2H) and w: == ..;1 Ïi. This yields two sets of relations between w and
ItH, as shown in figue 2. In group I, w2 = (1Ct + 1c2)C2 + wi, so that iwl ~ wi and in
the limit as IItHI  00, these high frequency waves have w = :iICHC. Thus they are just
nondispersive sound waves, and are referred to as pmodes in astrophysics. In group II,
w2 = C21cl,WV((ItJ¡ + 1c2)C2 + wll, so that iwl ~~. In the lit as IICHI  00, we have
w = :iWh which are simply gravity waves (caled gmodes).
We have considered only isentropic perturbations to the static state, neglecting thermal
effects. Since S = c" In p  c" 7 In p, i¡~ = 0 implies that
Dp pDp Po +ý Dp
Dt = 7; Dt = 7 Po + ¡I Dt (30)
To first order, then we have
Dp 2 Dp
c
Dt  Dt . (31)
27
This is a Lagrangian perturbation and it yields two famies of solutions as discussed.
A rough and ready approach that gives just the sound waves is ~; = 0 (an Eulerian
pert ur bation) .
As we are also interested in the velocity of such wave solutions, we could have elim
inated pI and p' from equations (23), (24) and (25). Rotating the:: axs into the di
rection of propagation (i.e. letting t1 = 0) and introducing w = vorticity = V x u and
X = compre.5.5ibility = V . u we obtain (Lamb, 1924)
a2X 2 2 ( dc2 ) aX. _ (32)&t2  c V iX +  ;g  + gz . V x wdz' az
and
a2w ¡dC' J
(33)&t2   d;+(¡l)g V x (zx.)
where
82 82V~ = 8z2 + 8z2' (34)
In the case of an isothermal atmosphere, these equations can be solved to yield hori
zontal traveling wave solutions of the form u = eC"fI)'./c2 I( ct  z) and w = O.
Remaining issues which may be important are the effects of nonlinearty and dissi
pation which have been neglected in this discussion. A further complication is the con
sideration of a 'polytropic atmosphere in which temperature Vaes linearly with height,
T(z) = ßz. In this case we find two famles of solutions which are somewhat similar to
the isothermal case, as shown in figue 3~
This is a very superfcial introduction to thes matter and more details can be found
in Lamb's book. In the case of the su such waves ar now being observed and the
measured frequencies are used in connection with some knowledge of the radial amplitude
distribution to lear much about the internal structure and rotation of the sun. Yet we
stil know rather little about these modes. How and when do they go untable when
dissipation is included? Why do they seem to be so weaky nonlnear? How do they
couple to the turbulent convection?
References
Lamb, H. 1924 Hydrodynamici, st"ed. (Cambridge: Cambridge University Press).
Notes submitted by Wendy Welch and Brian Chaboyer
28
v.
, ~~\
Figure 3: Schematic dispersion relation for T = ßz
29
Lecture III. Photogasdynamics
1 The equations of photohydrodynamics
The picture to be buit up involves thinkng of the outward diffusion of photons
through the outer layers of the star as a fluid of photon paricles moving outward through
a porous medium. The radiation field might be derived frm the fu equations of electro
magnetism. However, quantum fluctuations and relativistic effects are not important in
a typical stellar atmosphere, and a corpuscular pictur where the radiation is represented
as a fluid of photon particles will be described here. We Wi also ignore polaration,
that is spin.
The matter field is described by the velocity u, the density p, and the pressure
p, whie the radiation field is characterized by the ftux F, the energ density E, and the
pressure tensor, 'P. The equations of momentum continuity, and thermal energ of the
field of matter are (Le. Hsieh and Spiegel, 1976; Mihalas and Mialas, 1984)
Du IC+CT (1)p Dt  Vpgpz+pFc
.
Dp
pV.u (2)Dt 
DT Dp
plcc~Š  E) (3)pc" Dt  dt 
where It is the mean absorption coeffcient and CT is the scattering coeffcient. The velocity
of the matter is measured with respect to an inertial frame, preferably the one in which
the star is at rest, cal it the star frame. Then F, Š, and E ar the raation ftux, source
function and energy density of the radiation field meased in the local rest frame of the
matter. These are simply reated to F, S, and E, the same quantities measurd in the
star frame, and which are the ftux, source function and energ density that ar normaly
referred to in the radiative transfer theory of a static medum. The traformtion, '
equations relating these quantities measured in the two fres ar (to leadng order in
30
lul/c),
F = F  Eu  'P . u, (4 )
E 2u. F (5 ) E2 'C
Š  S=aT4 (6 )
In addition, the equations governng the field of matter (1)(3) require an
equation of state,
n.p = pT (7)
J.
and the specification of the absorption and scattering coeffcients (7(p, p) and It(p, p).
The fial term in the momentum equation (1) represents the momentum im
parted to the matter per unit time from the radiation field, i.e. a radiation pressure
gradient.
The fluid equations for the radiation are derived by takng frequency integrated
moments of the transfer equation. The radiative equations of motion are:
BE
at +V.F 
1 BF
c2 at +V.1'
, (K.  (7)pK.( S  E)  p ; u. F, (8)
(K. + (7) pK.
= p ; F + ;(5  E)u (9)
The source function 5 simply represents the emssion of radiation by the fluid, and E
the absorption. The final term in equation (8), which govems the evolution of the energy
density of the radiation field, is the rate of work done by the radiation upon the matter
field. Equation (9) expresses the evolution of the momentum density of the fluid (and
is the counterpar of conservation of momentum for the matter field). The fit term on
the right hand side of (9) represents the force of matter upon the radation fluid, whereas
the second term indicates how a net loss of energ by absorption crates an additional
force upon the radation fluid, through the concomittant momentum exchange.
The system of equations for the radiative fluid are completed when an approx
imation or closure for the pressure tensor l' is specified. Here we shal use,
'1 1 ( 2 1l' = EI +  uF + Fu (u. F)I +T3 c2 3 (10)
31
where I is the identity tensor and T is the radiative viscosity tensor satisfying Tr(T) = o.
The quantity in square brackets in (10) is a transformation correction due to the motion
of the material.
The isotropic part of 'P provides the dominant contribution to the pressure
tensor in regions where the photon mean free path is smal, i.e. the medium is opticaly
thick. Neglecting al but this term yields
1
'P =  EI
3 (11)
This isotropic ,approximation to 'P is known as the Eddington approximation. When
the photon mean free path is large, the medium is "opticaly thi" and the ansotropic
components of 'P may become important.
When the region where the radiation and matter fields interact is plane paralel
and the particle fluid is motioi��ess, the equations reduce to what has been caled the Mile
problem (Chandrasekhar, 1960). In this' case the ful radiative tranfer equations can be
solved exactly.
The derivation of the photohydrodynamc equation suggests that there may
be a more natural approximation to the problem than the Eddington approximation.
Essentialy, the Eddington approximation assumes that the pressure tensor 'P is isotropic
in the rest frame of the star. The derivation of the equations takes account of the tra
formation between this frame and the rest frame of the moving matter. There may be
a dierent frame, which one might cal the "rdiative frame", in which, if one assumes
'P to be isotropic, the exact solution is better approximated. If one considers general
tranformations in the Mie problem, and subsequently implements the Eddigton ap
proximation in these franes, one fids that there ext two fres, one subsonic and one
supersonic (with respect to c/ v') in which the exact solution for a particular moment,
is produced by this means. However, whether this result is unque to the Mie problem,
and whether either choice of one of these fram is sensble is not evident.
2 Hydrostatics and linear stability
As an example, consider a planeparalel atmospher in hydrstatic balance
F = Foz j' S = E = a: j P = !lpT (12)
32
dp
(~+ U) ( 13)dz

pg + ~ Fop = pg.
dE
(~+U) (14)dz  3p ~ Fo
A simple solution is obtained by neglecting variations in ionization (so that R = R/ J. is
constant) and supposing that the atmosphere is hot enough that u (scattering) dominates
~ (absorption). The distribution of temperature with height is ilustrated in Figure 1. As
   _.                          
I
z
Figure 1: The temperature distribution with height for the idealzed model atmosphere.
At depth, the structure is polytropic whie, high in ~he atmosphere, the temperature
declines exponentialy to To (note that the scale height is approximately KI/g.).
the radiation pressure gradient increases, the effective gravitational constant g. decreases.
The case g. = ~ corresponds to the Eddington limit. Beyond this point the atmosphere
levitates. The hydrostatic solution for g. 0( 0 has a density which increases with height.
A star above the Eddington limit would presumbly have its sudace layers blown off by
radiation pressure. Yet rare stars exist that appear to excee the Eddington limit.
The stabilty of this solution may be examned by perturbing the hydrostatic
equilibrium and solving the resulting linear equations for the perurbation. In this ex
ample, instabilties occur when we omit the viscous term and absorption. Though dissi
pation ma.y push such linear instabilties to nearly the Eddington llt, they may occur
nonlnearly in stellar conditions and produce fluid dynamcal activity, such as "photon
bubbles", to be discussed shortly.
33
3 The heat equation in the absence of fluid motion
If the material velocity vanishes, the photohydrodynamc equations for the ther
mal energy of the matter field, a~d the radiation field, become
aT
PK.c(S  E) S = aT4, (15 )PCv = ,at
8E
+V.F  PK.c(S  E), (16)at
1. 8F + ~VE

_p (K.: ~) F. (17)c2 at 3
Since the travel times for light across the distances of interet are large for star (one
says that c  (0), this can be simplified. We see, fit of al, that E ~ S. Moreover,
the term c2Ft is smal. In fist approximation it may be neglected and this gives an
approximati~n for F, which we use to approximate F t in this equation. Then we obtain
the radiative heat equation for T (Unnoand S., P.A.S.J., 1966), which, after we take
certain liberties, becomes
(4 T3 ) aT c; 'l2 aT 4acT3 V2Ta +pc"  v =
" at 3p(K. + ~)K. at 3p(K. +~)
In the limit where the photon mean free path is large  that is, pZK.(K. +~) ~ 1 
equation (18) reduces to
(18)
aT
a ex (T  To) (19)
where To is a constant of integration chosen as the equibrium temperature. In this
opticaly thin (or transparent) limit, disturbances decay according to Newton's law of
cooling.
When the mean free path is smal (pzK.(K. +~) ~ 1), (18) reduces to
aT ex V2T
at (20)
In the opticaly thick or opaque limit, the decay of disturbances is governed by a diffusion
equation.
34
4 Photon Bubbles
4.1 The analogy with ft uidized beds
Consider a collection of smal particles (e.g. sand) resting on a porous plate. A
fluid (e.g. air) is forced up through the plate and paricles.
When the flux of fluid through the plate is low, the particles are relatively
unaffected, and act like a porous medium. This can be thought of as analogous to the
diffusion of a photon gas through a stella. atmosphere. In fact, the equation governng
the motion of the fluid is of the form
Dp
 = ÀFDz (21)
where p is the fluid pressure, À is the porosity of the paricles, and F is the vertical fluid
flux. This is just D' Arcy's law, and it is qualtatively simiar to the equations that result
in certain pr:obl~ms in radiative tranfer.
As the flux of fluid increases, the drag on each paricle due to the diffusing
fluid also increaSes. 'When the drag per paricle exceeds the weight of the particle the
whole bed is levitated. Levitation expands the bed, alowing the :ßwd to move more freely
between the paricles which diminishes the drg. The bed is then said to be fluidized
(Davidson and Harson, 1963). Quicksand is a famar example of this phenomenon.
IT the fluid density, Pit is much less than the density of an individual paricle, PP'
bubbles of fluid appear and rise thrugh the bed of paricles. These bubbles presumably
are generated by instabilty of the fluidized bed.
The bubbles can ascend to collapse at the suace of the bed, which has the ap
pearance of a boilng liqwd. The bubbles in flwdied beds are kidneyshaped (see Fig. 2)
in vertical crss section. Overal, the motion of the bubbles mies the paricles. However,
the paricles with the highest drag are cared upward with the bubbles. This process is
caled elutriation, and may be exploited in industrial applications for separting paricles
with different drg coeffcients. Perhaps in stars we can thi of photoelutriation.
4.2 Photoconvection
"
The analogy with flwdized beds suggests that there be bubbles fied with photons perco
lating through a stellar atmosphere~ Even without the beneft of the analogy, astrophysi
cists have speculated on this possibilty in raatively dominated situations. Indeed, the
35
Figure 2: A kidney shaped bubble in a fluidized bed.
, '
analogy is not perfect; the instabilty of a fluidized bed is caused by the density depen
dence of the drag coeffcient. This'is not usualy a signficant process in the astrophysical
cases. Morever, the bubbles in the fluidized bed are maitained by iluid that typicaly
canot be absorbed by the paricles, which is not the cas for photons in the stelar case.
Numerical calculations suggest that liear instabilties in hot stellar atmo
spheres are suppressed by dissipative processes excet for star nea the Eddington limit
(Marzec, 1976). If liear instabilty may occu for £ = L/ L. in excess of some crtical
vaue, £c, it is possible that the bifurcation is sub critical and nonlear intabilties may
'occur for some reasonable stellar conditions with £ : £c that would give rise to bubbles.
.
36
4.3 Bubble theory
For the simple theory of photon bubbles we consider the fate of a spherical hole
carved out of a stellar atmosphere. For the case of pure scattering in the Eddington
approximation, we have F lX V E and, for radiative equilibrium, we assume, V . F = O.
For a coordinate system centered on the center of the hole and with (J measured from the
vertical symmetry axs, we find
F = F.V ¡cos (r  :~) 1 (22)
where TO is the radius of the sphere. We are here assuming that the Eddington approxi
mation holds also within the bubble, where the photons scatter from the surace and are
isotropized. We have neglected absorption.
The radiation field about the hole is distorted in much the same way as an
electric field is anected by a conducting' sphere in electrostatics. The originaly uiform
flux F = Foz sufers a dipole distortion. The distorted field lines produce an additional
force on the mat.ter. The external force densty f is
f = pgi + pCT F = pg.i  pVt/
c
(23)
where g. is the effective gravity defied by (13) and t/ is the potential of the dipole
distribution
.J _ Foc .!
'l 3~
CTo r
Figue 3 ilustrates the distorting field V q,.
The force f produces a fluid circulation which causes the bubble to rise, and to
deform. 1£ V z is the velocity of the bubble, let v = u  V z be, the flow field around the
bubble in a frame of reference translating with it. H we assume that the bubble
remains
nearly spherical, then we may take v to be the incompreble flow around a spherical
obstacle. Thus,
(24)
v=vv(%(i+i;)) (25)
The flow v is irrotational, and satisfies Bernoul's Law. This may be used to
estimate the upward speed V of the bubble. Bemoul's Law specfies that
h = _g.% _ q,  ~lvl2 (26)
37
Figure 3: The field lines for the additional dipolar force caused by the presence of the
bubble.
38
is constant along streamlines. Taking into account the relevant boundary conditions
(Spiegel, 1976) gives the result
2V = ïýgro (27)
On the lower hal of the bubble, the fluid dynamcal pressure Iv12/2 must be balanced by
an accompanying distortion of the bubble. This distorted shape is indicated in figure 5,
and in the fluidized beds, gives rise to the kidney shaped bubbles.
At the surface the bubble will burst with an efHuence of the photons contained
within it. Thus excess flux from bursting bubbles may produce shot noise in the stellar
luminosity. The ati;ospheric vortices that we shal describe below could even produce
bright spots. When the bubble bursts, paricles may become supersonicaly ejected. Such
phenomena may be the bass for the intense hydrodynamc activity of these early stars.
The problem is to confi the existence of photon bubbles.
5 The, effects of rotation
Hot star are fast rotators and the interplay of rotational and radiative dynami
cal processes is likely to be central to understanding their ßuid dynamcs. Rotation alone
is already a signficant modulator of stelar ßuid dynamcs. Its effect upon an initialy
spherical object is to decrease the polar radius R" and increase the equatorial radius
Re,. The difference between these rad is
ilR Re,  JL. 02
= l
Re, Re, 02 + Gp (28)
where p is some mean density.
The rotation induces a distortion of the suraces of constant temperature; the
pole and equator exbit a temperature difference of approxiately
ilT IT  ilRI R.
From the perfect gas law, we estimate the presure dierence to be
ilRilp  ~påT  PR' (29)
, 'This must drve an '''astrostrophic''zonal flow u, such that
ilp  pun. (30)
39
Therefore the magi��tude of tlus thermal wind is
gt1RuOR (31)
where the surface pressure p has been replaced by pgR according to estimate based upon
vertical hydrostatics.
When the rotation is very fast, as in many hot young star, equatorial cen
trifugal accelerations compare to g. Thus gin", RO = ~lf' Therefore the flow produces
an equatorial acceleration and, since the Reynolds numbers of such flows are large, in
stabilties are likely, We expect vortex formtion, as on major planets. Because of the
complications invloved in the radiative flows, these vortices may may serve as conduits for
rapid escape of radiations from within and produce powerful emergent beam (Dowling
and S., in press).
5.1 Vortices in hot atmospheres
Consider a vortex in a polytropic atmosphere. H the vortex is strong enough, we may
for a qualtative fist look, ask what it will be like without the effect of rotation. Under
hydrostatic balance the vertical and horizontal pressure grents of a steady vortex
satisfy dp dp pv2dz = pg, å: = 7' (32)
where z increases in the downward direction. H the specfic enthalpy is h, then
h(r,z) = gz + f(r), (33)
where f( r) must satisfy
df ,,2
dr = ;'
For a standard vortex of the form
(34)
v = J vorlro if r ~ ro
\ vorolr if r ~ ro
we have
, f  2 i r2 12r~  1 if r ~ ro Vo "/2" i'f
rô r" ,. ~ ro (35)
40
0
I
i
I
I
Î
I
1
~N0
~ 2"'
~
~'
3
4
4 2 o 2 4
(r Iro)
Figure 4: The isotherms in a vertical section thrugh the ax of an atmospheric vortex.
The dotted lines indicate the streamnes of theradativeßux as it is focussed into the
vortex.
The vortex deforms the stellar surace frm the plane z = 0 into
z = v~ J r2/2r;  1 if ,r : ro .9 1 rV2r2 if r ~ ro
Therefore there is a depression of the surface. The maxmum depth of the depression is
v5fg.
The isotherm h = c07atcint in a vertical section thrugh the axs of the vortex
are shown in figue 5. The ßuid at the center of the vortex is now cooler than that outside
it. The raative flux is now focussed into the vortex and consequently the depression of
the surace appears bright. Thus it form a "starpot". The focussig of the flux is al
shown in figu 4. Therefore, upon the surace of hot stars there may be bright spots, in
contrast to the dark spots observ,ed upon cooler star such as the sun.
, Finaly, we n~te that this mechansm may alow a star to exst above the
Eddington limit: the'vortices chanel photons along their axes and therefore reduce the
pressure of radation over the remainder of the sudace.
;,
lJ
~.
jo
~
h
41
References
1. Chandrasekhar, S. (1960). Radiative Tran$fer (New York, Dover).
2. Davidson, J.F. and Harson, D. (1963). Fluidized Particle$ (Cambridge University
Press).
3. Hsieh, S.H. and Spiegel, E.A. (1976). Ap. J., 207, 244.
4. Marzek, C.M. Thesis. Columbia University, Dept of Physics (1976).
5. Mihalas, D. and Mihalas, B.W. (1984). Radiation H1/drod1/namic$.
6. Spiegel, E.A. (1976). In Problem, in Stellar Convection, ed. E.A. Spiegel and J .P.
Zahn, Lecture notes in Physics 11 (SpringerVerlag, Berlin).
7. Dowling, ,T.E. and Spiegel E.A. (1990). In Nonlinear problem, in a,troph1/,ic$, ed.
R. Buchler and S. Gottesman (N.Y. Acad. Sci.).
Prepared by N.J. BalIiorth and G. Bell.
42
Lecture IV. Solar AFD
1 Introduction
A good white light photograph of the sun reveals that the solar sunace is covered with
a timedependent pattern of granules, a cellular arangement of bright patches. The
sizes of these granules range down to the resolution limit of observations, about 300
km. Individual granules may last ten minutes or more, depending on how deformed
they are alowed to get and stil be considered to be the original granule. Spectroscopic
observations show that the brighter (presumably hotter) portions of the granules are
rising. The granulation is normaly thought to be a mafestation of thermal convection.
The fims are the best way to get some feeling for the phenomenon. They were made
at the Pic du Midi Observatory (thans to T. Rouddier for providing that one) and from
the observations,made in space (blessings on G.W. Simon for that one). They have been
processed to bring out the granulation by the grup at, Lockheed (Alan Title, and his
colleagues), which means, in particular, that the vigorous acoustic oscilations of the solar
atmosphere have been fitered out. The speeds of the granules themselves (a few tenths
km/sec) are well below the local sound speed of close to 10 la/sec. The morphological
detais of the granulation are too complicated to repeat in these notes.
Spectroscopic, observations showing the velocity component toward the earth at each
point on the solar surace reveal other structur. The most signficant are the super
granules. These are celllie structures about 20,000 km acoss conssting of horizontal
motions flQwing from central upwellngs. The outflow velocity is about 0.5 km/ see, com
parable to the equatorial rotation speed of the whole sun But the lietime of one of these
supergranules is about a day, compared to the rotation period of one month.
Another sort of observtion made of the solar surace is the magnetic field, which
tends be quite ropy. The main concentrations of field ar at the vertices where the
supergranules meet. It is believed that the intersticial field strengths ru to 1700 gauss.
The magnetic field thus form a large pattern outlinig the superganules. Moreover,
emission by ionied calcium is particuarly strong where the field is strong (probably
because of plasma afects generated by the fields) and a calcium emssion network clearly
outlines the supergrulation.
There is a wealth of fuher structure in the solar obserations, but there is no time
, to go into such detai. This paricular selection was made becau~ of the belief that the
phenomena I have mentioned are direct manesations of the strong convection that is
thought, to power varous signs of fluid activity on the sun. One other observed process
ought to mentioned  the sunspots.
That same white light photograph of the sun exosed to bring out the granulation,
will often show dark spots (comparable in size to superganwes). ,'Other observt~on8
technques reveal that in these spots there are fields of a few thousand gauss. Such fields
can inhbit the convective motions and lowe the emergent heat flux. That wi aleady
cause some d��rkening, but there is more to, the story of spot structure. The degree of
t
T\
43
'Figure 1: Solar Granulation
spottedness of the sun varies on a time scale of 11 year, perhaps chaoticaly. We shal
come back to this at the end of the lecture.
2 Solar Convection
Energy is generated in the core of the sun (as in most star) and it appears largely in
the form of radiation. It takes a photon about 30 mion year to escape from the deep
interior of the swi. ' The hydrostatic equations, together with the trasport equation
for the radiation, then lead to a static model that gives the mah of the state varables,
temperature, density and specific entropy, through the star. For a penect gas, the specifc
entropy is
5 = a"log.!
p"
where C" is the specific heat at constant volum and 7 the ratio of specific heats. So, if
we use the equation of state for a penect gas, we get
~=~,(dT +L)dr T dr Cp
where 9 = GM,.lr2 and M,. is the mass interior to a spher of radus r.
This can be understood by displacing a parcel of fluid in the verical direction by an.
amount dr. Its energy change consists of an interal ener perurbation of CpdT plus a
change of potential energy amounting to gdr. So the total energ chage ags with TdS
and, when it is negative,.we have instabilty. Th crteron for the onset of convection is
caled the Schwarzschid criterion, after one of the many who derved it for themselves,
going back at least to the middle of the nieteeth cetur.
44
Starting fr?m the center, the entropy gradent from static models is positive most
of the way out to the edge of the sun, where it goes strongly negative, and stays that
way nearly to the edge. The final reversal back to a positive entropy gradient occurs a.t
about one photon mean free path from the outside. This is schematicaly shown in the
figure. I can produce only a schematic figue since I know of no modern calculation of
the specific entropy distribution of a static solar modeL. It appears that al program tha.t
calculate solar models contain an algorithm that replaces the unstable region and some of
the underlying stable region by an effectively neutral region. Any reasonable algorithm
that replaces the static entropy portions tha.t are either stable or neutral is bound to
produce a model in which the neutral zone is larger than the original untable zone.
All stars with Te less than about 8,OOOK have extensive outer zones of convective
equilibrium. The cause is the ionization of hydrogen, the major constituent of stellar
material. In the hot stars, hydrogen is completely ionized throughout. In the cool stars,
there is nearly ne~tral hydrogen close to the suace. The reatively few atoms that are
ionized release electrons that find themselves in a. sea of hydrogen atoms. An electron
near an atom polarizes it so that there is a weak attraction between them. The spectru
of interaction energies is almost always continuous. So anr passing photon can cause a
change in this energy and thus be absorbed or deflected. In other words, partialy ionized
hydrogen is very opaque. A strong temperatur gradient is therefore needed to force the
radiation through'it. At the same time, the abilty of the material to soak up energy into
ionization causes Cp to be large. So we get convection zones in the outer regions of cool
stars.
There is no means of calculating a conventiona Rayleigh numer without a good static
model, but estimates, from the fuly convective model suggest that the Rayleigh numer
is truy astronomical, probably in excess of 1020. So when the convection stars, it is
certainly turbulent. That is why only the local crterion is consdered. The idea then is
to replace the unstable region by one of convective neutralty where the entropy gradient
is localy zero. As I have aleady said, this device is bound to make the convective zone
deeper than the original unstable zone. The thickness of this zone (fied by the choice of
a free parameter in the algorithm) is adjusted to make the solar raus come out right.
Recent developments in acoustic sounding of the su have permtted fie tunng of the
models and there is now some confdence that the depth of the solar convection zone is
300, 000 km. Presumbly, there wi be bounda layers on this zone, especialy at the
top. Estimates of the depth of the upper boundar layer ar of the same order as the size
of large granules.
In fact, there is no general agreement on what physics deternes the lengh scales
and time scales of the graules and supergranwes. It is not even certain that the granules
are primary drven by buoyancy., The large scale shears in the supergranules may playa
role in their formation. Magneti,c f��edbacks may have a role in, determning the preferred,
sizes of either su,pergraules or granules. We do not know how these processes couple
to the large scale circwation seen on the solar surace. We can reach for analogies to
laboratory convection or to numerical simUlations, but even those are not realy under
~
,g
:~: ,
45
stood. However, at this moment, the simulations, experiments and solar observations are
al moving forward quickly, so this is a good time to be thinking of these things.
3 Solar Rotation
Observational evidence suggests that stars condense out of the interstellar medium. As
the protostar contracts, it spins faster. The object is probably turbulent and the angular
velocity is roughly constant throughout its bul. To become something resembling a star,
it must shed considerable angular momentum. This it does through a combination of
magnetic stresses and mass expulsion. There may be an ambient disk left behind in the
contraction from which a solar sytem may form. In the end, if there is to be an object
resembling a main sequence star formed, it' ought not to have an angular velocity too
much in excess of /2GM/ R3, corresponding to a rotation period of about a day for the
sun. It is a problem to get rid of the excess anguar momentum, so that most newly
formed (single) stars will be pressed up agait this liting anguar velocity. However,
most aged, cool solartype stars have rotation periods signcantly longer than a day and
it is concluded that they must have lost anguar momentUm sice arriving on the man
sequence.
There is ,a mass flow of about 1013 Me/yr. The deeper causes are not understood,
but something is heating the very outer, highly tenuous solar layers to a temperature of
2 x 106K. The leading candidate is plasma intabilty. In any cae, the material is so
ionized that it emits ineffciently and has trouble getting rid of the energy, so it is forced
to expand to avoid thermal ruaway. In fact, it exands right off the sun and makes a
thermal wind. The magnetic field lends a certai' rigidity to the' flow and it therefore
decouples from the sun at a greater anguar velocity than that of the surace. This makes
the solar wind effcient at removing anguar momentum.
Suppose you had a bucket of water suspended by a rope and spinng. If you put some
holes in the side of the bucket how fast would it slow down? Suppose that you stuck some
pipes into the holes so that the water had to go out a distance before leaving the system.
The water would leave at a greater distance from the side of the bucket and, even if the
rate of loss of mass were the same as before, the rate of anguar mOinentum loss would be
greatly enhanced. That may give you some qualtative notion of how the magnetic field
works in Truildng the sun slow down more quickly. If the su wer to somehow maitai
rigid rotation, then the half life of its anguar mometum would be about 5 x 10'yr. This
agreement with the age of the su may not be a concidence.
If the sun is approximately rigid, we can derive the law of its anguar velocity. Suppose
that the outflow remains nearly rigid out to some distance RA. This distance has to do
with the strength of the magnetic field that is puled out with the flow and is estimated
to be about 100Re. Then ' , .
MR2 dV = MR2y, 0 dt A
where Y = Ren. Though M  _1013 Me/yr today, we do not know whether it is
46
constant. It is almost surely afected by the magnetic field strength. The magnetic field
is likely to be produced by dynamo action that is dependant on rotation. So other things
being equal, we assume that
MR¡ = _ (v)n
MR2 a Ifo
where a has dimensions of (timejl. Tils gives
vn = Von1 + ant
This law of anguar velocity has a couple of free parameters, but for large t, we see that
it suggests that the half life of stellar anguar velocity is nit. This explains the agreement
we have aleady ,noted for the solar case if n ~ i.
Of course, the sun is not rigid. So we have a classical sort of spin down problem with
the sunace layers being slowed down, including the highly turbulent convection zone.
The remaining question is whether the momentum can be extracted from the interior
layers at the required rate. This stratified spin down problem with its attendant stabilty
problem has not been solved. Not even the formulation is generaly ageed upon. But
now, with new solar soundings, we have measurements of zonal Hows, so theories of the
solar circulation now have some reason!1bly hard facts to confont. The mai issue, one
that stil is not decided, is whether the whole sun has spun down to the sunace rotation
rate. This question was the center of controversy twenty year ago because a rapidly
rotating core could result in a smal quadrapole moment to the su's mass distribution.
This would modify the sun's gravitational field and have a sma effect on the precession
of the perihelon of Mercur. Thus the rotation rate of.the' sun's core might be relevant
to tests of the theory of general relativity. This is not a lively possibilty today, but there
are other interesting questions that are a.ected by the possibilty of fast rotation of the
solar core (a possiblitystil defended at Yale), such as the prospect of dynamo action.
4 Solar Cycle
In 1843 Schwabe suggested that the sunspot number vared with a period of about ten
years. The sunpot number as offcialy defied is a measur of the coverage of the sun by
relatively dark areas associated with strong magnetic fields. The solar magnetic vaation
is complicated in its details, but the mai feature is that the total vares on a time scale
of eleven year in a way that looks rather chaotic. If this is a chaotic process, it is alo
spatiotempora. Active regions of spoitedness fit appear at solar latitudes :l3T' and
these peak of activity drft equatorward.
The latitudinal bands of activity are dominated by large spot groups typicaly about
105km apart. The grups are mainly pairs of spots with opposite magnetic polarti~ as
if they were simply the two feet of an enormous magetic arch. Throughout each band,
the leadig spots wi tend to have the same magnetic polarty,' so we can imagie that
each band is the trace of a huge magnetic serpent girding the sun. The polarty of the
47

.
..
i,. ITI ... .. 1 I..
Figure 2: The annual mean sunspot number in the Bracewell style
leading spots is different in the northern and southern bands. About seven years after
these first appear, at about the same time in the north and south, and they have drfted
about a third of the way to the equator, the ban from the previous cycle arve at the
equator and app~ent1ydisappear. The bands, at midlatitude continue their equatorward
drift for another four years or so, when their "Successors appear at :i37��. Again the new
wave continues equatorward, and so forth, and the new bands have the opposite polarty,
wi th respect to leading and following spots, from their predecessors. The cycle could
be therefore said to have a twentytwo year time scale. On the other hand the waves of
activity take eighteen years to go from the latitude offist appearance to the equator. IT (as
Proctor and ¡. suspect) there are only four solitar waves, which ar virtualy indetectable
when they move away from the equator, the cycle consists of simple round trips in which
a wave of solar activity takes eighteen year to go to the equator and four to retur.
There are a number of questions that this zeroth order characterization rases about
the solar activity waves. What is their natur? What happens to them when they reach
the equator? What generates them? What determnes their characteristics, speed, width,
amplitude? These are questions about dynamo theory and they are at the hear of solar
research. In this introductory spirit, I shal not pursue furher detais, but tur to the
lumped case, which is the study of the tempora vaation of the total sunspot numer,
leaving the spatial behavior out of account.
In figue 2 we show.the annual mean' sunspot numer since 1700 in a representation
designed to alow for the magnetic cycle, where the negative of the numer of spots is
shown on every' other cycle, after a suggestion of Braewel. H we use information frm
before 1700, as Eddy has done, we get another view, that of figue 3.
We see that durng the time of Newton there were virtualy no spots. 'A conftion
of this result of historical scholarhip comes frm ' the study of the abundance of C14 as a
function of time using tre rings and cores from the poles. The idea is that, in times of
strong solar activity, the solar wind puls out magnetic field with it. This reaches out to
the earth and shields it frm cosmic rays, which would otherwse mae C14. So the periods
of high C14 abundance are timesoflow solar activity. At any ~te, the,data, so interpreted,
conf the Ma:under minimum, or Newton interssion. More than this, they suggest
that such intermssions are recurent, that is, that solar activity is intermttent.
Naturaly, there are some attempts to model such thigs, though the number of care
48
1211 C.....
G,~M""I""M S. 
i ~~~~___A_=__=_~_J
~ ;;f' ' ~'Ç~11100 1200 1:i 1'00 150 110 170 110 110
O..e
Figure 3: Variation of the concentration of 14C after Eddy.
Figure 4: Projection of the threed phas~ portrat of the solar cycle
fuly observed cycles seems a bit too sma for the purose. The most diect way to
proceed is to attempt to construct a phase portrait of the solar cycle. This can be done
from N(t¡), the number of spots as a function of time. Using a tabulation of these data,
kindly provided by J .A. Eddy, A.N. Wolf and I used the reconstruction trick proposed by
Ruelle and others. This is done by takng the colum of Ni and makng makng a copy
of it displaced by about three year. A third copy displaced by the same amount then
gave us a listing of triplets, loosely analogous to N, Ñ, Ñ, which could be plotted as a
trajectory in a threedimensional phase space. Such a plot, projected onto a plane, looks
like nothing more than a scatter diagram. But put it into a computer and rotate it around
(in thre space) til it looks like something, and you get figu 4. There does seem
to be
some structure; indeed too much to model. The daiy vaations in these data are too rich. , '
The idea is to try to model a simpler version, by smoothig the data. Indeed, you can
smooth the thing til it looks likea ,limit cycle. But, with somewhat less smoothig, say
on a time scale of a. year and half, using the Bracewel trick, you get something that has
the general shape öf a Lorenz attractor as shown. This does not have any intermttency
of the solar kind. . .
kl:
:r
49
The simplest reasonable kind of model seems to be one with about dimension five,
such as Childress used to model dynamos some year ago. But here is one that has the
bowtie look of a Lorenz attractor and gives the right kind of intermttency:
ž  _:z3  2xy + ).x  Eii:i3 2). .
Y  y :z + y  EIIY
,\  E(). + a(x2 + y2  1))
The idea of this model for the solar cycle goes back to discussions with D.W. Moore. We
thought that there could be two dynamo processes in the sun. One is a general solar
dynamo going on continualy, with a second, sub convective dyna.o generating the solar
activity, but strongly coupled to the convective zone. This model is a loose representation
of that picture. For vanshing :z and :i the system is just the wellknown Lorenz system
(transformed to the Walsh Cottage version). The reason for that was Malus' suggestion
that a dynamo could nicely be modeled that way (see also Ruzmakin's discusion of solar
dynamoes). The other two degrees of freedom ar meant to represent a suhconvective
oscilation. At any rate, we get from this primitive idea s~me strong intermttency and
some healthy chaotic oscillations (see the report by Platt, in this volume).
~ .
Iat,rmUat b,hawior
tor ( = 0.1, ø = 6.5, &Ad II = 4.125.
1
z
I.
.. .. T  II
References:
R.Kippenhah and A.Weigert, Stellar Structure and Evolution. SpringerVerlag, Berlin
1989.
M.Stix, The SunAn Introduction, SprigerVerlag, Berli 1989.
G.Newkirk Jr. and K.Frazer, "The Solar Cycle", Phy,. Today, 35, Vol. 4, pp.2533 (1982).
A. Ruzmaikin, Comm. A,trophy,. and Space Sci..
50
A ø~/I StnOofÍrJ, ( .v i ri )
SI! /lv Qi;1;Lt:t ci;i
.
51
Lecture V. Rotating Stars
1 0 bservations
The sun forms the best known example of a rotating star. Galleo was the
first to observe the drift of sunspots on the solar surface in the 17th century.
He noticed that a spot took about 14 days to cross the solar disc, and that
this was roughly the same whether the spot passed through the center of,
the disc or along a shorter path at some distance from the equator. The
rate of rotation also seemed nonuniform with motion appearing to slow as
the spot approached the solar limb. Galleo recognised this as an effect of
foreshortening which would result if and only the spots were near the solar
surface.
Scheiner, a Jesuit priest, observed that the sun took 27 days to complete a
rotation. He discovered the differential rotation of the sun by more accurately
mapping the passage of sunspots at different latitudes. Spots farther from the
solar equator were found to move with a slower velocity. It was also known
that the sun was tilted by ~ 5% to the earth's axs. In 164348, Revelus
managed to map the surfaces of the moon and the sun by engraving their
projected images on a tinplate. Such were the accuracy of these mappings
that modern techniques can be used to analyse the differential rotation. The
results differ from p:resent day readings.
For other stars, however, more sophisticated techniques are required. The
most commonly used is the analysis of spectral lines. For a.nonrotating star,
the presence of certain elements in the stellar atmosphere wil produce dark
absorption lines in the stellar emission spectrum.
If the star is rotating however and radiation is sampled from both receding
and approaching sides of the stellar disc, then the absorption linewil be both
red and blue shifted. As a result the absorption line wil broaden and flatten.
The width of the band is proportional to v sin i where i = inclination of
the stellar axs to the line of vision and v = equatorial velocity. Unfortunately
i is unknown. If the stellar surface is uniformly iluminated the absorption
line would have an ellptical shape. However the flattening at the bottom
indicates the darkening at the stellar limb.
52
Lumosity
atf
L(f)
L(f)
Red
slufted
f
Figue 1: ~ absorption di~
The Broadened
absorption lie
Blue
shifted
.., ,;,, I, ,, ', ,I ,, ,I ,, ,I ,I ,, ,, ,, ,, ,\ I
.. ;
f
Figure 2: doppler broa.dening
53
v .
equatonal
200
kmš1
FAST+
+t:t~+~rl tt+++~ + A add solar system
anguar momentu
to sun
20
SLQW
++ :
                 TTT~~~~+~T~TT~ 
HH. .
.... The Sun
log M ·
Figure 3: Rotational velocities vs log M
There are ways to obtain more information about the stellar rotation if the
surface exhbits nonuniformities. For example tracking a patch of dierent
chemical composition (or other irregularities that appear periodicaly) across
the surface can yield more specific information about the rotation speeds at
particular latitudes. However this approach is not without hazards. Care
must be taken to ensure that variations are not a result of luminosity changes
or of stellar pulsations. Sometimes frequency spliting due to the Coriolis
force may be seen, much like the zeeman effect.
Equatorial velocities tend to have two orders of magnitude; either fast
at 200kms1 or slow below 20kms1 (e.g. the sun corresponds to 2kms1).
Massive stars al are fast rotators whereas low mass stars are slow rotators.
It appears al star loose most of their angular velocity as they age via
combination of magnetic braking and direct mass loss. For our sun, it was
suggested earlier that the process had been predominantly mass loss in the
form of orbiting planets. The solar angular momentum consists of i~oth of
the total solar system value. If the sun was to retrieve the total angular
momentum of the solar system it would be a fast rotator. But observations
of stellar clusters do not support this explanation.
54
GO KO K5 ~O ~t:)
1,10
120
100 l
Ž ao
¡¡
:.
60
+
+ 0 Hll 625 A.'1D H1I73a, WITH
E(8V)=0,04 ASSUMED
+
+
++ +
+
+
+
. 0
. G
+
o SAME STARS, WITl E(BV)=
0,34, 0040 RESPECTlVEL Y
++
+
+ +
+ +
+
+
40
20
+ +
+
i.. + + .t .i ++..:; .i +.i + + '+l
i if i~*~ lH0lñift;t + f.. + + l ++ +
I1,5 2VI '
FIG. 6.Disiribution or spcclroscopic rotational velocities as a (unction or reddcning..orr~icd i" 1 color for the Plc'i.:uJes. HlI 6:!S ilnd HII 7JS arc shoD
l'Nice. corresponding to the two altcrn;uh"c reddening corrections.
o 2,5
. ,5
Figure 4: average age 10E8 years
A star cluster alows a large number (105/106) of stars, of simiar ages,
to be studied. A plot of v sin i versus an Irdicator of mass yields fig. 4 for a
cluster of average age 108 years.
Notice the high degree of scatter although there are the beginngs of
accumulation at v sin i ~ 20kmr1. An older cluster, with average age ~
5 x 108 years exhbits considerable slow down (note the change of vertical
scale ).
It seems that the "slow down" time scale is ~ 500 x 106 years. Nothing
can be said about the direction of rotation. The formation of stars is thought
to be such a turbulent process that any large scale galactic vorticity is lost
and hence no uniform star rotation direction should be expected. Calcium
emission lines can be used to infer direction however this requires much effort.
The orientation of the stellar axs to the line of sight seems such that sin i is
randomly distributed. "
Stars are found to reach a state where their rotational velocity is depen
dent only on their mass, regardless of initial conditions. A star typicaly loses
most of its angular moment��m in its premain sequence period.
The above HertzsprungRussell diagram demonstrates how well stellar
55
co KO KS MO
J5
JO x
6 RADICK
0 DUNCAN
+ SODERBLOM
X KRA
V KRA LIMIT
0 MilT
M,MT. vsim ( 10
o
25
X
o o~
8
20
, I
i
BV
Fie;, 2,Di.inb,uli~n of roiaiional veloe,;lie. in the Hyades as a funciion of 8  V color (or speciral iype), Speciroscopic roiaiional velociiies have
been correcied for inchnauon etrecis ,,unung a random dislnbuiioD of wal onenialions. Sian with roiaiional veloeiiv limiis of u sin i ~ 10 k Ibluer ihan B  V  1.3 have been onulled for clariy, . m s
is 0 ~
o 0
o
~~%o
x Ql+ O:
v w~6i
+ Õ&
O)~o 68 66
O~, , I , , , I , ,
o
10 6""~_
5
6 Ih 6 6
,6 ,8 1.2 1,4 1,6
Figue 5: average age 5x10E8 years
evolution can be predicted purely from the mass. In particular the fial or
long time (109 years) value for the equatorial velocity is independent of the
initial anguar momentum. The main process thought to produce this effect is
the brakng by magnetic stellar wids. Kawaler in "Angular Momentum loss
in LowMass stars" astrophysical journal vol 333 p236247 (1988) produces
a simple expression of angular momentum loss by this process, and applies it
to evolutionary stellar models. The basic assumption in magnetic brakng is
that escaping matter does not possess the angular momentum correspondig
to the stellar surace but to a distance, caled the Alfven radius r A, far above
the surface. In a simplified view, the magnetic lines of force act as a lever
arm, which out to r A, forces the escaping material to rotate rigidly with the
star. Beyond r A the field becomes too weak to enforce rigid rotation and the
angular momentum of the escaping matter is conserved. The loss of angular
momentum dJ corresponding to a spherical shell of mass dM crossing r A
during th~ time interval dt is
dJ 2 dM 2 (r A ) J 2
dt = .3"dt R n (ii rcdicl
56
...0 . .
""0" 0
.00 .....,1 ldyr
2
1

.!
..
~
..
to
o
..
". ".
, .....0
'12~ .
, .
,
,
, ,Ul
,
, 0
,
"
"
O.au "
,
" 0
"
"
,
, .fU
"
a
1
3.9 3.7
log(T.u)
3.63.8
o
.
. 0
.
..
10 Myr
. o
o
0.3~..
3.5
Figure 1. Rotational velocities oC lowmas premaln sequence stars in the lIR diagram. Open
i
and dark circles represent T Tauri stars with Her equivalent widths smaller and greater than 10 Å,
respectively, and the circle area is proportional to 1Iini in the range Crom les than 10 to about 100 km
sI. Solid lines: theoretical premain seuence evolutionary track for stars in the mas range from
0.35 to 3 M0 (from Cohen and ¡(uhi 1979). Dotted line,: isochrones corresponding to an age of 10.
and 10' years, respetively. DiuAed line: theoretical zeroage main seuence.
Figure 6: HertzsprungRussell diagram
57
for a radial field geometry and
dJ 2 dM 2 r A
a = 3dtR n(( If )dipole)
for a dipole field geometry. Here R is the stellar radius and r A is the Alfven
radius to which corotation of stellar wind is maintained. The field strength
determines the value 1t while the field geometry at r A governs the power of
1t in the equation. Kawaler uses:
dJ 2 dM 2 (( r A ) )n
a = 3yR n If radial
where n = 2 is radial field geometry and n = 3/7 for a dipole field. He also
assumes the simplification that the total field strength is proportional to the
rotation rate to some power.
Bo = kB(.! t2na
Ro
Combining
dJ = _kUln1+4an/3(.! )2n( m14)i2n/3( m tn/3dt . Ro mo
Here mi4 = d: in units of 10i4Moyri, and kUl,kv,kB,n&a are free pa
rameters alowing fitting of the modeL. He uses a = 1 & alows n to vary. To
relate J to veq he uses 2 cases A& B. In case A the star is assumed to be
uniformly rotating i.e. the angular momentum is evenly distributed, whereas
in case B a convective layer is assumed steadiy rotating and below this each
spherical shell retains its own angular momentum. The asymptotic result
veq '" t3an/4 is achieved for long time. His results are as follows:
The most interesting plot is, however, below.This clearly shows the in
dependence of the final equatorial velocity on the initial angular momentum
for both cases.
Kawaler shows that the simple formulation of angular momentum loss by
the magnetic winds, when applied to evolving pre mainsequence and main
sequence models, is adequate qualtatively to explore the spin down of low
mass stars. The rotation velocity, at times .~ 109 years, is independent of
the initial J.
58
log(Veq C 1)
log(Veq em  )
. ............... ... dJ/dt=O
........................
n=l
n=2
CASE A
1.0 solar mass
logt
Figue 7: case A
. . .. .. .. .. .. .. .. . ... dJ/dt=O
........................................................ ..
n=l ,
,~
i\
fu
~t:i
n= 1.5
CASE B n=2
logt
Figue 8: case B
59
10g(V 1 )
eq
J=20
..
..
..
..
..
.... "
..
.... ..
:::~::~
::~
CASE A
..
..
.... CASEB
J=Jo/2 mass is 1 solar mass
10gt
Figue 9: dieren.t angular momentum intial conditions
...........\ complicated
unown
10gL
..'
.
\..1.. ....:
....
t D
region ,
A birt
HETZPRUNG
RUSSELL
(1930s)
evolutionar trck
Main Sequence
4:
10gT
eff
Figure 10: Evolutionary track
60
In summary the evolution of a star appears to be adequately characterised
by only the mass for most of its lifetime (ABeD). One would expect depen
dence on the angular velocity n. However as ilustrated above, after the star
has lost most of its angular momentum (at the beginning of its lifetime on
the mainsequence) asettles to a value which only depends on the m��ssM
and not on the initial conditions. Thus if evolution depends on M & n then
Eevolution = E(M, n) = E(M, n(M)) = Ê(M). Eventualy the star leaves
the mainsequence having burnt most of its H2 to He. The resulting loss of
homogeneity produces a subsequent evolution too rich to be described simply
by the mass.
Low mass stars consist of a radiative interior, which is unmied and where
nuclear burning takes place, and a surrounding mied convective zone. At
tempts to model the radiative interior as a ID problem ( depth as the in
dependent variable ) have proved inadequate. It appears that differential
rotation, which is intimately coupled to the magnetic field, is a crucial pro
cess in the dynamics and must be included in any description.
t
'f;
61
2.1 General Properties and The Virial Theorem
In an inviscid, selfgravitating fluid with no associated magnetic field, the equation of
motion is
D
P Dt v = V p  pV ø
where ø is the gravitational potential, that is
(1)
g = Vø.
We define cartesian and cylindrical frames of reference in the following way
; ,~
iS
I
i
..~ !l ~
fig. 2.1: Cartesian and Cylidrical Coordinate Systems.
i.e. the Z¡ are the cartesian coordinates, and s, z, and ip are the cylidrcal polar coordi
nates with unit vectors as shown. We can also define
x = (zi,z2,Z3)T .
In terms of cartesian coordinates, (1) is
D 8 8
PV¡ = p  pø .Dt 8z . 8z .J ,J (1a)
We multiply this equation by Z¡ and integrate over the volume of interest, e.g. the whole
volume of the star. Considering the right hand side of (1a), we see
r pz'.!v' dV = r .!(pz'v. dV)  r p.!z'v. dV  r z.v..!(pdV)J v i Dt J J v Dt i J J v Dt i J Jv i J Dt '
where
dV = dzidz2dz3 .
By definition,
D
z'v.
Dt i  i,
62
and, by conservation of mass, since the mass of a parcel of fluid cannot change as it moves
with the fluid,
D
Dt(pdV) = O.
Thus
r px.Ev. dV = r E(px'v. dV)  r v.v.pdVJ v i Dt J J v Dt i J J v i J
= ~ i xivjpdV  2Kik ,
where Ki1c is defined as
1/2 i ViVjP dV .
(Note that K = Kii is the kinetic energy of the system).
Now, if the volume of integration V contains al the ,mass of the system, which we
assume is the case if the volume contains the entire mass of the star, then
lt = G r p(x',t) dV' ,Jv Ixx'i
where G is the gravitational constant, and dV' is analogous to dV above. Thus,
h a h h p(x t)p(x' t)x .(x .  x'.)pXiltdV = G " .' i J J dV dV'v ax j v v Ix=
Now, if we define
 h L p(x, t)p(x', t)Xi(Xj  xj)(X¡ xD ,
Wij = 1 over2G I 13 dV dV ,v v x  x'
then
r PXiaa ltdV = Wij and W = Wi¡ = 1/2 r pltdV ,Jv Xj Jv
W being the gravitational potential energy.
Finaly, the last term we need to consider is
,~
h:
i"
r xi~pdV =  r ~(XiP) dV  bij r pdVJv aXj JvaXj Jv
=  L Xipnj dS + b¡j lPdV
by Gauss' Divergence theorem, where n is the unit normal to the boundary S of the volume
V. But by definition p is zero there. Therefore, collating al the terms of the integration
of (la), we see that
~ I X¡VjpdV = 2Kij + Wij + bij IPdv (2)
63
where al the tensors on the right hand side are symmetric over their indices. Thus, if we
multiply across by the alternating tensor fijk, we see that the right hand side of (2) is zero,
and the left hand side, in vector notation, becomes
~ I x x vp dV = 0 . (3)
Physicaly, this equation states that angular momentum is conserved globaly.
In the presence of a magnetic field, there is a Lorentz force, and so (1) becomes
P ~t v = Vp  pVqi + j x B , (3)
where j is current and B is magnetic field. In this case
dd r x x vpdV = r x x (j x B)pdV ,t~ ~, (4)
i.e. in general there may be a nonzero torque, depending on the magnetic field structure.
Returning to (2), from the symmetry over the indices of the right hand side, we see
that
~ I XiVjpdV = ~ I XjVipdV .
Therefore
dID d r D 1d2 r
dt Jv Xi Dt VjpdV = dt Jv Xj Dt vipdV = 2" dt2 Jv XiXjpdV .
Thus, if we define the tensor
Iij = I XiXjpdV ,
we see that
Iii = 1= I1xl2pdV ,
the moment of inertia. Thus (2) can be written in the form1 d2 r
2" dt2Iij = 2Kij + Wij + bij JvpdV , (5)
which is known as the Virial equation.
If we take the trace of (5) we,.obtain
~~I=2K+W+3 r pdV2 dt2 Jv'
which expresses the energy balance of the system. We assume that the system of interest
is made up of an ideal gas in a steady state. In the stellar context, deviations from the
(6)
64
ideal gas law are often quite smal, and the evolution of inertia is quite smal compared to
times cales of kinetic energy such as the rotation period. Thus these assumptions are often
quite reasonable. For a perfect gas
p/p=RT,
where R, the gas constant, is defined as K,/iñ, where iñ is the mean mass of the particles.
Also, in terms of specific heat capacities at constant pressure (cp) and volume (cv)
cp  Cv = cv(cp/cv 1) == cv(, 1) = R .
Thus (6) may be written as
1 d.
2" dt2 1 = 2K + W + 3(,  l)UT , (7)
where
UT = 1 cvTpdV
is the total thermal energy, which must be a positive quantity. In the case of an approxi
mately steady state, the right hand side of (7) is assumed zero, and so
2K + W + 3(, l)UT = 0 . (8)
This equation shows the balance of energy in the star system. W is negative, and the
other two terms are positive (cp 2: Cv always). If we initialy ignore macroscopic kinetic
energy, we see that, under contraction, gravitational energy must change into thermal en
ergy. However, in fast rotating stars, UT may be considered smal, yet positive. Clearly
K can never be greater than W/2. But, under contraction, conservation of angular mo
mentum requires the star system to spin faster. Thus excess kinetic energy must be lost
in some way, i.e. radiated. Nevertheless, this is a gross simplification of the true picture.
An increase in thermal energy causes an expansion that wi vary both Wand K. So a
parameter of interest is the ratio of kinetic energy K to the absolute magnitude of the
gravitational potential energy IWI.
A rough measure of this for a star of radius Ro and rotation rate 0 is 102/IWI. Now
fRoI", 47rp 10 r4 dr = 3MR~/5
,
,t
J'
(M = 4/31rpR~) ,
and
/WI '" GM/2Ro .
Thus
102//WI '" 02 R~/GM ,
the oblateness of the star. This is usualy quite smal (in the case of the sun the value is
'" 2 X 105) but may approach higher values (of about 0.1) in fast rotators. If we include
a magnetic field,we also see that, in stellar terms, the magnetic energy is smal compared
to the gravitational energy.
65
2.3 Hydrostatic Equilbrium
Let us thus investigate the implications of the following assumptions:
i) the star is isolated in space, and rotates about a fied axs (which, without loss of
generalty, we assume to be the z axs) with angular velocity O( s, z);
ii) the system is stationary in an inertial frame, i.e. we have purely rotational motion
(often referred to as hydrostatic motion);
ii) dissipative forces may be neglected;
iv) no electromagnetic force is acting on the star.
With these assumptions, the equation of motion becomes
o = ~VP  Vø + ss02(s,z) .
P
(9)
If we now take the curl of (9), we eliminate the gravitational term and so
o = V~xVp+ V(s02(s,z))xs .
P
(10)
Thus, if 0 is purely a function of s the second term in (10) is zero. Thus, for (10) to be
satisfied, surfaces of equal pressure (isobars) and equal density (isopycnals) must coincide.
This is referred to as a barotropic state. In a barotropic state, since 0 is purely a
function of s, we may rewrite (9) as
1 ¡6O=VpVø+V s'02(s',z)ds',P 0 (9a)
where
16 s'02(s',z)ds'
is referred to as the centrifugal potential. Now we see that we can define a "total" or
effective gravitational potential 'I that takes into account the effect of rotation, where
'I == ø 16 s'02(s', z) ds', (10)
and
ge =  Vw
is caled the effective gravity.
(As a sideline, in the absence of a barotropic assumption, an effective gravity is often
defined as
ge = Vø+ss02(s,z) .
In this case, we see from (9) that effective gravity, defined in this way, is always orthogonal
to isobars.)
66
2.2 Differential Rotation and Kinetic Energy
We have aleady aluded to the effect of the principle of conservation of angular momentum
on the evolution of the rotational energy. Let us now consider this more closely in the case
of a sphericaly symmetric rotating star of given mass and radius. If we consider this star
in a cylindrical coordinate frame as defined above, the angular momentum J is given by
J = j, s20(s,z)pdV .
If we now perturb an iiutial constant rotation rate 00 to
O(s, z) = 00 + ��O(s, z) ,
we see that the kinetic energy of the new configuration is given by
K = 1/21 s202pdV = 1/21 s20~pdV + 1/21 s2��f12pdV + 00 I s2��OpdV .
But
J = i s200pdV + I s2��f!pdV = Jo ,
thus
I s2��OpdV = 0
and
K = Ko + 1/21 s2��f!2pdV .
~"
ic
"
f:
Thus any perturbation to uniform angular rotation can only increase kinetic energy, and
differential rotation may be considered as a storage of kinetic energy. This has an important
consequence, since any dissipation within the system wi tend to reduce the kinetic energy,
and hence we have a tendency to uniform rotation.
67
Now, in the barotropic case, we have
1
Vp=Vw
p
i.e. ~dp = dw .
p
(11)
Now, since on a surface of constant qi, d'I = 0, we see that dp = 0 there, i.e. the surface is
also a surface of constant p. We have aleady demonstrated that the surfaces of constant
p and p coincide. These relationships are usualy signified by writing p and p purely as
functions of qi, though of course, qi as a function of p conveys the same information.
Alternatively, if we alow the rotation rate to vary along its rotation axs (i.e. let !l
be a function of z as well as s) we have what is known as a baroclinic star. The curl of
(9) becomes
o = V~xVP+Ødd (,,2!l(s,z)) . (12)P z
The second term quantifies the effect of differential rotation on the hydrostatic equibrium
of the :fuid. In this case, isobars and isopycnals are orientated at an angle, which is a
characteristic of baroclinicity.
In summary, barotropes are distinguished by ~~ = 0, p('I), p('I), where qi is defined
by (10).
2.4 The Von Zeipel Paradox
In the stellar context, varations of rotation rate along z are often relatively smal, and so,
as a first approxiation, we investigate the implications of a state of thermal equibrium
on a barotropic star. If we assume that a barotropic star is in strict radiative equibrium,
the radiative :fux F and the nuclear reaction rate fN are related by
V.F = pfN . (13)
(This is just the evolution equation for specific entropy in thermal equibrium.) We wi
also assume that we are deep enough in the star so that it is opticaly thick. Then F is
given by the Eddington approximation, i.e.
F = XVT, (14)
where X = X(p, T) is the radiative conductivity. Let us assume that the star has uniform
chemical composition, so that the temperature is a function of p and p. Thus in such a
star, al dependent variables can be expressed in terms of 'I, the effective gravitational
potential defined by (10). Thus (13) can be rewritten as
o = V. ( X(qi) ~~ V 'I ) + pfN , (15)
or
¡d~ (x~~)(Vqi?J =¡(X~~)V2'I+PfNJ . (16)
68
If we now make the further assumption of uniform rotation, (i.e. assume n = no, acon
stant), we see that the right hand side of (16) is purely a function of W, and hence is
constant on level surfaces of W, since by definition,
V2lJ = 2n~  47rGp . (17)
However, effective gravity ge (= VlJ) is not a constant on level surfaces in general, and
thus (16) can only be satisfied if
dT
X dw = C a constant, (18)
and, from (17),
eN = 47rGC (1 2~~P) .
This solution is highly unphysical, since eN required to maintain radiative equibrium is
virtualy depth independent except near the surface, where it is a function of the rotation
rate. Also, the model breaks down in the limit of a surface of zero density. This is known
as Von Zeipel's Paradox, who first studied this problem in 1924. Varous extensions of
this paradox to more general rotation laws have been made, and so it is necessary to alow
a barotropic star to depart from thermal equibrium, I.e considering the specific entropy
equation,
(19)
\ ,
pT (Ð: +u.VS) = V.F+peN
= pen =l 0 . (20)
The rotation thus acts as if it generated sources and/or sinks Qf energy, with an energy
generation rate en per unit mass. The presence of this source term on the right hand side
of (20) implies either that the specific entropy wil be localy modified or that motions wi
occur, or indeed a combination of these two effects.
,;, 
1'1f
¡.
,
r
Notes submitted by Richard Kerswell and Colmcile Cauleld.
69
LECTURE VI
1 Governing Equations; Characteristic TimeScales
As shown in the previous lecture, we cannot have a rotating star in thermodynamic and
hydrostatic equilibrium. Therefore, we must solve the full set of equatioris:
p ¡ ~~ + (11 . V) 11J
ap + V . (p11)
at
pT ¡ ~~ + 11 . V s J
r aCi J
p L at + 11. VCi
VppVø+V.it (1)
o (2)
V . xVT + pEn + ¡viscous dissipation of heat) (3)
= sources + sinks + V . DV Ci (4)
where 11 = 3dimensional velocity relative to an inertial frame (i.e. including velocity due
to rotation of the star), it = viscous stress tensor, X = conductivity, En = nuclear reaction
rate, D = diffusivity, and Ci = concentration of substance i. (Thus there wil be an equation
of the form (4) for each substance.) In addition, we have Poisson's Law, describing the
gravitational field:
V2ø = 47rGp
and finaly, we have the equation of state:
(5)
p = p(p,T,Ci) (6)
and likewise:
En = En (p, T, Ci). (7)
To solve these equations, we need to separate scales, both spatialy and temporaly.
Spatialy, we have two natural scales: microscopic and macroscopic. For instance, in the
convection zone of a star, diffusion of energy occurs primarily on a microscopic scale (and
a fast time scale) and is governed by the following flux equation:
Fe = pCpDiT . V S. (8)
Temporaly, there is a wide range of scales, as ilustrated by Fig. 1. The smaler
time scales correspond to hydrostatic balance, rotation, and convection. We are interested
in the larger time scales, corresponding to large scale thermal and nuclear adjustments.
Thus we can neglect terms in the above equations which correspond to fast time scales,
i.e. ap/at and a11/at. This is because density and velocity changes are much faster than
70
anytng
(p,T....)
..
: ....
, 'Wick ~
MjIlSI/ll: r'fdiOt
(lt~dlsl4. ~ oJf)l~(¡l
l?o./MCt) :
1hr 1 day 1yr :
:.._..a.............................................................
we are interested in these time scales
thennal
adjustment
..""
..
..
..
..
.... "
nuclear
adjustment
'
1E7yrs 1ElOyrs log(time)
Figure 1: Time Scales of Motions in a Star
z
.f
y
x
Figure 2: Axisymmetric Coordinate System
entropy and concentration. (Entropy changes are related to average temperature changes
and concentration changes are due to nuclear reactions, both of which occur on slow time
scales, as shown by Fig. 1.)
Therefore, let us rewrite the above equation set. We assume hydrostatic balance and
axisymmetry (on these large time scales; see Fig. 1) and we make the anelastic assumption
(8p/8t = 0, as explained above). Furthermore, we split the velocity into meridional (��) and
azimuthal components: v = ��(s, z) + J (sO + u4i), where 0 = average rotation rate of the
star (considered constant). Using these assumptions we arrive at:
20pu4is
P ¡ :t (su4i) + Ü. Y' (S20 + SU4i) J
Y'. p��
pT ¡ ~~ + �� . Y' S J
r 8Ci .. \i J
p l åt + u. v Cï
 Y'p  pY'i + (Y' . l) m
Y' . (pvs2Y' (:4i) J
(9)
(10)
(11)
(12)
o
Y' . (XY'T) + PEn + visc diss of heat
sources + sinks + Y' . (DY' Cï) (13)
71
where"p = lt  lS202 and (V. l)m = viscous stress in the meridional plane.
This set of equations has only three timederivatives and thus should be relatively easy
to solve if the transport coeffcients of mass (D), momentum (v), and energy (X) plus the
nuclear reaction rate (tn) are prescribed. However, identifying these coeffcients can be
diffcult, because they must quantify transport in turbulent motion.
2 Waves
Before solving the set of equations for long time scales, let us investigate the effects of
leaving ap/at and av/at in place. The momentum equation then is:
ail.. 1
 + 20 x il = ~  V p  V ltat p (14)
where �� = 3dimensional velocity relative to a rotating reference frame and ñ = rotation
rate of the star. Note that we have neglected the stress term V . l of (1) because the
viscosity is assumed to be smal.
Now, taking the curl of (14) yields:
~ (V x il)  20 ail _ 20 (il . V p ) z =  V (~) x V p.at az p p (15)
We have used the anelastic assumption ap/at = 0 to give V.il = _1 (il. V p) and the vector'd . Pi entity: '
V x ñ x il = ��. vñ  fi. v�� + ñ (V. il)  il (V. ñ) (16)
where the first and last terms vanish because we consider ñ to be spatialy constant.
In the special case of zero stratification (V p = 0), equation (15) describes socaled
inertial waves. When V p =1 0, there is a strong coupling between those waves and gravity
(or internal) waves. Only purely horizontal waves do not feel the stratification (since then
Ü. V p = 0); we consider them next.
Writing (14) in spherical coordinates, projecting everything onto a horizontal plane (see
Fig. 3) yields:
aus
at  20 cos 9 Uq,
auq,
i + 20 cos 9 Us
1 ap

pra9
1 ap
pr sin 9 aø
(17)
(18)
72
z¿
y
x
Figure 3: Spherical Coordinate System
Eliminating the pressure then yields:
:t ¡ :0 (sin 0 U.p)  ~~ J  200 sin2 0 Ue = 0 (19)
Now we look for toroidal solutions in terms of Legendre Polynomials and Fourier modes in
the form:
Ue
imP;: ( cos 0) ( . (,L ))
. 0 exp i m.y  utsin
ap;:(cosO) (.(,L ))
 exp i m.y  ut
ao
(20)
U.p (21)
(Note that this velocity field is divergencefree.) The solutions obey the following dispersion
relation:
(un (n + 1) + 20m) P: (cos 0) exp (i (Ø  ut)) = 0 (22)
or: u 20
 = wave speed =  ( )m n n+ 1
These solutions are analogous to planetary waves in oceanography.
(23)
3 Convective Type Instabilities
Returning once more to the case oflong time scales (where we disregard ap/at and aif/at),
we recognize that we have the following feedback loop: Instabilities, then, playa key role
73
large time scale evolutions
(approx. lE8 years)
1
I
intabilties
(hours, days)
transport of average quantities;
feeds back to
large scale motions
1
turbulence
(smal scale motions)
Figure 4: Feedback Loop of Instabilties
in the long time scale motions of a star. Let us therefore consider the different kinds of
instabilities which can arise in the system.
3.1 Double Diffusive Instabilities
First, consider a positive entropy gradient in a nonrotating star. (See Figure 5.) If a
parcel of matter is disturbed from its original position A to position B, then it tends to go
back to A, since the surrounding environment has higher temperature and lower density.
Thus an oscilation develops in the system. If the star is homogeneous then the parcel will
oscilate with a buoyancy frequency:
2 i dB 9 âlnT âlnTN = d = H (( âl )adiabatic  âIn )Cp r p np p (24)
If N2 .( 0 then we have instability (negative entropy gradient), whie if N2 :: 0 then the
system is in a state of stable equilibrium. If the medium is chemicaly stratified (i.e. we have
a gradient of a molecular weight which acts as a restoring force) the buoyancy frequency
has two components
N2 = N~ + N; (25)
where Nf is as before and N; = gjHp(âlnpjâlnp). Here, p is the molecular weight.
Let us examine the effect of dissipation. In a system with entropy stratification, insta
74
S
specific
entropy
(to surface) r
Figure 5: Positive entropy gradient
Amplitude of
the motion
time
Figure 6: Stable oscilatory state
bility occurs if
N2 const(
tvtd (26)
where tv is a viscous diffusion time and td is a thermal diffusion time. If N2 ? 0 and N;
is negligible then there are two cases according to td ~ iv or td "" iv. If td ~ iv then the
system is in a stable oscilatory state. (See Figure 6.)
Iftd"" iv then the oscilations tends to die out. (See Figure 7.) Thus, i£thermal diffusion
is fast enough (i.e. td smal enough) then it weakens the restoring force, damping out the
motion.
The case when N2 ? 0 and Nj. and Ni have opposite signs is interesting because time
scales for diffusion of matter and temperature are quite different. This leads to a double
75
Amplitude of
the motion
time
Figure 7: Oscilations die out
Salty wann
9
Fresh cold
Figure 8: Salt warm water on top of cool fresh water
diffusive instabilty well known in Geophysical Fluid Dynamics, but generaly ignored in
Astrophysical Fluid Dynamics.
First consider Nj. ~ 0 and N; .: O. As an oceanographic example, suppose we have
salty warm water on top of cool fresh water. (See Figure 8.) If the salnity exceeds some
threshold value in the top layer then we have the famous salt fingers instability. There is
a stellar analogy with a binary star system in which one member, having burned al the
hydrogen in its core, throws off heliumrich matter to its partner. The companion star gains
heavier matter and experiences doublediffusive instability.
If Nj. .: 0 and N; ~ 0 (for example a destabilzing thermal gradient and a stabilizing
salnity gradient) then oscilations grow in the system and we have thermohaline convection
or overstabilitYi in astrophysics this may occur in the convection zone of massive stars, and
is caled semiconvection. (See Figure 9.)
76
Amplitude of
the motion
time
Figure 9: Overstabilty
s.n.?
Anguar
momentu
(from rotation axis) s
Figure 10: Positive angular momemtum gradient
3.2 BaroclInic Instabilities
Now, let us add rotation to the system. First consider an angular momentum gradient
(by analogy with the entropy gradient in the nonrotating case). (See Figure 10.) If we try
to displace a parcel of matter as shown above, the conservation of angular momentum wil
act as a restoring force. Similarly to the case of a nonrotating star, we define the Rayleigh
frequency by
N? = 2.~(s2n?o s3 ds (27)
As before we have stability if Nfi ) 0 and instability if Nfi .: O.
Now, we turn our attention to a system which has both rotation and stratification.
77
Assume 0 = O( s, z). Then taking the curl of hydrostatic equation:
o = ~V P + VlP + 05(802)
P
(28)
we arrive at
o = V(~) X VP + V(S02) X'o5
P
(29)
For a perfect gas we have
dp
Tds = CpdT  
p
(30)
Thus,
n
TdS = CpdT  dlnP
Cp
(31)
This implies 1 n
dS = (1  )dlnP  dlnpCp Cp (32)
Substituting this last equation into equation (29) yields:
1 VS
 x V P = V( 802) X oS
Cp P
(33)
We can approximate the above equation by
1 dS dP. 8
1111 sm a = (802)cpp dn dn 8z (34)
dS dP
where n is outward normal and a is a baroclinic angle between dn and  dn' Using
hydrostatic balance we obtain
N2 sin a = 88 (s02) = .! ddS sin aZ Cp r (35)
Note that if n increases towards the equator then V S points outward as shown in Figure
11. Rotating this picture and choosing surfaces of constant entropy and pressure we obtain
Figure 12. Two possible scenarios for the displaced parcel of matter are displayed. In case
(1) we have stability because dSjdr ~ O. In case (2) the displaced parcel of matter is
warmer and lighter than its environment and tends to move away due to buoyancy. Thus
the shaded region in Figure 12 is unstable. However, for the case of an axsymmetric
78
z 'CS _ VP
s
Figure 11: Rotation and stratification
o
1)
1('\
i
i
I p=const
VP
Figure 12: Rotation and stratification
79
..
Figure 13: Coordinate system
displacement, the system is stil stable if Nf, ~ 0, since type (2) displacements are ruled
out by the conservation of angular momentum.
If we add dissipation with v ~ "', the system is unstable when Nf, ;( ~N2, where N2 is
a buoyancy frequency described earlier.
Another instabilty occurs in axsymmetric system when N2~ ;( Is88~21. This instability
is caled the GoldreichSchubertFricke instabilty.
For nonaxsymmetric displacements, the. necessary condition for linear instability is
that anI ay = 0 in the domain where n is the potential vorticity:
, ,VBn=(20+Vxu)
p
(36)
and :y is the derivative in the latitudinal directionl (see Figure 13). Thus, we have insta
bility if there is an inflection point in the longitudinal direction of the velocity field. It can
be shown 2 that
an _ ß a2u _~~L pau
ay  ( ay2) pazN2( az) (37)
where ß = Oc:s6, f = Osin.O. Thus, the necessary condition for instability dnldy = 0 is
analogous to the Rayleigh criterion for paralel shear flows.
80
References
¡11 J. Pedlosky, Geophysical Fluid Dynamics, SpringerVerlag, 2nd edition, (1987)
¡21 J.P. Zahn, Instability and Turbulence in Rotating Objects, Cource 8, Astrophysical
Fluid Dynamics, Elsevier Science Publishers B.V., (1989)
Notes submitted by N. Platt and W. Welch
81
Lect ure VII.
Shear Instabilities
1 Introduction
Upon the main sequence, where they spend most of their life burning hydrogen, stars can
be roughly divided into two classes. The low mass stars (with masses less than 1.6 Mø)
have convective envelopes and mostly rotate slowly. Higher mass stars have convective cores
and usualy rotate comparatively quickly. Within these convection zones, the velocity of
the flows can become quite large, and in some cases approaches the speed of sound. Conse
quently the timescale upon which quantities are advected throughout the convection zone is
very short (of the order of months) in comparison to the typical times upon which thermal
or evolutionary adjustments take place. In the radiative regions of the star, motions gener
ated by the rotation are much slower. These motions (instabilties) can, however, transport
chemical elements and momentum and are therefore important for the slow adjustments and
evolution of the star. In fact their effect can be embodied within the transport coeffcients
(such as viscosity and diffusion) that occur in the fluid dynamic equations governing the
adjustments of the star.
Our primary concern are shear flows, since these are common in differentialy rotating
stars. Instabilties in shear flows were first investigated by Rayleigh (1880). He showed that
a necessary condition for the existence of a linear instabilty within the mean flow of an
inviscid fluid is simply that there be an inflexion point in the velocity profie in the direction
of the shear (as ilustrated in figure 1). This result was extended by Fjortoft (1950) who
showed that if there was a shear in the background How, then for linear instabilty, the
vorticity must be localy enhanced at the inflexion point.
y y
Crossstream velocity, U dU I dy
Figure 1. An unstable shear flow. Shown are the crossstream velocity profie and the vorticity.
82
k Stable
Unstable
Re
Figure 2: The range of linearly unstable horizontal wavenumbers as a function of the
Reynolds number of the flow Re. The background flow has no shear.
This flow is unstable to smal perturbations. If the perturbation has the dependance
eik:r upon the coordinate in the direction of the flow, x, then the perturbation is unstable
provided kd .( 0(1) where d is the characteristic scale of the velocity inflexion, and the
mean flow has no background shear. That is, there exist longwave instabilties.
When the fluid is viscous, then the instabilty must also compete against dissipation.
The range of linearly unstable wavenumbers as a function of the effective Reynolds number
Re = /dU / dz I d2 /U, is shown in figure 2. (In this example there is no background shearing
motion. )
2 PoIseuIlle Flow
Plane Poiseuie flow describes the flow of fluid between two planeparalel plates. The mean
velocity profie is parabolic and is strongly influenced by the presence of the boundaries. In
fact it is the presence of these boundaries that causes the flow to become linearly unstable
a.t a Reynolds number of R ~6000. Experimentaly the onset of instability occurs at critical
Reynolds number of Rc ~ 1000. This is indicative that there are finite amplitude instabilties
at lower values of R. Such instabilties can be examined by developing the velocity potential
of the perturbation to the mean flow U(y, t) as a set of discrete modes:
u = xU(y, t)  V x z'l(x, y, t), (1)
with
N
'l(y, t) = L 'in(y, t) ��nk:.
n=l (2)
This expansion has the advantage that the variation of the perturbation in the direction
of the shear can be analyzed in detail at the expense of retaining only a few modes to
describe the variation in the direction of the mean flow. Of course one stil has to specify
the wavenumber k, but it can be treated as a free parameter and varied to estimate its effect
83
0,03
10'
E
0,04
0,02
0,01
/0 ./...... /.. ./..0,5 1,0 1,5 2,0 IX
FIGURE 2. Ilustrating tho solution surface on which finite amplitude steady solutions aro found in the onomode approximation
(case I). The vertical coordinate is the disturbance energy E (cf. (2.22)). The linear instability loop and tho region of metastabilty
for case I shown in figuro i are indicated in the E = 0 plane.
Figure 3:
upon the solutions. A reasonable choice of its value might be that value that corresponds
to the critical Reynolds number. The nonlnear system derived for the case N = 2 was
investigated by Zahn et al. (1974). They found that this system does indeed exhbit finite
amplitude instabilties at Reynolds numbers below the critical value for the onset of linear
instabilty. In fact the critical Reynolds number was found to be reduced to f' 3000. The
range of unstable solutions with a given amplitude as functions of the wavenumber k and the
Reynolds number is indicated in the threedimensional plot shown in figure 3 (taken from
Zahn et al., 1974). The nonlnear solutions that figure 3 is derived from can in some cases
be described by stationary waves in some Gallean frame moving in the direction of the flow
with speed c. These are, however, not observed in the experiments, suggesting that even
these are unstable to some other solutions. A further analysis by Orszag and Kells (1980)
seems to confirm this; These twodimensional solutions are unstable to threedimensional
perturbations. This reduces the critical Reynolds number even further and brings tolerable
agreement between theory and experiment.
84
y II
I
I
I
I
I
I
I
I
I
I
y
D
i
I
I
I
I
I
Crossstream velocity, U dU / dy
Figure 4: Couette flow and a peruturbed Couette flow containing an inflexion point.
3 Plane Couette Flow
Couette flow describes the motion of a fluid with a uniform shear transverse to the direction
of the flow (figure 4). Theoreticaly this flow is linearly stable, but it is found to be unstable
in the laboratory. When the flow also contains a superposed inflexion point, as ilustrated
in figure 4, the flow un surprisingly becomes linearly unstable.
Lerner and Knobloch (1987) has considered the inviscid case and finds unstable wavenum
bers with k ( €j d, where € is the fractional increase in the vorticity. Since the growth rate
of the instabilty must exceed the rate of viscous dissipation one expects instability for
(J ~ IdUjdyl€ ). kvjd2. (3)
If the characteristic scale of the perturbation is L = 1r j k then these conditions can be
rewritten in terms of the effective Reynolds number of the flow,
Re = IdUjdyl€2L2jv). (Ljd)2, ¡~.~.j,
"
rand 1rd/ L ( €. These conditions define a region of the plane €  (dj L) in which instabilities
exist (figure 5). There exists a critical value of € for the flow which scales as Re1/3 (Dubrulle
and Zahn, 1990). A finiteamplitude analysis of Couette flow has yet to be performed.
4 Stabilization through Stratification
Under certain conditions shear flows can become stabilzed. When the fluid is stratified in
the direction of the shear, a necessary condition for linear stabilty can be written in terms
of the Richardson number N2 1Ri = IdUjdyl2 ). ¡, (5)
85
Unstable
Stable
Stable
d/L
Figure 5: The range of linearly unstable perturbations with a characteristic scale L shown
as a region in the f  (dj L) plane.
where N is the buoyancy or Brunt Väisälä frequency and the fluid is stratified in the
direction of y. Therefore stratification can loosely be thought of as providing a stabilzation
(but see Thorpe, 1969).
However, stratification is not always felt by the instabilty. If the characteristic timescale
upon which the disturbance is smoothed out by the diffusion of heat, K,jd2 (where K, is the
diffusivity), is shorter than the characteristic timescale ofthe perturbation, IdUjdyll, then
the stratification is not felt by the perturbation. Therefore some criterion concerning the
diffusion timescale must be introduced into equation (5). If the Peclet number is written as
Pe = IdUjdyld2 j K"
then the necessary condition can be written as
Ri = Id~;yI2 (Maæ(1,Pe)t1 ) 0(1).
If we introduce the Reynolds number R = IdUjdyld2/v, then where the Peclet number
exceeds one, the condition becomes
N2
/dUjdyI2(VjK,)R) 0(1).
Since instability wil only occur when the Reynolds number exceeds a critical value Rc, the
complete condition for stabilty is
N2
IdUjdyI2(VjK,)Rc) 0(1).
86
Another possible restoring force that may cause stabilization in stars is that provided by
rotation. The analogue to equation (5), the necessary condition for stabilty in the rotating
fluid would be
Ri = Id~~yI2 ~ 0(1),
where No is now the Rayleigh frequency. However, in contrast with a stratification, it
appears that the coriolis force is not capable of stabilzing the flow.
References
1. Dubrule, B., and Zahn, Z.P. (1990). Submitted to J. Fluid Meeh..
2. Fjortoft, R. (1950). Geofys. Publ., 17.
3. Lerner, J., and Knobloch, E. (1988). J. Fluid Meeh., 198,117.
4. Orszag, S.A., and Kells, L.C. (1980). J. Fluid Meeh., 96, 159.
5. Rayleigh (1880). Scientific Papers, 1, 474 (Cambridge university press).
6. Thorpe, S.A. (1969). J. Fluid Meeh., 36, 673.
7. Zahn, J.P., Toomre, J., Spiegel, E.A., and Gough, D.O. (1974). J. Fluid Meeh., 64,
319.
Prepared by N.J. Balforth and Brian Chaboyer
87
Lecture VIII.
Shear Flow Instability
and the Transition to Turbulence
1 TaylorCouette Flow
The TaylorCouette experiment provides another example of shear flow instabil
ity. In a classic paper, G. i. Taylor (1923) described experimental and analytical results
concerning the flow between two coaxal cylinders of rad�� R1 and R2, with R1 c R2, ro
tating with angular velocities 01 and O2, respectively. We begin our analysis of such an
experiment by looking for steady solutions. We wil use cylindrical coordinates where s de
notes radius, ø azimuthal angle and z displacement along the axs. The radial symmetry of
the problem suggests that we look for solutions where the pressure p and angular rotation
rate 0 depend only upon s, and where the other two components of the velocity are zero.
The sand ø components of the NavierStokes equations then simplify to
dp
ds pS02 (1)
1 d ( 3 dO)
S2 ds s ds
= 0 . (2)
Equation (2) has solutions of the form sn. The general solution is
k1
O( s) = 2" + k2 ,
s
(3)
where the constants k1 and k2 are determined by the boundary conditions 0(R1) = 01 and
0(R2) = O2, In particular
k _ 02R~  01Rl2 R2 R2
2  1
(4)
1.1 The Rayleigh condition for linear inviscid stability
If we ignore viscosity, there is a simple condition for instabilty of the flow (3) to
smal perturbations. This condition is the analog (in a cylindrical geometry) of the well
known Rayleigh inflection point criterion for planar shear flows. If the angular momentum
Is20( s) I increases with increasing s, then a fluid parcel displaced in radius experiences a
88
restoring force, otherwise it is pushed further away from its initial radius. Mathematicaly,
the Rayleigh condition (necessary for stabilty) is that
o ~ (S20) = 2s0k2 2: 0 , for al s in the interval Ri ( s ( R2 . (5)
Since vorticity w = djds(s20)js, this condition is equivalent to the statement that w(s)
and O(s) have the same sign everywhere in the flow.
If the cylinders are rotating in opposite directions, then O( s) changes sign be
tween them, and since k2 is constant (5) is not satisfied and thus the flow is unstable. If
the cylinders are rotating in the same direction (Oi 2: 0, O2 2: 0) the Rayleigh condition is
satisfied if and only if k2 ) 0, or 02R~ ) OiR~ . (6)
Therefore, when the inner cylinder is at rest (Oi = 0), the Rayleigh condition is satisfied
whereas if the outer cylinder is at rest (02 = 0), the Rayleigh condition is not satisfied and
the flow is unstable.
We note two important limitations of the above analysis. First, the Rayleigh
condition (5) is not suffcient for stabilty. The above analysis applies only to axsymmetric
disturbances, and the flow may be linearly unstable to more general types of disturbances.
Second, the Reynolds number of the flow was assumed to be infinite, and nonzero viscosity
may stabilze flows that do not satisfy (5). Despite these limitations, the Rayleigh condition
proves a simple rule of thumb for stabilty in the case of large Reynolds number.
1.2 Two types of instability
Figure 1 summarizes the stabilty regimes for TaylorCouette flow. The Rayleigh
criterion (6) gives the inviscid linear stabilty boundary 02R~ = OiRl. The region of linear
instabilty, including viscosity, was calculated from theory by Taylor (1923). This region
is eventualy entered as the velocity of the inner cylinder increases. Experimentaly, when
the flow enters the region of linear instabilty it undergoes a transition into a different type
of steady flow. Taylor vortices appear, which are radialy symmetric circulations similar
in appearance to convection cells (Figure 2). The agreement between the theoretical and
experimental linear stabilty boundaries is excellent. As the velocity of the inner cylinder is
increased further several more transitions occur until the motion is three dimensional and
eventualy turbulent.
By deriving a bound on the total energy of the flow, Joseph & Munson (1970)
showed that in the hatched region of Figure 1, the flow was stable to perturbations of
arbitrary amplitude. However, the flow is not stable to finite amplitude disturbances in
the entire region of linear stabilty. In fact, as early experiments demonstrated (Wendt,
1933), a completely different kind of instabilty from that seen by Taylor can occur. Wendt
89
Wendt eXPrients
Figure 1: Summary of stabilty regimes for TaylorCouette flow.
'i
.a:4 1( i.
i.
conducted an experiment in which the inner cylinder was at rest (in this regime the flow
(3) is always linearly stable). At he increased the speed of the outer cylinder, he observed
a transition near Reynolds number R = Rt to a turbulent flow which was intermittent in
space. When the outer cylinder was then slowed down the instabilty persisted until R
dropped below Rc, where Rc .: Rt. From this we can deduce that the instabilty requires
a finite kick, or is a sub critical instabilty, although Wendt himself did not give such an
explanation.
Coles (1965) obtained the sub critical instabilty by starting in the linearly un
:0
." ...
'. ......
".1 ��0
.Q, ....'. ,". .. .l.;:.:
rD
0.....'
. ..
0...
. .
..
Mu
Figure 2: Taylor vortices between coaxal cylinders.
90
stable region and then slowly varying the rotation rates f!¡ and O2 in such a way that
the instabilty was always present yet the inner cylinder eventualy was brought to rest.
Experimental results support the hypothesis that the subcritical instability is found only
above the critical Reynolds number Rc, which has been estimated experimentaly as
Rc = 02R2(R2  Ri) ~ 2000 .
v
(7)
It is interesting that this value is close to the critical Reynolds number for plane Couette
flow.
2 Geostrophic Turbulence
The standard view of three dimensional turbulence is that the energy of the
flow moves from large scales to smal scales where it is eventualy dissipated by viscosity.
However, the large scale motions of planetary atmospheres and oceans are to leading order
two dimensional and geostrophic (i.e. horizontal pressure gradient balanced by the Coriolis
force). The energy cascade of geostrophic flows differs greatly from the energy cascade
characteristic of three dimensional turbulence, and in this section we examine the relevance
of the theory of geostrophic turbulence (Rhines, 1975; Pedlosky 1987) to stellar dynamics.
In a two dimensional geostrophic turbulence, the peak of the energy spectrum
shifts in time towards a wavenumber 1/ Rd, where ~ is caled the Rossby deformation
radius and is defined by
NHRd=¡ (8)
Here N is the BruntVäisäla frequency and H is the pressure scale height. The parameter
f = 20 sin (), caled the effective Coriolis parameter, is the component of the planetary
vorticity OZ normal to the planetary surface at latit.ude (). The Rossby deformation radius
is very large near the equator and approaches zero at the poles, but for the midlatitude
regions of the earth, Rd f' 100 km for the oceans and Rd f' 1000 km for the atmosphere.
For many stars, the Rossby deformation radius at midlatitudes is on the order of the
radius of the star T (if the stratification is felt, i.e. P��clet number much larger than unity).
Thus the common oceanic and atmospheric approximation that f is constant (or linear)
over a smal portion of the surface wil be invald in general, and evidently we must include
from the start the full spherical geometry of the problem.
The argument of the last paragraph leads us to believe that if geostrophic eddies
exist in stars, their horizontal length scale L may be comparable to the stellar radius T.
Let U be a typical velocity in such an eddy. If the turnover time of the eddy U / L is greater
than a rotation period, the eddy can excite Rossby wavesthe characteristic low frequency
91
waves common in the earth's atmosphere and oceans. Presumably if U / L is comparable to
the frequency of a Rossby wave, the eddy wil disperse as a packet of Rossby waves. Thus,
a persistent geostrophic eddy must satisfyU ßl
 (L 12 + n2 + (L/RJ.)2 ' (9)
where 1 and n are the horizontal wavenumbers of the Rossby wave. The parameter ß is
defined by
ß = 20 cos 8 = ! 8 f .
r r 88
When Rd ~ r, ß is approximately constant in a region around the latitude 8 = 80 (this is
the case in atmospheric and oceanic situations). However, for a star Rd '" r, and ß varies
considerably over the problem. In particular, ß goes to zero at the poles, which are regions
where Rossby waves cannot exist. Once again, a complete treatment of the problem is
more complex as it must take into account the sphericity of the star, but the analysis here
at least gives us an idea of what may be expected.
(10)
2.1 Dissipation of Energy
We now address the problem of the dissipation of energy in the horizontal modes.
It is clear, though that eddies and waves dissipate energy differently. Where there are
eddies, one can form a diffusivity from the largest length and velocity scales and reasonably
expect this diffusivity to damp out the eddies. In waves, which can transport momentum,
however, dissipation is much weaker, and may also occur in wave breaking. Thus in order
to know how energy is dissipated at any given point in the How, we must know in which
regimes to expect eddies and in which regimes to expect waves.
A clue is given by the Rossby number Ro = u/201. The momentum equation
for a rotating Huid contains the sum of advection and Coriolis terms:
(u . V)u + 20 x u
Letting 1 be a characteristic length, we can eS,timate the magnitude of the advection term by
u2/1 and the magnitude of the Coriolis t,erm by 20u. Their quotient, the Rossby number,
then tells us the relative importance of advection vs. Coriolis forces for the How. For
example, on smal length scales Ro ). 1 and we expect the Coriolis force to be negligible.
In these regimes, then, the How should exhibit threedimensional character and an inertial
cascade leading to viscous dissipation. When Ro ( 1 rotation is important and the How
should posess anisotropic character and presumably waves. Hopfinger, Browand, and Gagne
devised an experiment which demonstrates these different How patterns on different scales
92
l
Figure 3: Schematic of the experimental setup of Hopfinger, Browand, and Gagne with
results.
Îzo 1
Q.~ . ;)
JnA~J
V ~'co C J
D
\A,.
.l "=
!. J\..:L: " ,
13
, Oscilati
grd
1: ranspa,r,e,n t
cover
'.. " ,c.i ' ')
Cj '�� r" ~)c.
o u 2 . s i (.J )
I
't ' .
Et . Laser .
l "an,e~o~ete,r
e 'u
o
'.00
e
U'
, '"
j;
t.
.4: em,
pI
in a rotating fluid. Their setup, diagrammed with their findings in Fig. 3, consisted of a
circular cylindrical tank which could be rotated about its axs. They introduced energy
into the system by oscilating a grid close to the bottom end of their tank. They found
turbulent eddies near the oscilating grid and oscilatory disturbances toward the other end
of the tan, since the turnover rate scales as ulZ '" Z2, the Rossby number decreases
with height. The transition from eddies to waves occurred at the critical Rossby number
ROcrt = 0.2, with eddies in the region where Ro ~ 0.2 and waves in the region where
Ro.c 0.2 which agrees with our previous arguments.
2.2 Stabilizing Effect of a Composition Gradient
Before we resume our discussion of stars, let us briefly discuss an experiment by
Stillnger, Mellard, and van Atta. They maintained a salnity gradient in a rectangular
tank which they disturbed at one end with an oscilating grid (Fig. 4). Recal that for our
purposes in stars, a salnity gradient has the same characteristics as a molecular weight
gradient provifded by elements heavier than hydrogen.
Fig. 5 summarizes their results with the following scales also plotted: dissipation
scale, characteristic length scale, and the Ozmidov scale, which is the scale for which
ulZ = N"" where N", is the buoyancy frequency. Their results show the usual progression
of eddies near the grid, a transition region with fewer and fewer eddies and more and more
wave characteristics, followed by waves only beyond a certain point. What is surprising,
93
 î
 \. '" ~ L ' \ ' t' �� ��
 t..J~ '

~
\1 ( /) v. ¿t j
~v ø\  ' ) ~ L'..) J L? L )' ~ t ) ' ç
, L 'j L L.
' J '\ .) �� L I
' .J
' L. '
L. L
Figure 4: Schematic of experimental setup of Stillnger et. al., with qualtative results.
/) ~JJ
r r wLt.h
lL  1'
 = ¡.
t.
10
l\ 1.4 LRt \ .
. ~ \,
r\,i L \\
,
Completely
active
tubulence
Transition
region In ternal
waves
L (em)
I ""\, ¡ , j ,
i Upper bound due to' _   ' '
r :~tiOOS~;~~~1Ienlcel~' ~ ~if; ~"~~¡,nl ~
~\ p~~~~~mp¡n. duotn ri":¡:'
.£ .. c L I:/:J',,'/"l ,\, . (,..
? ec .le.
i
10010
xnJ
Figure 5: Evolution map for homogenous turbulence in a stably stratified fluid with Lx ~
LR initially.
94
1i ~
b
ß
C\: ",i ': V
b: \.:51
~
Co.. _4 3
: e.J t
A: 3 I) r~��ol\
B: 2  D "~io'"
Åc�� ~
Figure 6: Regions of eddies of size Z and velocity u.
though, is that the eddyonly tltrbulent region is so short, and that the waveonly region
begins so soon. Thus, the experiment shows that a salnity gradient (i. e. mass gradient
in stars) has an anisotropic effect on the flow. Indeed the group reports that the velocity
u '" Zl which means that (since length 1", z) ulZ '" Z2. '
2.3 Vertical Diffusivity in Rotating Stars
Let us now use these ideas to begin to, buid a theory of rotating stars. We
wi begin by determining the eddy diffusivity. Fig. 6 is a graph showing the distribution
of eddies of size 1 and velocity u. Line a, alongi which ul = v, we wi cal the viscous
dissipation line. We are interested in the region above the viscous dissipation line where
Re ). 1. Line b, along which ull = n, we wi cal the Rossby line. The Rossby line
divides the region above the viscous dissipation line into two regions: region A above the
Rossby line where Ro ). 1, and region B below the Rossby line where Ro c 1. In region
A we expect threedimensional turbulence obeying u3 II = fti where ft is the generation
rate of turbulent kinetic energy. This relation, which is equivalent to Kolmogorov's k5/3
law, is in Fig. 6 as line e, which we wi naturaly cal the Kolmogorov line. In region B,
where Coriolis forces make the motion twodimensional, we expect geostrophic turbulence,
or waves. It is clear from the above that the velocity u and the length 1 are the important
parameters. The eddy diffusivity should be their product when they are largest in region
A. This is clearly at the intersection of the Rossby line with the Kolmogorov line which is
marked as a in Fig. 6. At a, u = lftln and 1 = lft/n3. The vertical diffusivity, nv, is
thus
ftnv='
n2
95
In the presence of a molecular weight gradient, the Rossby line is replaced by a similar line
where u/Z ='N;, when N; ~ 02, and the vertical diffusivity is then
ft
Dv = N2'
~
3 Thermal Imbalances in a Barotropic Stars
Now that we understand how rotation affects instabilties, let us begin to consider its
effect on the star as a whole. Recal that although rotating stars can be in hydrostatic
equilbrium, since rotation acts as if it generates sources and sinks of energy, radiative
equilibrium cannot be achieved, and therefore
pT(~~ + u. VS)  V . F + pfn
pfn =l 0
(11)
where F = XVT is the heat flux and fn is the energy generation rate per unit mass.
We wi compute fn in the manner we have done before, except that we wil restrict our
attention to the barotropic case since this is the only case in which we can express fn in
closed form.
For a barotropic star, the Laplacian of the total potential '\ is
V2'\ = 47rGp + ~.!(02S2)
s ds
(Recal that in the barotropic ca.se S, p, X, fn, etc. are functions of'\ only). Furthermore
F = X(~~)VT.
Recalng that 0 is a function of s only (see Lecture 1, Hydrostatic Equilbrium), we now
split the rotational term above into a mean part (02) over a level surface, which is a function
of W only, and its fluctuating part (02)', which depends also on the latitude:
~.!(02S2) = 2(02) + 2(02)'.
S ds
In the same way we shal expand the square of the effective gravity:
(V'\)2 = (92) + (l),. (12)
96
Likewise, the right hand side of eq. 11 can be written
 V . F + pf ( V . F + pfn) + (  V . F)'
(d~ (X ~~) (l) + (X ~~)( 47rGp + 2(02)) + Pfn)
((92)' d ( dT) (2) (2)'( dT))'+ 92) d'i X 'i 9 + 2 0 X d'i .
(13)
Note that the X(dT jd'i), which was constant before, is now in the barotropic case a function
of 'i, which we wil cal c('i). We thus deduce the vector equation
X('i)VT = c('i)V'i
of which we take the flux over an equipotential surface ~('i). The integral of the left hand
term is
f k X('i)VT dB = L('l),
where L('l) is the luminosity, i. e. the total energy traversing that surface, and that of the
right hand term is
c('l) f f V'i . dB 
1:
c('i) f f f V2'i dV
V
c('i) ( 47rGM('l) + 2(02)V('l)),
where M(w) and v(w) are the mass and the volume contained by the equipotentialsurface.
Hence L 1
c('I) = 47rGM 1  A
where L, M, and A = i~~~ (the oblateness) are functions of W. With this value for c('I),
and recalng that the mean part of eq. 13 is zero, we get the following expression for the
energy generation rate fO which is due to the rotation:
fO = (92)' (L (1 .J~~;) _" fn) _ (02)' ~~.
(92) M 1  A, 27rGp M 1  A (14)
Let us examine the sign of fO in a uniformly rotating star ((0)'). In most of the star the
nuclear reaction rate, fO, is negligible. The sign of eq. 14 is then governed by two terms:
the (92)' term which is varies with latitude, and the 1 (02) j27rGp term which is a function
of depth alone. Since gravity is stronger at the poles and weaker at the equator, it is clear
from eq. 12 that (92)' :: 0 near the poles and (92)' .. 0 near the equator (since (92) is
97
Q.

"'
'\
\
+
+
+
 
+
+ +
t
Figure 7: Regions of sources (+) and sinks () of rotational energy within a rotating
barotropic star. At Te, (!V) 127rGp = 1.
constant). Thus, at a fied depth, fn changes sign as one moves from pole to equator.
Deep enough within the star the term (!V) j27rGp is smal, and so 1  (!V) j27rGp ~ O. As
one moves outward to the star's surface the density decreases, and at a certain distance
from the center, Te, (lt2) 127rGp = 1. For T ~ Te, 1  (lt2) j27rGp .. O. This is summarized
schematicaly in Fig. 7 where "+" represents sources and "" represents sinks. Since the
radial component of the meridional circulation is proportional to fn, the star is divided into
two separated circulation cells: an inner one and an outer one. This separation is indeed
exhibited in the results of a numerical study conducted by G. G. Pavlov and D. G. Yakolev
(Fig. 8).
And finaly, let us remark that the term, (lt2) 127rGp, whose sign was responsible
for the division of the rotating barotropic star into inner and outer circulation cells regions
(eq. 14), gives rise to a singular perturbation: It is inversely proportional to density p, and
is second order in the oblateness A. Since (92)' is first order in A, the 11 p dependence of
fn means one must be extremely careful at the star's edge, since a first order theory in A
wil not exhbit the outer convection cell. It is unfortunate that a whole literature missedthis phenomenon. %
;. 
"
G
¡,
l:
FIG.8. Mendlonal circubtion in a
s:ar In the upper pan o( the main se
quence with 0 = canst. The dashdot
line denotes the flow interface p = P.:
at the dead poims Qieavy marking)
the flow, velocity equals zero: C) con
vecuve core.
98
References
. Coles. JFM 21 1965 p. 385.
. Hopfinger, Browand, and Gagne. JFM 125 1982 p. 505.
. Joseph, D. D. and Hung, W. 1971,Arch. Ration. Mech. Anal. 44, i.
. Joseph & Munson. JFM 43 1970 p. 545.
. G. G. Pavlov and D. G. Yakolev, 1978, Sov. Astron. 22, 595.
. Pedlosky, Geophysical Fluid Dynamics, SpringerVerlag 1987.
. Rhines. JFM 69 1975 p. 417.
. Stilnger, Mellard, and van Atta. JFM 131 1983 p. 91.
. Wendt. Ing. Arch. 4 1933 p. 577
Notes submitted by George Bell and Eric Won.
99
Lecture ix.
Flow Between the Sun's
Convection and Radiation Zones
and Transport of Chemicals
1 Introduction
In the last lecture we started the discussion of the large scale behavior in space
and the long time behavior of stars and pointed out that their basic driving mechanism is
the Stokes term and that we deal with a situation of nonthermal equilibrium. Here we want
to discuss the influence of stress where we have in particular the sun in mind. Remember
that in the convection zone nearly al heat transport is carried by convection. Radiation
plays a negligible role. The differential rotation (i.e. that the equator rotates about 30%
faster than the poles) in the convection zone is enforced by Reynolds stresses. We know
today from helioseismology that differential rotation goes through the whole convection
zone and that in contrast the radiation zone appears to rotate uniformly. This information
comes from measurements of the eigenfrequencies of acoustic waves. The splitting of the
eigenfrequencies due to Coriolis forces can nowadays be measured with depth and latitude.
The latter indicates that no differential rotation takes place in the radiative core of the
sun. Thus between the convection zone and the radiation zone there must be a layer
in which the switching to differential rotation in the sun's convection zone takes place.
Within this boundary layer, which is just below the convection zone, additional circulations
have to compensate the jumps in angular momentum. Since the of the convection zone is
slower than of the radiation zone over most latitudes, one has to expect Ekman layer like
circulation between both similar to Taylor vortices as shown in Fig.I.I.
From observations follows that in this boundary layer seems to be a jump in
the angular velocity, but that the average angular momentum varies smoothly. Thus there
seems to be no net torque, only a differential torque imposed by by the convection zone. In
this lecture we first discuss in more mathematical detai the processes leading to flows in
the boundary layer between radiation and convection zone. After that we shortly discuss
the transport of a chemical and close the lecture with an application of these results to the
Lithium transport.
100
Fig.!.!. Convection and radiation zone and meridional circulations between
both.
.D
"ractilltiolA
~Okt.
,.
fig. 2.1: Spherical Coordinate System.
rL ~
J
'tl
101
2 Circulations below a Convective Envelope
As before, we make the following assumptions:
i) the fluid is in hydrostatic equilbrium;
ii) the fluid motion is axsymmetric (i.e. motion is independent of azimuthal angle);
iii) the anelastic approximation may be made, i.e. the partial time derivative of the
density in the equation of mass conservation may be neglected, so that it becomes
V'.(pu) = 0 ;
iv) the Rossby number of the fluid motion is much less than one. (The Rossby number,
Ro, is a measure of the effect of rotation on a fluid motion of a certain scale. The
simplest case to consider is that of a body of fluid rotating ~ith constant period ni.
A fluid motion scale L and characteristic velocity U has a natural timescale, or period,
LjU. If this timescale is much less than the timescale of the rotation (1jn), then the
rotation wil play a very important role in the evolution of the system. Thus, if we
define Ro as the ratio of these two timescales, i.e.
URo = 2nL '
then in the case of Ro ~ 1, rotation effects are likely to be important).
Above we described the flow in a cylindrical coordinate system. A perhaps more
natural system is that of spherical polar coordinates r, 9, and azimuthal angle ø (see fig.
2.1).
We assume that the oblateness of a spheroid of interest (e.g. the sun), is suffciently
smal to alow us to treat the radial direction as perpendicular to horizontal surfaces
without introducing significant errors. We are interested in stationary solutions. In this
case, we split al the dependent variables into two parts, one part that is a pure function
of r, that we know a priori, and an (assumed) perturbative part that is a function of both
rand 9, Le.
p(r,9) = po(r) + tpi(r,9) ,
p(r,9) = po(r) + tpi(r,9) ,
n(r,9) = no(r) + tni(r,9) ,
T(r,9) = To(r) + tTi(r,9) ,
(t ~ 1) .
In a steady state of constant rotation, the fluid velocity is defined by the relation
u=rxO,
and since
o = f(n cos 9)  O(n sin 9) ,
then the unperturbed velocity has only one nonzero component, which is in the azimuthal
direction and is given by the following relation:
Uq, = rn sin 9 .
102
Thus, in the perturbed case, if we wish to consider smal deviations from solid body
rotation, we consider a velocity vector
U = f(Ur,Ue,uq,)T ,
and we assume that, to first order, uq, has an analogous form to the unperturbed flow, i.e.
Uø = 1'f!i (1', 8) sin 8 .
We now consider, in spherical polar coordinates, the axsymmetric equations of motion
for such a smal velocity perturbation. Remembering that we have assumed axsymmetry,
and hydrostatics, the equations of motion and mass conservation, assuming viscosity re
mains constant, can be expanded in powers of f. We make the further assumption that
the scale over which the vertical velocity changes is very much smaler than the scale over
which the density changes. In this case we treat density as constant except in the buoyancy
term in the momentum equation. This is known as the Boussinesq approximation. The
only zero order terms in f occur in the radial component of the momentum equation, and
express hydrostatic balance, i.e.
8
81'Po = pog . (2.1)
For simplicity, we shal ignore al terms except the n2U terms in the viscous drag, and
we assume that the viscosity v is constant. In this case, the governing equations, reduce
to a linear set. The momentum equation, when decomposed into 1', 8 and ø coordinate
directions respectively, becomes, remembering that al terms are independent of ø,
. 2 1 8 Pi
200011' sIn 8 = Pi + gpo 81' po
V (8 ( 2 8) 1 8 (. 8 )J
+ 1'2 81' l' 81' Ur + .sin 8 88 SIn 8 88 Ur
200011' sin 8 cos 8 = ~ 88nP1
po1' vV r 8 ( 2 8) 1 8 (. 8 )J
+ 1'2 L 81' l' 81' Ue + sin 8 88 SIn 8 88 Ue
. rsin88 (8 )20o( Ur SIn 8 + Ue cos 8) = v l 81' 1'4 81' (Oil
1 8 (. 3 8 )J
+ 1'sin2 8 88 sIn 8 88 01
(2.2)
(2.3)
(2.4)
The equation for the conservation of mass becomes
1 8 ( 2 ) po 8 ( . ) ,
8 l' POUr + : 811 SIn 8ue = 0 .l' l' sin v v (2.5)
For closure, we also require a balance of heat. We assume that conductivity or opacity
is constant, and also that we can ignore the effect of fN, the nuclear reaction rate as a
103
source of heat. If furthermore, there are no sources or sinks of heat of any kind, there
must be a balance between the advection and diffusion of heat. We define a characteristic
lengthscale of entropy advection
((alnT) (aInT)Ji
H = Hp a In P ad  a In P ,
where subscript ad denotes an adiabatic motion, and Hp is the pressure scale height.
Then, to first order in e, heat balance requires
Tour K, (a (2 a) 1 a (. a ) J
I = r2 ar r ar Ti + sin 8 ô8 SIn 8 ô8 Ti . (2.6)
Equations 2.26 constitute a set of five equations in the five unknown perturbative parts
of the dependent variables, i.e. Ph Ti, n, Ur, and ue. We then look for separable solutions
for r near to R but within the radiative zone (i.e. r ~ R) of the form (in the example of
pressure)
Pi oc P( 8) exp( ke) ,
where
e=Rr,
and k, the inverse of a "scale height" is assumed "large" in some sense, defined as
Rk~ 1. (2.7)
In general, for a variable V of this form,
ô
arV=kV.
We see simple scalng for balance in (2.5) implies that
Rkur rv Ue .
Thus
Ur ~ Ue . (2.8)
We now investigate the implications of scalngs (2.7) and (2.8). In (2.6), since Rk is
large, we may ignore the second term on the right hand side, and so we have a balance
Ur k2 Ti
H = K, To' (2.9)
Now, if we have a perfect gas, to first order we may approximate Pi/ po as Ti/To. Since in
(3), the effect of differential rotation must be suffciently strong to balance the latitudinal
pressure gradient, the effective balance is
2noniRsin8cos8 =  p~R:8Pi . (2.10)
104
Therefore, the CoriolIs term is negligible in (2.2), which thus reduces to simple hydrostatic
balance, i.e. 1 Ti '
kPi = gr; .po .L 0 (2.11)
Only in the equation in the azimuthal direction (i.e. (2.4)) is viscosity important at
this order. Remembering (2.7) and (2.8), we see that the most significant balance in (2.4)
is
2UBOO cos8 = vk20iRsin8 . (2.12)
Collating al these results, we see that the problem reduces to solving the mass con
servation equation for the 8 dependence of UB, i.e.
Rkur + si~ 8 :8 (sin 8UB) = 0 , (2.13)
with the scalng relations (2.912). Now (2.9) and (2.11) imply
",PHur=Pi'
pog (2.14)
If we differentiate (2.13) with r~spect to 8, and use (2.14), we see that the terms in Ur
cancel, and
8 ( 1 8. J ",k4 RH 8
88 sin 8 88 (SIn 8UB) + pog 88 Pi = 0 .
But, using (2.10) and (2.12), we see that
(2.15)
(2.16)
Thus combining (2.15) and (2.16),
8 r 1 8 J r 4",k2 R2 H02 J
88 L sin 8 88 (sin 8UB) + L vg 0 cos2 8UB = 0 ,
or 8 r 1 8 ( . )J 2 2
88 L sin 8 88 SIn 8UB + À cos 8UB = 0 (2.17)
where
À2 = 4",k2R2HO~ = (20okR)2vg N Pri/2
105
and Pr is the Prandtl number, (the ratio of molecular to thermal diffusivities), and N is
the Brunt Vaisäla frequency.
If we make the substitution cos B = :i, and sin BUB = l(:i), we see that we impose the
boundary conditions that l(:l1) = 0 anddId 1 d
d:i =  sin B dB =  (1  :i2)1/2 dB '
and (2.17) reduces to
( 2)1/2 d ( d) ).2:i2 _ 1  :i d:i  d:i l + (1 _ :i2)1/2 l  0 ,
or Q, ).2:i2
d:i2 l + (1 _ :i2) l = 0 ,
(2.18)
with
l(:l1) = 0 . (2.19)
Now, (2.18) is an equation for the B dependence of UB. We have aleady assumed that
the r dependence is of the form exp( ke), (e = R  r). So, from (2.12), we obtain an
expression for the differential rotation fh, namely.
2UBOO cos B ' :i01 = Rk2' B = ~Oexp(ke)l 2l(:i) ,v SIl  :i
using (2.18) and defining
(2.20)
200
~O = vk2 '
the scaled "varabilty" in the rotation rate.
Now from (2.13), we see that
RUr =  k s~n B ~ (sin BUB) ,
1 d
= k d:i l exp( ke) ,
= v(kR) (~O) exp( ke)!! l2 00 d:i '
= 202 HK(kR)3 (~O) (_kC)!!lg).2 00 exp ~ d:i '
using (2.17) and (2.21). Simiarly, we may write
R sin BUB = R exp( ke)l(:i) ,
= v(k2R)2 (~~) exp(ke)l(:i) ,
= 202~~~kR)3 (~~) exp(ke)l(:i) .
(2.21 )
(2.22)
(2.23)
(2.24)
(2.25)
106
Let us now consider the physical implications of this solution. Firstly, the entire
solution is driven at e = 0, and by the definition of e, we only consider solutions penetrating
downwards into the radiative interior. This model does not attempt to describe the complex
motions in the convective zone, although if the porosity is large, matching with models
util sing such concepts as eddy viscosity is at least consistent along the boundary, since
from (2.2425), radial velocities can be considered to be driven by either viscosity, v, or
thermal diffusivity, K.
If we turn our attention to the form of the solutions for 1 i.e. the () dependence of the
velocity field satisfying (2.1819), we start from the assumption that there is an equatorial
plane of symmetry. In that case we investigate odd solutions for 1. Without going into
the ful solution, we see that, for smal z, the solution approaches
À2Z51(z)"z.20
We may roughly sketch the solution curve thus:
~
~
'X
+1\
fig 2.2: Sketch of Solution Curve.
Thus we have a circulation near the interface with the convective zone, which may be
considered to be a thermal boundary layer. However, closer to the core, motions driven by
heat sinks and sourc.es ( see the EddingtonSweet circulation above) overcome the effects
of differential rotation. We can compare times cales for the two separate circulations. We
aleady know that the timescale for the EddingtonSweet circulation can be taken as
tES " tKH (i~3) ,
where tKH, the KelvinHelmholtz timescale is R2 / K. Thus
gR N2 RH
t E S " K!l2 " KfP (2.26)
107
From (2.23), a typical timescale of the motions due to differential rotation is
i g).2 (ßO) 1 d
RUr = 202HIt(kR)3 00 exp(ke) dzf ,
~ (N2 RH) (00 exp( ke) ~ f)1t02 ßO( kR)3 RH dz '
(00 exp( ke) d )
~ tES ßO(kR)3RH dzf
In the Sun ßO
n~O.3.
Thus, when e is smal, this process due to differential rotation may be important. But
in the above equation, kR is large (as assumed in the model), and hence the effect of
the circulation driven by the differential rotation rapidly drops. Higher modes, i.e. those
modes that occur with larger )., have even narrower shells of effect. Thus a cellular rotation
structure wi be restricted to a region very close to the interface between the convective
and radiative zones. At larger depths, the EddingtonSweet circulation must dominate.
However, this model is extremely simplistic. The differential rotation is unstable, and
the turbulent motions generated by such instabilties give rise to an eddy viscosity which
depends on depth. This problem is then no longer linear. Although the (J dependence
can stil be obtained by the above method, the ful equations must be solved in r for the
vertical dependence, assuming, as usual, that viscosity does not vary with latitude. This
approach is, nevertheless, beyond the scope of these lectures.
lOB
3 Transport of a chemical
When inhomogeneities develop in a star, one can ask for the effects of varying
chemical composition. The mass balance' equation for a chemical with concentration c
reads
a
a/pc) + \7 . (pcU) = \7 . (Dtp\7c) (3.1)
Here the second term on the left hand side represents the advection of the chemical, the
right hand side the difussion with Dt being the total diffusivity. A concentration flux due
to advection through the surface ~
ø =h pcu . dS (3.2)
can only be nonzero if c varies over~. Let ~ be a horizontal surface at radius r. Then the
total concentration at a radius r can be decomposed in the mean concentration eo(r) and
fluctuations 6c(r, fJ)
c(r,fJ) = eo(r) + 6c(r,fJ) (3.3)
representing the vertical changes of c( r, fJ) at radius r. In general, horizontal and vertical
anisotropies due to concentration are possible. In the sun the horizontal transport is likely
to be much more effective than the one in the vertical, due to the instabilities generated by
differential rotation. The basic problem in the following wi be to estimate 6c. We follow
along the lines of G.!. Taylor in his discussion of the diffusion in a pipe, but in contrast to
that work we have here no mean velocity. The equations for the mean vertical transport
equation following from concentration field equation is given by
a 1 d 1 d 2deo
a/Peo) + r2 dr (pU6c) = r2 dr (D",pr ;) (3.4)
Here d", denotes the vertical diffusivity and the brackets denote averaging over a level
surface. The second term on the right hand side of eq. (3.4) represents the advection
controlled by the transport of 6c. We know from Lecture 8 that
ftD I'  (3.5)
'" 02
where ft is the turbulent viscous dissipation, which depends on the amount of differential
rotation 60. To estimate the strength of the advection term in eq. (3.4) we need an
estimation of & which can be obtained as follows:
One multiplies eq. (3.1) written in terms of & and carries out lateral averaging.
Then one finds1 a d
2 at (p(&)2) + (&pU,. dr eo) + (6cpU . \7&) = (6c \7. (Dtp\76c)) (3.6)
Assuming that
109
. horizontal diffusion is very strong, i. e. that Dt can be replaced by DB,
. I SC I" deo/dr, and
. quasistationarity holds,
the second term on the left hand side of eq. (3.6) is equal to the right hand side of eq. (3.6).
Replacing V by h leads finaly to
deo R2
1 SC I'"  1 U,. I b DB
where 1 ... 1 denotes the maxmum. The important point is that I Sc 1 is proportional to
the radial concentration gradient. Inserting the above formula into the advective part of
the mean vertical transport equation gives
(3.7)
1 d 2 ( ) 1 d ( * 2 dco)'
r pU Sc ~  D pr r2 dr r2 dr dr (3.8)
with
D* '" I U,. 12 R2 . (3.9)
DH
Thus the transport of a chenucal can be written in form of a diffusion equation with a
diffusion constant D*, which shows that the Eddington Sweet circulation actualy becomes
a diffusive process. For more details concerning the estimation of the maxmum value of
Sc we refer to the fellow contribution of Brian Chaboyer.
4 Transport of Lithium
Let us finaly discuss an experimental result which supports the existence of
transport below the convection zone. Lithium is burned typicaly at temperatures around
2106 K, thus well in the radiation zone of the sun. If there would not exist a motion below
the convection zone, whose bottom is at temperatures of 106 K, would keep its original
abundance. But the observations show that it is depleted by a factor of about 103. The
same mild transport seems to occur in other stars. In Fig. 4.1 the abundance of Lithium for
Hyades dwarfs is shown as function of the surface temperature ( or the mass of the stars).
Also there Lithium is transported to the surfaceprobably due meridional circulation. The
deep well for Tell about 6700K is one up to now not well explained feature.
110
3.0

..,o \\ HYADES
"
0.0
7:000 6500 6000
Teff (K)
5500
fig. 4.1. Lithium abund:ice (on the scale o( log .v(R) = 12.00) (or die Hyades dwarfs as a funcuon of etri:ctive temperature, Tne open cicles and ~pi:n
triangles correspond to detections and upper limits. ri:spec:uvely. The crosses are the G dwarf d:it:i of Cayrel et al. (1984). The sm:i open squ:ie, :ie
:ibund:ices from equivalent \\idths o( Dunc:i :id Jones (198j) from spectr:i t:ien at Lick Observtory.
Notes submitted by Stefan Linz and Colmcie Cauleld.
III
Lecture X. ,Vibrational Instabilities
1 Unstable Stars
Though most stars are changing slowly and can be considered hydrostatic, they also
fluctuate about the hydrostatic state with diverse time scales and amplitudes. Probably,
every star h~ some varabilty that can be detected by sufciently careful and extensive
observations. Astronomers have qualtatively classified may kinds of varability, and the
members of each class are caled Xtype varables, where X is the name of the star that
is the prototype of the class of varables. We would require too much time to go into al
the classes and what they do, so we wi simply mention some of the favorites. These are
best indicated by their place in the HR diagr.
Figure i is a sketch of the HR diagam showig the zero age man sequence (ZAMS)
which represents the end of the contraction phase and the star of nuclear burng in the
core. The point l, the Kumar limit represents the lowest possible mass (.. O.05Me) for
which the temperature at the core is hot enough to initiate nuclear burg of hydrogen.
There may exist stars below this limit, and thei possibie exstence is ,suggested by the the
dashed lie, which possibly continues on to JUpiter. Point 2 maks the place on the mai
sequence where stars fist encounter the Eddigton limit. At the critical mass represented
by point 2, (.. 70Me) the outward radi~tive force on the matter at the surace of the star
balances ~he ~avitational attraction. If more matter is added to the star, the lumnosity
is raised and the excess matter is thought to be blown away by the radiative force.
The most' important stellar varabilty occu in the nearly vertical strip shown in
the figure. Thi is caled the Cepheid instabilty strip, afer the Cepheid vaables that
lie in the upper portion of the instabilty strip. These star va nearly periodicaly
in lumnosity and surace velocity, as a reult of an intabilty that we shal describe
presently. The Cepheid vaables, whose perods are measurd in days and weeks, have
played a very important role in the subject because their lumosities are wel correlated
with their periods of vaation. They ca be used to gauge the distance to any stelar
system in which they can be observed, once the perodlumosity law is calbrated. That
tured out to be a dicult matter becus the star in the lower par of the strip, with
shorter periods, are not the same as the classica Cepheids, but they were the only kid
near enough to us for a lumosity to be wel detered There were some fa steps
made on this account that had racations for estimates of the scale of the unverse. It
is also intereting that the strip reaches down to the location of the sun in the diagram.
Indeed, the su alo is subject to overstabilties but, the su's gravest modes are not
untable, in contrast to the case of the classical Cepheid vaables.
The star wel above the main sequence and cooler than Cepheids alo tend to be
varable, but they are not periodic. Much less is understood about these semiregar and
irreguar vaables than about the Cepheids. There ar sever other kids of vaation
throughout the diagram For example, the star in the neighborhood of the Z in Z AM S
vary for unown causes. These ß Cans Majoris star are especialy interesting in GFD
112
f
,

2 ~,,~
'~
' '
 /d~
 '
rA~
/ /
~
G
".
, '
\
l
"
"",
l'
~ lf4 T(J ff
Figure 1: The HR digram and the instabilty strip
because they have an oscilatory doublediusive instabilty in progress, with helium as
the slow diffu~er.
Another evident kind of varabilty is that assoåated with the solar cycle, and it is
interesting in having a time scale measured in deces. Hits of simiar varations have
been detected in cool stars. It is not yet clear what fies the time scale of the solar
varabilty. This is in stark contrast to the sipler vaation of the Cepheids, whose time
scale is the 'travel time of a sound wave acoss the star. The solar varabilty has a time
sca.e that may be set by the travel time of the wave of activity that moves from mid
latitudes to the equator. Proctor and I believe that this wave is an envelope of overs table
dynamo waves, which suggests that its propagation spee is controlled indirectly by the
turbulent processes that produce the sub convective dynamo.
.
2 The Kmechanism.
When a star exands a bit, it cools, but it is not clear whether the increased area or the
decreased temperature wins in determg its modied emerent radation. Depending
on thermal effects or phases, this could go either way. H the star radates more when it is
cool, the perturbation wi die away, but if ener is put in durng the hot phase, that is
destabilzing. Of course such remarks ar too vae to be of h~lp in understandig stelar
instabilty, and ,there is no very simple physical arguent for exPlaing it. We shal
instead sumarze a pedagogical model of N.H. Baker (1966) that clares the physical
mechansm responsible for the pulsational oversabilty in Cepheids. The mechansm
113
arses in fluctuations in the opacity It and was proposed by Eddington. ,
In a spherical distribution of matter, the mas M.. contained in a sphere of radius r is
a monotonicaly increasing function of r. If the star remains radial, M.. is a Lagrangian
coordinate and we wil use it as independent varable with 0 c: M.. c: M. The equation
describing how r varies with M is
ar i
=,
8M.. 41rr2p
where p(M.., t) is the density. The equation of motion for a thin shell of mass dM.. is
(2.1)
8p i (GM.. 82r)
8M.. =  41rr2  + at2 (2.2)
where p(M.., t) is the gas pressure. For a. ideal gas, the energy equation for the fluid
between M.. and M.. + dM.. is
8L = c aT +! 8p
8M.. II at pat
where L(M.., t) is the lumnosity, T(M.."t) is the absolute temperature, eii is the specific
heat per unit mass at constant pressure and
(2.3)
~ _ _ (81np)a  8lnT'ii
The condition for radiative equilibrium is
64racD'lT3 aT

31c 8M,.
where a, c, D' are fundamental constants and IC is the opacty. Equation (2.4) is a good
approximation only in the opticaly thick reon of the star and under conditions of local
thermodynamc equibrium. We wi assume (2.4) to be vad even in the outer envelope
as we are only interested in a qualtative understandig. Lastly, we have the ideal gas law
L= (2.4)
Iep = pT (2.5)
p.mIl
where the mean molecular weight p.is assumed constant and k is the, Boltzman's constant.
2. i Onezone approximation
Equations (2.2) to (2.5) can be solved under suitable boundar conditions when :l = 0
and this gives the static state(s) of the star (Po, Po, To,ro), with the lumosity Lo. We
assume smal perturbations from the static state:
r = ro(l +r') (2.6.1)
114
~
~
:e.\~~e \. ~,~t' /

~
, 
.
....
1COR E'
Figure 2: The onezone modeL.
T = To(l +T')
p = Po(l + p')
P = Po(l + p')
£=£0+1'
(2.6.2)
(2.6.3)
(2.6.4)
(2.6.5 )
Normaly, these equations are solved by fite dierencing, each element in that scheme
corresponding to a shell of mas of the star. In Baker's qualtative model, one simply
writes the equation for a single mass shel as if it were the whole envelope. The instabil
ity mechansm in Cepheids arses due to opacity varations in a relatively shalow outer
envelope containing a very smal frction of the total mas of the star. Since the motion
is almost entirely in this envelope we wrte our equations only for the envelope, assum
ing that the region below the boundar 1 is motionless. Al, the varation of p', p' and
T' with M,. is neglected in the envelope. We canot however neglect the vaation of I'
because as: is realy the term that drves the osciations. Therefore we wi make the
approximation
81' l  118M,. = 2m (2.7)
where l is the lumnosity fluctuation at some "halay point" in the envelope, li is the
value of l at the boundar 1 and m is the total mas in the envelope. Since the core does"
not paricipate in the osciations, the lumosity at the boundar 1 does 'not change, so
that 11 = 0 . Therefore (2.7) becomes
81' l

8M,.  2m (2.8)
115
2.2 Perturbation equations
On substituting equations (2.6) into (2.2) and retaining only term linear in the primed
quantities we get
d2r/dt2 =g(4r/+p/) (2.9)
where 9 = ~~ and we have used the fact that the static pressure vanishes on roo Tils says
o
that the dynamics of the envelope is drven by pressure fluctuations and perturbations of
gravitational potential. To understand how this works, we need to look at the coupling
to the thermodynamcs.
In the onezone approximation (2.7), linearzation of (2.3) gives
i' dT' dp'
2m = ToCp dt + D;.
(2.10)
Linearization of (2.4) gives
l = _ "p P' _ ("T _ 4) T'Lo "0 lt (2.11)
where 8"
"p = 8p (Po, To)
8"
"T = aT(po,To)
"0 = "(Po, To).
The linearzed form of (2.5) is
(2.12)p' = p' + T'
and of the continuity equation (2.1) is
3r'  p' = o. (2.13)
If nothing else, this set of equations shows why it is not simple to characterize the
conditions for vibrational instabilty of'a star. The thermodynamc interdependences are
just too nch. However, (2.11) is the key; it shows how the couplings are mediated by
the dependences of ~ on p and T. There is a cerain loose analogy to intabilties due to
negative dierential resistivities in al this.
Let ri be the the charactenstic radus of the envelope, and set
¡.
,
r
1"' :: 1"1/2 (2.14)
where ri is the outer radius. Similarly (2.13) gives
, 31"1p=2 (2.15)
116
Equations (2.15), (2.12), (2.11), (2.10) and (2.9),can be combined into a single equation
for r~:
d3ri A d2ri B dri Gr' = 0
dt3 + dt2 + dt + i (2.16)
where
A  Lo ( "'T 4)
 2m(S  ToGp) ,K. 
B = as  11ToGp
2( S  ToGp)
Lo ( "'p ( "'T ) )G = , 3 + 11   44m( S  ToGp) K. K.
If we look for solutions in the form Ti '" e.t then (2.16) gives
S3 + As2 + Bs + G = 0 (2.17)
Equation (2.17) has three roots. As one crosses the Cephied instabilty strip due to the
specific way the opacity derivatives Itp, K. change, the two complex roots move across the
imaginar axs to the real positive hal of the complex plane . This represents the onset
of overstabilty as envisioned by Eddington (1925). However, for a reliable numerical
solution of the Cepheid instability problem, forty years were needed.
3 Convective Overstabilities
The onset of overstabilty is usualy a subtle proceSs, but the one that is crucial to the
Cepheid instabilty is too complicated to be rea fu. A simpler case of 9verstabilty~occurs
when the energy source is differential buoyacy. The:ft instance was magnetoconvection
(studied by W.B. Thompson and by S. Chandraskhar) and lat~r came the rotating case
(Chandrasekhar). The simplest example is in double disive convection as suggested
by Melvin Stem and de~onstrated by George Vernis. The mechansm in this kind of
overstabilty is generic in convection with restoring forces (Moore and Spiegel, 1966)¡ the
irreversiblity of the osciations that CowlIng spoke of in tht; a:d case plays a role in ~,cases. , , '
Consider a fluid parcel of mass m movig through the star, but neglect the effect of
the parcel on the star. Assume that there is a mechasm such as Coriolis force or a
stable concentration grent that makes for an oscilatory behavior. IT we alow for this
mechansm by a generic restoring force, r per unt mass, the equation of vertical motion
1S
J;z
m dt2 = g(m  mol +mr(z).
where the' mas displaced by the parcel is
(3.1)
mo = mpolp (3.2)
11 7
and Pa( z) is the prescribed background density.
We use the Boussinesq equation of state,
P = p.(1  aCT  T.)l (3.3)
where p. and T_ are constants. After some manipulations using the Boussinesq ideas and
set ting
9 = T  To(z)
we reduce the equation of motion to the form
(3.4)
d2z
dt2 = ga9 + r(z). (3.5)
This is somewhat easer to grasp than the equivaent equation for stellar pulsation (2.9)
parly because there is less going on in this case, but alo because of the simplifications
alowed in the Boussinesq case. For a fuler treatment, we should include frction.
We adopt Newton's law of cooling to describe the thermal effects:
dT
di = q(T  To). (3.6)
We have dT d d dB dz
dt = dt (T  To + To) ~ dt (9 + To) = dt  13 dt
where ß(z) '= dTo/dz. Then (3.6) becomes
dB dz
dt =/3 dt qB
(3.7)
(3.8).
The third order problem posed here has the same linear stnicture as that for the stellar
pulsation case. However, in this simpler cas, we have included nonlear effects in the
zdependences of 13 and r. Using the leadg order ter in thei Taylor series, we get a
nonlnear model for the development of intabilty. With sutable choices of parameters,
we fid a chaotic behavior (Moore and Spiegel, 1966). In a siar, way, the analysis of
the previous section can be extended to the nonlear ree (Baker, Moore and Spiegel,
Astron J., 1966) with chtic results. Thus the occurence of aperiodic stelar oscilations
may have a natural explanation. But we nee to be able to exract the nonlear equations
more relably. We tur to that question nex.
¡~
~
I'h
f\
4 Amplitude equations
Let the state of the star be denoted by a colum vector U whose elements are the densty
p, the temperature T, the components of velocity and whatever fi~ds are necessar to
unquely specify, the conditions in the star. The genera equation desribing the star can
generaly be wrtten in the form
8eU = F(U,8) (4.1)
118
where t is time, B denotes spatial derivatives and F expresses the ful stellar dynamcal
behavior  it is a flow in state space. It is understood that boundary and initial conditions
are attached to F and at, respectively.
A steady state of the star is given by Uo(x) where
F (Uo(x), B) = o. ( 4.2)
4.1 The linear problem
The perturbation equation about the static state can be wrtten in terms of the new
variable
u(z, t) = U(z, t)  Uo(z).
We split the right hand side of (4.1) into linear and strictly nonlnear parts:
( 4.3)
Beu = ,C(a,'\)u+N(u) (4.4 )
where 1: is a linear operator and N is a nonlnear operator (which can in general depend
on space derivatives) and ,\ represents al the parameters describing the star (such as its
total mass). It is useful to fist study the associated linear problem obtained by omittng
N(u) from (4.4):
8eu = 1:u. (4.4a)
We look for solutions of the form
u(x, t) = v(x)ell. (4.5)
Then (4.4a) gives an eigenvaue problem for determning s and v.
1:v = "v. (4.6) t
We assume here that (4.6) has a discrete spectru "h "2.. . . with eigenvectors 'li, 'l2,' ... IT
1: is selfadjoint then the eigenvectors form a bass. (In cas of degeneracy one can construct
an orthogonal bass in the subspace corrsponding to the degenerate eigenvaue.) However,
1: is in general not selfadjoint and there may exst a root(s) of algebraic multiplicity n,
with m, the numer of eigenvectors corrsponding to it, less than n. In such a case, we
need additional vectors to span the subspace (Friedman, 1956).
To fid additional basis vectors, we look for solutions of the form
i
u(z,t) = L 'l1i(z)tlieIlAIi
"=0
( 4. 7) ,
where L :5 n. As ilustration, conSider a cas with n=2 and m=1. Suppose we have aleady
found the one eigenve~tor 'li (z) such that 1:'li = ".,i. We look for a solution in the form
u(z, t) = tPo(z)ed Ao + Øi(z)tell Ai (4.8)
119
where ,pi (:c) is unknown. Substituting into the associated liear equation we find
(3  Loo)Ao  AitLio = 0
AoLoi + Ai(tLii  ts  1) = 0
(4.9)
( 4. i 0)
where
( Loo Lo1 ),'Lio Lii
is the matrix representation of C. in the basis ('lo,r/¡).
The terms with different powers of t vansh separately, so from (4.9) we fid Lao = 3
and Lio = O. Simiarly, from (4.10), we get L11 = s and Ai = .4Lo1. Therefore there is a
solution (4.8) of the form
u(:c, t) = .4ed(iPo + Loit'li),
which, on substitution into the linear equation, gives us the additional information that
.4£',po = Aos'lo + AiiPi.
The matrix representation of £. is then of the form
(~ :)
where q = Ad .4.
This procedure has an obvious generalzation to the cas m, n :; 1. If one succeeds in
applying this procedure to each degenerate subspace then the matri representation of .c
. is in general
( Bi 0 0 . .. )
o Bi 0 . . .
. ~ . . ~ . ~~ : : :
where Bi, B2 etc. are blocks each having the form
(; f ~ ~ J
One can fid another repreentation of the above matri caed the companon form
or the JordanArnold form, that is sometimes more convenent. In the 3 x 3 cae it has
the form
(~ i !)
120
Figure 3: Distribution of eigenvaues a.t marnalty
I:~
i~l
,. ,
A 'i,____ ,_... .
i~rr mode(
4.2 Center manifold theory
l"A)i i"',~ ff 0 J ~ (
If you have a mi of reacting chemical or nuclei, the rapidly reacting species go quickly
to the equibrium abundances dictated by the local instantaneous conditions. This equi
librium,is then evolved according to the behavior of the slowly reacting species. In the
case of instabilty theory, a simar situation preva. H you exand the state vector U
in term of the normal modes, you get a set of ordiar dieretial equations for these
modes and these are of the form of reaction equations. The slowly evolving modes wi
control the situation here just as the slowly reting speces dictate the nature of the
equilibrium~
The slow modes in this case ar those with low grwth rates and low damping rates.
These are modes that are nearly marnal Ths statemt can be made precise when we
have modes that are exactly magial for some vaues of the paraieters. Suppose that
the distribution of eigenvaues .s in the complex plane ha the following special structure
for some vaue(s) of the pareter(s) À = Ào: a fite numer n of the modes are on the
imaginar ax whie al the others are bounded away frm the in1agiriar axs for sma.
IÀ  Àol. The modes away from the imnar ax ca, for may puroses be on either
side of it, but we shal consider the siplest ca here wher there are slow modes with
R( s) = 0 and fast modes with R(.s) c: a, wher a ~ o.
We denote the fast modes by lP; and the slow modes by 1/;, and make the decomposition
u(z, t) = L B,tP,(z) + L: A;(t)1/;(z).
;
Substituting (4.11) into, (4.4) we fid the equations for At 'and B;
dA.
dt = M,;A; + f¡(A,B)
(4.11)
(4.12)
121
dB.dt = ICi;BIc + gi(A, B) (4.13)
Here the matrices M and IC are given by linear theory and Ii and g; are strictly nonlnear
functions of the amplitudes in the expansions. By the very construction of this situation,
we know that IC is invertible, so (4.13) can be rewritten as
B = 1c1I3  IC1g(A,B). ( 4.14)
We expect that, after a short transient, the fast modes wi have equilibrated and
thereafter the system will change slowly. Hence, we take advantage of this to construct a
series of approximations based on (4.14). In fit approximation, we have
B =0. (4.15)
This is caled the Galerkin approximation. To get the next order we put B = 0 on the
right hand side of (4.14). Then
B = ic1g(O,B). ( 4.16)
This is caled the adiabatic approximation. One can continue this iteration procedure
substituting (4.16) into (4.14) to get the nex order of approximation, and so on. We can
then in principle determine a function '
B = B(A). (4.17)
This function defies an invaant subspace of the state space (coordinatized by the coef
ficients in its exansion in eigemunctions) in which the system moves after the tranients.
This is the center manfold.
If we substitute (4.17) into (4.13) we have
Å = MA + r(A,B(A)). (4.18)
Thus, the dynamcs of the system is reduced from an infte diensional state space
down to motion in a subspace of fite diension. Ths is what we did in the previous
sectioi:, but the present approach is more reable. Equation (4.18) is sometimes caled
an amplitude equation. '
4.3 Normal forms
The liear portion 9f (4.18) is determned only by the number of intabilties and their
degeneracies. In each case, we want to use (4.18) for a neighborhood of parameter space
near to .\. The forms of the matrix M for each cas are standard, so we need only to
compute the 'nonlnear terms. Can we not alo fid a standar form for the nonlnear
amplitude equation? If so, før each confguation of instabilties, we shal have a single
equation to study to get some idea of the temporal behavior.
122
Supose that after the reductions to the center manfold, we have derived an amplitude
equation like (4.18) with conceivably complicated nonlnear term as in (4.18). Suppose
that this can be wrtten as
B =MB+r(B) (4.19)
where r is arbitrary, but we shal assume that it is analytic.
We want to put (4.19) into a standard or normal form by means of transformation
B = X + 'P(X) ( 4.20)
where 'P is a strictly nonlnear function. This wi not change the linear part of (4.19),
but it will modify the nonlnear term. Let us assume the new equation is
x = MX+G(X) ( 4.21)
where G is some standard form of the nonlear term. We wi see that the form that
are alowed are determned by the linear theory.
Substituting (4.21) in (4.19) we find
:F'P = r(X + 'l)  G8,rll  G ( 4.22)
where
:F = MX8,r  M. ( 4.23)
The operator :F is determned by the liear problem and it is in fact the Lie derivative
with respect to the vector field MX.
We shal not give detais (see Spiegel, 1985) but sketch the mai idea, which is that
if we want a paricular G we can see whether that choice maes (4.22) soluble so that
there is a 'I that does the job. To che��k this we neethe soed solvabilty conditions
on (4.22). For this we need an inner product on state space, which we shal assume has
been defined. Then we can defie an adjoint opertor Fl by
(1='1,~) = (qt,F~) ( 4.24)
where (. , .) denotes the inner product. Now any 'Í such that
F'Í = 0 (4.25 )
is orthogonal to the left side of (4.24) and we conclude that
(r(x + '1)  G8,r'1  G, 'Í) = o. ( 4.26)
The conditions (4.26) (there wi generay be sever adjQ4it nul vectors) can be used
to fi G. Normaly, 'one proceeds perturbatively, exandig in X and gets G and 'l term
by term. Thus; fot' any stabilty, confgution, provided the degee of intabilty is not
pronounced, the form of the amplitude equation ca be studied
123
4.4 The end
For the case of convective iiistabilties such as we encounter in cool star atmospheres,
these amplitude expanions wil not be very helpful by themselves. The instabilties are
too strong and the linear modes are far to the right of the imaginar axs. We have to
imagine that the turbulence has renormalzed the situation back to a slightly unstable
state, in the large. Then the motions that develop against the turbulent background can
be studied with amplitude equations. '
The simplest situations have' only one instabilty, the direct instabilty that giv?s the
socaled pitchfork bifurcation and overstabilty that flowers into the Hopf bifurcation. For
double instabilties, there are four simple possibilties: two direct instabilties, two over
stabilties, one of each, and an overstabilty whose frequency ca vash as the ma"ginal
condition is approached. This last gives rise to what may be caled the Bogdanov bifur
cation. Al these cases produce either periC?dic or quasperiodic behavior.
When three instabilties occur, then chaotic behavior can develop as in the example of
the oscilating fluid parceL. The norm form for these cases have al been wrtten down
to leading order.' Buchler and collaborators have bee systematicaly applying them to
stellar studies.
5 References:
��(11 Ameodo A., Coulet P.B., Spiegel E.A.,1985, Geophys. Astrophys. Fluid Dynamcs,
Vol.31, page i.
(2) Baker N., 1963, Simplified model for cepheid intabilty, Plenum Press, Stelar Evo
lution.
(31 Car J., Applications of Centre Manfold Theory, SpringerVerlag, Applied Mathemat
ical Sciences 35 (1981)
(4) Coulet P.B., Spiegel E.A., 1983, SIAM J. Appl. Math., Vol. 43, No.4, page 776.
(5) Eddington A.S., 1882, The intern constitution of the star, Cambridge University
Press.
(6) Friedman B, 1956, John Wiley and Sons, New York, Priciples and technques of
applied mathematics.
(71 Kippenhah R., Wiegert A., SprigerVerlag, Stel structure and evolution.
(81 Moore D.W., Spiegel E.A., 1966, Ap.J, Vol 143, page 871.
(9) Spiegel E.A., Cosmic Arrhythmas, in Cbaotic Bebavior in Astrophysics, R. Buchler,
J. Perdang and E.A. Spiegel, eds. (Reidel, Dordrect), 91, 1985.
124
An introduction to solar MHD (Summary)
Stephen Chidress
1 Basics
Astrophysical :fuid dynamics must generaly include the possibilty of elec
tromagnetic fields since the :fuid is often in the plasma state, currents can
flow, and magnetic and electrostatic forces may be important. Astrophysical
systems are also large, so that Reynolds numbers of the form U L/71, where
U and L are characteristic velocity and length scales and 71 is the diffusivity
of some physical quantity, are typicaly very large. The turbulent phenom
ena which result involve complicated interactions between fluid and magnetic
field, solar magnetism being the most directly observable product of these
interactions.
The solar magnetic field is characterized by two complementary prop
erties: On the one hand, there is a global structure connected with the
wellknown 22 year cycle. Every period of roughly 11 years produces a new
round of magnetic activity (most notably in the appearance of sunspots) and
culmnates in a reversal of polarity. This periodicity implies an active pro
cess of renewal of magnetic activity, with production of fresh field (at least in
the observable surface layers) its main consequence. We note that this solar
dynamo cycle is only roughly periodic, and indeed essentialy turned off for
the 70 year "Maunder minimum" beginning in 1645, and in other similar
epochs.
On the other hand, the magnetic field is, in the smal, "rough " and
intermittent. Perhaps the most expressive adjective is "fibriated". Instead
of being a smooth vector field, the magnetic field B(x, t) in the photosphere
(and presumably also in the convection zone) has a fiamentary structure
where intense tubes of :fux are embedded in regions of much smaler field.
As a result, peak fields on the Sun reach several thousand gauss, whie the
spatialyaveraged field is only about one gauss.
;'
i
, ß
T';
'.: :
125
We thus see two lines of enquiry into solar magnetic phenomena, one in
the large and dealng with the cycle as a whole, the other in the smal and
focusing on specific magnetic features such as the fibril structure or definite
objects such as sunspots, :Hares, and prominences. Solar :Hares, in particular,
are noteworthy for the energy released. Flares are apparently magnetic in
origin and have a variety of configurations. Parker has suggested that smaler
versions of :Hares, in the form of tangential discontinuities in the magnetic
field, may playa significant role in the heating of the solar corona up to 106
degrees Kelvin.
The minimal system needed to discuss these phenomena mathematicaly
are the equations of one:Huid, ideal magnetohydromagnetics (MHD). These
equations describe a perfectly conducting, generaly compressible gas. To
alow for resistive effects (such as solar :Hares), we can relax the condition of
infinite conductivity. Then Ohm's law takes the form, in the simplest MHD
model,
J = O'(E + u x B) (1)
where we have introduced customary symbols for current, electric, and ve
locity fields, and 0' is the electrical conductivity of the material. We also
have the equations of Amp��re and Faraday, and the solenoidal property of
the magnetic induction field:
v x E = 8Bj8t,
pJ = V x B, V. B = O.
(2)
(3)
Here p is the magnetic permeabilty. For most problems of interest in astro
physics the displacement current may be neglected in Amp��re's law (the time
scale of events being large compared to the transit time of light across the
system), an approximation we have aleady included in equation (3). This
has the effect of fitering out the electromagnetic radiation.
These equations combine to yield, for any given velocity field u(x, t), a
kinematic equation of the magnetic field,
8Bj8t  V x (u x B) ljRmV2B = O. ( 4)
Here we have gone over to dimensionless variables and introduced the mag
netic Reynolds number Rm. In the solar photosphere and generaly in astro
physical MHD, Rm is a large parameter ( with values generaly greater than
126
106). Here we refer to estimates based upon the molecular, not the turbu
lent value of the diffusivity. Thus equation (4), which is the MHD induction
equation, is in astrophysical problems highly singular in the distribution of
resistive effects, and indeed, and we have aleady noted, one sees this physi
cal property in the fibrilation of the solar magnetic field. We may take the
existence of flux ropes and sheets as evidence of smal dissipation.
In MHD the principal new dynamical effect is the Lorentz force which
appears on the right of the momentum equation
p(dujdt) + Vp = J x B. (5)
Using a vector identity, we see that this new force is a sum of a pressure
gradient and the effective tension in the lines of force,
J x B = JL1(B. VB  VB2j2). (6)
This effective tension results in ne.w wavelike phenomena, typified by the
shear Alfv��n waves.
If diffusive effects are expelled from equation (4) (by setting Rm = 00) we
then recover the kinematic equation for a magnetic field carried by a perfect
conductor. One usualy then refers to the magnetic field as "frozen" into
the conductor; indeed it is only in this ideal case that one has a firm grasp
of magnetic lines of force as entities which are moved about in a prescribed
manner. In this case the evolution of the magnetic field reduces to the geom
etry of material lines under the Lagrangian map determined by the velocity
field u. Lagrangian coordinates x( a, t) are defined by
dxjdt = u(x(a,t),t), x(a,O) = a. (7)
If Jiji = âzijâaj is the Jacobian of the Lagrangian map, then the magnetic
field transforms according to
Bi(x(a,t),t) = JijB;(a,O). (8)
A consequence of this is the conservation of flux: if S is a material surface,
then
dj dt J B . dS = O. (9)
127
This alows field intensity to be increased by compression or by stretching of
fluid elements. In two dimensions, such stretching also leads to folding and ul
timately to highly sheared magnetic structures which are rapidly dissipated.
In three dimensions, however, models such as the "stretchtwistfold" map
suggest that field reinforcement can occur and a dynamo process realzed.
Even for systems with a large Rm the effects of dissipation thus remain
localy important. A good example of this is the "flux expulsion" from 2D
eddies and from other (even 3D) fluid motions which tend to produce highly
sheared structures.
Most models of solar processes rely heavily on the approximation of
frozenin fields. The early work by Babcock on the solar cycle, and sub
sequent improvements by Leighton, suggest a global geometry for the solar
magnetic cycle, but a detailed understanding of the dynamo process is lack
ing, owing primarily to the ambiguous status of certain physical mechanisms
(especialy the "alpha effect") which are often invoked in kinematic dynamo
theory, but can be mathematicaly analyzed only in a few models.
2 Fast dynamos
The Lagrangian viewpoint is useful in describing unsteady kinematic dy
namos consisting of sudden fast movements interspersed within epochs of
zero fluid motion. The epochs of "stasis" alow diffusion to smooth out the
field up to some smal length scale. Using this technique for maps of the unit
cube into itself, we can construct "fast" dynamos which amplify magnetic
field exponentialy at a rate which, for large Rm, is independent of Rm and
therefore of the order of the "eddy turnover time" LjU.
These fast dynamos are also thought to occur in steady threedimensional
flows containing regions of Lagrangian chaos. The reason for this is the
exponential line stretching which occurs in the chaotic parts, as measured e.g.
by the Liapunov exponent for the flow. This stretching amplifies magnetic
field in the ideal limit. The question then arises: is the folding up of the field
so severe that shear dissipation cancels the amplification process? In one
example, Andrew Gilbert has found that dynamo action survives robustly.
This example utilzes a "chaotic web" to extract a simple map along the
separatrices of a periodic array of eddies. Equation (8) is used to compute
the flux of field through a smal element of the chaotic region, as a function
128
of time. Exponential growth is observed in the chaotic region but not in
the regions of integrable flow. The field becomes highly intermittent and
fibriated as time progresses and there is thus good reason to suppose that
true turbulence wi always generate a field such as we see on the Sun.
In summary, there are some special cases where we can compute the evolu
tion of magnetic fields in the kinematic sense, which suggest that a magnetic
field such as we see on the Sun is very likely in astrophysical turbulence.
However little is known about the physics of smalscale structure, about the
corresponding dynamical problems, or about the global organization of these
weakly dissipative processes.
References
Chidress, S. and Strauss, H.R. , Lecture Notes on Solar MHD, Courant
Institute, Spring 1990.
Parker, E.N. Cosmical Magnetic Fields, Clarendon Press, Oxford, 1979.
Priest, E.R. Solar Magnetohydrodynamics , Riedel Publishing Company,
1984.
129
V Observing volcanic eruptions on the porch of Walsh Cottage
130
Relativistic FluctuationDissipation Theorems, Radiative
Hydrodynamics and Galaxy F'ormation
James L. Anderson
Stevens Institute of Technology
Hoboken, New Jersey, ,07030
ABSTRACT
The origin of structure in the universe continues to be a major subject
of research. The odginal suggestion of Newton that structure arose in an
initially uniform distribution of matter from gravitational instability, now
called Jean's instability, is still considered to be the basic underlying
source of this structure. The main difficulty with applying this idea to the
problem of galaxy formation is that, in an expanding universe; instabiUties
grow only as a power ( roughly 2/3) of the time rather than exponentially due
to the competition between the collapse of a fluctuation that exceeds the
Jean's mass and the expansion of the universe. Thus thermal fluctuations that
arise after the era of decoupling of matter and radiation will not have enough
time to evolve into galactic concentrations 1,n the time between decoupling and
the present. At the same time, it was diffi.cult to see how the s1gnHicantly
larger density fluctuations (on the order of percents) needed to make galax:Les
on that time scale could have arisen naturally.' One possibility which has
been put forward is that density fluctuations arose 1n the very early universe
due to quantum fluctuat1,ons 1,n the Higg' s field present during a hypothetical
"inflatio��ary" phase of the universe's evolution. However; that proposal has
run into the dHficul,ty that the fluctuations predicted by the Coleman
Weinberg potential used in the "ney lnflationary" scenario are 45 orders of
magni tude too big.
An alternate proposal was put forth by Saslay (1968) who suggested that
there might be a s1gnUicant enhancement of thermal fluctuations due to the
longrange nature of the gravitational interaction. In particular, Saslaw was
able to show; uS1,ng thermodynam1,c arguments; that these enhanced fluctuations
have a pronounced peak in their spectrum at the 'Jean 's yavelength. FOllowing
on Saslay' s suggestion, Simon (1970) made use of a fluctuationdissipation
theorem for the NavierStokes equations derived by Landau and Lifshitz (1963)
to shoy that such a spectrum yould arise if one ,added fluctuating forces to
these equations. However, both Saslaw' s and Simon's treatments were non
relativ1.stic and hence could not be applied to fluctuations that yould have
arisen before the era of decoupling. While such fluctuations might have had
enough time to evolve into galaxies at the present time, they yould have had
to survive a period of acoustic damping just prior to decoupl1ng. Weinberg
(1971) invest1.gated this damping and found that if the dens1.y of the present
universe 1.s 1030 g/cm3 then galaxY sized fluctuations would be damped by a
factor of 1071 and that only cluster sized fluctuations and larger would
survive through the decoupling era. In a someYhat denser universe however;
galaxy sized fluctuations might just manage to survive if they were
sufficient 1 Y large to begin with. However; Weinberg offered no mechanism for
such fluctuations prior to decoupl1ng.
As a preliminary to an investigation of the origin and growth of
fluctuations prior to the decoupl1ng era, ye have developed a relativistic
generalization of the LandauLifshitz fluctuationdissipation theorems for a
;
"
~
¡,
l
131
onecomponent relativistic fluid (1990). In particular we have determined the
relation between the fluctuating forces that have been added to the
relativistic NavierStokes equations and the coefficients of bulk and shear
viscosity and thermal conductivity appearing in these equations. Since
however, prior to decoupl.ing, radiation played an important role, it is
necessary to extend these results to a mul tj,component system. We have used
the equatioris of radiative hydrodynamics derived by us (976) for this
purpose. We hope to use these results to study whether the density
fluctuations produced by these fluctuating forces could have survived beyond
the decoupl1ng era to becom galades.
References
Anderson, J.L. 1976,Qe. Rßl..& Crav. 7, 53
Anderson, J.L., Nowotny, E. 1990, Ebysica A 163, 501
Landau,L.D., Lifshitz,E.M. 1963, Fluid Mechanics, Pergamon Press, Oxford
Saslaw, W. C. 1. 968, ~NQ Roy. AstrOl .c. 141, 1
Simon,R. 1970, Astron. & A��~hi~ 6, 151
Weinberg, S. 1971 , AstrophiiL. 168, 127
132
SOLAR ACOUSTIC OSCILLATIONS
N.J. Balorth
Institute of Astronomy,
University of Cambridge,
Madingley Road, Cambridge, CB3 OHA,
England.
The measurements of the oscilations of the sun have now reached con
siderable precision: literaly thousands of individual oscilations have been
identifed and many of their frequencies have been determied to accuracies
of less than a percent.
Theoretical modellng of the linear, adiabatic pulsations of solar mod
els has reproduced these frequency measurements to a surprising level of
accuracy. There are smal, but nevertheless sigiuficant differences (e.g.
ChristensenDalsgaad, 1989). Tils seems to suggest that these theoretical
adiabatic pulsations describe solar oscilations faily well. Indeed, it appears
that the dierences between the observed frequencies and the theoretical
adiabatic pulsation frequencies are largely caused by differences between
the basic structure of the sun and that of the theoretical model intended to
represent it.
The formulation of the linear, adiabatic pulsation problem can be used
to develop inversion technques to iner how the structure of a particular
theoretical model diers from that of the sun. Tils has been very successful
in helping to refie the modellng of solar structure. Using these methods,
the agreement between observed and theoretical pulsation frequencies has
been improved. The observed frequencies, however, do not compose a com
plete mode set. Tils leads to resolution and unqueness problems in the
theoretical inversion. The observed frequencies also correspond only to pul
sations that do not propagate substantialy into the central regions of the
sun. Tils severely restricts the depth to wruch information can be extracted
about the sun. Extensive observations are planed in the coming decade and
may aleviate these problems (for example, the "SOHO" space mission and
the "GONG" groundbased observatories). Another theoretical problem is
133
the error that is introduced when the pulsations are treated as adiabaticaly
propagating waves in the turbulent superadiabatic boundary layer of the
solar convection zone. hi fact, tlus is probably where much of the remainder
of the discrepancy between observation and theory has its origin.
hiversion methods can also be applied to iner inormation about the
rotation rate of the sun. hi a sphericaly symetric star the axal symmetry
gives rise to a degeneracy in the azimuthal order of the pulsation modes. Any
nonaxaly symmetric phenomenon that afects the oscilations wil break
tlus symmetry and remove the degeneracy. Tlus is exactly analogous to the
instance where a magnetic field lits the degeneracy of the energy levels of an
atom and produces Zeeman splitting. The rotation of the sun is responsible
for lifting tlus degeneracy in the solar oscilations and produces a splitting
of the mode frequencies. Since the extent of the splitting depends in some
integral fasluon upon the depthdependent rotation curve, the inormation
contaied in the pulsations that penetrate to different depths inside the sun
can be used to measure some features of tlus rotation curve.
The power of the solar oscilations is concentrated witlun an envelope in
frequency about 3 mHz. Tlus envelope appears to be largely independent
of the horizontal wavenumber of the modes (with the possible exception
of modes that propagate predomantly horizontaly throughout the sun 
ChristensenDalsgaad and Gough, 1982; Libbrecht et al., 1985; Rhodes,
1990). Tlus suggests that oscilations are generated where the modes prop
agate alost verticaly, i.e. near the surface of the sun in the turbulent con
vective boundary layer. Under the conditions that prevai in these regions,
it is extremely diffcult to model nonadiabatic linear pulsations, principaly
because there exists no reliable description of turbulent convection. Never
theless, there are indications that the acoustic oscilations are intrinsicaly
stable, but driven stochasticaly and nonlearly by turbulence (Stein, Nord
lund and Kuh, 1989; Kumar and Goldreich, 1989; Balorth and Gough,
1990): the modes become the manestation of acoustic noise, generated by
turbulent convection, in a resonant acoustic cavity. If tlus is true, then
the modes react lie damped simple harmonic oscilators under the inu
ence of a temporaly random forcing. The power spectrum of the oscilators
is approxitely Lorenztian, and the hal widths at hal maxum of the
peaks are just the damping rates of the modes. These line widths have been
measured for the sun (Libbrecht, 1988), and are qualtatively sinular to re
cent theoretical nonadiabatic liear pulsation calculations (Balorth and
Gough, 1990).
Most recently it has been discovered that the solar oscilation frequencies
134
are changing with the solar cycle. The dependence of the frequency change
upon the absolute frequency of the modes is indicative that the oscilations
are being afected by changes in the turbulent convective boundary layer
(Libbrechtand Woodward, 1990). Therefore the frequency change may be
used as a diect probe of the principal effects of the solar cycle upon the
stratification of the sun. This effect is surrisingly superficial. It can also
be modelled by simple theoretical calculations that change the effcacy of
convective energy transport, which is a potential consequence of the build up
of magnetic :fux in the convection zone. However, these calculations diectly
contradict the observations of the change in the solar lumnosity over the
solar cycle (Woodward and Hudson, 1983). Thus the solar structure does
not appear to be undergoing a global (latitudinaly invariant) change over
the solar cycle, uiess the direct effect of the magnetic forces is responsible
for changing the pulsation frequencies.
REFERENCES
1. Balorth, N.J., and Gough, D.O., 1990. To appear in the proceedings
of the conference held in Versaies, May, 1989, entitled Inside the sun;
t�� be published in Solar Physics.
2. ChristensenDalsgaad, J., 1989. In Seismology of the sun and sunlike
stars,ed. V. Domigo and E. Rolfe, ESA.
3. ChristensenDalsgaad, J., and Gough, D.O., 1982. Monthly Notices
Royal Astronomical Society, 198, 141.
4. Libbrecht, K.L., 1988. Astrophysical Journal, 334, 510.
5. Libbrecht, K.L., Popp, B.D., Kaufan, J.M., and Penn, M.J., 1985.
Nature, 323, 235.
6. Libbrecht, K.L., and Woodward, , 1990. Nature, in press.
7. Kumar, P., and Goldreich, P., 1989. Astrophysical Journal, 342, 558.
8. Rhodes, E., 1990. In a presentation at the Institute for Theoretical
Physics, Santa Barbara, as part of the programe on Helioseismology,
January 1990 to July 1990.
9. Stein, R.F., Nordlund, l., and Kuh, J., 1989. In Seismology of the
Bun and sunlike stars,ed. V. Domigo and E. Rolfe, ESA.
10. Woodard, M.W., and Hudson, H.S., 1983. Nature, 306 ,589.
135
Inviscid models associated with vortex
reconnection
Stephen Childress
The question of global regularity of 3D Euler flow has implications for
the kinds of mechanisms avaiable for viscous reconnect ion of vortex lines.
Conversely, studies of vortex reconnect ion might be suggestive of the most
singular Euler flows. A simple model equation studied by Constantin, Lax,
and Majda has the form Wt = wH(w) where H is the Hilbert transform
and w is a function of x, t. This equation, whose initialvalue problem may
be solved exactly, supports "dipolelike" singularities across which w changes
sign. Thus in this onedimensional analog the singularity is closely associated
with viscous cancellation described by Wt = w2/2 + IIW:i:i where W = H( w) +
iw.
In real 3D Euler flows no clear evidence of a singularity exists. Using
the vortexreconnection criterion, the TaylorGreen initial condition is not
especialy appropriate, since the early phase of the motion compresses vortex
lines connecting the eddies into layers of one sign. Vortex reconnection would
be enhanced for the initial velocity ('ly, 'l:i, 0) where'l = sin x sin yf(z) near
a double zero of f(z). On the other hand the work of Pumir and Siggia on
singulanty development from antiparalel vortex tubes deals with essentialy
a vortex reconnection geometry. The inviscid development of the vortices
involves considerable 2D distortion of the cores. Their recent computations
utilzing mesh refinement have found at most exponential growth of vorticity,
however. Viscous simulation of reconnection at tube Reynolds numbers of
order 1000 shows /brid a weaker compression of the vortices toward thin
cores. The "bridging" process occurs in a layer formed by the compression
of the cores onto the plane of symmetry. Saffman has devised a model for
the "snapback" of the bridged or rejoined tubes, which accounts for the
enhanced straining induced by the axal flow along the tubes away from
136
the reconnection region. This nonlinear feedback is caused by the" viscous
annihilation of the axal lowpressure cores. The time for viscous reconnect ion
depends upon the flow causing it. Simple strainingflow models of viscous
reconnect ion suggest that, with at most exponential of vorticity, the time
grows with Reynolds number Re like log Re. For singularities consistent
with the BealeKatoMajda estimate e.g. maxmum vorticity growing like
(t*  t)I, reconnect ion is compl��te by time t*.
A model for singularitiesjreconnection utilzing evolution of a bilayer of
antiparalel vorticity, has a certain attraction since core distortion tends to
eliminate tubes as discrete objects with welldefined circulations. A thin
layer model based upon evolution from perturbations of a triangular jet
(u,v,w) = (O,V(~),O),V(~) = wo(I~ljf+XO) was developed. The thin
layer limit involves fied 0(1) variables ~jf, Zjf for smal f. Letting~, z
now denote these stretched variables, and u,w simiarly denoting stretched
velocity components, the resulting system is
Dv = 0, U:i + vy + Wz = 0, (1)
(2)D(w:i  uz) + v:iWy  vyw:i + uzvy  vzuy = O.
On the boundary ~ = ::X(y, z, t) of the vortical region we have
U = Xt + wxz,u = lP:i,v = O,w = lPz, (3)
where lP is harmonic in ~, z in the exterior and vanishes at infinity. The two
dimensional version of this problem leads to the wellknown nonlinear wave
equation for x: Xt + WoXXy = O. Various thinlayer initial conditions have
been considered. Two antiparalel ellptic patches were simulated, using
contour dynamics, by David Dritschel. The patches develop into a highly
distorted "T"shaped structure, resembling the distorted vortex cores in the
PumirSiggia modeL.
The 3D structure of the above thinlayer model is stil intact, and we have
focused first on a truncation which assumes
v = w(y,z,t)(I~I + X(y,z,t)),w = w(y,z,t), (4)
and that w = H(u) on the boundary, where H is the Hilbert transform in z.
This model reduces to the system describing volume conservation,
Xt + (VX)y + (WX)z = 0, (5)
137
where V = wX/2 is the average yvelocity over the layer, an assumed equation
for w,
~
Wt + VWy + wWz; = WWz;, (6)
and the Hilbert connection between u on the boundary and w. Since the
last two equations imply l' + 2VVy + w Vz = 0, we see that the system
admits an integral V = \' =constant. By examining the linear stability
of the triangular jet in the various approximations, we find that the thin
layer model perserves the "varicose", neutraly stable waves, but that the
simplified model, restricted to the integral surface V = \', expels the 2D
KelvinHelmholtz waves which distort the cores of vorticies. We thus obtain
a system
Xt + (WX)z; = O,w = H(xwz; + \'Xy). (7)
David Olson has studied the evolution of pertubations of Xu in this modeL.
Although the results are preliminary and have been obtained only at modest
resolution, the indication are that highlow pairs form and propagate. " The
low steadiy deepens whie the high remains at about :fed amplitude. Since
2\' = wx is fixed, singularities now correspond to zeros of X. We think of
the vicinity of such a point as a "hot spot" were reconnection is heightened.
We thus envisage introducing in such a region a local viscous boundary layer
on the plane of symmetry, where the extent of reconnection can be assessed.
, T
,!
~'i
138
Rossby Wave Radiation from Strong Eddies
Glenn R. Flierl
MIT
Models of steady nonlinear eddies in geophysical flows often require cunditions which
seem not to match well with eddies which have been studied in the atmosphere or ocean.
For example, consider a two layer quasigeostrophic model with mean flows U i, U 2 in the
upper and lower layers respectively. For a steadily propagating eddy, the motions in the
far field are a linear combination of the solutions to
V2~ = (K2 / R~)~
with the two values of K2 satisfying the equation
(K2 + ßR~  (~  c)/(1 + 6))(K2 + ßR~  (U.  c)6/(1 + 6)) = (l)2Uic U2 C +6
with 6 being the ratio of the upper layer depth to the lower layer depth. Analysis of
this equations shows that solutions are confined (both K2 values positive) for c :; U i
and c :; U 2. Thus isolated eddies must travel eastward with respect to both mean flows.
This is not generaly observed. Also, isolated eddies must satisfy have vanishing angular
momentum J J 6r x Vi + r x V2 = O. Again, it is hard to justify this as applying to Gulf
Stream Rings or features such as the Red Spot.
Thus, we must consider cases in which one or more of the exterior fields have a wave
like or radiating character. We then would not expect to find steady state solutions, but
we can look for situations in which the exchange of energy between a strong eddy and the
Rossby wave field is relatively weak. Then the eddy would be relatively longlived and
might be thought to be a steady state solution. It is important to understand the rate
at which energy is lost from the eddy structure and how the wave field might affect the
evolution of the eddy.
As a first example, we consider the case in which one mode is trapped and one wave
like  in the absence of mean flows, this would correspond to a situation in which the eddy
is moving westward with a velocity c ~ ßR~, so that the baroclinic mode is not wave
like, but the barotropic m��de is. We look first at just the baroclinic mode in isolation.
The vanishing net angular momentum theorem implies that either the speed is exactly the
long wave speed or the baroclinic angular momentum must vanish (J J 4iBc = 0). In the
former case, there is a solution with streamfunction decaying as ri sin () which includes a
strong axsymmetric monopolar component. (This solution has, of course, no net angular
momentum because the flow is purely baroclinic and therefore 6vi + V2 = 0.)
But this eddy does not satisfy the two layer equations exactly: the nonlinear interac
tions generate barotropic flow and that, in turn, alters the baroclinic flow field. Analysis
of the equations in the limit where 6 is smal show that the barotropic streamfunction
satisfies
(V2  ß)4iBT = (V2  ~  ß)4iBCC R cd
139
which is the equation for topographicaly forced waves. The right hand side vanishes in
the exterior (streamlines connected to infinity) but not in the interior. Since c c 0, the
barotropic equation has radiating solutions; therefore, we must satisfy radiation conditions.
This leads to an asymmetry with most of the barotropic field extending to the east of
the eddy. The asymmetric part of the field has a southward flow near the eddy (for an
anticyclonic baroclinic circulation), forcing the eddy to move southward. In addition, the
baroclinic energy decays as
~ ~ l l(IV4iBcI2 + I4iBCI2 IR~) =  (1 ~ 6)2 l l(:X 4iBT)(ß  cV2 + (cIRd)2)4iBC
The right hand side has no contribution from the exterior part of the eddy, but has a
negative contribution, proportional to the energy itself, from the interior region. Thus we
have decay of the baroclinic eddy on a time scale inversely proportional to the layer depth
ratio, 6.
As a second example, consider a barotropic vortex pair oriented so that it wil move
westward. In this case, the only mode present is radiating, rather than trapped. When the
flow speeds in the vortex pair, which are proportional to the eddy velocity, U, are large
compared to the Rossby wave speed, ßt2, we can solve by expanding in a smal parameter
E = ßt2IU. We solve by matched asymptotic expansions: in the far field, the flow varies
on a large scale, X = E1/2Z, and the dominant balance is linear. We look at the solution
which is singular near the origin (and therefore wil match with the decaying field of the
interior dipole) together with the free waves necessary to satisfy the radiation condition.
As we approach the eddy, the far field reduces to the r1 sin 8 characteristic of the dipole
plus a weak r2 sin 28 strain field arising from the Rossby wave field.
The interior equations are then examined at various orders in E  in particular, the
order 1 and E terms lead to a steadily propagating solution which can be matched to
the singular part of the far field. The E3/2 terms bring in a slow time dependence and a
linear equation for the perturbation to the dipole streamfunction. We use two solvabilty
conditions to demonstrate that the enstrophy in the dipole is conserved in the presence of
the strain field, whie the energy decays. We add an assumption that the eddy maintains
the form of a Batchelor I Lamb modon; this seems necessary because we cannot solve the i~
perturbation equation in detai. Presumably a ful solution would alow us to determine the ':J
evolution of the functional relationship between the potential vorticity and streamfunction.
With this assumption, we can show that the speed is fied, but the radius decays and the
amplitude decreases as t1/2.
Finaly, we consider the problem of an equivalent barotropic monopolar eddy. Here,
neither of the approaches above works. The eddy is radiating in the only avaiable mode,
but is not coupled weakly. Nor is the wave scale significantly different form the eddy scale;
both are order of the deformation radius. The integral theorem suggests that the speed
should be near the long wave speed, as do numerical experiments. If c = ßR~, the system
is at the boundary between trapped and wavelike; however, it can stil radiate because of
adjustment to the initial state. Analysis of the resultant radiation pattern (using an ad
hoc and unjustifiable linearization) suggests that an anticyclonic vortex should have a
southward velocity proportional to t for smal times and to lit for long times as the wave
field is gradualy left behind.
140
'
,In summary, we have explored a number of models of strong eddies which decay
because of loss of energy to radiating modes. In some cases these losses could be made
up for by other mechanisms, such as the absorption of incoming waves or vortices; in
other situations, the waves simply provide a natural (and non viscous) decay mechanism.
While steady state solutions provide much information about the dynamics, these unsteady
solutions may be more relevant to many strong oceanic and atmospheric phenomena.
CONVCTION A! CHOS
Andrew C. Fowler
University of Oxford
Abstract. We discuss two types of chaotic behaviour exhibited by
high Prandtl number convection, that is 'phase chaos' and plumes.
In mantle convection, these differing aspects of the motion find
their expression in the migration of subduction zones and hot
spots, respectively. The analysis of plumes in a single
convection cell can be attempted in the framework of Howard's
'bubble' model of convection, using an asymptotic analysis based
on a similar method applied to the Lorenz equations. In the
partial differential equation case, this leads us in principle to
an approximate Poincare map for the flow. However, we find that
Howard's assumption of differing time scales for the processes of
growth and flow instability is in error, and (for a single
developing plume) the thermal regime is likely to be periodic.
For the case of' a large aspect ratio, where multiple plume
development can take place, the corresponding Poincare map should
lead to a chaotic distribution of plumes in space and time. '
For cellular convection, we give a synopsis of the recent
scaling theory of the Chicago group. , There is a mean flow in the
cell, fuelled by thermal plumes which erupt from the boundary
layer as they are advected across the cell. The theory is
strictly appli~able, however, only to Prandtl numers of 0(1).
Cellular (' phase') chaos at large Rayleigh number can be
modelled using a set of ordinary differential equations for
variables which describe the size and location of slowly varying
convection cells. The differential equations are parametrised
using quasistationary boundary layer theory. The same method
can in principle be extended to three dimensions, and represents
a paradigm for the study of timedependent motions of the earth's
Ii thospheric plates.
141
On thermonuclear convection
Sandip Ghosal
Abstract
Dile and Gough (1972) suggested that some gmodes in the sun might be over
stable, triggering periodic mixing of the solar core. This according to the authors
might account for the observed deficiency of neutrinos coming from the sun and also
explai the ice ages. Unno (6j, with the help of a quasi adiabatic analysis using the
work integral, concludes that I f' 3 or 4 and n f' 1 are the modes most likely to be
unstable. histabilty of these modes have been looked for numericaly but no fim
conclusion has yet been reached. Unno's quasiadiabatic analysis is vald provided
the time scale of gmode oscilations is much shorter than the thermal time scale.
This condition is clearly satisfied for the low i low n modes. However, for the low
i high n modes, the horizontal wavenumber is smal and the gmode oscilation pe
riod approaches zero. We consider a two layer model with the lower layer contaig
temperaturedependent heat sources (representing the layer of H e3 in the solar core.)
This model should always be stable according to the quasiadiabatic analysis. But
it is found that in the lit of large horizontal scales, thermal instabilty afects the
spatial structure of the gmodes (which in this limit have long time scales) and induce
a monotonic instabilty.
The degree of the temperature dependence of the heat sources is described by a
parameter E. The critical E for the onset of thermonuclear convection depends on
the boundary conditions assumed, but it is found to be in the same general range
as the values of E in the solar core. A nonlnear analysis is made and it is found
that the onset of thermonuclear convection is described by a sub critical bifurcation
in E. This kid of qualtative analysis suggests that there might be long and narow
"shellular" convection patterns in the solar core confed mostly in the region of
the He3 layer. If this is true, it wil have the following effect on our calculation of
the neutrino lumnosity of the sun: the rate of generation of heat in the layer we
are considering can be represented as c Tm :; where m is some integer f' 5 and
c :; denotes horizontal average. c T :; is the temperature distribution computed
from the sphericaly symmetric standard solar modeL. Now c Tm :; is greater than
c T :;m, therefore the lumnosity of the new model with the shellular convection is a
bit larger than that of the standard solar modeL. hi order to bring the lumnosity into
agreement with the observed lumosity the central temperature of the model must
be decreased. Since the neutrino lumnosity Lv of the high energy neutrinos come
from a region very close to the centre, Lv f' T: where Tc is the central temperature
and n '" 16. Therefore a smal change in Tc produces a significant change in Lv. A
numerical estimate shows that in order to reduce Lv to hal its value one needs about
a 40 percent variation of the temperature over the length of a cell (for the 1=1 modes.)
The possible effects of rotation have not been studied yet. It is important to know
whether this kind of convection that has been demonstrated qualtatively is realy
present in the sun even if it turns out that the solar neutrino problem has a dierent
origin; complete understanding of the macroscopic aspects of the problem is essential
if one for example wants to set some bounds on the neutrino mass.
142
References
¡I) Dile F.W.W. and Gough D.O. Nature 240, no.5379, p262, Decl 1972.
¡2) Defouw R.J., thesis (1970)
¡3) Defouw R.J., Ap.J 160, 659 (1970)
¡4) Field G.B., Ap.J., 142, 531 (1965)
¡5) Unno W.,Osaki Y.,Ando H.,Saio H.,Shibahashi H., "Nonradial oscilations of stars",
University of Tokyo press, 1989.
¡6) Unno W.,PubL. Astron. Soc. Japan, 27, 8199, (1975).
¡7) Boury A., Gabriel M., Noels A., Scuflaire R., Ledoux P., Astron. and Astrophys. 41,
279285 (1975)
¡8) ChristenDalsgaard J., Dile F.W.W., Gough D.O., Mon. Not. R. Astron. Soc. 169,
429445 (1974)
¡9) Defouw R.J., Siquig R.A., Hansen C.J., Ap.J. 184, 581586 (1973)
¡10) Dziembowski W., Sienkiewicz R., Acta Astr. 23, 273 (1973)
¡ii) Saio H., Ap.J. 240, 685 (1980)
¡12) Rosenbluth N.M., Bahcal N.J., Ap.J. 184, 916 (1973)
¡13) Saio H., Cox J.P., Hansen C.J., Astron. and Astrophys. 85, 263264 (1980)
(14) Schwarzschd M., Harm R., Ap.J. 184, 58 (1973)
(15) Shibahashi H., Osaki Y., Unno W., PubL. Astron. Soc. Japan, 27,401410 (1975)
(16) Spiegel E.A. and Veronis G., Ap.J., 131,442 (1960)
(17) Parker E.N., Ap.J., 117,431 (1953)
(18) Merryfeld W.J., Toomre J., Gough D.O., Ap.J. 353, 678697 (1990)
143
ON FAST DYNAMO ACTION IN STEADY CHAOTIC FLOW
A.D. Gilbert, D.A.M.T.P., Cambridge University, U.K.,
S. Chidress, C.I.M.S., New York University, U.S.A.
& U. Frisch, Observatoire de Nice, France.
It is now understood how the motion of a conducting fluid can amplify weak magnetic
field (see, for example, Moffatt 1978 for a review). Although kinematic dynamo theory is
generaly believed to explain the existence of terrestrial and solar magnetic fields, many
fascinating questions remain. One interesting problem is to understand the rapid rate of
magnetic field generation in the sun. This led to the following distinction (Vainshtein &
Zeldovich 1972): a dynamo is caled "fast" if the magnetic energy grows on a convective
timescale when the magnetic Reynolds number is very large, as is the case for the sun.
If the growth of the field occurs on a diffusive timescale, which is very large in the sun,
or on some intermediate timescale, the dynamo is caled "slow". Chaotic flows are prime
candidates for fast dynamo action since they stretch vectors exponentialy; in the absence
of magnetic diffusion, the magnetic energy grows exponentialy on the convective time
scale. However a chaotic flow folds as well as stretches field, and this generaly leads to
intense dissipation offield when weak diffusion is introduced. This can dramaticaly reduce
the growth of field and even ki it completely, an example being when the flow is planar.
We have considered dynamo action in a steady chaotic flow which is modelled on the
ABC flow for A = B = 1 and C ~ 1 (see, for example, Dombre, Frisch, Greene, H��non,
Mehr & Soward 1986). The model flow is constructed in such a way that the motion of
particles and magnetic field vectors can be calculated exactly when there is no diffusion
(Gilbert & Chidress 1990). The model flow contains a chaotic web and field is stretched
exponentialy. We calculated the evolution of field and found evidence that field is folded
constructively in the sense that the average field in a volume grows exponentialy in time.
This provides evidence for fast dynamo action since the principal effect of weak diffusion is
to smear field over space, destroying fine structure whie leaving mean field behind (Finn
& Ott 1988). We have also obtained preliminary results for the original ABC flows for the
case A = B = C = 1, which indicate constructive folding of magnetic field and suggest
fast dynamo action. Research is now underway to include the effects of weak diffusion in
our calculations.
References
Dombre, T., Frisch, U., Greene, J.M., H��non, M., Mehr., A. & Soward, A.M., J. Fluid
Meeh., 167, 353 (1986).
Finn, J .M. & Ott, E., Phys. Fluids, 31, 2992 (1988).
Gilbert, A.D. & Chidress, S. Phys. Rev. LeU., submitted (1990).
Moffatt, H.K., "Magnetic fluid generation in electricaly conducting fluids," Cambridge
University ~ress (1978).
Vainshtein S.1. & Zeldovich, Ya.B., Sov. Phys. Usp., 15, 159 (1972).
144
ARTIFICIAL BOUNDARY CONDITIONS
FOR WAVE PROBLE IN UNBOUNDED DOMAINS
by
Dan Givali
Dept. .of Aeraspace Engineering
Technian, Haifa 32000, Israel
ABTRCT: When salving numerically a wave prablem in a
unbaunded damain, .one first has ta make the camputatianaldamain finite by intraducing an artificial baundary. Then an
apprapriate baundary canditian must be impased an this
artificial baundary, sa that waves caming aut .of the
camputatianal damain are transmi tted thraugh the baundary
withaut giving rise ta spuriaus reflectian. A cansiderable
amaunt .of wark has been dane ta devise such nanreflecting
artificial baundary canditians. In this talk the variaus
appraaches are reviewed, usually leading ta lacal appraximate
baundary condi tians . Then a new methaq. is presented, which is
the result .of jaint wark with Jaseph B. Keller. In this methad
we chaase the artifcial baundary ta be a circle .or a sphere,
and we derive an exact nanlacal baundary canditian an this
baundary. Thus, the .original prablem in the unbaunded damain
is replaced by anather prablem in a finite damain which has
exactly the same salutian there. The finite element presented
which demanstrate the superiarity .of the exact nanlacalbaundary candi tian aver appraximate lacal .ones.
145
Very High Resolution Solar Xray Imaging
Leon Golub
Smithsonian Astrophysical Observatory
60 Garden Street, Cambridge MA 02138 USA
1. Magnetic Fields and Coronal Emission
Study of the solar corona has a relatively short history  only 150 years  with a handful
of major milestones along the way (Table I). Briefly, following the rather late recognition that
the Sun has a ~orona at all, the big breakthrough came with the realization that the coronal
temperature is of order 106 K. This high temperature explains the possibilty of a corona extend
ing to heights of several Solar radii above the photosphere; it also has the further implication
that, because thermal conductivity is so effcient, an unconstrained corona wil expand outward,
producing a solar wind; this was subsequently observed when satelltes were placed into orbit
and could measure the particle flux directly. The last major milestone was the realization that
there exists a close connection between regions of enhanced emission in the lower corona and
locations on the solar surface at which strong magnetic field regions have emerged from the
interior.
Table 1. Milestones in Coronal Physics
1842: Total eclipse across S. Europe  first serious study of corona
1850's: Photography of corona and its spectrum
1930: Development of coronagraph
1939: Identification of 'x6374 A as Fe X
1940's: Solar radio astronomy begins
1946: Availability of UV, XUV and xray observations from space
1958: Unconstrained hot corona wil expand  solar wind
1960's: Connection between enhanced coronal regions and magnetic fields
The connection between coronal activity and magnetic fields is most easily seen when the corona k"~,~,;,.,,,'
is viewed ondisk, as can be done in Xrays (Figure 1). It is clear that the corona is brightest ,;
when there is newly emerged magnetic flux and weaker when the field has diffused across the sur 'i':
face. The open, unconstrained corona, which is connected with highspeed solar wind streams,
Bartels Mregions, and recurrent geomagnetic substorms, is associated mainly with coronal holes
(Krieger, Timothy and Roelof 1973), which are large areas of the Solar surface dominated by a
single magnetic polarity and within which the magnetic field is open to interplanetary space.
The relation between B and Xray emission is more than just qualitative; the view which
developed out of the Skylab analysis was that the magnetic field played an active role in the
coronal heating process (Rosner et al. 1978). From this view it is possible to obtain scaling
relations among observable quantities (Golub et al.i 1980) which can then be compared to the
actual data. There is, however, no unanimity about this view, and theories which consider the
Bfield to have a more passive role are stil being explored. For instance, a popular idea in the
60's was that closed loops would trap Alfven waves, and this view is once again being explored
in a quantitative manner today;
146
Figure 1. Skylab soft xray image of the corona compared with KPNO magnetogram
on the same day. Bright regions of enhanced coronal emission are clearly associated
with the surface locations at which strong bipolar magnetic fields have emerged.
No matter what heating mechanism is invoked, it must explain the strong correlation be
tween the intensity of coronal emission and the strength of the magnetic field. However, the
correlation is not a simple one because, as we wil show below, high temperature plasma is not
seen above sunspot umbrae, which are the strongest magnetic field regions. Clearly, there is
more involved than just the strength of the magnetic field or the length of the loop, and we must
consider the footpoint boundary conditions as well when we attempt to model the formation
and heating of coronal structures (Rosner and Golub 1990).
2. Xray Detection and Recording Methods
The two basic instrumental challenges involved in Xray observations of the corona are forma
tion of an image and recording of the image. Each of the tasks can be divided into two major
methods: Xray imaging has been accomplished for many years by grazing incidence optics,
which have the advantage that they are proven reliable, that they can be used at short wave
lengths and that they work over a broad wavelength range. The alternative technique which we
are using, is multilayer coating of figured optics, which permits Xrays to be reflected at nor
mal incidence. This technique has the advantage that image quality, in terms of optical figure
quality, aberrations and scattering, is substantially improved. Also, the optics are relatively low
cost and lightweight, and they work over a narrow wavelength band, thereby providing some
spectroscopic capability free of charge.
For image recording one can either use photographic emulsions or electronic (TVtype)
array detectors. The latter have progressed significantly in recent years (Kalata and Golub
1988), but for an object like the Sun they still are not wholly adequate. This is because the
Sun has a large angular diameter and we are now achieving subarcsecond resolution. In order
to fully utilize the information in a 1/2 arcsecond resolution image, one needs a pixel size of 1/4
arcsecond (actually 0.22 arcsec), With an angular diameter of c: 2500 arcsec if we want to image
portions of the corona projecting above the limb, the number of pixels needed is 10,000x10,000,
i.e" 108 pixels per image or 109 bits per image.
147
There are some methods available by which this large image format could be achieved,
but their costs far exceed the budgetary constraints of a rocket program or even most astron
omy satellite missions. We are therefore using film to record the image, while also developing
an electronic Xray imaging detector which operates at broadcastquality TV resolution, For
photographic emulsions, the sensitivity is roughly inversely proportional to the square of the
resolution; there is thus a tradeoff between resolving power and exposure time, In a rocket flight
the observing is about 5 minutes, so that exposure time is a major consideration; the number
of photons per second reaching the focal plane translates directly into a limiting resolution for
the experiment.
Figure 2. Xray image of the Solar corona obtained by the NIXT sounding rocket
payload, 11 September 1989; data are recorded at 63.5 A in the coronal emision lines
of Mg X and Fe XVI.
148
The NIXT payload uses a 25 cm diameter, f/8 mirror at prime focus, coated to reflect
63.5 A xrays. The passband includes coronal emission lines of Fe XVI and Mg X, formed at
temperatures of 3Xio6 and 1X106 K, respectively (Golub et al. 1990). This passband permits
imaging of the quiet corona as well as active regions; also, as we describe in sec. 4 below, plasma
at flare temperatures is recorded by this instrument in a particularly effective manner.
3. Loop Atmospheres
The new high resolution data (Figure 2) reveal three major features of importance for models
of coronal formation and heating (Golub et al. 1990): i) the coronal loops which were observed
in previous grazing incidence studies are now resolved into more numerous thinner, i.e., higher
aspect ratio ("spaghetti" model) loops; ii) the xray loops seen in the corona maintain a fairly
constant crosssection for most of their length, then taper rapidly at their endpoints, terminating
in small patches of bright chromospheric emission, either in regions of enhanced network or in
penumbral brightenings; iii) there are no hot loops terminating within sunspot umbrae: the
corona directly above sunspots is dark and is surrounded by hot, bright loops which originate
in strong magnetic field regions outside of the spots themselves.
4. Flares
The multilayer coated mirrors which we used in obtaining the new data have the property
that they wil reflect and focus any 63.5 A xrays which enter the telescope. The central
passband wavelength was chosen because of the Fe XVI emission coronal emission line, so that
active regions wil be most effectively imaged. However, the high temperature (107 K) plasma
produced in flares wil emit continuum at all soft xray wavelengths, and the portion of this
continuous spectrum which falls within the NIXT passband wil be imaged just as are any other
63.5 A photons. We are thus able to observe not only the active regions which produce flares,
but also the flares which occur in those regions.
Moreover, the fact that it is continuum rather than line radiation which is imaged has,
by chance, the desireable property that the amount of radiation in the multilayer passband
is, per unit emission measure, sharply reduced. This effect largely cancels the observationally
problematic fact that, in a flare, the emission measure is greatly enhanced at all wavelengths,
reaching a maximum at the higher temperatures. However, because the NIXT response is lower
at high temperatures, the flare is only an order of magnitude brighter in our images than are
active regions, even though the emission measure in the flare is over a thousand times greater
than that of active regions. We are thus able to observe both the flare and the surrounding
fainter regions in the same image; in this we are also helped by the extremely low scattering of
multilayer mirrors, which keeps the bright emission localized away from the fainter features. It
is for these reasons that flares have the beneficial property of looking distinctly unspectacular
in the NIXT data.
149
There are two striking new features visible in the xray data: the main body of the flare
consists of a single bright arch of emission, comprising about most of the total; a 3D display of
the flare intensity is shown in Figure 5 to ilustrate this point. The second feature is that there
is bright xray emission at the location of the two Ha ribbons, and each of the xray ribbons
contains structure within it. Examination of the development of the flare in Ha explains the
latter observation: each of the flare ribbons is itself observed to be a small tworibbon event,
so that what we are observing is a pair of tworibbon flares within the larger tworibbon event.
Since we have only this one observation it is too soon to tell whether this type of complication
is characteristic of flares in general or peculiar to this one event. However, the Ha development
of this flare was not particularly unusual, so it appears that our new ability to see the fainter
xray structure simultaneous with the brighter structure is really yielding a new view of flares.
During the brief flight of the NIXT payload, a second flare started in an active region at
the west limb. In Ha the flare is seen as a small spray of material beginning at about 16:36
UT. As often happens in such events, the material observed in the Ha ejection disappears as
it leaves the Solar surface; however, since the event is transverse to the line of sight, it is not
likely that the material is Doppler shifted out of the passband. Instead, we are now able to
establish through the xray observations that the material disappears from Ha because it is
heated: in xrays we observe an event which is cospatial and nearly cotemporal with the Ha
event. The main difference is a small time delay between the two, in that the evolution of the
event in xrays is about 1/2 minute behind the Ha event. Moreover, ,the xray ejection appears
to be a hollow cone of hot material, as would be the case if a layer on the outside of the ejected
chromospheric material is being heated as it rises. A detailed study of this event is currently in
progress (Herant et al., in preparation).
references
Golub, L., Herant, M., Kalata, K., Lovas, 1., Nystrom, G., Pardo, F., Spiler, E. and Wilczynski,
J.: 1990, Nature 344, 8.ll.
Golub, L., Maxson, C., Rosner, R., Serio, S. and Vaiana, G.S.: 1980, Ap. J. 238, 343.
Kalata, K. and Golub, L.: 1988, Proc. SPIE 982, 64.
Krieger, A.S., Timothy, A.F. and Roelof, E.C.: 1973, Solar Phyi. 29,505.
Rosner, R. and Golub, L.: 1990, J. Geoph. Rei. (in press).
Rosner, R., Golub, L., Coppi, B. and Vaiana, G.S.: 1978, Ap. J. 222, 317.
Spiler, E.: 1990, Opt. Eng. 29, 609.
iso
Coherent Structures and Statistical Theory of Turbulence
Jackson Herring
National Center for Atmospheric Research'", Boulder, Colorado 80307, U.S.A.
The statistical theory of turbulence (in the form of equations of motion for the two
point covariances, such as Kraichnan's, 1959 DIA) may be viewed as the logical avenue
to singlepoint equations for Reynolds stress and energy dissipation (Leslie, 1973; Her
ring, 1973; Yoshizawa, 1980). For example, far from boundaries a typical closure for the
Reynolds stress (UiUj) == Rij(x, t) is (Hankalic and Launder, 1972):
DRij I Dt = ( c¡fr )(Rij  (2/3)6ijE) +CDßk(E I EH(RknRij,n + RjnRik,n +RinRjk,nl+'"
(1 )
In (1), DIDt == ßt + (Ui)ßi, Eis the kinetic energy = (1/2)Rii, and r the turbulence
timescale E I E, E being the energy dissipation whose equation of evolution begins as
DEI Dt = c€€/r + . . . (2)
, Equations (1) and (2) leadin their simplest formto mixing length, or K,E estimates
for heat and momentum transport. Coeffcients (C¡,'CD,C€) specify rates at which the state
of homogeneity (CD), isotropy (C¡), and timescaling (c€) are approached in the absence
of forcing. They may be "derived" from hyopoint closures by making the usual WKB
approximation, assuIlng R ~ Xi + X2) slow, and p == (1/2)(xi  X2) fast. Effects of the
fast variables then show up in (1) and (2) in de~ermining the coeffcients that are in reality
spectral integrals (see Yoshizawa, 1980). Hence, it is important to assess under what
conditions such equations may have suffcient validity to be useful guides in inferring heat
and momentum transport. Generaly, consolidating of thefi��w into isolated structures is
inhospitable to the underlying nearGaussian assumptions made to arrive at the twopoint
formalsm. This talk, then, mainly concerns some examples of flow in which such structures
may play an important role.
We examine first the simple problem of the decay of isotropic turbulence starting
from initial conditions in which the energy (and enstrophy 0 == It p2 E(p )dp) is mainly
at large scales and for which viscous ~ffects are negligible. The question is whether the,
flow wil remain of bounded variation for a finite time, or wilas predicted by some of
the simpler statistical theories (such as the eddydamped Markovian quasinormal theory,
e,g., Lesieur, 1988, pp.9294)explode, with 0  00 at a time tc '" 60(0). Recent
numerical simulation results for random initial conditions are discussed, which suggest
that 0 increases only exponentially. We argue that such regularity suggests quasitwo
dimensional small scales (�� la Pumir and Siggia, 1990), and that the tendency is toward
an exotic state unrelated to real turbulence. For example, an exponential growth of O(t)
implies an exponential decrease of ((ß'uIßx)3)/(((ßuIßx)2))3/2 == S(t). This follows from
the simple isotropic relation (see e.g., Lesieur op. cit. p. 93, equation (75)):
dO(t)ldt", S(t)03/2 (3)
'" TheN ational Center for Atmospheric Research is sponsored by the National Science
Foundation.
151
Experimentally S rv .5, for developed turbulence. The explanation of the failure of the
closure in this case lies in the spacial separation of regions of strong vorticity (w) from
strong strain S. This is readily seen from (3), and the relation (see again Lesieur, op. cit.
p. 93, equation (72))
dO(t)jdt = ((WiSijWj)) (4)
Some numerical data is presented that ilustrates these points.
Next, we examine the question of "Eulerization", by which we mean that there exist
patches in the flow in which the nonlinear terms are strongly reduced from their nominal
Gaussian value. (The discussion is drawn from Chen et aL., 1989). Such "Eulerized "
regions are thought to be related, but perhaps not synonymous with, coherent structures.
One measure of the importance of such a region is
(( 8uj 8t)2) j (( 8uj 8t)2) ��a
where ('��a means that the moments of u entering (Ôtu? are evaluated as if u were
Gaussian. The comparisons of DNS with closure (in the form of the DIA, Kraichnan,
1959) are in reasonable agreement, both predicting a reduction by about a factor of 2 in
the meansquared Eulerian acceleration, as compared to its Gaussian evaluation.
Finaly we consider twodimensional turbulence, which exhibits an extreme tendency
to organize itself into isolated vortices if left to decay. We examine such structures and
the failngs of the statistical theory, as described in Herring and McWiliams (1984). We
may get at the issue of separateness of vorticity and strain by examiningin the mean
field approximationthe short time development of the twopoint Reynolds stress in the
presence of a structured mean field, which representsin an' admittedly vague waythe
large scales of the flow. The spectrum of the Reynolds stress is specified in a compact
representation in which Rij(k) ~ ((I Ui + iU2 12) = Uo(k), ((ui  iU2)2) = U2(kH. (These
are just rv the first two angular harmonics of the stream function, Herring, 1975). Then
for U 0 (k ), U 2 ( k ):
(DjDt + 2v(k))Uo(k) = 8*(1 + k8j8(4k))U2(k) +...
(DjDt + 2v(k))U2(k) = i(U2 + (lj2 + k8j8(4k))Uo'"
8 == (8iui + 82U2 + i(82Ui + 8iU2))
( = (82Ui + 8iU2)
(5)
(6)
(7)
(8)
If we now assume an additional term in the RHS of (6) representing a linear return
to isotropy ( ¡i'u 2 (k )), and that this term balances the mean vorticity "tumbling" term,
we may eliminate U2(k) in (5), and find:
(DjDt + 2v(k))Uo =1 S 12 /1j(/12 + (2)(1 + kôjô(4k))(1j2 + k8j8(4k))Uo(k) (9)
We remark that setting the RHS of (9) zero ~ Eo(k) = 27rkUo(k) rv k3. Notice that
regions of excess strain 8 (over vorticity ( imply rapid transfer (to ever smaller scales),
whereas regions where I ( 1??1 8 I are stable. This is similar to the "Weiss" criterion
(Weiss, 1981) for discriminating zones of stabilty in two dimensional flow. Weiss' criterion
152
is simply S2 .. (2) implies stabilty. In three dimensionsj it is the second invariant of S,
proportional to the pressure variance: this point has been explored by Hunt (1988) and
by Nelkin and Tabor (1990).
If we now try to integrate (7) into homogeneous turbulence theory, we have a funda
mental problem in that for homogeneous flows, there is no way to discriminate between
strain and vorticity, at least at the level of secondorder moments. Thus, the mechanism
for discriminating between stable and active regions is lost, unless we have equations that
involve more than secondorder moments and associated Green's functions.
References
Chen, H. D., J. R. Herring, R. M. Kerr, and R. H. Kraichnan, 1989: NonGaussian statistics
in isotropic turbulence. Phys. Fluids A. 1(11), 184454.
Hanjalic, K., and B. E. Launder 1972: A Reynolds stress model of turbulence and its
application to thin shear flows. J. Fluid Meeh., 52, 609638.
Herring, J. R., 1973: Statistical turbulence theory and turbulence phenomenology. In
Proe. Langley Working Conf. on Free Turbulent Shear Flows. NASASP321, Langley
Research Center, Langley, VA (available from NTIS as N732815415GA).
Herring, J. R., 1975: Theory of twodimensional anisotropic turbulence. J. Atmos. Sci.,
32, 22542271.
Herring, J. R., and J. C. McWillams, 1985: Comparison of direct numerical simulation of
twodimensional turbulence with twopoint closure: The effects of intermittency. J.
Fluid Meeh., 158, 229242.
Hunt, J. C. R., A. A. Wray, and P. Moin 1988: Eddies streams and convergence zones
, in turbulent flows. In Studying Turbulence Using Numerical Simulation Database II;
Proceedings of the 1988 Summer Program, Center for Turbulence Research. Report
CTRS88, December 1988, NASA Ames Research CenterStanford University.
Kraichnan, R. H., 1959: The structure of isotropic turbulence at very high Reynolds
numbers. J. Fluid Meeh., 5,497543.
Lesieur, M., 1987: Turbulence in Fluids, Martinus and Nijhoff Publishers, Dordrecht, The
Nether lands.
Leslie, D. C., 1973: Developments in the Theory of Turbulence, Clarendon Press, Oxford.
Nelkin, M. and M. Tabor, 1990: Time correlations and random sweeping inb isotropic
turbulence. Phys. Fluids, 2, 8183.
Pumir, A. and E. Siggia, 1990: Collapsing solutions to the 3D Euler equations Phys.
Fluids A 2, 220241.
Yoshizawa, A., 1980: Statistical theory for Boussinesq turbulence. J. Phys. Soc. Japan,
48,647652.
Weiss, J., 1981: The dynamics of enstrophy transfer in twodimensional hydrodynamics.
La Jolla Institute Report, La Jolla, CA. 123 pp.
153
A MODAL ~~DYNAMO IN THE LIMIT OF
ASYMPTOTICALLY SMALL VISCOSITY
Rainer Hollerbach
Inst. of Geophys. and Planet. Physics
Scripps Institution of Oceanography
La Jolla, CA 92093 USA
The macrodynamic equilibration of ~1dynamos in the limit
of asymptotically small viscosity is considered. In the mom
entum equation, the primary balance of forces is between the
Lorentz force and the Coriolis force. However, if one con
siders the integrated Lorentz torque on geostrophic contours,
the Coriolis force is incapable of balancing this torque, and
so if there is such a net torque it must be balanced' by vis
cosity. As a result, in the limit of asymptotically small
viscosity, the geostrophic flow induced by this torque can
become quite large. The resulting scaling of the field,
namely as some positive power of the viscos~ty, means that in
the limit of vanishing viscosity one has no dynamo.
Taylor (1963) proposed that there simply is no such net
torque, that the flow distorts the field in precisely such a
way that there is no torque, and showed that this requirement
in fact uniquely determines the flow. Thus, viscosity is not
relevant or even present in this framework, and so the equili
bration is inviscid, as geophysical scaling arguments suggest
it ought to be. The difficulty with this approach is that in
general, that is for most choices of~, the linear kinematic
eigensolution does generate a net torque, and so at least for
some small range of supercritical forcing viscosity must be
important.
However, as the forcing becomes more and more super
critical and hence the nonlinear coupling between the flow and
the magnetic field becomes more and more important, it is
conceivable, as hypothesized by Malkus & Proctor (1975), that
the flow will tend toward just the "eigenflow" required in
Taylor's development. That this approach to the Taylor state
does in fact occur, at least in an infinite planar g~ometry,
was subsequently verified by Soward & Jones (1983). It is
this transition from the viscously controlled regime to the
inviscid Taylor regime, and the subsequent equilibrätion, that
we explore in this work, this time in a spherical g~cmetry.
154
The dynamo presented here is an expansion in the free
decay modes of the magnetic field. The momentum equation is
expanded as an asymptotic series, making use of a simplified
functional form for the dissipation, and yields e~~licit
expressions for the leading order geostrophic and next order
ageostrophic velocities. With these expressions substituted
back into the induction equation, a set of modal amplitude
equations is derived and solved for a variety of chpices of ~.
, All 0(' s investigated yielded Taylor solutions. For some
choices of ~ the Taylor state is approached in a smootn pro
gression starting from the linear eigensolution, but for other
choices it requires a finite amplitude jump. In the latter
case the solution track leading to the Taylor state can be
either stable or unstable, and this affects where the transi
tion from the viscous regime to the inviscid regime takes
place. In the asymptotic limit the subsequent equilibration
is indeed independent of viscosity, as envisioned by Malkus &
Proctor, and depending on the choice of 0( it can be either
steadystate or os£lJle��tory.
REFERENCES
Malkus, W. V. R. and Proctor, M. R. E., "The macro��ynamics of
O(effect dynamos in rotating fluids", J. Flui�� Eech. 67,
417443 (1975).
Soward, A. M. and Jones, C. A., "CX2.Dynamos and Taylor's
constraint", Geophys. & Astrophys. Fluid Dynam. 27,
87122 (1983).
Taylor, J. B., "The magnetohydrodynamics of a rotating fluid
and the Earth's dynamo problem", Proc. Roy. Soc. London A
274, 274283 (1963).
,
J;
,~
L
a'
155
~DYNAMICS OF LOCALIZED STRUCTUES
WITH GALILBA INVARIAHCE
Christian Elphick, G. R. Ierley, Oded Regev and E. A. Spiegel
ABSTRCT. We consider a nonlinear partial differential equation
having both translational and Galilean invariance arising in the
Kapi tza problem (Benney, 1964). Under sui table conditions, Hopf
bifurcation in extended systems , to leading order, is
represented by this phase equation. We study the interaction of
the localized structures that are formed in such systems both
numerically and by means of an effective particle approach. The
dynamics of a pair of interacting localized structures produces
an interesting scattering problem with the possibility of capture
into one of an infinite number of discrete bound states. For the
manybody problem the localized structures organize themselves
into patterns displaying spatial chaos.
156
Fast reaction, slow diffusion,
curve shortening and harmonic mapS1
Joseph B. Keller
Departments of Mathematics and Mechanical Engineering
Stanford University
Abstract
The reactiondiffusion. problem
Ut = c.6u  ciVu(u), u(x,O,c) = g(x), anu = 0 on an
for a vector u( x, t, c) is considered in a domain n E Rm. Ail asymptotic solution is con
structed for c smalL. It shows that at each x, u tends quickly to a minimum of V ( u ). When
V has several minima, u tends to a piecewise constant function. Boundary layer expan
sions are constructed around the resulting surfaces of discontinuity or fronts. Each front is
found to move along its normal with a constant velocity determined by the discontinuity
(VJ in V across it. When (VJ = 0, the front's normal velocity is cK, where K is its mean
curvature. The moti()n of fronts in this manner is studied for arcs in the plane which are
normal to an at theit endpoints, and for fronts which are closed curves. It is shown a front
can shrink to a point in a finite time or tend to a locally shortest diameter of n. In the
latter case, a nonconstant steady state u(x, 00, c) results.
Then the asymptotic behavior of u is determined for c small when j( u) = 0 on a
connected manifold M of stable equilibrium points. It is found that u tends rapidly to
M, being driven by reaction. Then u evolves slowly by diffusion restricted to M. It tends
ultimately to a limit that is a harmonic map of n into M. Next, the case where j(u) has
stable equilibrium points on two manifolds Mi and M2 is treated. In this case a front
develops in n. It separates the regions where u is close to Mi from the regions where u
is close to M2. For j ( u) = Vu ( u) a boundary layer solution is constructed for u near the
front, and the velocity of the front is found to be proportional to the jump in V across it,
to leading order in c. When V(u) has the same value on Mi and M2, this term is zero
and the front velocity is c times its mean curvature. The case of a spherically symmetric
potential V(lul) and the case M = 8i are presented to ilustrate the results.
,
1'1
"
¡.
,
f,
;
1 This lecture is based upon the following two papers which were published last year:
Jacob Rubinstein, Peter Sternberg and Joseph B. Keller, SIAM J. Appl. Math. 49, 116
133, 17221733 (19~~)~
157
POL YTROPES AND THEIR PERTURBATIONS
Norman R. Lebovitz
The University of Chicago
The classical theory of poly tropes is that of spherically symmetric,
selfgravitating figures of equilbrium in which the pressuredensity
relation is given by the formula p=Kp 1+1/n. This theory is summarized and
the limitation to the range O:n:S is derived in a simple manner via an
identity due to Pohozaev.
For n?S the figures tend to infinite central condensation, Le., the
ratio of central density to mean density becomes infinite. This limit
therefore models the late stages of stellar evolution. For this purpose a
perturbation theory based on £=Sn as the perturbation parameter, wherein
the singular model of infinite central condensation is the unperturbed
solution, is described. The formal theory is shown to have certain
convenient properties, but the relation of the formal, asymptotic solutions
to the exact solutions remains unsettled.
Rotational perturbations of poly tropes have been discussed in the
literature over a long period of time. Here the idea is that of using the
angular velocity as a small parameter to obtain approximate solutions of
the equations governing rotating stars. First some generalities regarding
axially symmetric rotating masses are discussed, and then the theory of
rotating poly tropes is summarized. Defects in this theory are noted and
certain possible modifications are suggested to try to correct them.
Among these are a coordinate stretching, as implicitly employed in
Clairaut theory, and a systematic version of an idea of Monaghan and
Roxburgh employing different approximations in inner and outer regions,
which might be able to extend the perturbation theory to large rotation
rates.
158
Interface Dynamics: Playing with Symmetries
A. Libchaber
The Research Insutute
University of Chicago
Abstract. The diectional growth of a nematic phase into the isotropic one of a liquid
crystal is presented The lengthscales define the problem:
ID=DIV , diffusion length, where D is the impurty diffusion coefficient and V the
velocity of the interface. Ths is the destabilzing term.
Ii=(m t1C)/G, thermal length, where G is the thermal gradient in deg cm1 and
mt1C the temperature jump across the solidusliquidus region (m is the slope of the
liquidus).
1c=1L, capilar length, where r is the interface tension and L the latent heat.
IT and Ie are stabilzing terms. The onset of a wavy pattern corresponds to ID IT.
When ID or IT become comparable to Ie the interface restabilzes. One thus defines and
measures a stabilty tongue.
.12e
ri
cellu lo.r
,'(\ t erta.ce r / a t J' (\ C e r f'c c e.
() ~c.i .f T
For small velocities the bifurcation to a cellular interface is supercritical. Secondar
instabilties develop beyond this critical velocity. They correspond to a spontaneous
breakng of translational invarance, leading to travellng waves with sources and sinks.
For larger velocities, drops of tilte cells appear, breakng the party invarance X7 x. For
even higher velocities time symmetr is broken.
We have presented an overview of this problem, where a rich varety of
bifurcauons can be studied. A simple model with two coupled modes of wavenumber 9
and 29 mimic the observations. '
159
Some Aspects of Convection in Binary Fluid
Mixtures
Stefan J. Linz
FR.l!.1 Theoret. Physik, Universitaet des Saarlandes
D66 Saarbruecken, WestGermany
The onset of convection and the weakly nonlnear convective behavior of binary fluid
mixtures in a RayleighBenard setup has attracted growing interest in the last few years.
In this seminar we reviewed some of our results (18) obtained in the last years. The system
we consider is a horizontal layer of a binary fluid miture enclosed between two paralel,
impermeable plates. The gravitational field acts verticaly. The system is described by
the field equations for velocity, temperature, and concentration of the lighter component
of the miture. Temperature and concentration field are crosscoupled via the Soret effect
(temperature fluctuations can drive concentration fluctuations) and the Dufour effect (con
centration diffusion currents can drive temperature fluctuations). Between the boundaries
there is a vertical temperature gradient.
Focussing on liquid mixtures (where the Dufour effect is ignorable) we derived a gen
eralzed Lorenz model (1) assuming idealzed freeslip boundary conditions for the velocity
field, but realstic impermeable boundary conditions for the concentration field. The model
based on a Galerkin approximation alows twodimensional standing and propagating roll
patterns. The stabilty analysis (1,2) shows that, near the codimensiontwo point where
stationary and oscilatory instabilties compete, there is a gap in the critical wave numbers
and consequently the zero Hopf frequency limit cannot be reached. The weakly nonlnear
solutions were discussed: steady overturning convection (1,2,4), traveling wave solutions
(3,4), standing wave solutions (4) and their stabilty. Beyond that, transport properties
of traveling wave solutions were elucidated (3,4): they generate heat, concentration, and
mass currents andvia Reynolds stressesa mean flow in the horizontal direction. These
properties were later confirmed in a numerical simulation (9). The changes of the stabilty
thresholds caused by realstic noslip boundary conditions were presented (7).
The influence of nonOberbeckBoussinesq effects for smal Soret coupling and the effect
of barodiffusion on the stabilty of the conductive state (5) were discussed.
We estimated that in gaseous mixtures the Dufour effect is no longer ignorable (6). As
an aside we presented a slightly different system where the temperature at the top and
bottom boundary are equal and a vertical concentration gradient is applied. There the
Dufour effect can generate stationary and oscilatory instabilties for large enough Dufour
coupling (6). This seems to be possible in gaseous mitures. Returning to the Rayleigh
Benard like system we studied the changes of the stabilty of the conductive state caused
by the additional influence of the Dufour effect in gaseous mitures (8).
Finaly we reviewed briefly some of our results (4) on the convective behavior if there is
in addition a porous medium between the boundaries. In particular we showed that there
are no supercritical traveling wave solutions.
160
(1) S.J.Linz,M.Luecke,Phys.Rev.A35,3997 (1987).
(2) S.J.Linz,M.Luecke,Springer Series in Synergetics 41,292 (1988).
(3) S.J .Linz,M.Luecke,H,W .Mueler,J . Niederlaender ,Phys.Rev .A38,5727 (1988).
(4) S.J.Linz,Ph.D. Thesis,Universitaet Saarbruecken (1989).
(5) S.J.Linz,M.Luecke,Phys.Rev.A36,3505 (1987).
(6) S.J.Linz,Phys.Rev.A40,7175 (1989).
(7) W.Hort,S.J.Linz,M.Luecke, in Nonlinear Evolution of spatio temporal Structures in Dis
sipative Continuous Systems (F.H.Busse,L.Kramer eds.), NATO ASI Series B (1990) and
in preparation.
(8) O.Lhost,S.J.Linz,H.W.Mueller,submitted for publication.
(9) W .Barten,M.Luecke,W .Hort,M.Kamps,Phys.Rev .Lett.63,376 (1989).
161
HYDROMAGNETIC INSTABILITY DUE TO ELLIPTICAL FLOW
David W. Hughes
Department of Applied Mathematics and The��retical Physics
University of Cambridge
and
Wilem V. R. Malkus
Department of Mathematics
MIT
An unbounded region of ellptical flow gives rise to threedimensional instabil
ities which are exact solutions, (Bayly, 1986). The simplest and fastest growing
of such solutions is the "spinover" mode, (Waleffe, 1988, 1990). This "spinover"
mode is observed to be a principal instabilty in bounded laboratory flows (Gledzer
et aI., 1975, Malkus, 1989). The search for hydromagnetic consequences of these
instabilties include the G.F .D. study by Brazell 1987 and the work of Craik 1988.
In both of these latter papers it was concluded that an initial magnetic field was not
amplified by the growing threedimensional mode, but merely advected. An appar
ent exception to this finding was the special case of constant electric current parallel
to the constant vorticity of the ellptic flow. This basic state permitted exponential
growth of both the fluid spinover mode and its magnetic counterpart, for any de
partures of the basic state from magnetickinetic energy equipartion (Bush 1988).
The energy sources and equilbration of this unique solution are considered here.
The complete equations for the disturbance vorticity wand disturbance current j
are written
w = D : (w  aj)
j = R : (j  aw) + ! (w X j),
2
where D is the ellptical strain matrix, R the rotation matrix, and a is the value
of the ratio of current to vorticity of the basic state. Bush showed that parallel
disturbances, wand j which were orthogonal to the basic vorticity are exponen
tially growing, exact solutions to the above equations. It is found here that this
special solution is unstable to slight variations in the parallelness of wand j. The
figure below exhibits the growth and equilbration in time of the component of j
antiparallel to the basic current. In contrast to the initial exponential growth of
the other j components, the antiparallel component grows superexponentially, due
to the nonlinear term, until it cancels the basic current. Although the disturbance
vorticity continues its growth unabated, this study establishes that the disturbance
magnetic field has drawn its energy from the basic magnetic field and has not ex
hibited dynamo action. Further studies should include the role of boundaries. The
effect of a bounded domain is known to couple the spinover mode to its basic field,
and may couple it to the magnetic field as well.
162
o 50 100 150 200
0.30
0.00
'0.05
0.10
0.15
N
0.20
0.25
0.35
The superexponential growth in time of the disturbance magnetic field, Z, anti
parallel to the basic state.
REFERENCES
1. Bayly, B.J., "Threedimensional instability of elliptical flow", Phys. Rev. LeU., 57,
2160, (1986).
2. Brazell, L., "The instability of an ellptical flow in the presence of a magnetic field" ,
WHOI GFD (1987), Fellow's report.
3. Bush, J., "Hydromagnetic instability of ellptical flow", a M.LT. 18.357 term paper,
outlined in Malkus, 1989.
4. Craik, A.D.D., "A class of exact solutions in viscous incompressible magnetohydro
dynamics", Proc. R. Soc. A, 417,235 (1988).
5. Gledzer, Ye. B., Dolzhanskiy, F. V., Obukhov, A.M. and Pomonarev, V. M., "An
experimental and theoretical study of the stability of motion of a liquid in an ellptical
cylinder". Izv. Atmos. and Oceanic Phys., 11, 981 (1975).
6. Malkus, W. V. R., "An experimental study of global instabilities due to the tidal
(elliptical) distortion of a rotating elastic cylinder", GAFD, 48, 123 (1989).
7. Waleffe, F., "3D instability of bounded elliptical flow", WHOI GFD (1988) Fellow's
report.
8. Waleffe, F., "3D instability of unbounded ellptical flow", Physics of Fluids, A2(1),
76, (1990). '
163
The Free Energy .Princip leN, Negat i ve Energy Modes
and Stability
P. J. Morrison
Department of Physics and Institute for Fusion Studies
University of Texas at Austin
Austin, TX 78712
The Free Energy "Principle" is a conjecture about stabi 1 i ty in
Hamiltonian dynamical systems1, such as those that describe the motion
of ideal fluids and plasmas2. The' conjecture proceeds from S2F, a
functional that measures the energy difference between a dynamically
accessible perturbed state and an equi librium state. This quantity is
easily derived for all equilibria of ideal fluid and plasma models, e.g.,
Euler's equation, the ideal magnetohydrodynamics equations, and the
Maxw ell Vlasov equat ions3. When S2F is definite an equil ibrium is
stable. This is a generalization of the LagrangeDirichlet theorem of
mechanics, and di ffers from previous work regarding Liapunov stabi lity4.2
in that S2F is the energy di fference restricted to the constancy of all of
the kinematical or Casimir invariants. When S2F is indefinite, either the
equil ibrium is linearly unstable, or we have the interest ing situation
w here a linearly stab le equi librium does not correspond to an energy
extremum. In the latter case the system possesses a negative energy
mode. This definition of a negative energy mode (NEM) is a
generalization of that commonly used in plasma physics that is based on '
the dielectric function5. Finite degreeoffreedom Hamiltonian systems
have NEM's when the linear normal form is stable with indefinite
signature; for example, when the linear Hamiltonian in actionangle
variables (J,a) has the form S2F = r w¡J¡, where some of the frequencies
w¡ are negative. It is conjectured that systems with NEM's are
generically nonlinearly unstable to infinitesimal perturbations, in spite
of the fact that they are linearly stable. This is exemplified by an
eXample due to Cherry6, which due to linear resonance demonstrates
explosive growth. Three or more degreeof  freedom systems with NEM's
are thought to be unstable without linear resonance, due to a mechanism'
known as Arnold diffusion. It is tempting to speculate that generically,
infinite degreeoffreedom systems with NEM's are also unstable. In
164
add i t ion to non linear inst abil i ty, sy stems with NEM's are often
structurally unstable when dissipation is added to the dynamical system.
Physically this is an appealing intuitive idea since when energy is
removed from an NEM its amplitude must grow. This is a generalization
of the Kelvin Tait theorem. In summary, the Free Energy "Principle" is
the conj ecture that sy stems with NEM's are generica iiy unstab Ie, either
nonl inearly or structurally.
References
1. P. J. Morrison and M. Kotschenreuther in Proceedings of the 4th
International Workshop on Nonlinear Turbulent Processes in Physics,
Kiev, USSR, 1989 (World Scientific, Singapore, in press 1990); M.
Kotschenreuther et aL., in Plasma Physics and Controlled Nuclear
Fusion Research 1986 (IAEA, Vienna, 1987), VoL. 2, p. 149.
2. P. J. Morrison and S. Eliezer, Phys. Rev. A 33,4205 (1986); P. J.
Morrison and D. Pfirsch, Phys. Rev. A 40,3898 (1989).
3. P. J. Morrison and D. Pfirsch, Physics of Fluids B 2, 1113 (1990).
4. M. Krus,kal and C. Oberman, Phys. Fluids 1, 275 (1958); C. S. Gardner,
Phys. Fluids 6, 839 (1963); D. Holm, J. E. Marsden, T. Ratiu and A.
Weinstein, Physics Reports 123, 1 (1985).
5. P. A. Sturrock, J. App1. Phys. 31, 2052 (1960). See also R. C.
Davidson, Methods in Nonlinear Plasma Theory, (Academic Press, New
York, 1972).
6. T. M. Cherry, Trans. Camb. Phi 1. Soc. X X I I i. 199 (1925). See E. T.
Whittaker Analytical Dynamics. (Cambridge. London. 1937) Sec. 182,
p. 142; and A. Wintner. The Analytical Foundations of Celestial
Mechanics, (Princeton Univ.. Princeton, 1 947) Sec. 136 bis, p. 101.
165
Radiatively Driven Stellar Winds
Stan Owocki
Bartol Research Institute
University of Delaware
Newark, DE 19711
The massive winds from hot, luminous stars are thought to driven by
linescattering of the star's continuum radiation flux. This
summary emphasizes the physics of this linedriving mechanism and
what it implies for basic wind properties, as well as for wind
structure and variability. Linear perturbation analyses of the basic,
CAK model for a, steady linedriven wind indicate that such winds
are in fact highly unstable. Numerical simulations of the nonlinear
evolution of small ampliude' (1 %), periodic perturbations at the
wind base show that this instability leads to high speed
rarefactions which terminate in strong reverse shocks. Subsequent
work indicates that such variability can have an intrinsic character
that persists even in the absence of explicit perturbations, and it
now seems that this is a direct consequence of a degeneracy of the
steadystate solutions. Current efforts are aimed at determining
how this degeneracy and intrinisic variability are affected by
including the dynamical effects of the diffuse, scattered radiation
field. Future work wil focus on generalizing the model to 2D (or
3D) with rotation, and on modeling the unsteady flow energy and
ionization balance with radiative terms.
A more complete discussion of these points can be found in my
recent reviews, "Winds from Hot Stars" (1990; Reviews of Modern
Astronomy, Vol. 3., Springer: Berlin) and "Theory Instrinsic
Variability in Hot Star Winds", (1990; Proc. of IAU Colloq. #143, K.
Van de Hucht, ed.)
166
CHOS AND NOISE IN DYNAMICA SYSTEM WITH SLOW
INVAJ SUBS PACES _
Michael R. E. Proctor
Dept. of Applied Mathematics & Theoretical Physics,
University of Cambridge, Silver st., Cambridge CB3 9EW, England
ABTRCT _ There are many dynamical systems of interest in
mathematical physics which possess the feature that there is aninvariant subspace in which one or more of the variables
vanish. If parts of this subspace are attracting and parts
repelling with respect to orthoginol perturbations, then thetraj ectory may return repeatedly to a neighbourhood of the
subspace. If in addition the dynamics in the subspace takes
place on an asymptotically long time scale, then it is possible
to represent the dynamics by compositions of maps, all of which
are independent of the time scale ratio. The method is
particularly useful when the slow subspace is one dimensional.
A particular example is given for the equations describing
threewave resonance, and it is shown that the dynamics may be
reduced to a map of the interval, given in simple analytic
form.
During the evolution the orthogonal variables can become
extremely small, and so any small perturbation that destroys
the invariant plane has a dramatic effect on the dynamics. Inparticular, if the ratio of time scales is b" ((.(. .1) then
perturbation~ have( an O( 1) effect when their size E: is such
that \ lf.1n ' I ~o i). If �� is larger than this the dynamics is
dominated by the noise, and the evolution takes the form of a
'noisy periodic orbit' whose (mean) amplitude and period depend
crucially on ��. Dynamical systems of this type may provide a
new paradigm for intermittency in disordered flows.
References: D.W.Hughes and M.R.E. Proctor 1990. A loworder model of the shear instability of
convection: chaos and the effect of noise.
Nonlin~aritv l, 127153.
D.W.Hughes and M.R.E. Proctor 1990. Chaos and
the effect of noise in a model of threewave
mode coupling, Phvsica D, in press.
167
Calculating Transient Coronal Loops '
David W. Rose
12 July 1990
Magnetic flux tubes, which emerge and then reenter the photosphere, occur
on a wide range of length scales in the corona of the sun. Most are fairly stable
and persistent, lasting on the order of days to weeks. In 1972, E. N. Parker pro
posed a mechanism by which the extraordinarily high temperature of the solar
corona could be caused by Ohmic heating of electrical currents contained ,within
these flux tubes. In order that suffcient heat to produce the temperature differ
ential be produced, based upon rough measurements of magnetic field strengths
and flux tube volumes, it was necessary that very intense concentrations of cur
rent be formed in these flux tubes. Parker surmised that very strong and highly
local current sheets would form as the magnetic flux lines became twisted up
through motions in the photosphere. He argued that while the magnettc field
would be driven to a forcefree state within the corona, that the condition of
its flux lines tied to photosphere plasma motions would cause an increase in
the amount of line braiding until forcefree fields would cease to exist. Then
current sheets would form, causing local reconnection of the magnetic field, and
simultaneously coronal heating.
The second part of this hypothesis, that current sheets would form, is stil
unknown. The diffculties are that the mathematical system is diffcult, being
inherently a threedimensional nonlinear problem. Analytically, there is no
theory for the formation of singularities from such boundary motions, and while
it is true that force free fields cease to exist, there is no theory of what happens.
Even numerically, the task is diffcult. One may reduce the MHD equations
into a simpler system which presumably retains the essential terms necessary
to determine current sheet formation. This simplification was performed by H.
Strauss in 1975. In its simplest form, it is obtained by renormalizing the MHD
equations with the density and rescaling the velocity and magnetic induction as
~ )
i~
i.
fi
~
B + Z + ��B(:i, y, ��Z, ��t)
u + ��U(:i,y,��Z,��t)
(1)
(2)
and taking as equal the coeffcients of powers of ��. The effect of this is to
decouple components tangent and transverse to the zaxis. It also has the effect
of enforcing incompressibility on the transverse flow velocity. '
168
At present, I have produced a numerical code which performs pseudospectral
approximation in the transverse space variables and centered finite differences in
z. With time advance using a twostep method, a numerical instability results
in the transverse direction originating at the location of maximal gradient of
the magnetic potentiaL. For a simple case, this instability is effectively of the
type due to decoupling in space, coupling in time. A proposed solution to this
problem is to "split" the Jacobian terms of the form (A, U) = AzUy  AyUz
so that the second term occurs implicitly in the next time step, while the first
occurs explicitly.
Blocking A Barotropic Shear Flow
Melvin Stern
Florida State University
The "upstream influence" on an inviscid shear flow around a semicircular cape (ra
dius=A) is computed for an undisturbed (x = 00) velocity profie: uo(y)=y (for 0 :: y
:: 1), 11 (y) = 1 + (2 Y  (2 (for i :: y c (0), where (2 is a constant vorticity. Linear
theory for A + 0, (2 = 0, Time = t + 00 gives a "weak" upstream infuence, in wmch the
upstream area of the boundary layer (vorticity =  1) increass as t1/2. For A ). 1 contour
dynamcal numerical calculations for the piecewise unform vorticity flow show "strong"
blocking or "complete" blockng, in which either a fraction or none of the boundar layer
flux passes around the cape. A semiquantitative critical condition on ((2, A) no upstream
influence is developed; The relevance of tms simple barotropic model to the control of
oceanic coastal and strait currents is suggested.
169
A Fluid Mechanicists Introduction to
Lie Symmetry Groups and Partial Differential Equations
by Rick Salmon
Symmetr group methods are attractive because they apply to general nonlinear
equations. Goo references include the books by Bluman and Cole (1974), Olver (1986),
and Bluman and Kumei (1989). The lectures summarze below are a gentle introducuon
to Olvets chapters 2 and 3.
Lecture 1. Symmetry Groups
Given a differential equauon, the idea is to find transformations of the dependent and
independent varables for which the equation is unchanged. For ordinar differenual
equations, each such transformauon leads to a reduction, by 1, in the order of the
differential equation. That is, applied successively, transformation groups can lead to
quadrtur. For paral differenual equations, the trnsformation grups lead to a general
family of invarant (similarty) solutions.
Given a paral differential equation of (say) second order in u(x,t),
F (x, t, u, uz, u" U;a, U;r, un) = 0 (1)
we want to find a solution,
G(x,t,U)=O (2)
Here, F and G are ordinar functions of their respective arguments. The general situation
is that F is a given function and G must be found. Prom a geometrc viewpint, (1) is a
7diensional surace in the 8diensionaljet space with coodiates
x, t, U, uz, U" U;a, Uxi' un (3)
A solution (2) is a 2dimensional surace in me 3diensional base space with coordiates
x,t,u (4)
The generazauon to more varables and higher derivatives is obvious.
We consider transformations of the varables from "old" coordinates (x,t,u) to "new"
coordinates (x',t,u'). Under certain assumptions such trnsformations form a group. If
me grup depends conunuously on a pareter s , then it is caled a Lie group :
x '= f (x , t, u; s)
t= g (x , t , u; s)
u'= h(x, t, u; s) (5)
It is conventional to let s=O corspond to the identity element of the grup. Then
i 70
x = f (x , t, U ;0)
t=g(x,t,u;O)
u=h(x,t,u;O) (6)
One way to generate such a grup is as the solution to equauons of the form
~' = ç(xl , tl , ul),dt' (' I ')ds=11x,t,u,du' "'( I I ')ds='tx,t,u,
X' (0) = x
t (0) = t
u i (0) = u
(7)
It is then useful to think of (x',t,u') as the "locauon" at "time" s of a parcle initially at
(x,t,u) that moves always with the "velocity"
v = (Ç(x, t, u), 11(x, t, u), q,(x, t, U)) (8)
The simplifying feature is that the "velocity field" (8) is "steady", i.e. s independent.
Thus, (8) is everywhere tangent to the basespace trajectories that define the
trsformation. '
The "velocity field" (8) deterines a coresponding "velocity field"
( d(u.) d(u,) d(u",) d(uii) d(U")Jpr v = ç, 11, q" ds ' , ds ' ds ' (i (9)
in the jet space with coordiates (3). (The notation pr v stands for "prolongation of v," a
good terminology.) The f1ist thee components of (9) are the same as (8). The last five
components of (9) can be expressed in terms of ç, 11, q, and their derivatives. It is obvious
that such expressions must exist, from the simple fact that the formula for the tranormation
of a function implicitly determines formulas for the transformations of all its derivauves.
As an example, we wil calculate d(ut)/ds.
The Taylor expansion of (7) is
xl= x+sç(X,t,u)+...
t= t + s11 (x , t , u) +. . .
u'= u + sq, (x, t, u) +. . . (l0)
and (lOab) revert to
x = x' sç(x', t, u') +. . .
, (' I ')t = t  s11 x , t , u +... (11)
So
171
au' (aX a cJ a) (2)
iJ i = a ax + aa U + sø (x, t, u)) + 0 s
= ~ + s~D,Ø  u"D,ç  u,D,11 1 + O(S2) (12)
where
D, æ ~ + u,~ (13)
Therefore
d(u,)
l = D,Ø  U" Diç  uiDI11 (14)
Returing to our differenual equation (1), we seek transformations of the form (7) for
which the equauon takes the same form in new varables as in old, i.e.
F( , , " , , , ')  0X ,t ,u ,u " U" U ",,,,,U "'I"U 1'1'  (15)
where F( , , , , ) is the same function of its arguments in (15) as it is in (1). For small
s, (15) is
F(x + Ç(X, t, u)s + O(S2), t + 11(x, t, u)s + O(S2), ...........)(16)
Thus, subtracting (1) from (16) and letung sO,
aF ' aF d(un) aFç (x, t, u)"d + 11 (x, t, u)e1 +. . . + '"di = 0
n (17)
(provided that the paral derivatives of F ar not al zero.) By changing the definitions (8)
and (9) slightly to
V æ ç(x, t, U)%x + 11(x, t, U)~ + Ø(x, t, U)%u (18)
¡
J,
.~
t
4
and
a a a d(u,,) apr v æ ~(x, t, U)ax + 11(X, t, U)ai+ Ø(x, t, U)au + ds au" +. .
d (Un) a
...+ds au" (19)
we can rewrte (17) in the compact form
(pr v) F = 0 (20)
172
The tangent vectors (18) and (19) are simply the "advecuve derivauves" associated with
the "velocity fields" (8) and (9). Equauon (20) just states that the jetspace "trajectory"
must lie in the hyprsurace (1) corspondig to the differential equauon.
We can now solve (20) and (1) to determine the components of the "velocity field" that
defines the trsformauon. This involves equaung the coefficients of like powers of (3) to
zero in (20), after using (1) to remove one of the coordinates. There results a set of linear
differenual equauons in the funcuons ç, 71, and t/x,t,u). These equations, which are called
the determining system of the transformation, are linear (cf. eq. 14), even when the
original differenual equauon (1) is nonlinear. This is what makes the method so usefuL.
For highorder nonlinear equations, the determining system may contain hundreds of
equations. Fortunately, there now exist symbolic manipulauon progrs that do nearly al
the work of setung up and solving these systems.
As a simple example, we consider the heat equauon,
./
F=uu =0I ic (21)
for which the general solution of the determining system turs out to be
ç (x, t, u) = ci + c.t + 2cst + 4ci;t
71(x, t, u) = c2+.2ci + 4c6t2
Ø(x, t, u) = (c3 csx 2c;  Ci;2)U + a(x, t) (22)
where the Ci are arbitrar constants and alx,t) is an arbitr soluuon to the heat equation.
We thus say that the heat equauon is invarant to the trsforauons generated byd dv = ax' v 2 = a'
d d
v 5 = 2t ax  xu au '
d
Va = a(x, t)du
d d d
v 3 = U au' V 4 = x ax + 2tad 2d (2 ) d
v 6 = 4tx ax + 4t a  x + 2t u au '
(23)
The transforation generated by vI is
x'= x + s, t= t, u'= u (24)
Let
u=f(x,t) (25)
be a parcular solution of the heat equation. The vitrsform of (25) is
u'= f(x' s, t) (26)
But we alady know that the primed varables also satisfy the heat equation. We therefore
conclude that if f(x,t) is a solution to the heat equation, then so must be f(xs,t) for any
constats. RepeaUßg this logic for all of the generator in (23) we conclude that:
i 73
If u=frx.i) is a soluuon to the heat equation. then so must be:
v I:
v 2:
v 3:
VA:
V s:
V 6:
va:
u = f (X  s, t)
u=f(x,ts)
u = e8 f ( x, t)
u =f(e8x, e28t)
2
u = e8%~ 'f(x  2st, t)1 J  sr) (X . t )
u = VI + 4st eXi\1 + 4st f 1 + 4st ' 1 + 4st
u=f(x,t)+sa(x,t) (27)
where s is an arbitr constat and a(x,t) is an arbitr soluuon to the heat equauon.
The abilty to trsform soluuons into other soluuons is someumes useful by itself; the
transformation of even trvial solutions (e.g. constants) can yield nontrivial results.
However, it is in the determinauon and classifcation of invarant soluuons that symmetr
group methods realy show their muscles.
Lecture 2. Invariant Solutions
\ '
In the first lectue we drw an analogy between a trsformation grup of a differenual
equation and the parcle trajectories in a steady flow. The generators of the group are
analogous to the velocity field of the flow. Knowing the generators is equivalent to
knowing the trnsformation group, but, as the fluid mechanical analogy would suggest, it
is usualy much easier to deal with the generators than it is do deal with group.
The generators form a Lie algebra with mathemaucal properties that reflect the
underlying group. The most interesting of these properues is this: If Va and Vb ar any
two generators, then their Lie bracket defined as
(v G' v bl = V G Vb  V b V G (28)
¡.
. !T
fis a linear combination of al the generators. For example, consulting (23), we find that
(v 2' v 6l = 4v 4  2v 3 (29)
This closure propert of the Lie algebra is a consequence of the correspondence between
generators and transformauon grups: It can easily be shown that the commutator (28) is
the generator of the composite transformation consisting of an infinitesimal displacement
along the trajectory corrsponding to Va, followed by an infinitesimal displacement in the
diection of Vb, followed by backwars displacements in the diections of va, and then Vb.
This composite transformation is certainly a member of the general group of
transformations, and therefore its generator, (28), is some linear combination of the basis
vectors Vi.
We now tu to invarant (similarty) soluuons. Recall that Cpr v) is, by hypothesis,
tagent to the soluuon surace F=O in the eightdimensional jetspace. However, v is not
174
necessarly tagent to an arbitrarly chosen solution surace G=O. That is, (pr v) F=O but
vG# O. In fact, it is the "flow" across soluuon suraces that cares soluuons into other
solutions, as in (27).
Now let v be a particular generator and consider the special soluuons for which vG=
O. The generator vi in (23) offers a trvial but prototyical example. VI has a component
in the xdirection, but no components in the t or u diections. Thus VI G= 0 only if
G=G(t,u). That is, soluuons invarant to VI tae the form u=u(t) and ar independent of x.
(Tese soluuons tur out to be trvial indee: they are just constats!)
For an arbitrar generator v, the invarant solutions are found by a method that
amounts to finding the special coordinates in (x,t,u)space for which that generator takes
the canonical form (24). This is most easily done by the method of characteristics: one
determines the funcuons J11(X,t,U) and J12(X,t,U) , called diferential invariants, whose iso
suraces intersect to form the trajectories of v. The similarty solution then takes the form
J11 =g(J12) where g is a function to be determined by substitution into the original
differenual equauon.
As an example, we calculate the similarty soluuons corresponding to the generator
CVI +V5 of the heat equauon, with c an arbitr constant. The charcteristic equauons
dx dt__du2t + c = 0  xu (30)
yield the dierenual invarants
II = t '" J1  u ex ( .! x 2 )ri 2  f\ 2t2 + c (31)
so that the similarty soluuon taes the form
( _.! x2 )
u = g (£) eXI 2t : c (32)
with g(t) left to be determined by substitution in the heat equation.
To study the most general similarty solution of the heat equauon, we must use the
genera generator
v = ci VI + c2 V 2 + c3 v 3 + c4 v 4 + Cs v s + c6 v 6 + v a (33)
where Ci ar arbitr constants. Unfortunately, the characteristic equations corrsponding
to (33) are very difficult to solve. However, this task can be cirumvented by a procedur
that forms the slickest par of the whole theory.
The essential idea is very well ilustrted by the example (3032). To obtan (32) we
can use a combination of VI and V5 as above; or we can use V5 by itself to obtain (32) with
c= 0, and then use the property (27b), obtaned from V2, that any soluuon can be ume
trslated That is, we can use a more restrcted generator to obtain our similarty solutions
i 75
if we combine the results with the rules for transforming soluuons into soluuons. It can be
shown, in fact, that if c5;t in (33), then we can have ci=O with no loss in generality.
The special geometrcal relationship between vi, v2 and vs that allows this can be
explained as follows: If the trajectories tangent to vs are subjected to a coordinate
trsformation corresponding to V2, then the trsfored trajectories are tangent to a linear
combination ofvS and VI. We can regard the trajectories as material lines in a perfect fluid
with velocity V2. The material lines ar cared along by the fluid, and the tagents to these
lines ar Lie dragged by the velocity field V2 in the same way as the vorucity or magnetic
field vectors in a perfect fluid. In complete analogy with the vortcity or inducuon
equauons, the evoluuon equation for the tagent, v, is
dv
ds = (V 2' V) (34)
with iniual condition v(O)=VS. The Taylorseries solution of (34) is
v(S)=vs+S(V2,vs)+ls2(V2,(V2'VS))+'. .
=vs+ S(2Vi)+iS2(V2,2vi) +. ..
=vs+2svi
Thus, a transformation ("advecuon") by V2 drags vs into vs +2s VI.
Stang with the general generator (33), we employ all possible draggings to eliminate
as many components of (33) as possible. Each elimination requires an assumpuon about
the arbitrar constats in (33) (typically, that a parcular Ci is nonzero), and the converse of
each assumption must be separtely examined. The final result is an optimal subset of
generators, each very much simpler than (33). All the similarty solutions obtainable from
(33) can then be obtaied from this optimum subset (whose characteristic equauons are
much easier to integrte) plus the rules for transforming solutions into solutions.
(35)
For the heat equation, the optimal subset and corrsponding forms of the similarty
soluuons tu out to be:
v4+ cV3 u = t g( .J )
u = (1 + 4if1l4exp.f  tr 2  ~ arta(2i) 1. g (oJ x )1. 1 + 4t J 1 + 4t2
u = expfxt + ~il g(x + l)
u = eel g(x)u = g (t ) (36)
V 2 + V 6 + CV 3
v2VS
v2+ cV3
v i
All the other similarty forms can be obtained from (36) by the transformations (27). The
varous cases in (36) yield ordinar differential equations for g whose soluuons can be
wrtten down as parbolic cylinder functions, Air funcuons, or trgonometrc functions.
Rainer Hollerbach and I have applied the above methods to the thermocline equation, a
paricularly gresome fourtorder nonlinear paral differential equation of interest to
oceanographers. The results, which include many previously unnoticed similarty forms,
wil be reported in a fortcoming publication. These lectues were a precursor to our
summer on "Geometrcal methods in fluid dynamics" in 1993.
i 76
VORTICES IN TIGHT EMBRACE
by
E. L. Schucking, Deparent of Physics, New York University
and
E. A. Spiegel, Deparment of Astronomy, Columbia University
ABSTRACf. We study simple flows of an incompressible fluid of density p and
constant kinematic viscosity v in a hypersphere S3 of radus Â. The S3 is imbedded into
a Euclidean space of four diensions and described by the equauon
z z z
I zil + I zzl = Â, zi = x + i y , Zz = Z + i u.
We divide the S3 into the th regions I, II, and III by
l O~lzil~ riÂ
II Âri ~ IZil ~ rzÂ
III Ârz ~ IZil ~ Â.
The regions I and III are solid tori which show at ume t=O the velocity field of rigid
rotation:
l dZil .dt =imizi,
1 =0dZil .
dt = i m3z i'
1 =0
;1 =0;1=0
dZzl
dt =0
1=0
III
with m) and m3 constant, Le. independent of r= IZ)I!Â,
In region II between the two vortex cores we assume at t=O a pseudopotenual flow
which has vanishing divergence of its shear with velocity field
dZil .
dt = i m (r)z l'
1 =0
dZzl
 =0,dt
1=0
m( ri) = ml' m( rz) = mz'
All three flows are stationar solutions of the NavierStokes equauons at t=O for a
continuous velocity field. The jumps in velocity gradients at r) and ri lead to a time
dependent solution of the NavierStokes equations for m(t,r) in terms of Jacobi
polynomials in r with timedependent coeffcients.
Flows in S 3
The S3 of radius Â is given by the equation
z z z
IZil + izzi = Â (1)
177
with
zi==x+iy, Z2 == Z + i u (2)
where x, y, z, and u are Caresian coordinates of a Euclidean R4.
We parameterize the S3 by the coordinates r, q, and t¡such that, with r ranging from 0
to 1,
zi = À r ei ~, 0 ~ q" t¡ ~ 2n , Z2 = À.J 1 r2 /". (3)
The dierenuals ar
dZi = À(dr + i r dq,)ei~, dZ2 = À( ';lt: r + i .Ji r dt¡ )ei iy.
(4)
We get then for the line element
dsl = Idzt + Idzl = À2 ( 1 ~2r2 + rdq,2 + (1 r)dt¡2).
(5)
We shal study flows for which rand t¡ are constat. This gives for the velocity v
ds dq,
di = v = Âre == Mm. (6)
The anguar velocity m may be a funcuon of the two varables r and t,:
m = mer, t). (7)
We first write down the condition that the flow has vanishing divergence. If we
number coordinates r, q" t¡ as xl, x2, x3 the velocity vector vi has the components
U=1,2,3)
,;, 
"
~.
dxi = v j _ ,,_~ j _ dq, ~ j
dt   WCi  dt ui
t
r
(8)
The fom��ula for the divergence of a vector field vi is given by
" 1 ( ")VI, == div v = .yvl;1 .. . j (9)
where the semicolon denotes the covarant derivative with respect to xi U =1,2,3) and the
comma the ordinar paral derivauve. The determinant of the metrc tensor is called g. In
our case we have from (9)
i 78
r2
~J (10)( 1
2 1 r2
Ilg,,11 = Å ~
o
o
and thus
g = detllgji:/I = 'lr2. (11)
The flow has vanishing divergence if
. 1 (3 )V / ,== div v = ~ Å rm = O.;/ Å r ,2 (12)
Since we assumed that m did not depend on x2 =Ø the divergence vanshes.
The shear tensor for the flow is dermed as1 ) 1 I l( )
Sjl:= ï(Vj;1: + v.t;;  ïgj.tV ;1 = i Vj;1: + VI:;j
(13)
where the last equalty follows from the vanishing of the divergence in (12).
According to the definitions
I
v. == g.LV.l V.,L == v. L  r'L v, = v. L  r.L ,v',/ I.. I,.. I. .. I.. I , .. i...
r  1'L I = 2( g 'I .t + g I: , g '.t I)i... I. .¡ I. (14)
one derives that
l( I I I )S '.t = ï g '.t ,v + g/lv ,+ g,iv L'J I . .1 I ." (15)
With (8) we get
1Sj.t= ï(gjk.2m + gu w,j + gj2 w,.t)' (16)
As the first term vanishes since the metrc is independent of tf2 , the only nonvanishing
term of the shear for our flow (8) is
1 '12 2Òl
S12 = S2I = i A r ar'
(17)
With an m independent of r we obtai thus rigid rotation.
i 79
The viscous stresses enter the NavierStokes equations though the divergence of the
shear tensor. We calculate
l l 1 (_r; l) 1 lls,.. =S'Ol=V vgso 2'gll'sJ". / . g / ./
. l (18)
from the defmiuons of covarant derivatives. The last term in ths expression is zero since
g12 vanishes.
We have from (10) for the contrvarant (inverse) metrc
Ilgilll = ;2
1 r
o
o
o
1
r
o
o
o
1
1 r (19)
Thus frm (17)
2 :1 1 dmSI = sl2 g = 2' dr ' S 1_ S gll _lr(l r2)dm2 21 2 dr' (20)
All other components vanish. We obtan thus from (18)
suol = 0, S3l;l =0, S2l;l = 2~ f( r3(i  r)d; J.
(21)
The divergence of the shear tensor vanishes if
dm _ _ 2B _ _ 2B ( 1 + r2 + r )
dr  r3(1 r2)  r 1 r2 ' B = const. (22)
Ths integrtes to
m = mo + B(In i r + :2)' mo = const.
(23)
The flow consists of a rigid rotation with constat angular velocity ~ and the "pseudo
potential" flow given by the angular velocity
m = B( In 1~i'2 + ~).
(24)
The expression"pseudopotential" derives from the fact that in the limit Å~oo the
logarthmic term goes out and we obta the potential flow
1vL' L= Âr (25)
180
where L denotes the distance from the axis of rotauon. However, while the potential flow
in R3 has vanishing vorticity we have for the pseudopotenual flow in S3 with (24)
2
) 2 d 1 ,i 2BÂvI 2  v2 1= Â( ,i(i =  BÂ dr In ,i = ( ,i) ,",I r 1 (26)
while the other components of curl v vanish.
Going back now to the expression for the pseudopotential flow in (24) we observe that
(i becomes singuar on the axis at r=O and also at the largest distance from the axs at r=1.
For posiuve B the angular velocity (i decreases monotonously from +00 at r=O to.,oo at
r=1.
A model flow
We want to study the development of a Couette flow without any free or rigid suraces
that is finite and has no singularues. To this end we sandwich the pseudopotenual flow
between two rigidly rotaUßg vortex cores.
W Wi
.'r i,~,'d
'Y~lI' Di
~~~;, øtPl; le
~.)O
W3
"l i t'o(
~tA .j ht
~C ~ ~ ~
Figu 1. Intial distrbution of angular velocity at tie t =0 for B~.
Ths eliminates the singularties at r =0 and r=1. The angular velocity  and thus also the
velocity  is a conUßuous function of the radial distance r. At tie t =0 we star with a
stauonar solution of the NavierStokes equauons that has been pieced together at r=rj and
r=r2.. At these two suraces we have finite jumps in the radial velocity gradients. These
disconunuiues wil create a time dependence of the flow and it is to be expected that for
t~oo a uniform rigid rotation wil result with an angular velocity determined by the initial
(conserved) total anguar momentum.
The NavierStokes equations ar given by
J~
,!t..i\k
J'"
Dv, 1 j
Dt = Pp,¡ +2v s¡;j (27)
181
where p is the constant density of the fluid, p its pressure and v its kinematic viscosity
which is assumed to be constat here. The lefthand side of the equauon is given by
Dv ¡ dv ¡ k dv¡ k 1 k
Dt == a + v ¡;k V = a + (V ¡ ,k  V k, ¡ ) V + '2( V k V ) ,
,J (28)
Since we take
v = V = 01 3 (29)
for our flow we let the pressure p depend only on r and t. The II 3 "component of the
equauon is then idenuca1ly fulfilled as we see from (21) which also tells us that the II 1 "_
component becomes
DV1 2 1 ( 2 2 ) "12 g.2 1 dp
Dt = V2,1V + '2 V2,1V + V2V ,1 = IL rw = p dr' (30)
This means that the pressure gradient wil balance the centrugal force. The "2"equation
becomes with (21)
DV2 = dv2 = x2 ram = 2v s k = lø r3(1 r2)am JDt at at 2k ; r ar dr (31)
We introduce now a new dimensionless Ufe varable 'C by
'C = 4vt
X2 (32)
and indicate paral derivatives with respect to 'C by a dot
;., = amUJik (33)
Our equation (31) beomes
rm = iø r4(i  r2)i am J2r ar 2r dr (34)
Instead of r we introduce
ç == r2 (35)
as a new varable also raging from zero to one. We denote paral derivatives with respect
to ç by a prime
m' = am _ lam
 aÇ  2r dr (36)
182
Equauon (34) becomes then
çw =(ç2(1_ ç)co'J'= ç2(1_ ç)co"+ ç(23ç)co'. (37)
Ths gives the paral diferenual equation
w = ç(1 ç)co"+ (2 3ç)co'. (38)
Separauon of varables gives
co = f ('r)g (ç) (39)
and
L. g" g'
= ç(l ç) + (2 3ç) = af g g (40)
with constant a. We obtai thus the equations
f + af = 0, ç(1 ç)g "+ (2  3ç) g'+ ag = O. (41)
The second equation poses an eigenvalue problem with eigenvalue a in the interval from 0
to 1 with the boundar conditions that g should be finite. The soluuon is given by the
Jacobi polynomials J n(2, 2, x) which solve the hypergeometrc diferential equauon
ç(l ç) g' '+ (2 3ç)g '+ n(2+ n)g =0 (42)
Ths means the eigenvalues a are given by
a= n(n + 2), n = 0, 1,2,. ... (43)
We wrte here
J,,(2,2,x) == tP,,(x) (44) t
t!and have as genera solution
co = !c"tP,,(x)e I
( ,,+2)i'
,,=0 (45)
The eigenfunctions tPn ar given byl
1 A simple formula is tP,,(x) = (n ; 1)! tf"Jx"+l(1 x) "1
183
"4',,(x) = 1 + n ~ 1 L( 1) k(~)(n +nk + l)xk.
k=1 (46)
The first ones ar given by
4'0 = 1, 4'1 = I  t x , 4'2 = 1  4x + i~ x 2
4'3 = I  ~ x + 15x 2  ~ X 3. (47)
The 4'n(x) form a complete system of orthogonal polynomials on the interval from zero to
one with weight function x. We have
i
f x4'm(x )4',,(x)dx = O"m 1 3'o 2( n + 1) (48)
To solve the iniual value problem we have to determine the coefficients Cn. We have from
(45) for i=0
if Cmxw (0, x) 4'm(x )dx = 3o 2(m+1) (49)
The function m(0,x) is given by
w( 0, x) = wi for
w( 0, x) = w2 for
O c: c: 2_x'i
2 c: c: i
'i_x (50)
with constants Wj and CO. In the range
2 c: c: 2
'i  x  r2 (51)
we have
( I 1 x )w(O,x)=B X+In x (52)
with
( 2)1 1 'iWi = B i + In 2 'ri ri ( 1 I  r22)w2 = B i + In 2 .r2 r2 (53)
All integrals in (49) ar elementa. A full discussion of the problem is sull needed.
184
oe W  GACTIC DYNAMOS
A.M. Soward, Department of Mathematics and Statistics,
The University of Newcastle upon Tyne, NE1 7RU, U.K.
ABTRCT. A simple representation of a galactic dynamo
consists of a thin disc of electrically conducting fluid with
the region external to it a vaccuum. Differential rotation inthe disc provides the ~effect which stretches out meridional
magnetic field into the azimuthal direction. Small scale
turbulent motions are responsible for diffusion but more
importantly the ~ effect, which produces meridional magnetic
field from the azimuthal field so completing the dynamoprocess. When the ~ effect is antisymetric about the
equatorial plane, modes of two symetries can be distinguished
namely even (dipole) and odd (quadrapole). As the magnitude of
Dynamo number, which provides a dimensionless measure of the
product of the øl and 4) effects, is increased .the growth rate of
modes generally increases. Dynamo action often occurs when the
real part of the complex growth rate vanishes. Four marginal
modes can be distinquished. The are the steady dipole, steady
quadrapole, oscillatory dipo��e and oscillatory quadrapole.
The galactic disc is characterise. by two length scales,
the disc radius R and the disc thickness b. Together they
define the aspect ratio€ = b/R. The dynamo modes themselves
have a length scale L, which is generally intermediate between
the disc radius and its thickness: b~~L~~R. In this limit the
dynamo can be modelled locally by a slab model, first
investigated extensively by Parker (See Parker, 1979). In this
lecture multiple length scale proc��dures are described, which
use solutions for the slab model as the first approximation to
the complete solution. A test for the validity of the
procedures is obtained by comparing results with stix's (1975,1978) numerical results for a particular model in an oblate
spheroid. The asymptotic results of Soward (1977) for the
first steady dipole (sometimes called the "forgotten" mode) are
described. It is explained how the exterior potential magnetic
field is a more potent mechanism for linking field amplitudes
at distant parts of the disc than lateral diffusion within the
disc, contary to the assumption made by some authors. New
resul ts are described for the oscillatory dipole. This model
is unusual in as much as the radial dynamo length scale L is
comparable to the disc width b. A comparison is made with
stix's results. Local results for the oscillatory quadrapole
have been obtained. In this case the most unstable mode occurs
185
on a long length scale L (~~b) .
satisfactory solution of the amplitude
that length scale has been found.
As yet, however,
modulation problem
no
on
REFECES:
Parker, E. N. ,
Clarendon.
1979,
4
Cosmical Magnetic Fields,
Soward, A. M. , 1977, "A thin disc model of the
galactic dynamo", Astron. Nachr. 299, 2533.
Stix, M. 1975, "The galactic dynamo", Astron, &
Astrophys. 42, 8589.
Stix, M. 1978, "The galactic dynamo", Astron. &
Astrophys. 68, 459.
THE CATASTROPHE STRUCTURE
OF THERMOHALINE CONVECTION
Olivier THU AL
NCAR, Po Box 3000, Boulder, Co, 80303
We apply a surace forcig to the 2D Boussinesq thermohalne convection in a rect
anguar box, by imposing a fied cosine temperature and salty flux. Multiple steady
states are found numericaly, alowing the competition between a thermaly and a salty
drven symmetric ciculations, as wel as a one cel' asymetric ciculation. In the control
space of the two forcig parameters these equibria form a double cusp catastrophe. This
catastrophe can be nimiced on simple box model connecting stired reservoir through
pipes. These low order models might give hints for an asymptotic exansion which could
catch the catastrophe phenomena on the Boussinesq equations.
186
Simulations of Solar Convection
by Bob Stein and ike Nordlund
Michigan State University
We have developed a threedimensional hydrodynamic code for studying solar
convecuon. The code is written in modular form. It solves the equauons of mass,
momentum, and internal energy conservation using In p, u, and e as varables. This
incrases the accuracy of vertcal hydrostatic equilbrium (since In p is nearly linear while
p is nearly exponenual). Derivatives are calculated using cubic splines and FFs. The
bottom boundar is a node for horizontally uniform motions (to faciltate analysis of p
mode drving). Inflowing material has a uniform entropy and the pressure is uniform
across the boundar layer. An extra large zone is placed at the top, across which the
vertical derivative of the velocity and deparre from hydrostatic equilbrium is zero and at
the top of which the energy is held fixed.
A realistic, tabular, equation of state is used and radiative energy exchange is
calculated by solving the nongrey, LTE transfer equation using a fourbin opacity
distrbution funcuon.
The code has been tested for advection. Discontinuities less than six grdpoints
produce ringing. An arficial viscosity, proportional to the fluid velocity, is used to spread
such discontinuities and eliminate ringing.
Shock tubes with a pressure jump of 10 (Mach number 1.55) were run. A
viscosity proportonal to the velocity jump is used to spread shocks over four grd zones
and dissipate their energy. Excellent agreement between numerical and theoretical profi��s
for velocity, energy, and In p show that the code has good conservation propertes.
Linear waves drven by a piston show that there is little damping and very small
phase speed error for wavelength greater than four grd zones, and the daping and phase
speed error increase sharly for smaller wavelengths. Twodimensional acoustic and
internal grvity wave pulses in a stratifed atmosphere ar dispersive, and we are checking
the group and phase velocities against theory. Wave reflection is noticeable, but smalL.
Finally, we are comparng the results of a simulation of threedimensional
convection in a box of an ideal gas using our code and a ppm code. Preliminar results
show that the spectrm of varous quantiues are the same at large scales, but our code is
more diffusive at scales less than five grd zones.
We have found that the flow topology for convection in a stratified medium (sun)
consists of a warm, smooth, slow upflow with embedded cool, fiamentary, fast,
downdrafts. This can be seen from the velocity field in twodimensional slices, from
following fluid parcels in time, and from transparnt views of the vertcal kineuc energy
flux.
The horizontal flow pattern is cellular. At the surface the upflow breaks up into
grules. Overpressure in the centers drves the flow horizontally, cooling radiatively as it
goes. Eventually it runs into the expanding flow from neighboring granules and is
stopped in intergranule lanes. In the process it has lost entropy, so gravity pulls it back
down into the interior. As the downflows descend they converge into narow fiaments at
the vertces between granules.
The horizontal flow at large depths is also cellular but with a larger scale 
corresponding to mesogranule scales. There is often a swirling motion about the
downdrafts, so there is vertical vorticity and helicity. We compared two runs, one with
twice the number of grdpoints as the other, both starng from the same initial state after
1.52 largescale turnover umes. The large scale upflow and downdraft strctue is similar
in the two rus and the high resolution ru developed much more small scale strctu.
187
Finally, we have compared the emergent surface intensity with observed solar
granulauon. The size spectrm has similar shape. However, the simulauon which has
been smoothed by atmospheric seeing and a telescope modulation transfer function has
more power at small scale than the observations. A video with superiposed observations
frm Pic du Midi and simulauon results was shown, as well as a video showing how fluid
parels move trking the changes in the grulation pattern.
Instability of Flow with Temerature
Dependent Viscosity:
A Model of Magm Dyamcs
J. A. Whitehead and Kal R. Helfrich
Woods Hole Oceanogphic Institution
Woods Hole, MA. 02543
ABTRCT: In a material whose viscosity is very temperaturedependent, flow from a chamber through a cooled slot can
develop a fingering instability or time dependent behavior,
depending on the elastic properties of the chamber, the
viscosity temperature relationship, and the geometry of the
slot. A laboratory experiment is described where syrup flows
from a reservoir through a tube immersed in a chilled bath to
an exi t hole at constant pressure. Flow is either steady, or
periodic, depending on the temperature of the bath and the flow
rate into the reservoir. A theory indicates that the
transition from steady to periodic flow depends on
nonlinearities in the steady state relation between pressure
and flow rate and a general stabililty criterion is advanced.
Parameters governing the oscillation period are determined.Theory also indicates that flow through a slot would develop
fingerlike instabilities under certain conditions.
Qualitative laboratory experiments with paraffin spreading over
a cold plate reveal the fingering.
"
,
¡,
r
188
A LONG LARATORY SALT FINGER
by George Veronis
Kline Geological Laboratory
Yale University
ABTRCT: The initiation of salt fingers from an initial two
layer configuration is well understood and the essential physical
balances necessary to maintain long quasisteady salt fingers are
also known. However, the evolution from linear stability to long
fingers has received little atte~tion, primarily because the
process is very strongly nonlinear and therefore not amenable to
analysis. The best hope for developing a model for this
evolution is to study the case of weakly driven salt fingers in
the presence of strongly stable stratification. In two layers
this case corresponds to R=oc~T/ ..tlS ).). 1, where o¿ is the thermal
expansion coefficient, 8 the salinity contraction and .. T, AS theimposed temperature and salinity differences across the
interface. 6S must be small enough so that shear instabililty of
adjacent rising and sinking fingers does not occur. this model
is currently being analyzed in order to understand the evolution.
During a recent visit with Jorg Imberger at the Centre for Water
Research of the University of Western Australia, I learned about
a simple laboratory experiment, originally proposed by G. I.
Taylor, which seems to be a simpler version of the same problem.A long tube filled with water opens up into a reservoir of salt
water at the top. Taylor predicted that a long salt finger wouldpenetrate downward to a finite depth and then stop. Except for
slow molecular diffusion there would be no further penetration of
salt. Imberger's experiment confirmed Taylor's prediction.
This seemed to be a nice simple problem for a proj ect for one of
the summer fellows at this year's GFD program but the latter
had all started on their projects by the time that I arrived. I
ended up talking to Joe Keller about the problem and out of
curiosity we wrote down the basic balances and derived a solution
for a long quasisteady 2D finger. The model is very similar to
the one for infinitely long steady salt fingers in Howard and
Veronis (1987) with modified boundary conditions. We also
reproduced Imberger' s experiment.
A model for the downward penetration of the salt tongue assumes
that the horizontal salt balance is between horizontal diffusion
of salt and vertical advection of the mean salt gradient but that
the amplitude of the concentration (or vertical velocitythe two
are proportional) is a slowly varying function of time and the
vertical coordinate. Two equations for this t, z dependence
emerge from the horizontally integrated vorticity balance and
sal t conservation. The horizontal structure in these integral
189
balances is taken to be the same as in the zeroorder model.
Numerical integration of the equations leads to a relation
between time and the vertical penetration of the salt tongue that
is qualitatively the same as the behavior observed in the
laboratory experiment, viz., a rapid penetration is followed by
balance between penetration distance and the logarithm of the
time and finally the tongue slows and stops.
REFERECES:
Howard, L.N. and G. Veronis (1987) The salt finger zone.
JFM, 183, pp. 124.
190
Applications of compactly supported wavelets
to
the numerical solution of partial differential equations.
John Weiss
A ware, Inc.
Suite 310
124 Mount Auburn Street
Cambridge, MA 02138
Introduction
Compactly supported wavelets have several properties that are quite useful for rep
resenting solutions of PDEs. The orthogonality, compact support and exact repre
sentation of polynomials of a fixed degree allow the effcient and stable calculation of
regions with strong gradients or oscilations. For instance, we have applied wavelets to
problems of shock capture and turbulence. The general method is a straightforward
adaptation of the Galerkin procedure with a wavelet basis. Among the equations
studied so far are Burgers equation, the equations of Gas dynamics, and the Navier
Stokes/Euler equations for an incompressible fluid in two dimensions.
The compact wavelets have a finite number of derivatives and the derivatives,
when they exist, can be highly oscilatory. This makes, say, the numerical evaluation
of integrals diffcult and unstable. We have found methods for the evaluation of func
tionals on wavelet bases. Comparison with standard numerical results demonstrates
that these procedures are critical for the wavelet methods, especially as applied to
nonlinear problems.
Compactly supported wavelets
Ingrid Daubechies defined the class of compactly supported wavelets (1,2). Briefly,
let ip be a solution of the scaling relation
N
ip(x) = L akip(2x  k).
k=O
The ak are a collection of coeffcients that categorize the specific wavelet basis. The
expression ip is called the scaling function.
191
The normalization r c.dx = 1 of the scaling function obtains the condition
¿ ak = 2.
The translates of c. are required to be orthonormal
J c.(x  k)c.(x  m) = Dk,m.
From the scaling relation this implies the condition
N
¿ akak2m = Dom.
k=O
For coeffcients verifying the above two conditions, the functions consisting of trans
lates and dilations of the scaling function, c.(2j x  k), form a complete, orthogonal
basis for square integrable functions on the real line, L2(R).
If only a finite number of the ak are nonzero then c. wil have compact support.
Smooth scaling functions arise as a consequence of the degree of approximation of
the translates. The conditionm that the polynomials 1, x,." , xp1 be expressed as
linear combinations of the translates of c.( x  k) is implied by the condition
¿( i)kkmak = 0
for m = 0,1,..., p  1.
The WaveletGaler kin Method
For a PDE of the form
F(U, Ut, . . . , Ux, Uxx, . . .) = 0
define the wavelet expansion
U = ¿ Ukc.(x  k).
An approximation to the solution is defined by
NÛ = ¿ Ûkc.(x  k).
k=M
In effect, the solution is projected onto the subspace spanned by
~(M, N) = fc.(x  k) : k = M,... ,N;.
To determine the coeffcients of this expansion we substitute into the equation and
again project the resulting expression onto the subs pace ~(M, N). This uniquely
determines the coeffcients Uk.
192
The projection requires Ûk to verify the equations
1: r.(x  k)F(Û, ût, Ûx," .)dx = 0
for k = M,. .., N. To evaluate this expression we must know the coeffcients of the
form
J r.(x)r.x(x  ki).. .r.xx(x  k2).. .dx.
Our original expansion is over the space dependence of the solution. If the equation
has a time dependence the resulting equations for the Ûk wil be a system of ordinary
differential equations in t.
Burgers' equation
Burgers' equation is
Ut + UUx = uUxx,
where U(x, 0) = Uo(x) and u is the viscosity.
We use the exact formula l3) to check our numerical results. The dynamics of the
Burgers' equation for small viscosity cause the formation of steep gradients and, in
the limit of null viscosity, jump discontinuities or shocks l3).
To apply the WaveletGalerkin method to Burgers' equation we use the D6 scaling
relation
5
r.( x) = E akr.(2x  k)
k=O
and the scaling function expansion
MÛ = E Ukr.(x  k).
k=M
The differential equations for Uk
Uk,t  uAk,mUm + Bk,m,IUmUI = 0
are defined from the coeffcients
Ak,m = 1: r.x:i(X  k)r.(x  m)dx
.
Bk,m,1 = 1: r.x(X  k)r.(x  m)r.(x  l)dx.
The semiimplicit time differencing for Burgers' equation
(Un+1  Un) /6it + UnUn+1,x = uUn+1.xx
193
is used throughout and the WaveletGalerkin equations are
(Uk,n+1  Uk,n) /.6t  O"Ak,mUm,n+1 + Bk,m,IUm,nUI,n+1 = 0
Here Uk,n is the wavelet space and semiimplicit time discretization of solution for
Burgers' equation.
We compare the finitedifference, spectral and waveletGalerkin methods. Each
has the semiimplicit time differencing and 64 modes. These results were presented
by Latto and Tenenbaum in reference (4).
After calculation the solutions are smoothed with a three point averaging
Ûk,n = (Uki,n + 2Uk,n + UHi,n) /4.
We compare the numerical solutions, smoothed and unsmoothed, with the exact
solutions evaluated using the ColeHopf transformation.
In summary the results are (4)
1. The WaveletGalerkin (WG) method appears to be stable for all viscosities,
including null viscosity.
2. The WG appears to be close to exact for large times and small viscosity.
3. The three term smoothing of the oscilatory WG appears to be close to the exact
solution for all times and small viscosity.
4. The WG method appears to handle error in a way that the information content
(the exact solution) is not destroyed and can be recovered from the approxima
tion.
Acknowledgement
This research was supported in part by the Advanced Research Projects Agency of
the Department of Defense and was monitored by the Air Force Offce of Scientific
Research under Contract No. F4962089C0125. The United States Government is
authorized to reproduce and distribute reprints for governmental purposes notwith
standing any copyright notation hereon.
;
lf
References
1. 1. Daubechies, " Orthonormal Bases of Compactly Supported Wavelets", Commun.
Pure Appl. Math, 41 (1988) 909996.
2. G. Strang, " Wavelets and Dilation Equations: A Brief Introduction", SIAM
Review, 31, (1989), 614627.
3. G. B. Whitham, Linear and Nonlinear Waves, John Wiley and Sons, New
York, NY (1974).
4. A. Latto and E. Tenenbaum, " Les ondellettes�� support compact et la solution
num��rique de l'��quation de Burgers", Aware Tech, Report AD900307.1.2, March
(1990). To appear, Compte rendus Acad. Sci. France.
194
Traveling Waves and Oscilations in Compressible Magnetoconvection
Nigel Weiss
Department of Applied Mathematics and Theoretical Physics
University of Cambridge, England
Sunspots are dark owing to partial suppression of convection by a strong vertical
magnetic field. A large sunspot occupies only 0.1% of the visible disk but starspots
may cover 50% of the surface of more active stars. These features have motivated
studies of Boussinesq and (more recently) compressible magnetoconvection (Proctor
& Weiss 1982; Hughes & Proctor 1988). The pioneering paper Hurlburt & Toomre
(1988) dealt mainly with steady motion: this lecture describes studies of time
dependent convection carried out by Derek Brownjohn, Neal Hurlburt, Michael
Proctor and myself (Hurlburt et aL.1989; Weiss et aL. 1990).
We consider twodimensional convection in a compressible layer with an imposed
temperature difference l:T in an initially uniform magnetic field Bo. The relevant
parameters are the Rayleigh number Rex l:T, the plasma ß = p/(Bo 2/2J.o) ex Ql
and the ratio ( of the magnetic to the thermal diffusivity. The system is assumed
to be periodic in the horizontal direction with a dimensionless wavelength .À. As R
is increased the static solution may lose stabilty either in a pitchfork bifurcation
or (if ( c 1 and Q is suffciently large) in a Hopf bifurcation. In the latter case
branches of standing wave and traveling wave solutions emerge from the bifurcation.
Normal form equations indicate which solution is preferred but stability may be
transferred from one branch to the other via an intermediate branch of modulated
wave solutions. With a horizontal field, for example, traveling waves are preferred
in the Boussinesq limit but we find for ß = 32 that standing waves are stable near
the Hopf bifurcation at R(o), while modulated waves appear for 4 R(o) ~R~ 16 R
(0) and traveling waves eventually gain stability at R=32R(0).
In a shallow layer with a vertical magnetic field magnetic pressure fluctuations
become important when ß = 0(1). We find that standing waves are always stable
near the Hopf bifurcation but for .À=2 and 32~ ß ~ 6 there is a transition from
a 2roll standing wave solution to a 4roll traveling wave solution.'; The traveling
waves have a triangular structure with a prograde jetstream and propagate with
a velocity v:: 1/2 .À Va as in the Boussinesq limit. Apparently the motion builds
up fields so strong that magnetic pressure has to be balanced by inertial terms
rather than gas pressure. For.À = 1 there is a straightforward transition from
standing waves to traveling waves but for .À=2 the change of scale is achieved via
an intermediate mixedmode solution that is quasiperiodic. At higher values of ß
the steady solutions are unstable and periodic solutions with largescale streaming
motion can be found.
Traveling waves only appear when ( :: 0.1. In a sunspot there is a transition
from photospheric layers, where ( :: 0.003, to levels below 2000 km depth where (
195
~ 30. We expect oscilatory behavior above and steady overturning motion below.
This can be modelled by taking a deep stratified layer with a density contrast
Pbottom/ ptop = 11 and setting ( oc p. For 0.2 ~ ( ~ 2.2 and À = 4/3 we find that the
initial bifurcation is a pitchfork leading to steady 2roll convection. As R is increased
counterrolls appear and grow. Eventually a complicated sequence of secondary and
tertiary bifurcations leads to stable periodic oscilations. In these solutions there
are four rolls which are modulated periodically in space. At the base of the layer
the sense of motion is unchanged throughout the oscilation but the velocities at the
top reverse as alternate rolls become proiinent and penetrate towards the upper
boundary. This form of modulated oscilation provides a natural explanation for the
sporadic bright features, called umbral dots, that are observed near the center of
sunspots. Timedependent convection of this type must be responsible for supplying
the energy radiated from sunspots, which remains significant in spite of the magnetic
field.
References
Hughes, D.W. & Proctor, M.R.E. 1988. Ann.Rev. Fluid Mech.20, 187
Hurlburt, N.E., Proctor, M.R.E., Weiss, N.O. & Brownjohn, D.P. 1989. J.Fluid
M echo 207, 587
Hurlburt, N.E. & Toomre, J. 1988. Astrophys. J.327,920
Proctor, M.R.E. & Weiss, N.O. 1982. Rep.Prog.Phys..45, 1317
Weiss, N.O., Brownjohn, D.P., Hurlburt, N.E. & Proctor, M.R.E. 1990 Mon. Not.
Roy. Astron. Soc., in press
196
Modeling Mesogranules And Exploders On The Sun
Nigel Weiss
Department of Applied Mathematics and Theoretical Physics
University of Cambridge, England
Observations reveal three different scales of convection on the surface of the sun.
Supergranules, with a characteristic diameter of 30 Mm, have horizontal velocities
of around 0.5 Km S1 which are correlated with the magnetic network. Granules
have a typical diameter of 1 Mm and velocities of 1km S1. By tracking the proper
motions of granules it has become possible to derive intermediatescale velocity pat
terns and the existence of mesogranules, with diameters around 5 Mm and velocities
of 0.5 km ç1, has been confrmed (Simon et al. 1988). Highresolution whitelight
observations also show exploding granules (exploders); these are granules that ex
pand rapidly to a diameter of about 3 Mm, forming a ringlike structure with a dark
core and then fragmenting. Nearby granules are swept aside with velocities of order
2 km S1. The fact that exploders occur preferentially near the center of mesogran
ules raises the question: Are mesogranules just the timeaveraged consequence of
recurrent exploders (Title et al. 1989)?
George Simon, Alan Title and I have used a simple kinematic model of the hor
izontal velocity in the photosphere in an attempt to answer this question. Sources
are represented by axisymmetric outflows that can be derived from potentials of the
form
'
lP(r)=~ VR exp ((rjR?) (1)
It has ~eady been shown that such sources provide a good description of meso
granular flows observed from Spacelab 2 (Simon & Weiss 1990). We now suppose
that a mesogranule can be represented by sources distributed with a probability
1/(ro)=(7r p 2)1 exp((rj p?) (2)
that there is a source distant ro from the origin. Then the expected value of the
potential has the same form as (1) with a radius R=(R2+p2) 1/2 and a velocity
V=(RjR)3 V.
We have compared mesogranules (Rm= 2 Mm, V m=1 km S1) with exploders
(Rg=1 Mm, Vg=8 km s~ lifetime ßt=10 min), distributed normally about the
mesogranule center. As diagnostics of the flow we introduce passive test particles
(corks) travellng with local fluid velocity. Then we compare the cork patterns pro
duced by different flows after periods of 16 hr have elapsed. In addition we obtain
a quantitative description of these patterns by computing their factal dimensions.
Figure 1(a) shows cork patterns generated by randomly distributed mesogranules.
197
The region shown has dimensions 40" x 40" (30 Mm x 30 Mm) and the average
distance between mesogranule centers is 7 Mm. After 2 hr the mesogranule centers
are cleared and corks move gradually into a network which is clearly apparent after
5 hr. The corresponding patterns produced by explodes are shown in Figure l(b).
The network forms more rapidly but has a ragged appearance. The patterns are
qualitatively similar and it is not possible to distinguish between them from the ob
servations. However it is possible to rule out other models, with exploders dropped
randomly over the region or distributed randoiiy about their predecessor so as to
follow a random walk.
Figure 2 ilustrates the effects of combining mesogranular and supergranular ve
locities. We deposit these supergranules (Rs=10 Mm, Vs=l km S1) in the region
and allow mesogranules to travel with the local supergranular velocity; if meso
granules approach too closely or escape from the region they are replaced and the
cork supply is replenished to compensate for those escaping from the region. The
patterns produced by mesogranules in Figure 2( a) can be compared with those gen
erated by exploders in figure 2(b). The magnetic network between the supergranules
is distorted by the smallscale motion.
These results can be compared with a 3hour observational run made at the
PicduMidi (Frank et al. 1989). This sequence shows mesogranules drifting across
supergranules, with an apparent lifetime of 3 hr. A preliminary inspection suggests
that exploders develop from granules that appear at the center of mesogranules
and drift outwards as they expand. This supports a picture in which a systematic
outflow combiries with random exploders to provide the averaged mesogranular
velocity. FUrther work is needed to relate these surface features to the underlying
threedimensional dynamics.
References
Frank, Z. et al. 1989. Preprint.
Simon, G.W. et al. Astrophys. J. 327, 964.
Simon, G.W. and Weiss, N.O., 1989. Astrophys.J.345, 1060.
Title, A.M. et al. 1989. Astrophys.J. 336, 475.
198
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("
Fluid Mechanics and Melting.
Andrew W. Woods,
Scripps Institution of Oceanography.
In this presentation, we considered the role of melt convection upon the melting of
a solid into a binary melt. At a horizontal interface six different convecting regimes may
arise, depending upon the relative composition of the melt and the solid and whether the
melt underlies or overlies the solid. These are summarised in the figure below.
First we considered the case of purely diffusion driven melting described by Woods
(1990a). When the melt temperature far exceeds that ofthe melting temperature, the solid
melts at a rate determined by the thermal diffusivity; in contrast, when the melt is of the
same temperature or colder than the solid, but of a different composition, the solid may
dissolve into the melt at a rate determined by the compositional diffusivity. The crucial
difference between these two regimes is that in the melting regime, the compositional
boundary layer becomes embedded in the melt, whie in the dissolving regime, it is always
located at the melt/solid interface; this difference can lead to some differences in the style
of convection that results.
Next, we reviewed the situation in which the liquid which is melting the solid is in a
state of thermal convection, as discussed by Huppert and Sparks (1988). This situation
arises when the liquid underlies the relatively buoyant melting solid; owing to the density
difference due to composition, the molten solid remains above the liquid, and each layer
may convect thermaly, as in a doublediffusive system (Turner, 1979).
We then considered the problem of melting a relatively buoyant solid by an overlying
layer of dense liquid (Woods (1990b)). The molten solid is buoyant and rises into the
liquid, producing vigorous convection whicli mies the melt uniformly. A global model of
the melting is able to predict the evolution of the melt temperature and composition as a
function of the depth of melted solid. Experimental data suggest that in this wellmixed
convection regime, the flux of solute at the interface scales as Fc fV ÂC5/2. This is a
result of the complex coupling of the convective heat flux, which drives the melting and
thereby drives the convection; this nonlinear coupling produces a compositional flux which
200
exceeds that which would arise due to RaleighBenard type compositional convection, in
which the convective flux would scale as tiT4/3 when the boundary compositions are
fied. The difference arses because the convection is driven by the compositional flux at
the boundaries whose magnitude is determined by the rate of melting.
We considered application of these results in the geological context of magma cham
bers; hot, dense molten basalt may intrude a crustal chamber and melt the wals, floor
and roof. Typicaly the molten crust is less dense and therefore compositional convection
in the chamber due to the floor melting wi arse, in conjunction with thermal convection
from the melting at the roof.
References. '
Huppert, H.E. and Sparks, R.S.J., 1988, Melting the roof of a turbulently convecting
chamber, J.F .M.
Woods, A.W., 1990a, Melting and dissolving a binary aloy, submitted.
Woods, A.W., 1990b, Fluid ming during melting, submitted.
201
o VQ. :/'1'~~J .l0fI _
TemperatureI I I iI I ITemperJtur. I I I II I I I II I II I I II i I I II I I i I Far~ri.id melt compositIon I I II I I II I I I I Is Ins than the Interfacial I I I II I I II I I I I composItion and therefore I I II I I the InterfacIal melt IS I I II I I I I I stable. leading to il purelyl I II I II I I I dirruslon governed proc,?, I II I I I Riolon or comøosltlon I I I II I I drIven convectIon
I I I II i I II I I I I I I II II l i I I I I II II I I lints Of constant density I I I IInini."" ~ II I I I I II I I I I/~ i I I II I I
Solid lies on (ti I lines 0' consllll density I I I
~ oj :dus I In the milt II
comøoSltlon and
temperature or the
Interracial melt
Composition
eutectic IIn,
iuiie tic IIn.
Composition
comDoslllon and
temperature Or the
Interfacial melt
The composition and temperature of the overlying melt
determine the nature of the convection driven by the
melting of pure Ice (COmposition' 0), Note the
Interface composition Is lower than that In the
rarfleld,
The composition and temperature of the overlying melt
determine the nature of the convection driven by the
melting of pure Ice (composition' 0)
U",d~/Î ;': ¡1 d l
i I TemperatureI II
I ITemperalÙre I I II I I I II I I II I I I II I I II I I I I I Farrleld melt comPOSition I I I II I II I I I I Is less than til Interhclal I I I II I I II I I I I composItion and tllrerore I I II I I Ihe Interficlai melt Is I II I I I I I unl""e. leadIng 10 I I I II II I I I convict lv' OYtrturnlng I I I II I I I Region of dOUle I I I III I I I diffusive conviction I I I Ii II I I I I I II I I I II I I II I I I I I II I I llnei or contant ",Illy I I I II In tII mill ~ II I I I I I I
I I I I I I
I /" i I I II I I I ISolid IIn on thr I I 11_ .r conllani delly I I I IsolleJus I I '.111 NI II I II
I
compOSItion Inc
ttmøerltllt of thl
Inter' It III milt
ComPOiltlon CompoSH Ion
eutectIc lint
comPOSition an
temperature or Ihe
InterflClal mtlt
The composition ancl temøerature Or tile underlYI"9
melt determine the nature or tile convectIon drlvin by
tile meltIng of pure Ice (CompOSItion  0), Noti tili
Interface compOSlt Ion IS lower tllan tllat In tili
farfield,
Tile composition and temperaturi of tile unrlying
melt determIne tile nature of the convection driven by
tili melting of pure Ice (composition' 0),
202
CONVECTIVE PENETRATION IN STELLAR INTERIORS
JeanPaul Zahn
1. Introduction
It has long been recognized that the penetration of motions beyond the classical
boundary of a convective core set by the Schwarzschild criterion (i.e. where the
temperature gradient becomes subadiabatic again) would have a crucial impact on
stellar evolution. Such penetration would also modify substantialy the temperature
profile at the base of a convective envelope. But stellar structure models rarely
include that effect, and there is stil debate on whether such penetration even occurs.
One reason is that astrophysicists too often neglect to confront their problems
with those of their colleagues in geophysics and fluid dynamics, who study simiar
cases, and often decades ahead. Another reason is the faiure of the minglength
treatment, which proved so powerful when dealng with convective envelopes (largely
because it reduced one's ignorance to a single parameter), but cannot come to grips
with penetrative convection.
Our purpose (in a Paper submitted to Astron. Astrophys.) is to convince the
astrophysicists that convective penetration indeed exists, by recalng the relevant
observations, experiments and computer simulations that are available. Further
more, we give an estimate for the extent of penetration that is inspired by such
observations and calculations, and does not rely on the minglength treatment.
2. The evidence for penetrative convection
There is overwhelmng evidence for penetrative convection: in the atmosphere,
the oceans, the laboratory. The dynamics of the planetary boundary layer, for
instance, is entirely governed by it: every morning the nocturnal stable stratification
is replaced from below by a growing unstable layer due to surface heating.
In the laboratory, penetration has been observed whenever a convectively
unstable layer is imbedded in a stable stratification of fluid. One famous example
is the icewater experiment, which uses the pecular property of water having a
i
r
203
density maxmum at 4oC¡ it was suggested by W. Malus and performed here in
Woods Hole by Furumoto and Roth (1961), and was later repeated by Townsend
(1966) and by Adrian (1975). In al experiments, one observes a substantial amount
of penetration, sometimes comparable with the thickness of the unstable layer.
What are the theoretical predictions? According to linear theory, some weak
overshooting should occur outside the unstable region, but that result cannot be
extrapolated into the nonlinear regime, since the functional form of the solutions
is deeply modified by their feedback on the thermal stratification. This has been
recognized aleady by Veronis (1963), who did the first nonlinear analysis of pen
etrative convection (and published it in the Astrophys. Journal)! Since, various
fully nonlinear simulations have been carried out. All those which are based on
firm physical ground confirm that there is indeed much more penetration than
what could be inferred from the linear theory (Musman 1968¡ Moore and Weiss
1979¡ Zahn, Toomre and Latour 1982¡ Massaguer, Latour, Toomre and Zahn 1984).
The early investigations dealt with severely truncated equations, and as a
result the solutions were stationary. The penetration depth was found comparable
with the thickness of the unstable layer¡ it depended on the degree of stabilty of the
outer medium and on the aspect ratio of the cells. Furthermore, threedimensional ;
cells were seen to penetrate much more than horizontal rolls, in the direction of
their net kinetic energy flux.
More recently, penetrative convection has been simulated in two dimensions
with much better spatial resolution, in a compressible fluid (Hurlburt, Toomre and
Massaguer 1986). The solutions are no longer stationary, and they show vigorous
concentrated downdrafts which penetrate rather deep into the stable region below.
Those incursions strongly couple with gravity waves, as was observed by Townsend
(1966) in the laboratory. Such downdrafts appear to be a genuine and important
property of stratified convection, since they are also present in al threedimensional
simulations (Graham 1977, Nordlund 1984, Stein and Nordlund 1989, Cattaneo,
Hurlburt and Toomre 1989), where they behave more like longlived plumes than
as thermals which would just traverse the,domain.
To summarze, al numerical simulations exhbit substantial penetration of
the convective layer into its stable environment, irrespective of the approximations
made, provided the stratification is alost adiabatic in the unstable zone¡ that
nearly adiabatic stratification extends well beyond the limits of the unstable region.
Moreover, the flows show very similar behavior in the two parts of the domain: the
only obvious manifestation of the change in the entropy slope, from an unstable to
the stable one, is the reversal of the correlation between the vertical velocities and
the temperatu,re fluctuations, which switches the direction of convective heat flux
from upwards to downwards.
204
3. Subadiabatic penetration at the bottom or a convective envelope
We consider convection that is effcient enough to establish an adiabatic stratifica
tion and to transport most of the thermal energy produced in the core of the star.
This requires that the P��clet number wf.1 K be substantialy larger than unity (f.
and w are the typical size and velocity of the convective motions, K = xl pCp the
thermal diffusivity, and X the radiative conductivity).
~
(ft 1
dT
dz
Ftotal
I
A B i C i D
I
0
depth
o Lp z
Figure i
The structure of a star at the bottom of a convective envelope is sketched
in Fig. 1. Due to the increase of the conductivity with depth, the radiative flux
Frad rises until it equals the total flux Ftotal at the level zo. If there were no
convective penetration, this would be the edge of the convection zone, as predicted
by the Schwarzschid criterion; below, the energy flux would be carried only through
radiation, and the temperature gradient would then be (dTldz)rad = Ftotai!X. But
the motions penetrate into the stable region and they render it nearly adiabatic,
provided their velocity w is larger than some critical value (given below in Eq. 5.5).
The motions decelerate through buoyancy until thermal diffusion becomes more
205
important than advection, in a thin boundary layer where the temperature profie
settles from the adiabatic to the radiative slope.
We thus identify four regions at the bottom of the convection zone. The first
two have a quasiadiabatic stratification: the superadiabatic domain, which is the
seat of the convective instabilty (A), followed by the stable subadiabatic region
(B). Below, after a shalow thermal boundary layer (0), heat is transported only
through radiation in the radia,tive interior (D).
We now use this schematical picture to estimate the extent Lp of the subadia
batic penetration region (B). In that region, the convective flux is, to a fairly good
approximation,
Fconv =  (~~) 0 (~~) ad Z , (3.1)
where z is counted from the depth Zo where the Fconv = 0 (the subscript 0 refers to
that level).
We shal assume that the fraction f of the area is occupied by downwards
directed motions which transport most of the convective flux, as observed in the
laboratory and in the computer simulations. We express that convective flux to
lowest order in terms of the horizontal temperature fluctuations and the vertical
velocity of these motions, which we assume strongly correlated:
Fconv = fpCpW hT. (3.2)
To estimate the penetration depth, we follow the downdrafts from z = 0,
where their velocity is Wo, until they stop at the base of the penetration zone, at
z = Lp. Their deceleration is 'described to first order by
1 dW2 hp hT
2~ =g¡; = gQT'
with the usual assumption of pressure equilibrium, and Q = (8Iogpj8IogT)p
being the expansion coeffcient at constant pressure. Elimination of hT yields the
following expression for the depth Lp of the sub adiabatic region
(3.3)
L ¡ J 1/2
l = w:3/2f1/2 ~ Q K VHp 0 2g XP ad , (3.4)
with XP = (8Iogxj8IogP)ad' We can go a step further by using the property
that in al convection theories the velocity scales as Fconv = const pW3, in the
fuly developed regime; the proportionalty coeffcient is of order unity, and several
prescriptions are avaiable for it, which are roughly equivalent. Ours is
2/* Hp pW3Fconv = 3" A QVad' (3.5)
206
involving again a fing factor r (which could a priori be different from that in
the penetration zone), and A being the minglength. Since the convective flux
nearly equals the total flux over most of the unstable zone, we may approximate it
by Ftotal. We thus obtain an alternate form of expression (3.4):
Lp = (l~) 1/2 = (1.) 1/2 ,Hp af* xp xp
with a = AI Hp and ø = ¡ lar. Let us stress that this estimate can only be applied
to stars possessing a deep enough convective envelope, in which most of the heat is
carried by convection.
As could be anticipated, the penetration depth (in units of the pressure scale
height) only depends on the steepness of the conductivity gradient xp, and' on
the ratio of the effciencies of the convection in the unstable and stable regions, as
measured here by ø = ¡ lar. Judging by the computer simulations that have been
performed, this parameter ø should be close to unity, reflecting the similar behavior
of the flows in the stable and unstable parts of the convective domain.
Much in the same spirit, it is possible to evaluate the thickness of the thermal
boundary layer (C):
(3.6)
wi'th td = (Hgp) 1/2 .Lt ~ (ØXp)1/8 (Ktd)I/2 (3.7)
Below the solar convection zone, where the dynamcal time scale is td ~ 300 sand
the thermal diffusivity K = xl pCp ~ 2 107 cm2 Is, the thermal boundary layer
which terminates the penetration zone has a thickness of the order of 1 km. That
thin boundary layer plays no role in the dynamics: the convective motions have
aleady been slowed down to a velocity Wt ~ 1mls when entering the layer. Thus
the boundary of the whole convection zone (regions A and B) is sharply defined,
although it probably ondulates somewhat due to the velocity dispersion of the
impinging downdrafts (as ilustrated in our planetary boundary layer when it is
delineated by clouds).
4. Penetration of a convective core
The derivation of the extent of penetration of a convective core into its stable
surroundings is very similar to that for a convective envelope. The only difference
is the cause of the convective instabilty: in a stellar core, it is the steep increase
of the nuclear energy generation rate ��, as one approaches the center of the star,
which is 'mainly responsible for the onset of convection.
The penetration length scales as
( )1/2 ¡ ( )J 1/2
Lp = w:3/2¡1/2 ~~ QKV!! _ p�� ! ~H 0 9 H 9 ad   + 3XP H 'p p p ~ p
L
r
(4.1)
207
with the same notations as before, and p, pe being the mean p, pe in the superadi
abatic core of radius ro.
We shal again use the value of Wo given by the minglength treatment
(Eq. 3.6), with
A = ro min(1,aHpjro) , (4.2)
and assume that most of the energy is carried by convection in the superadiabatic
core. Then the alternate form of expression (4.1) is
( )1/2 ( ) 1/2Lp lP. p pe 1 ro =  min(1,aHpjro) =  == + Xp ,ro 6 p pe 3 H p (4.3)
lP = I j 1* being here the ratio between the fing factors in the stable and unstable
regions. It predicts an increase of Lpj Hp with the size of the unstable core, and
such a trend has actualy been noticed by Maeder and Mermiod (1981).
5. The temperature gradient in the subadiabatic penetration region
In our derivation of the penetration depth, we have assumed that the convective
motions are effcient enough to enforce a nearly adiabatic temperature gradient when
penetrating into the subadiabatic domain. We shal now establish the condition
which must be fuled for this to occur.
We start from the specific entropy equation
88 88 8u¡ 8F¡
pT 8t + pTu¡ = ll¡j   ,8z¡ 8zj 8z¡ (5.1)
whereu¡ is the velocity field, F¡ = x8T j 8z¡ the radiative heat flux and the first
term on the r.h.s. represents the heat produced by viscous frction, which we shal
neglect from now on. We restrict ourselves to quasistationary motions, i.e. to flows
whose organization lives longer that the travel time across the penetration region,
and we linearize the variables around their horizontal mean
8=8+81, T=T+T1 and F¡=Frad+(F¡h.
Assuming again that the fluctuations T1, 81 of temperature and entropy are strongly
correlated with the vertical velocity w, at least in the downdrafts which occupy a
fraction I of the area, we get after some manpulation and horizontal averaging
d r(dT) dTJ ( d, 1 )
 c dz Frad + C p L dz ad  dz p W = X dz2  a2 ST. (5.2)
Here W is the r .m.s. of the vertical velocity in the downdrafts, and c the triple
  1/2
correlation coeffcient pw3 = 2c pw2 (w2) .
208
Depending on the strength and of the geometry of the motions, the temper
ature fluctuations wi be controlled either by diffusion or by advection. In the
advective case, the r.h.s. of Eq. (5.2) is negligible, and we have
c~ (xdT) = pCpW ((dT) _ dTJ .dz dz dz ad dz (5.3)
Assuming that the temperature gradients V = Ô In T I ô In Pare approximatively
constant, we obtain the following approximate expression for the sub adiabatic gra
dient in the penetration region:
V
Vad  V
f WHp
cxp K (5.4)
We see that, although the temperature gradient is less adiabatic when it is
achieved by downdrafts that do not :f the space, as one may expect, the departure
from adiabacy wil stil be negligible as long as
f WHpK ~1.cxp (5.5)
This defines a critical value which the vertical velocity W must exceed to enforce
such a quasiadiabatic stratification.
At the base ofthe solar convection zone, where WHpl K = 2106 and xp = 1.3,
a fing factor of about 105 is suffcient for quasiadiabatic penetration to occur
(for c:: 1/10).
At the boundary of the convective core of a 9 M0 ZAMS star, where the P��clet
number Wrol K is of the order of 3 106, fing factors larger than about 105 wi
again ensure such quasiadiabatic penetration.
6. Conclusion
Our main result is that the conditions for sub adiabatic convection are certainly
fuled in the interior of star, both at the bottom of a deep convective envelope
and at the edge of a convective core, due to the high effciency of the convective
heat transport. Even if that flux is carried by a rather sparse network of plumes,
as suggested by the recently performed threedimensional simulations, the stratifi
cation is nearly adiabatic throughout the convection zone, including the region of
subadiabatic penetration, and the departures from adiabacy are confined within in
a very narrow thermal boundary layer.
This property alows the use of the integral constraint that has been proposed
by Roxburgh (1978, 1989) to calculate the size of a convective core. He showed
that, provided the departures from the adiabatic gradient are suffciently smal, one
209
can integrate Eq. (5.1) (divided bt T) over a whole convective core, to reach the
condition
fr 1 dT
Jo (Lrad  Ltotal) T2 dr dr = 0 , (6.1)
(with L = 47rr2 F). The applicabilty of his result to actualy predict the extent of
penetration was seriouly questioned on the ground that the entropy gradient is not
necessary smal everywhere (Baker and Kuhfuß 1987). But since the departures
from adiabacy are restricted to a very thin boundary layer, provided condition
(5.5) is fulfied, the integral condition above is satisfied to a fairly good degree of
approximation.
It is more problematic to use Roxburgh's prescription to predict the extent
of penetration below a convective envelope. The reason is that the integral above
would have to start at the surface of the star, that it would include a region where
the departures from adiabacy are high and whose weight would be very large in the
integrant, due to the strong temperature contrast between top and bottom. It then
seems preferable to use our  admittedly much cruder  estimate of Eq. (3.6), and
to turn to the observations for the calbration of the parameter ø.
Let us stress that our results have been obtained by making several simplify
ing assumptions. An implicit assumption was that we ignored the Coriolis force,
although it plays an important role in al convective cores and at the base of the
convective envelope of most stars. It is not clear whether our basic scalngs wi be
affected by rotation; presumably the law Feon" ex pW3 stil holds, but the propor
tionalty coeffcient wi be somewhat modified by the rotation, and hence also the
penetration depth.
We believe that stellar structure theory has matured to a point where convec
tive penetration must now be taken in account when constructing interior models.
One may object that this would require yet an other parameter ø, which cannot be
derived from first principles. But with the advent of helioseismology, the diagnostic
of the solar interior has improved so much that one wi be able in the near future
to evaluate the extent of penetration at the bottom of the convective envelope, by
measuring the value of the temperature gradient at the edge of the adiabatic zone,
and to derive from it the value of ø, which should not vary much from star to star.
In the case of a convective core, the situation is even simpler, since one can use
Roxburgh's prescription to predict the depth of penetration.
References
Adrian, R.J.: 1975, J. Fluid Mech. 35, 7
Baker, N.H. and Kuhfuß, R.: 1987, Astron. Astrophys. 185, 117
210
Cattaneo, F., Hurlburt, N .E. and Toomre, J. 1989, in Stellar and Solar Granulation,
ed. R.J. Rutten and G. Sverino (Kluwer, Dordrecht), p. 415
Furumoto, A. and Rooth, C.: 1961 Geophys. Fluid Dynamics. Woods Hole Oceano
graphic Institution Rep.
Graham, E.: 1977, in IA U Coll. 38, Problems of Stellar Convection, eds. E.A. Spiegel
and J.P. Zahn (Springer, Berlin), p. 689
Hurlburt, N.E., Toomre, J. and Massaguer, J.M.: 1986, Astrophys. J. 311, 563
Latour, J., Toomre, J. and Zahn, J.P.: 1981, Astrophys. J. 248, 1081
Maeder, A. and Mermillod, J.C.: 1981, Astron. Astrophys. 93, 136
Massaguer, J.M., Latour, J., Toomre, J. and Zahn, J.P.: 1984, Astron. Astrophys.
140, 1
Moore, D.R. and Weiss, N.D.: 1973, J. Fluid Mech. 61, 553
Musman, S.: 1968, J. Fluid Mech. 31, 342
Nordlund, Å: 1984, in Smal Scale Dynamcal Processes in Quiet Stellar Atmo
spheres, ed. S.L. KeIl (National Solar Observatory, Sunspot), p. 181
Press, W.H.: 1981, Astrophys. J. 245,303
Renzini, A.: 1987, Astron. Astrophys. 188, 49
Roxburgh, I.W.: 1978, Astron. Astrophys. 65, 281
Roxburgh, I.W.: 1989, Astron. Astrophys. 211, 361
Schmitt, J.H.M.M., Rosner, R. and Bohn, H.U.: 1984, Astrophys. J. 282,316
Stein, R. and Nordlund, Å: 1989, Astrophys. J. Letters 342, L95
Townsend, A.A.: 1966, Quart. J. Roy. Met. Soc. 90, 248
Veronis, G.: 1963, Astrophys. J. 137, 641
Zahn, J.P., Toomre, J. and Latour, J.: 1982, Geophys. Astrophys. Fluid Dyn. 22,
159
k
!i
i,
'.f:
211
THE STRUCTURE AND STABILITY OF RAPIDLY ROTATING POLYTROPES
N.J. Balmforth
Galaxes such as the Mily Way appear to have a very thi disclie structure. hi many
instances this disc is the setting for very beautifu spiral features. These "spiral arms" seem to be
the location of amplied star formation and presumably have an enhanced density over the rest
of the disc.
For the Miy Way itself the disparity of the radius of the disc with its thickness is typical:
the disc radius is of the order of 10,000 parsecs, whist its thickness is only about 500 parsecs. If
we take the ratio of these two numbers and cal the result the dimensionless number N, then we
fid
N '" 20.
For bodies that are inftely thi N  00, whilst stars and ellptical galaxes are characterized
by values of N that are of order unty. We have defied this parameter N in anticipation of
a characteristic, large parameter that shal occur in the equations governing the structure and
stabilty of the disc. We shal exploit the magntude of this parameter to develop the solution by
an asymptotic technque.
If the disc of a galaxy contais al of its mass then the highly flattened aspect of the galaxy
must indicate that the gas has been spun out by rotation and that there is a balance between the
selfgravity and the centrifugal acceleration. hi the direction of the rotation axs, however, the
effects of the centrifuge are not felt; the structure is determied by the stratifcation of the gas
under the balance of selfgravity and pressure. This leads to a characteristic thickness that is the
Jeans length, kil = 47rGpjc2, of the confguration, if it has the density p and sound speed c.
The Jeans length is more well known from studies of the instabilty to gravitational collapse
oflarge expanses of gas (termed the Jeans instabilty; Jeans, 1928). There, the gas is unstable to
disturbances with wavelengths that exceed the Jeans length. hi fact, this forms the basis for the
idea that the mass wil undergo a successive hierarcy of fragmentations until the characteristic
length scale is less than kil. This ilustrates how localzed clumps of matter could form in an
iitialy unorm unverse.
The Jeans length is therefore singuarly important for the structure and breakup of self
gravitating gaseous masses.
Clearly this conclusion is modied when one introduces anguar momentum into the gas, and
as we have noted, the gas can be spun out and become extremely elongated in the directions
perpendicular to the axs. This is presumably the reason why many galaxes form discs.
The detailed equation of state of the matter in the disc wil presently be considered unpor
tant. histead a barotropic, and in particular a polytropic equation of state,
p = Kpr = Kp(n+i)/n, (1.1)
where p is the pressure, p is the density, and K and r are constants, shal be assumed.
The work here is based largely on an unpublished manuscript by Howard and Spiegel (1970).
That paper develops the asymptotic technque and solves the equations for the equilibrium struc
ture of the disc. Since this paper is unpublished, section 1 gives a brief account of the work, and
ilustrates some of the results. The liear stabilty analysis of these structures is then detailed in
section 2.
212
1. THE EQUILIBRIUM STRUCTURES
1.1 The equations of galactic structure
A disc is characterized by two dierent kinds of dynamcal balance. hi the vertical we have
hydrostatic balance between the pressure support and the self gravity. If we use cylindrical coor
diates then this balance is ôp ô~
+p=Oôz ôz
where ~ is the gravitational potential. Horizontaly this is modied by the centrifugal acceleration
(1.2)
ôp ô~ y2
ôr + P ôr = P: (1.3)
but at leadig order, the pressure gradient must disappear. Y is the swirlig velocity. Finaly we
have Poisson's equation
V2~ = 41fGp . (1.4 )
We now nondiensionalze the equations of motion. We let Pc, the central density, be the unt
of density, and K Per the unt of pressure. As a unt of speed we adopt c = (rpc/ Pc)1/2 and as
a unt of length, we choose the radius of the disk, a. With these unts equations (1.2) and (1.3)
remai unchanged except for the appearance of a factor r1 in the pressure derivatives (though
strictly a different notion is caled for). Equations (1.4) becomes
Ô2~ 1 ô~ Ô2~
ô2+ô +ô2=N2p.r r r z (1.5)
where
N2 = 41fGpc a2
c2 (1.6)
This is just the parameter that we have aleady aluded to: the radius of the disk measured in
unts of a Jeans length k;1, where
k2 = 41fG Pcc c2'
The equations of motion have the integral
(1. 7)
npl/n + ~ = F(r) (1.8)
¡,
:lwhere F(r) is the centrifugal potential, and, in terms of F, the swirlig velocity is given by
y2 = rF'(r) . (1.9)
1.2 Solutions for N + 00
We shal now consider asymptotic solutions for large N. Since the disc is thi, vertical changes
must occur very rapidly. Therefore, it is the vertical derivatives in Poisson's equation that balance
the right hand side of equation (1.5) at leadig order. This simply reflects the fact that a is not
a natural unt for z and so it is convenient to introduce the stretched coordiate
(= NZ , (1.10)
213
where ( is 0(1) inside the disk. Equation (1.5) then becomes
82H + Hn =..i. (r8(FH))
8(2 nn N2 8r 8r (1.11)
where the enthalphy, H, is given by
H = np1/n = F  Cl. (1.12)
Our notation here might at fist seem a little strange. In particular the specific enthalpy is
normaly referred to as h. To avoid confusion, however, we have chosen to use upper case lettering
for the quantities describing the equibrium structure and lower case lettering to represent the
perturbations to it. The density and pressure wil be the only exceptions.
We now consider the asymptotic expansions
1H = H1 + NH2 + ...,Cl = NClo + ..Cl1 + ...,F = NFo + F1 +...
The leadig order terms in equations (1.12) and (1.11) give
(1.13)
Fo = Clo (1.14)
and 82 H1 Hr _ 0 ( )8(2 + nn  . 1.15
The boundary conditions are that H1(r,0) = H10(r) and (8H1/8()c=0 = 0, where H10 is
an arbitrary function whose importance in characterizing the structure of the disc shal become
apparent. A fist integral of equation (1.15) is then
(8H1)2 = 2 (Hn+i _ Hn+i)8( (n + 1 )nn 10 1 (1.16)
The solution of this equation can be obtaied by elementary quadratures or it can be expressed
as incomplete Betafunctions. It is convenient, however, to express it in terms of a fuction Cn
defied by
i1 d8:1= .. 1 .!0..(2) 8..(1  8 .. )1/2 (1.17)
Then,
P1 (r, () = P10( r )Cn(X( r)() (1.18)
where ¡ ..1 J 1/2
X( r) = 2np;¡n+1
The value :10 such that Cn(:io) = 0 is
(1.19)
11 d8 (mr1/2) r(..)_ _ n+1:10  ..1 .!   ..o 8..(18.. )1/2 n+1 r(2n+2) (1.20)
214
The matchg conditions then give
Fo = 100 'lo(k)Jo(kr)kdk (1.29)
and
( 2n ) 1/2 !i 100 P1t.. (r) =  k'lo(k)Jo(kr)kdk.n+ 1 0
Eliating the fuction 'lo from these equations leads to the integral equation relating Fo( r) and
P10( r),
(1.30)
11 ( 2n )1/2 W.Fo =  10 n + 1 P10" (s)K(s, r)sds, (1.31)
where the kernel
K(s, r) = 100 Jo(kr)Jo(ks)dk
is expressible as an ellptic integral. For future reference we also note that
(1.32)
00
K(r,s) = ~)2k + 1)ÀIeP2Ieh/i=;2)P2Ieb/i=";2),
1e=0 ( (2k)! J 21r
ÀIe = (22Iek!)2 2' (1.33)
where the P21e (z) are Legendre polynomials.
At leadig order, then, we are left with only one independent arbitrary fuction, which may
be either P10 or Fo.
The surface density of the disc is
~l(r) = 2 ie P1(r,()d( = 2 (~p~~+1)/nJ10 n + 1
which is just twice the function in the integrand of equation (1.31). We shal choose to use this
quantity instead of P10 to parameterize the radial structure of the disc.
(1.34 )
4. Some ilustrative density and rotation laws
The solution can be characterized by the surface density, ~1, and the function Fo which are
related by
1 rFo(r) = 2 10 ~l(s)K(r,s)sds
where the kernel, K( s, r), is given by equation (1.32).
The models we shal consider here have the form
~1 = 2(n2; 1 )1/2(1_ r2)M+l ,
(1.35)
(1.36)
where M is an integer. Near r = 0 this form gives ~1 '" 1  (M + l )r2 and so M is a measure of
the central concentration. For this surface density
( ) _ (~)1/2r(M + l)! . . 2Fo r   n+ 1 (M + 1)! '2F1(2,(M + 1), 1,r ) (1.37)
The surace density and rotation rate 00 = Vir for the indices M = 0, 1, 2, 3, 4, 5 and 10 are
shown in figue 1.
215
where r is the Gama function. Therefore the hal thickess, 0 of the polytropic disk is
r( i) (!lIn) 1/2 1..0(r) = n+1 .. p"T (r) .
r(!) n + i 102n+2 (1.21)
Typicaly P10 wil decrease from the centre to the edge of the disc, and from equation (1.21)
we see the curious shapes of poly tropes of dierent indices. For n = 1 (r = 2), the models are
discs of unorm thickness; for n :; 1 (r c 2) they flare toward their edges, whie for n c 1 (r :; 2)
they are lenticular.
1.3 Matching to the external potential field
To complete the solution to leadig order we must consider the potential outside the poly trope,
~ext. This satisfies Laplace's equation
v2~ext = 0 . (1.22)
The solution of this equation must match smoothly onto the potential in the interior, hence
~ext(r,N10) = ~(r,N10(r)), ~~xt(r,N10(r)) = ~z(r,N10) (1.23)
where subscript z denotes dierentiation with respect to z.
The asymptotic sequences for the interior solution suggest that we seek futher sequences of
the form
.;ext _ N.;ext + .;ext +
~  ~o ~1 ..., io = 01 + N O2 + ..., (1.24)
hence, at the upper edge of the disc,
~ext(r,N10) = N~~xt(r,O+) + 01~~~t(r,0+) + ~ixt(r,O+) +...
~ext(r,N10) = N~~xt(r,O+) + 01~~~~(r,0+) + ~i~t(r,O+) +.. (1.25)
The potential in the disk is given as a fuction of (, and at ( = 0, since ô/ôz = Nô/ô(, we have
~ = N~o(r, 01) + 02~odr, 0i) + ~l(r, 01) +...
and
~z = N2~odr, 0i) + N02~0,,(r, 0i) + N~ldr, 01) +.. (1.26)
Accordig to equation (1.14), we have ~o = Fo, and so the leadig terms of the matchg give
Fo(r) = ~ext(r, 0+),
~lC( r, 0) = ~~~t( r, 0+) . (1.27)
To complete the matchg we need only to solve equation (1.22) whose general axsymetric
solution for z :; 0,
~ext = 1000 1/( k )elez Jo( kr )kdk,
where we may write the unown fuction 1/ as
(1.27)
1/ = N 1/o + 1/1 + ... (1.28)
216
In figue 2 the shapes of the equibrium confguations are ilustrated for four dierent values
of the polytropic index (n = 0.5,1,1.5 and 3) and for the model with the surface density index
M =2. The edge evidently flares outwards to inty for the models with the larger values of n as
the density in the midplane becomes vanshigly smal. This iregularity occurs because towards
the disc's outer edge, the radial derivatives in the Poisson equation (1.5) become large and so the
asymptotic expansion breaks down. This indicates the presence of a boundary layer at the edge
that, when treated correctly, wil modify this singuar behaviour. Nevertheless, to some extent
the flare wil stil survive. An alternative method to avoid this flare is to recognize that outside
the disc the density may not be zero, just very smal. The edge then occurs at a fite density
at which the flare is less pronounced. The external gravitational potential is stil approxitely
given by equation (1.29), and so the explicit form of the solution is not appreciably modied.
This procedure bears some resemblance to the "patchig" technque of Monaghan and Roxburgh
(1965).
Model surface densities
1.0
0.8
0.6
,.
..
w
0.4
0.2
0,0
0.0 , 0.2 0.4 0,6
Radius r
0,8 1.0
Model rotation rates
1.8
1.6
1.4
1.2
,.
..
C'1.0
0.8
0.6
0.4
0,2
0.0 0.2 0.4 0,6 0.8 1,0
Radius r
,
::!~
i\
~
'I',
Figure 1: The surface density and rotation rate of polytropic models characterized by the index
M of the surface density law given by equation (1.37). The six curves correspond to the indices
M =0, 1, 2, 3, 4, 5 and 10. Since the polytropic index n appears only in a multiplicative factor
in the surface density and rotation laws, the curves are exactly those for any polytropic index
provided this factor is included. The curves are drawn for the case n = 1.
217
A very dierent way to avoid this behaviour is to introduce an external, halo potential of a
parabolic form IaN2(2, where a is an arbitrary parameter (it is actualy the ratio of the halo
density to the central density of the disc), at the outset. This modies the stratifcation from that
predicted by equation (1.18) to
Pi(r,() = PiO(r)Sn(x(r)Ç;JL(r))
where the integral
(1.38)
((n+1))l/2JL(r)=a ()PlO ril d8Z = ..I!t l'5..(;:;1£) 8"(1  8" + JL(l  8;))1/2 (1.39)
defies the fuction Sn( Z; JL)' This changes the shape of the poly trope and substantialy reduces
the flare. The dotted and dashed lies in figue 2 indicate the shape of the poly tropes for the
values of a of 0.1 and 1.
Vertical structure of the n=O.5 poly trope

...................
Vertical structure of the n=1.5 poly trope
...................eO' _ _ _ _ _
..:; 
":.=:  .:.::.:::.::.::. :         :..~..::...u..a..
Vertical structure of the n= 1 poly trope
,.'
............_...._.....................
 
"' ..
..:" ,,"
: "'
.: ,
...'. ..
.. ..
.........,~,

.. "
.. "
,~
;'.;' :;' :
~.:./
'"
.....
........................................
Vertical structure of the n=3 poly trope
Figure 2: The shapes of the poly tropes of index n =0.5, 1, 1.5 and 3. The index of the surface
density law is M = 2. The dotted and dashed lies indicate the shapes when an external potential
field laN2(2 is also present. The dotted lie has a = 0.1 whist the dashed lie is for a = 1.
218
2. LINEAR STABILITY ANALYSIS
Spiral ars seem to be the manestations of local enhancements of density and amplications
of star formation withi the disc of the galaxy. They are not material arms since radial spokes of
matter become very quickly sheared out and wound up on short times cales because of the relatively
large shearing flow of rotation. Therefore they may be rotating patterns of gravitationaly induced
instabilties at fite amplitude. Such speculations have been asserted in the past, notably by
Goldreich and LyndenBell (1965) in the context of gaseous discs, and by Shu (1970) in the theory
of the stellar dynamcs of sheets. Toomre (1969) has pointed out that such liear instabilties
simply propagate towards the centre or edge of the disc and become anated very quickly. Here,
however, we take the point of view that this may not happen at fite amplitude; nonlear waves
may form a steady rotating structure without dissipating. This view seems also to have been taken
by Norman (1978), Qian, Spiegel and Proctor (1990) and Qian and Spiegel (1990).
2.1 The equations in the limit of rapid rotation
The linear, nonaxsymetric perturbations of the disc satisfy the equations of motion,
(!. +imO) u  20v = _!. (Ø + h)ôt ôr
(!. + imO) v  2Bu =  im (ø+ h)ôt r
(2.1)
(2.2)
and
(:t + imO) w =  :z (Ø + h) (2.3)
where the velocity of the perturbation is (u, v, w), the fluctuations in the enthalpy and gravitational
potential are h and ø and the anguar dependance ��mØ has been assumed. The anguar velocity
of the mean undisturbed motion is 0 and
1 dO
B = 0 + r
2 dr (2.4)
is one of Oorts constants.
The continuity equation is
( ô .) 1 ô im ôôt + imO p' + ; ôr (rpu) + :pv + ôz (pw) = 0 , (2.5)
and Poisson's equation becomes
~!. (r ÔØ) _ m2 ø + ô2ø = N2 ,r ôr ôr r2 'ôz2 p (2.6)
The density fluctuation p' is related to the enthalpy perturbation by
(H)ni _ ,h p.n (2.7)
219
, Th~ equations of motion (2.1) to (2.3) can be combined to yield
u=  ((~ +imo)' HOBr ¡(~ +imO) :~ + 2i~/J,
. = ((~ + imo)' HOBr ¡2B:~  i~ (:1 + imO) iJ
(2.8)
(2.9)
and
( lJ . )1 lJi
W =  lJt + imO lJz' (2.10)
where 1 = h + fl. When these equations are substituted into the continuity equation, this new
equation and Poisson's equation become two coupled, secondorder equations for the perturbations.
In accordance with the asymptotic solution of the equlbrium model, we agai rescale the
vertical coordiate z = N(. Moreover, it is evident from the equations of motion (2.1) to (2.3)
that, if the time derivatives are not to vansh at leadig order, the characteristic timescale for the
motions is just the rotation period, which is short in the lit of rapid rotation. Therefore we
shal use the resealed time T = N1/2t.
We shal consider the normal modes of the disc, for which the dependance on time is of the form
er¡T, where r¡ is the characteristic growth rate. The two equations determg the perturbations
are then
(H)n1 1 (1 m2 jlJJfl   (f  fl) =  lJ,.(rlJ,.fl)  fl.. n N2 r r2 (2.11)
and
N2lJ,(plJd) = NlT2p'  l!lJ,. J pr (lTlJ,.1 + 2im(N1/20) ij �� imlTp (2(N1/2 B)lJ,. _ imlT i)r �� K, r J rK, r
2' pu2 J 1 ( P) (2im(N1/20) ( PO) m2j ��
= NlT P  K,2 �� ;lJ,.(rlJ,.) + lJ,. log K,2 lJ,. + rlT lJ,. logK,2  ; i J' (2.12)
where
IT = r¡ + imN1/20. (2.13)
H r¡ were purely imaginary, the motion would have the charactistic epicyclic frequency iK" with
K,2 = lT2 + 4N10B = 1 + Ll, (2.14)
where Ll is what could be described as the Rayleigh discrimant of the mean flow of rotation.
Provided r¡ is real, K, does not in general vansh exactly.
2.2 Asymptotic expansion and solution
The asymVtotic development of the variables into a series follows simply from that used for
the equibrium state: we set
fl = Nflo + fl1 + no i = N 10 + fi + ...,
220
for the perturbations,
11 = 1Ii + Ni1l2 + ..., "( = "(i + Ni"(2 + ...,
for the characteristic growth rate and its square, and
0= Ni/2(Oo + NiOi + ...), U = Ui + NiU2 + ..,
and
K = Ki + NiK2 + ..., il = 4NiOB = ili + Ni il2 + ...
for the other subsidiary variables.
H we introduce these expansions into the equations (2.11) and (2.12) then the leadig order
equations are trivialy satisfied by
10 = lo(r) tPo = tPo(r) tPo = 10' (2.15)
At next order equation (2.12) gives
ô,(Piôdi) = 0, (2.16)
which has the integral Pi Ô, fi = constant. However, Pi vanshes at the edge of the disc, and so if
fi is bounded, then
li = fi(r).
The equation of the correspondig order from equation (2.11) gives
(2.17)
ôlhi + (H¡fn)nihi = O. (2.18)
This equation can be reduced to quadrature. For the moment, however, we shal ignore the vertical
structure of the disturbance since we are more interested in its liear stabilty.
The O(N) terms of equation (2.12) give2f ( ) ¡2" (n) 2j J2' Ui rpi ' im HoPi mô,(Piô,h) = UiPi   ô,. rô,.lo + ô,.   Pi2" 10 ,r Ki Ui Ki Kir (2.21)
Integrating this equation verticaly, from the centre to the edge of the disc gives,
( ii )ei 2' uf f (rlh ) ¡2im (Oo~i) m2j JPiu,h 0 = Ui~i   ô,. Tô,.o + ô,.   ~i2" 10 ,r Ki Ui Ki Kir (2.22)
where ~i is once more the surface density of the disc, and
rei
~~ = 2 10 P~ d(
is the perturbation induced in it.
A thi layer of fluid, symetric about the plane z = 0, can support motions that are either
symetric or asymetric about this plane. The latter are sinuous modes and correspond to
corrugations of the layer. A simple corrgation does not lead to a local enhancement in the
density, and consequently the destabilzing effect of selfgravity is smaler for these modes than
221
for the symetric or varicose modes. H we particularize to the varicose modes, then 8, f2 = 0 at
, = 0, and so the left hand side of equation (2.22) vanshes. Thus
, 1 (r~i ) ¡2im (Oo~i) m2 J
~i = 8., 8.,/0 + 8.,   ~i 22 /0r "i O'i r "i "i r (2.23)
Moreover, integrating equation (2.18) over the vertical extent of the disc gives
(8,Øil'=0i = ~~~. (2.24)
These two equations, together with the formula derived from the boundary conditions upon the
perturbed gravitational potential (see the following section), constitute an eigenvalue problem for
0'1, with the eigenfunction /o( r).
We can derive analogous equations to (2.23) and (2.24) for the terms of 0(1). These give the
O(Ni) corrections to the eigenvalues. Explicitly they are
~~ = ~8., ((~2 + 8.,/i _ "12 +26.2) r~i28., /0)r ~i 8"'/0 "i "i
+ 2imOo f (/i _ 0'2 + Oi) 8., (~iOo) + 8., ((~i~o)( ~2 + Oi _ "12 +26.2)) 1.
O'ir 1. /0 O'i 00 "l "i ~i 00 "i f
_ m2~i (/i + ~2 _ "12 + 6.2) (2.25)
r2"l /0 ~i "l '
and 1 0i(8,Ø2l'=0i = 2~~  8., (r8.,o) . (2.26)
2.3 Matching the solution
Outside the disc the perturbation to the gravitational potential øezt is once more a solution
to Laplace's equation:
øezt = 1000 1/'( k )Jm( kr )e1ez+im6 kdk. (2.27)
To match this solution to the perturbation inside the disc we develop the fuction 1/' into a series
of powers of Ni, just as in the case of the equibrium modeL. The continuity of the potential
perturbation and its derivative then yields
/0 = 1000 1/~(k)Jm(kr)kdk (2.28)
and
(8,Øil'=0i =  1000 1/~(k)Jm(kr)k2dk, (2.29)
at leadig order.
Eliating 1/~( k) results in the equation
/o(r) =  1000 (8,Øil'=0i Km(r,s)sds, (2.30)
222
where the kernel is now
Km(s,r) = ioo Jm(kr)Jm(ks)dk.
At the following order, we have
(2.30)
f1(r) =  i1 (0,.4J2),=0i Km(s, r)sds
+~ (01~~  i1 sds i1 tdt~~(t)01(S)Km(s,r) Loo k2dkJm(kt)Jm(ks)). (2.31)
If 01 were independent of s (as it is in the case n = 1), then the fial term in square brackets
would vansh exactly.
2.4 The eigenvalue problem
Equations (2.23), (2.24) and (2.28) can now be combined to give an integral equation for fo:
1 i1
fo =  ~~Km(s, r)sds,2 0 (2.32)
or
fo =  (1 J !O. (S~10;fo) + r im O. (00;1) _ m:~~ J fo 1. Km( s, r )sds. (2.33)10 1. s 2K,1 L S0'1 K,1 2s K,1 J
These equations are clearly analogous to equation (1.55) of the static problem, and reflect the
leadig order balance between the galactic centrifuge and the self gravity of the confguation.
Equation (2.33) is an integral, eigenvalue equation. The eigenvalue 171 is explicitly contaied
in 0'1 and K,1, and fo is the eigenfunction. These correspond to the varicose normal modes of the
disc.
In principle, the O(N1) corrections to the eigenvalue and eigenfuction (12 and fi, respectively)
can be derived from equation (2.30). In practice, however, it is dicult to treat such a complicated
equation for general polytropic index and nonaxsymetric disturbances. The fial term on the
right hand side appears to be an unecessary complication: It vanshes for n = 1 and must be
smal for highorder modes (for which 01 ~ constant). If we assume that it is negligible, that
equation becomes
1 (1 (1 01(s)fi(r) = 2 10 ~~(s)Km(s,r)sds + 10 O.(sO.fo)Km(r,s)sds, (2.34)
with equation (2.25) determg the O(N1) correction to the perturbation to the surface density
~~.
2.5 Axisymmetric modes
If we particularize to the axsymetric case, m = 0 and equation (2.30) becomes somewhat
easier to analyse. We have
r (S~10.fo)fo(r) =  10 o. 2K,l Ko(s, r)ds. (2.35)
223
where K o( s, r) is identical to the kernel considered in the static modeL. This kernel can be decom
posed into an inte sum of Legendre polynomial and so
~ ~ 1i (s"£i8.fo) ~fo(r)=L.ÅIc(4k+1)P21c(v1r2) 8. 22 P21c(V1s2)ds.1c=0 0 ~i (2.36)
In addition
~l = 17l + 400Bo = li + ai. '
H we defie the function
(1 r2)i/28"fo(r)9i(r) = li + ai(r) , (2.37)
and change variables to Z = (1  r2)i/2 and y = (1  s2)i/2, then the eigenvalue equation can be
written in the more obvious form,
¡li + 5i(z))Gi(z) = f ÅIc(4k + 1)pilc(z) (i B2(y) Gi(y)Pilc(y)dy, (2.38)1c=0 10 y
where B(y) = "£(s), Gi(z) = 9i(r), 5(z) = a(r) and Pilc(z) is an associated Legendre polynomial.
In the previous section, a selection of equilibrium models were computed with the surface
density law "£i(r) = (1  r2)M+i/2. H we introduce this parameterization of "£i into equation
(2.38) then the eigenvalue equation equation becomes00 (i
(li + 5i(z))Gi(z) = t; ÅIc(4k + 1)pilc(z) 10 y2MGi(y)pilc(y)dy. (2.39)
H we further decompose the fuctions Gi (z ) and 5i (z ) into a sum over the even associated Legendre
polynomials,
00 00
Gi(z) = L Gi,jPij(z),
j=O
5i(z) = L5i,jPij(z),
j=O
(2.40)
then equation (2.34) is reduced to an intediensional matrix eigenvalue problem:
00 00 00
li L Gi,lcpilc (z ) + L L Gi,i5i,jPii( z )pij( z) =1c=0 i=O j=O00 00 i '
L ÅIc(4k + 1)pilc(z) L Gi,j 1 y2MPij(y)pilc(y)dy.1c=0 j=O 0
Setting Pli( z )Plj( z) = Ek:o CIijlcPilc( z), and comparing the coeffcients of Pilc( z) yields
(2.41)
00 00 00 (i
liGi,1c + L L CIijIcGi,j5i,i = L ÅIc(4k + 1)Gi,j 10 y2M Pij(y)pilc(y)dy. (2.42)i=O j=o j=o 0
This is an equation that is of the form
liGi = AiGi, (2.43)
224
where 1 00
A1,kj = Àk(4k + 1) 1 y2Mplj(y)plk(y)dy  Laijk61,i.o i=O (2.44)
(aijk can be seen to be the thid ran tensor that describes the coupling, in the presence of a
mean dierential rotation, between the eigenfunctions of a unormly rotating disc.)
This fial equation can be solved explicitly in the case of the rigid rotator (M = 0), and this
is detaied in the following section. It can also be solved numericaly.
When m = 0, the O(N1) corrections determed by equation (2.34) satisfy the simpler
equation
(1 f (sL1 ) ((L2 12 + .d2) sL1 J 1f¡(r) =  10 18. 2K.l 8./1 + 8. L1  K.l 2K.l 8./0 f K(s, r)ds
+ i1 018.(s8./0)K(s, r)ds. (2.45)
The correction to the surface density L2(r) is determined by specifying the equilibrium confg
uration. If there are to be no corrections to the surface density law L = (1 r2)M+1/2 at O(N1),
then this function is identicaly zero. Thus
(1 (SL1) (1 ((12 + .d2) sL1 Jf¡(r) =  10 8. 2K.l 8./1 K(s, r)ds + 10 8. K.1 28./0 K(s, r)ds
+ i1 018.(s8./0)K(s, r)ds.
Just as for the leadig order equation, we shal defie the function
(2.46)
G ( ) _ () _ (1 r2)1/28,.1(r)2 z  92 r  A () .
11 + ~1 r
(2.4 7)
Then equation (2.46) becomes
00
("1 + 61)G2(Z) = L Àk(4k + 1)plk(Z)
k=O
ti1 y2MG2(y)plk(y)dy  i1 (y2M (~:: ~:) + ~ ("1 + 61)) G1(y)plk(y)dy l' (2.48)
with 01 (y) = 01 (s). Decomposing the fuctions G1, G2 and the 61 on the left hand side of equation
(2.48) into sums ofthe PMz) gives
("1  A1)G2 = (A2 + 12B)G¡, (2.49)
where
i1 ¡ 62y2M 01 JA2,kj = Àk(4k + 1) 6 + ("1 + 61) Pij(y)Pik(y)dyo 11 + 1 Y
and
i1 y2M
Bkj = Àk(4k + 1) 6 Plk(y)Plj(y)dy.
o 11 + 1
225
The matrix Ai can be symmetrized by using the vector G~,i = \1( 4k + 1 )ÀIeGi,i instead of Gi,i
as teh eigenfunction. Therefore there must exist a vector Gt that satisfies the equation
(Gt? Ai = (GnT')i. (2.50)
Taking the product of this vector with equation (2.49) gives an expression for the O(Ni) correc
tion to the eigenvalue
')2 = (Gt)T A2Gi(Gt)TBGi' (2.51 )
In fact, since there is an infte spectrum of these eigenvalues, there is an infte set of equations
(2.51 ).
2.6 Axisymmetric instabilties of the rigid rotator
In the case of a rigidly rotating disc, M = 0 and 5i = 11. In this case the eigenvalue equation
(2.43) becomes 00 /i
(¡i + 1I)Gi,1e = L ÀIe(4k + l)Gi,; 10 pl;(y)plle(y)dy;=0 0
= À1e2k(2k + l)Gi,le. (2.52)
Thus the kth eigenvalue is
2 J ( (2k)! J 2 1.
')i,1e = 1Ji,1e = 11 1.2k(2k + 1) (2Iek!)2  1 J' (2.53)
This eigenvalue increases monotonicaly with k, and so it appears that, at this order, there is no
high wavenumber cutoff to the range of unstable modes. This is not true ofthe Jean's instabilties
of an inte, plane, polytropic slab of gas (Ledoux, 1951; Qian, Spiegel and Proctor, 1990).
Therefore, it is clear that there must be some stabilzing effect that enters into the equations at
O(Ni) and eventualy, for large enough k, dominates the instabilty determed at fist order to
create a fite range of unstable wavenumbers. This is physicaly plausible since at next order the
effects of pressure support must enter, and these oppose a local collapse of the disc.
The eigenfctions of the unforuy rotating disc are simply the associated Legendre poly
nomials Plle( Ý1  r2). The k = 1 solution is a toroidal perturbation of the disc, whist higher
order solutions wil look like concentric anul of fite thickness. For asymptoticaly large k, the
eigenfunction is
(4k)i/2Gle "" 1Ir cos(2ksini r + 11/4) (2.53)
well away from r = O. As the order increases, the amplitude of the mode becomes concentrated to
the centre of the disc; towards the edge, it oscilates strongly. Some eigenfunctions are sketched
in figure 3. Equation (2.51) determies the O(Ni) correction to ')i. For the rigid rotator this
equation becomes
(4k+1)(¡i,Ie+1I)2 /i (ll¡(y) 52 J i i
')2,1e =  2k(2k + 1) 10 Y + (¡i + 5i)2 P21e(y)P21e(y)dy, (2.54)
for the correction to the ith eigenvalue.
226
Eigenfunctions of the rigidly rotating disc
12
10 k=10
8
k=5
6 ;,I. ,, \/: \
. 1 : I
'I : I: 1 : I; 1 . i,f/ \ \!:1 '
4Q)
i
;��
a 2
E
""
k=1
:1
: i
, I: I
: I: I: I
: :l.. \ .::: ,.: I: I : j'
. I, i: ''1
I ; , ;1 ~::/ I:::1 \: i,:1 :1: j::1 : I ' ¡ ..
/ : \ ': /: "i' ij:\ : I : . .
,",/ :
o
2
4
6
8 '
0.0 0,2 0.4 0.6
Radius r
, 0.8 1.0
Figure 3: The axsymetric eigenfctions or ring modes of the rigidly rotating disc. hi particular
the modes of radial order k = 1, 5 and 10 are shown.
The fuction 62(z) can be expressed as a sum of the basis fuctions Pllc(z). Nevertheless,
when 11 is large (k ).). 1) it is a smal correction. If we ignore it, then
_ _ (4k + 1)(i1,lc + 11)2 r 91(y) 1'1 ( )1'1 ( )d12,lc  2k(2k + 1) 10 Y 2lc Y 2lc Y y.
For the poly trope with n = 1, 9 = 11/2 and so
_ _ (4k + 1)(11,lc + 11)211 I12,lc  2k(2k + 1) 2 lc,
(2.55)
(2.56)
where
1_(11'1 1'1 dY_2((2k1)!!J2~~ (k)j(k)i(k+l)j(k+l)ilc  10 2lc(Y) 2lc(Y) Y  lck! ~tt (i + j 1)(i + j)(l)j(l)i(i 1)!(j 1)!
(2.57)
and (2k  1)!! = (2k  1)(2k  3)..3.1.
The lowest eigenvalues, to O(N2), for various values of the large parameter N are plotted in
figue 4 for the poly trope with this value of n. It is clear that as the rotation of the disc grows
227
it becomes increasingly susceptible to instabilties. This is not because rotation promotes the
instabilty, but rather that as the rotation increases, the effects of pressure support which provide
the stabilzing infuence become more and more negligible.
Eigenvalues far M=O
50 )'1
o ..
N=1000
50 N=200
..
Q)
:;õ'
~c
Q)
.2' 100
w
150 N=100
200
o 5 10 15 20
Mode order
25 30
Figure 4: The axsymetric eigenvalues for the rigid rotator. The eigenvalue to fist order,
1'1, is indicated by the dashed lie and the D(N1) correction to it, 1'2, by the dotted lies for
three values of N. The solid lie is the eigenvalue correct to D(N2), i.e. 1'1 + N11'2.
2.7 Nonaxisymmetric Instabilties
For the nonaxsymetric modes, the analysis presented above becomes more complicated. In
particular, the kernel Km(r,s) no longer has the eigenvectors P2n(Z). Instead we have
00
Km(r,s) = L À::ip::(z)ip::(y)
n=O
(2.58)
where the orthonormal eigenvectors
ipm(z) = rem +!) r (2n)! (n + m + i) J 1/2 (1 z2)m/2cm+1/2(z)n 2 L 7r r(2 + 2m + 1) n (2.60)
228
and m 2 r r( n + l) j2 ( )Àn = (2n)!r(2n + 2m + 1) L r(n + m + 1) 2.61
The functions C;:(z) in equation (2.59) are Gegenbauer or ultraspherical polynomials of index m
and order n (e.g. Magnus, Oberhettinger and Soni, 1966).
The construction of the matrix eigenvalue equation now proceeds as before. For the rigidly
rotating disc the matrix is once more diagonal indicating that the eigenfunctions, Gl, are just the
functions 'Pl (z ). The dispersion relation is now more complicated. It reduces to a pai of cubic
equations for every 0'1,1c, and each cubic in the pai corresponds to one of the two possible choices
for m, namely ::lml. The solutions to these cubics are
(1) 2i ,i .0'1,1c = ::~V ~(m, k)  11 sinß,
0'~~k3) = ::v~(m, k)  11 coshß (1:: ~ tanß) , (2.62)
where
_ (2k + 2m)! rr(k + l)j2
~(m,k)2 (2k)! l(k+m)! (2k(2k+2m+1)+m1 (2.63)
and
ß ! an1 r ÀlmOo J
 3 t L v(~ 11)3/27 + (ÀlmOO)2 .
When ((~  11)/313/2 ).). IÀlmOol, these become
0'(1) /" :: 2iÀlmOo1,1c  ý'(~ _ 11)'
0'(2,3) ~ ::v'~  11 r 1 :: iÀlmOo j.1,1c L v'( ~  11 )3/2
These is related to the growth rates by
(2.64)
(2.65)
(i) _ (i) . n
'11,1c  0'1,1c  imuo. (2.66)
".
,
The modes now oscilate in time in addition to the exponential growth or decay. This oscilation is
caused in part by the simple precession of the mode (as indicated by the second term in equation
(2.661) and also because of the azimuthal motion itself (which leads to the imaginary parts of
equation (2.651).
The presence of six eigenvalues is not inconsistent with the results of the axsymetric case in
which we have noted only one eigenvalue. For the m = 0 case the eigenvalue that we have plotted
in figue 4 is actualy a pai: O'lc = ::1'��lc11/2 = ::1~(m = O,k)1I11/2. In addition, we have omitted
a possible solution '�� = O. In actual fact al three of these eigenvalues are doubly degenerate, and
when m =l 0 al six modes appear with nontrivial timedependence. This is shown in figue 5.
t
I!
229
Re
1m
    ~..     ,
,
,
f'
    (~     ,
m=O  Iml::O
Figure ��: An ilustration showing the double degeneracy of the eigenvalues for m = 0, and how
this is lifted when m l O. The picture is an "energy level diagram" showing the real and imaginary
parts of O'i,1c for the six modes.
3 CONCLUSIONS
We have constructed models of polytropic gaseous masses using an asymptotic techique in
which the disparity of the vertical and horizontal length scales led to a characteristic, large param
eter that was exploited to develop the asymptoticaly vald solution. These models have intrigug
shapes and vary substantialy with the polytropic index.
Our analysis of the liear stabilty of the normal, varicose modes of the disc takes account of the
global nature of the disturbances, instead of treating the perturbation purely localy. Moreover, by
constructing the equibrium structure intialy, the liear stabilty analysis is completely internaly
consistent.
It is clear that we have uncovered a wide range of instabilties in the rotating disc. Fortunately,
the disc is probably not catastrophicaly unstable, and the stabilzing inuence of pressure support
destroys the instabilty quickly at moderate wavenumer, provided the rotation is not excessively
large. For the Miy Way, with N rv 20, only the lowest order modes may be unstable.
The ring instabilties found here may not be purely mathematical artifacts: some galaxes do
indeed possess ringlie structures, suggesting, perhaps, the presence, at fite amplitude, of these
m = 0 modes.
The detaied analysis of the stabilty of the nonaxsymetric modes is currently underway.
Perhaps this wil shed light on the selection mechansm behind spiral structure.
Aclcnowledgements: I would lie to than Prof. E. A. Spiegel without whom this project would
have been unsurountably dicult. I would also lie to than the fellowship commttee for an
enjoyable sumer at Woods Hole.
230
REFERENCES
1. Goldreich, P., and LyndenBell, D. (1965). Mon. Not. R.A.S., 130, 125.
2. Howard, L.N., and Spiegel, E.A. (1970). Unpublished manuscript.
3. Jeans, J.H. (1928). Astronomy and Cosmogony (Cambridge unversity press).
4. Ledoux, P. (1951). Annales dAstrophys., 14,438.
5. Magnus, W., Oberhettinger, F., and Som, R.P. (1966). Formulas and theorems for the special
functions of mathematical physics (SpringerVerlag, New York).
6. Monaghan, J.J., and Roxburgh, I.W. (1965). Mon. Not. R.A.S., 131, 13.
7. Norman, C.A. (1978). Mon. Not. R.A.S., 182, 457.
8. Qian, Z., and Spiegel, E.A. (1990). Preprint.
9. Qian, Z., Spiegel, E.A., and Proctor, M.R.E. (1990). Instability and Applied Analysis of
Continuous Media, 1, in press.
10. Shu, F. (1970). Ap. J., 160, 99.
11. Toomre, A. (1969). Ap. J., 158, 899.
231
The Nonlinear Evolution of a
Perturbed Axisymmetric Eddy
George i. Bell
Woods Hole Oceanographic Institute
Woods Hole, MA 02543
1 Introduction
Numerical simulations of two dimensional and geostrophic turbulence (Babi
ano et al. 1986¡ McWillams, 1984) demonstrate that an initially turbulent vorticity
distribution evolves into one where the vorticity is concentrated into approximately
circular patches surrounded by regions where the vorticity distribution is fiamentary
and on average an order of magnitude lower. Robust eddies, relatively isolated from
the surrounding fluid, are also commonly observed in geophysical flows (eg. Gulf
Stream rings), as wel as laboratory experiments.
The problem of the linear and nonlinear evolution of a nearly axsymmetric
eddy is therefore of considerable interest. This research is motivated, in part, by
the numerical simulations of Carton & McWillams (1989). In the context of a two
layer model, they considered an initialy axisymmetric eddy with potential vorticity
in the i'th layer, ai, a given function of radius. The evolution of a slightly perturbed
and linearly unstable eddy was calculated using a pseudospectral numerical code.
They found that the instability was often nonlnearly stabilized at finite amplitude,
and that the eddy appeared to undergo a complicated oscilation about a nonlinear,
steadily rotating eddy which was approximately elptical. The suggestion that the
nonlinear dynamics of a perturbed axsymmetric eddy are quite complex, possibly
even chaotic, is the driving force for the present research.
Before we embark on our search for chaos we should remark about the recent
discovery of Polvani & Wisdom (1990) about one type of chaos which occurs in the
vicinity of a Kida vortex. A Kida vortex is simply an ellpse, filled with a uniform
distribution of vorticity and placed in a background strain field. When the strain
is zero, the elpse rotates uniforiny, a fact that was discovered by Kirchoff (1876).
232
When a straining flow is added, Kida (1981) showed that the problem is stil solvable
exactly. For smal strain rates the ellpse merely oscillates periodically. Although
the flow at any fied point is also periodic, particle trajectories outside the ellipse
may be chaotic (Polvani & Wisdom, 1990). This type of chaos has been termed
chaotic advection (Aref, 1984), or Lagrangian chaos. In contrast, our search is for
a vortex whose actual dynamics are chaotic. By this we mean that the ellpticity
of the vortex (or some measure of its degree of distortion from the axsymmetric
state) varies chaotically with time. Since the ellpticity of any stable Kida vortex is
a periodic function of time, clearly Kida vortices do not have chaotic dynamics.
In Section 2, we introduce the simple twocontour eddy, and in Section 3 cal
culate it's linear stability properties. Finite amplitude behavior is calculated using
a contour dynamics code (courtesy of Steven Meacham). In Section 4 it is demon
strated that an eddy which begins in a linearly unstable region of parameter space
either breaks up catastrophically or equilbrates. In Section 5 we document the be
havior of large amplitude perturbations to a linearly stable eddy, and by focussing
in on a specific example we demonstrate that it is possible to duplicate the nonlin
,ear interactions using a four degree of freedom Hamiltonian system. This reduced
Hamiltonian system appears to show chaotic behavior in this parameter regime. The
implications of this work are presented in Section 6, together with the suggestion that
by uncovering the Hamiltonian structure of contour dynamics it may be possible to
derive the nonlinear theory directly from the contour dynamics algorithm.
2 The twocontour eddy
The problem may be simplified considerably by supposing that the distribution
of vorticity with radius is piecewise constant. The geometry of our model is depicted
in Figure 1. For the unperturbed problem, let the potential vorticity be qi + q2 inside
an inner circle of radius ri, q2 inside an outer circle of radius r2, and zero outside
the outer circle. The reason for including two circles, or contours, is that it is the
simplest model which allows for linear instability.
Flierl (1988) showed that a barotropic twocontour eddy with q :; 0 everywhere
is always stable to ellptical (m = 2) perturbations. This suggests that a two layer
model is necessary for the complicated dynamics observed by Carton and McWillams
(1989) near an unstable m = 2 mode. However, we can avoid the complication of a
two layer model by alowing the lower layer to be infinitely deep and stationary (a so
caled equivalent barotropic model). The reason for using such a model, as we shal
see, is that the elliptical mode can be unstable (even when q :; 0 everywhere).
If the circular contour r = r¡ is perturbed into r(8,t) = r¡ + 17¡(8,t), then the
233
.i
1
,
I
i
I
Cf=O
r:~~
,
..
.. ..
Figure 1: The perturbed twocontour eddy.
stream function ., for the How in the upper layer is given by
£,., = :E a;0(ri + 71i(8, t)  r)
i=l,2
(1)
where 0 is the Heaviside step function and £, = V2  1 is the operator connecting
the stream function and potential vorticity distribution. Equation (1) is in nondi
mensional form, where time is measured in units of q1 and distance is measured in
units of Rossby deformation radius.
The motion of each contour is specified by the kinematic condition
871i _
at 
1 d
dLl .,(ri + 71i(8, t), 8, t)ri + 71i Q (2)
which specifies that the contour 71i moves inward or outward in response to the radial
Huid velocity at its location.
3 Linear theory
The linear theory for problems of this type has been performed for similar
specific cases (Chidress 1984; Flier! 1988). The perspective presented below was
chosen because it extracts the mathematical structure of the general case (with n
234
potential vorticity interfaces), and contains most previous results as special cases
involving other choices for the operator 1:.
We suppose first that the perturbation is small (compared to r j)
1/j( 8, t) = O( f) (3)
this implies by (1) that "p may be written as
"p = "po(r) + f"pi(r,8,t) (4)
Substitution of (3) and (4) into (1) and (5) leads to.
i:"po = L qj8(rj  r)
j=I,2 (5)
i:,,i = L qj1/j5(rj  r)
j=I,2 (6)
r.a1/j = _ a.,l(r.) _ a1/ja.,O(r.) (7)3 at a8 3 a8 ar 3
A further reduction is achieved by Fourier transforming 1/j in azimuthal angle,
in other words looking for normal modes
11 .(8 t)  B. eim(Bcit)3 ,  3 (8)
The unperturbed azimuthal velocity V�� = a"po/ ar and perturbed stream function .,I
can then be written in terms of simiar operators:
, ~ (I:  :2) V�� =  .L qj5(rj  r)3=1,2 (9)
(I:  ~2) .,I = ,L qj1/j5(rj  r)3=1,2
while the kinematic condition (7) reduces to
(10)
wrj1/j = .,i(rj) + 1/jV��(rj) (11)
Let Gm(rj,r) be the solution to
(i:  ~2) Gm(rj,r) = 5(rj  r)
(12)
235
From the form of equations (9) and (10), we can see that Vo and 'li can be represented
as sums over the Green's functions Gm. Substituting expressions for Vo and 'li in
terms ofthe Green's functions into (11) yields an eigenvaue problem for the frequency
w. The possible values of ware the eigenvalues of the matrix M, where the elements
of M are defined by
Mij = Dij r E ajGi(Tj,Ti)J _.! E ajGm(Tj,Ti) (13)
Ti lj=i,2 Ti j=i,2
~ ~
~
Vo(r¡)
Note that for the equivalent barotropic case we are considering, we have
~ a 1m(b)Km(a) if b i a
Gm(a, b) =
a1m(a)Km(b) if b ~ a (14)
where 1m and Km are modified Bessel functions of order m. Because the matrix M
defined in (13) is 2 by 2, it is easy to write down a criterion for the stability of a
particular azimuthal perturbation.
Figure 2 shows the regions of linear instability for azimuthal modes m = 2, 3
and 4. Higher modes have similar instabilty regions which pile up against the line
T2 = 1 = Ti. Perturbations in the m = 0 mode are not alowed because they change
the area enclosed by one of the contours, violating vorticity conservation. The m = 1
mode is equivalent to a displacement of one of the initial circles, is always linearly
stable here and gives rise to translations of the eddy (see Flierl1984). If both of the
potential vorticity jumps have the same sign, or a2/ ai )0 0, then a form of the well
known Rayleigh stability criterion may be invoked to verify that the perturbation is
neutraly stable (we'll not discuss the dynamics of such eddies further).
One interesting aspect of Figure 2 is that there are large regions of parameter
space where the eddy is linearly stable to al modes. This is rather surprising if
one considers the analogous planar problema barotropic shear layer with nonzero
vorticity within a strip d i y i d. Such a shear layer is always unstable to a
perturbation of wavenumber O(1/d). In the limit where T2 is near Ti; we would
expect that the curvature of the two interfaces is negligible compared with (T2 Titi
and the planar shear layer would be approximated. From Figure 2, we can see that
in this region, the eddy is likely to be unstable to a perturbation with a large m,
confirming our intuition that it must always be unstable. As T2 increases, it is the
quantization of m that leads to the linearly stable regions. For large d, unstable waves
in the shear layer have very long wavelengths and an integral number of wavelengths
do not fit around the perimeter of the circular eddy. This explains why the eddy is
linearly stable to all modes when T2 is large enough.
236
0.0 \ ~\\ ~
,~ \ ~
"" ~ \
"" ~\~ m=3\ ~
\ ~ lÁlA\~ble. \'\ m = 4 \ "" "' \~",'e. ~\ 1.~w.(e. \ "" \ ST
o \ \ \ \ \"0;
~e ~~\ \~( ~ \~
.. \ ~~ . I' ~.. '5to.ble.
'j . '\_ .. (a Lliv\t"\ \': ';,.";.\ P ~. '/
'.:\ \" ~;' \
'..
~tetble. (Ctll M ��
0.5
1.0
1.5
2.5
S'r.l
r,. .
. ¡I
'Ct (
a
' 2.0N
a
3.0
3.5
4.0
1.0 1.5 2.0 2.5 3.0 3.5
r 2
Figure 2: Regions of linear instabilty for the perturbed twocontour eddy (ri = 1).
4 Finite amplitude behavior in the region of lin
ear instability
The weaky nonlnear calculations of Flierl (1984), for the barotropic case,
indicate the type of bifurcation that occurs when the parameter r2 moves from a stable
region into an unstable region. Although Flierl proved the result for the barotropic
case, and only at specific points in parameter space, from simulations of the full
equations of motion (2) and (1) using the method of contour dynamics, the following
hypothesis may be drawn. If the instabilty region is entered by an increase of r2 (and
la21 :; lail), then the bifurcation is supercritical, and the solution oscilates around
a finite amplitude, steadily rotating solution. On the other hand, if the instability
region is entered by decreasing r2, then the bifurcation is sub critical, and the weak
nonlinearity further destabilzes the eddy.
What happens when an eddy lying wel inside one of the unstable regions is
perturbed? It turns out that there are two basic types of scenaro, depending on
whether la21 :; lail or la21 .. lail. A hint regarding the nonlinear development comes
237
from the conservation of angular momentum. In terms of the potential vorticity
a, conservation of angular momentum is equivaent to the statement that (1/2)ar2
integrated over the potential vorticity distribution is constant in time. In other words,
1 12'1 I; +'1; 1 12'1
 :E a; r2 r dr dB =  :E a; (r; + "1;)4  r; dB = c2 ;=1,2 0 r; 8 ;=1,2 0 (15)
The idea is that for large deformations which have grown from smal perturbations
the integrals in (15) are positive, whie c itself is smal. In order that the sum in (15)
nearly cancel, it is necessary that the deformation of the interface with the smaller
value of lail be greater. IT the linearizing assumption of small
"1; is made together
with the decomposition into normal modes (8), equation (15) may be rewritten
e2mw¡t :E a;r~IB;12 = c'
;=1,2
(16)
where Wi is the imaginary part of the frequency w. If the mode is linearly unstable
then the left hand side of (16) wil grow exponentially unless the expression in the
sum is identically zero. Evidently, c' = 0 and
Ir1B11 = j_ a2
Ir2B21 a1 (17)
Equation (17) is an exact identity satisfied by any linearly unstable mode, and spec
ifies that the perturbation of the weaker vorticity interface is proportionately larger.
Dritschel (1988) has extended (17) into the nonlnear regime by looking at a
linear combination of the conserved quantities angular momentum and area. Essen
tialy, he is able to derive a conserved quantity similar to (15) where the integrand is
positive definite in each "1;. His result demonstrates that even in a nonlnear sense,
the deformation of the weaker contour must be greater than the deformation of the
stronger contour.
4.1 Break up of a twosigned eddy
Unstable eddies with la21 .c la11 are composed of potential vorticity of both
signs, and the potential vorticity jump across the outer contour is weaker. In the
linearly unstable region of parameter space, by (17) we expect that the perturbation
in the outer contour will be of larger amplitude than the inner, and wil fiament first.
This is confirmed by a ful numerical simulation (Figure 3), showing the break up of
an eddy into a trip olelike structure. For some recent experimental results on the
formation of tripoles and dipoles from nearly axisymmetric vorticity distributions,
the reader is referred to the fascinating photos in Kloosterziel & Van Heijst (1989).
238
(9 ~.~ . cg~, cg.~ ~..ot;..~
cg cg cg (Q
(Q CQ cø Ø)
C!
Figure 3: Nonlnear evolution of a linearly unstable eddy with la21 c lail. Parameters
: Ti = 1, T2 = 2.0, ai = 1 and a2 = .5.
(C;x'O ((
~.o ~
(�� (�� 8 (0
v
Figure 4: Equibration of a linearly unstable eddy with la21 ~ lail. Parameters:
Ti = 1, T2 = 2.7, ai = 1 and a2 = 2.
239
4&2 Equilibration of a linearly unstable eddy
The behavior of linearly unstable eddies with la21 ). lai I is quite different. Such
eddies have an annular vorticity distribution, with potential vorticity gradients of both
signs, but are composed entirely of potential vorticity of one sign. Since the inner
contour is weaker, we expect fiamentation to occur first in that contour. Figure 4
shows the evolution of such an eddy. The inner contour fiaments and these filaments
are then wrapped around the center of the eddy. In Figure 4, contour surgery has
clipped off these fiaments where they drift into the region between the two primary
contours. The outer contour, in contrast, does not fiament.
One can think of the behavior of Figure 4 as a relaxation of the eddy into
a parameter regime where it is linearly stable, or at least marginally stable. The
filamentation of the inner contour has two important consequences. First, it reduces
the area of the inner contour. When Ti decreases, by recalculating Figure 2 it may
be shown that the instabilty regimes move upward. The effect is to move the eddy
toward a parameter range where it is marginaly stable. Second, if we imagine viscos
ity ming the fiament into the annular region between the contours, both lai I and
la21 wil be reduced and by the same amount. Since la21 ). lai I, this wil cause the
ratio lail ai I to increase. This also has the effect of moving the eddy into a region of
parameter space where it is linearly stable.
5 Finite amplitude behavior in the region of lin
ear stability
The calculations of the previous section lead us to believe that an eddy which
begins in the linearly unstable region with la21 ). lail wil evolve (in an approximate
sense) into the linearly stable regions between the instabilty "spokes" of Figure 2. In
this section we use the contour dynamics code to investigate finite amplitude behavior
of eddies in this linearly stable region.
We initialze the run with a fairly large (10% to 20%) perturbation in both
the m = 2 and m = 4 modes. As the evolution proceeds the inner and outer contours
deform, sometimes appearing nearly ellptical (m = 2 mode) and at other times more
square (m = 4 mode). To view the nonlnear interactions, we decompose the result
at each time t into the normal modes dictated by the linear theory:
¡ l1i(8, t) J 00 .
= L (Ami(t)umi + Am2(t)Um2l e,m8112(8, t) m=i (18)
where Umi and Um2 are the eigenvectors corresponding to the two normal modes (the
2tlO
eigenvectors of M as defed in (13)). We then plot the amplitudes IA;I'versus time.
IT the liear theory were exact, the amplitude of each mode would be constant in
time. Thus, any osciations of the modes result from nonlnear interactions.
Figue 5 shows a typical plot of mode amplitudes versus time for a fairly large
initial perturbation. The area enclosed by each contour (which should be constant
0.18
0.16
0.14
0.12
=0.10
4:
0.08
0.06
0.04
0.02
0.00
~~
ai.
ModLt.
ll i ",..
o 100 200 300 400 500 600 700 800
t
Figure 5: The nonlnear interaction between modes calcuated from the output of the
the contour dynamcs simulation. Here qi = 1, q2 = 2, 1"i = 1 and 1"2 = 2.3. The
eddy turover time is about 16 time unts, so the entire ru represents about 54 eddy
turnovers.
by vorticity conservation) was monitored to ensure the accuracy of the calculation.
In order to keep the changes in area under 1 % over the course of a ru, a smal time
, step (tit = .03125) was necessary and the ru of Figue 5 required nearly 24 hours of
cpu time on a Sun Sparkstation. The priary interaction is between the two m = 2
modes and one of the m = 4 modes. However, one of the m = 6 and m = 8 modes
are excited nonlnealy (and, as we shal see, caot be ignored). The odd numbered
modes are al nearly zero as there is no way they can be excited from even haronics.
Despite appearances, none of the cues in Figure 5 represent noise, as a reduction of
241
the time step and increase in the number of nodes per contour produces only slight
differences in every curve at the end of the run.
Although the evolution of the mode amplitudes in Figure 5 is complicated, it is
diffcult to determine if they are in fact chaotic. The Fourier transforms of the signals
are quite broad banded, however the problem does not appear to be very sensitive
to initial conditions. After shifting the initial phases of the modes by 10%, the new
mode amplitudes are stil highly correlated with their old vaues through the entire
run, with amplitude changes of only about 10% at the end. Sunaces of section may
be produced from the output, however they contai at most 50 points which is far too
few to uncover the dynamcs. Effectively, we are solvig a system of 500 differential
equations (125 z and y coordinates for each of the two contours). What is evidently
needed is a reduction in the number of degrees of freedom in the ,problem.
/
0.06 0.030
ii
0.05 0.025
i.l
0.04 0.020
'..
: 0.03 0.015
0.02 0.010
"(
0.01 0.005
tt
0.00 0.000
0 'a). 200 400 0 200 400
t t
Figure 6: Mode amplitudes versus time where only the 22 and 41 modes are nonzero
initialy. Parameters are the same as in Figure 5, and initial amplitudes of .05 and
.025 are shown.
We begin by simplifyng the number of interacting modes in the contour dy
namcs output. In Figure 6, we begin with only the 22 and 41 modes (it is evident
242
in Figure 5 that these modes interact strongly). Note that in Figure 6, if the initial
amplitude of the modes decreases by a factor of two, the nonlnear oscillations de
crease by a factor of four. Such a decrease indicates that the oscilations are due to
a quadratic nonlnearty. A Fourier transform of the 41 mode (Figure 7) shows that
four frequencies predominate, which may be written as sums and differences of the
linear frequencies (see Table 1).
211"  W..,l
104
Q)1: 105:J
+
0.
I~
E
1060 \
107
w" .. Wu. .. w"
l w'l"l"'~i
'"
..\,
r 
0.0 0.2 0.4 0.6 0.8 1.0
frequency
Figure 7: Fourier transform of the 41 mode of Figure 6 with initial amplitude .025.
Dotted curve is the Fourier transform of the 41 mode in the approximating Hamto
nian system (30) (Figure 8), displaced downward by a factor of 10.
Mode Frequency eigenvector Mode Frequency eigenvector
ii Wi; B1 B2 ii Wi; B1 B2
21 .131104 .853079 .521782 22 .223434 .988270 .152718
41 .111897 .999946 .010361 42 .636545 .108970 .994045
61 .066436 1.000000 .000851 62 1.318470 .008994 .999960
81 .249875 1.000000 .000107 82 2.040765 .001094 .999999
Table 1: Frequencies and eigenvectors for the normal modes for the linearly stable
eddy qi = 1, q2 = 2, Ti = 1 and T2 = 2.3
243
5.1 A low order nonlinear theory
Although it would be nice to derive a nonlinear interaction theory directly from
the equations of motion, we adopt a cruder approach by writing down the form of
the nonlinear interaction, and choosing the free parameters to match the output from
the contour dynamics program.
Consider the two normal modes A22(t)e2iB and A41(t)��iB with small amplitude
(IA.;I = O(€)), as in Figure 6. A quadratic nonlinearity may be introduced by
multiplying pairs of normal modes together. However, there is only one product
for each mode that has the proper phase (since the m = 4 mode is an azimuthal
harmonic of the m = 2 mode). This suggests that a first approximation to the two
mode interaction of Figure 6 is
A22 iW22A22 + a1A4iA;2 (19)
(20)A41  iW41~1 + a2A~2
where ai and a2 are arbitrary complex constants. Equations (19) and (20) may be
thought of as an asymptotic expansion in mode amplitudes (IAi;1 = O(€)) which
is truncated at order ,;. Note that we make no assumption about the size of the
constants ai (in fact, they wil turn out to be 0(1)).
Since the 2D Euler equations have a Hamitonian formulation (Morrison 1982),
it is reasonable to require that the equations (19) and (20) form a Hamiltonian system.
The conversion to Hamtonian form may be accomplished by the substitution
A.; = V Ji;(t)eiB¡j(t) (21)
In terms of the J's, the system may be written
. 8H
J22 = 
8(J22
. 8H
J41 = 
8(J41
. 8H(J22 = 8J22
. 8H(J41 = 8J4i
where
H = W22J22 + W41J41 + Cl J22.¡ cos((J41  2(J22 + aii) (22)
The complex constants ai and a2 are related to the real constants Ci and aii by the
relations:
Ci = lail = 21a2\ (23)
(24)
*
ei(rPi +w/2) = ai = _ a2
lail la21
244
Effectively, the requirement of a Hamiltonian system restricts the choices of the con
stants ai.
For IJij(O)1 small, the solution to the Hamtonian system (22) is Jij(t) = Jij(O)
and 8ij(t) = Wijt. As IJij(O)1 increases, the effect of the third term in (22) is to
introduce a modulation of IJij(t)1 with frequency W4i2w22. Note that this frequency
is exactly the dominant frequency that appears in Figure 7. From Figure 6, we can
see that j22 = 0 at t = 0, which implies that cPi = O. The constant Ci determines the
amplitude ofthe modulation in IJij(t)l, and we calculate from Figure 6 that Ci ~ .35.
With cPi = 0 and Ci = .35, the Hamtonian system (22) reproduces perfectly
the nonlnear modulation of the 22 and 41 mode amplitudes. The system (22) admits
an invarant, I = J22 + 2J4i (25)
and thus the two degree of freedom Hamiltonian system is integrable. To reproduce
the higher order modulations present in Figure 6, we must include higher order non
linear terms. This will lead to a more complex Hamiltonian system which is not
integrable.
5.2 A higher order nonlinear theory
The idea behind the higher order nonlinear theory is simple: we proceed in
our amplitude expansion (assuming A22,A4i = O(e)), but now include terms up to
order tf. Unfortunately, there are a very large number of possible terms of order tf.
However, by looking at Figure 6, we can see that many of these terms are smal. For
example, we see that the 21 mode is excited to a much lesser extent than the 61 and
81 modes, and the 42,62 and 82 modes do not appear to be excited at alL. The reason
that these last three modes are not excited is that their eigenvectors are skewed very
strongly towards the outer contour, while the 22 and 41 modes are skewed towards
the inner contour (see Table 1). Thus, it seems reasonable to include only those O(tf)
terms involving the 22, 41, 61 and 81 modes. Such a truncation yields
A22 = IW22A22 + ai~iA;2 + a3A6iA:i
+ A22(a41A2212 + a5IA4112) + ßiAsiA;i (26)
~i IW4iA41 + a2A~2 + a6A6iA;2 + a7AsiA:i
+ A41(as IA4112 + a91A2212) (27)
(28)
(29)
Áii IW6iA6i + aioA22A4i + ß2AsiA;2
IwsiAsi + allA~i + ß3A22A6iAsi
245
Note that the forced modes 61 and 81 have amplitudes 0(£2). The terms ßi, although
formaly of order ��, are included because they link the 61 and 81 modes together,
and (with hindsight) the coeffcients ßi are large. Applying the transformation (21)
we find the associated Hamitonian
H = W22J22 + W41J41 + W61J61 + WS1JS1 + a1J:2 + a2J:1 + a3J22J41
+ C1 J22¡i COS(841  2822 + cP1)
+ C2 VJ22J41J61 COS(861  841  822 + cP2)
+ C3 J41 ¡¡ cos(8S1  2841 + cP3)
+ C4 V J22J61JS1 cos(8S1  861  822 + cP4) (30)
where once again the real constants ai, Cï and cPi are al related to the complex
constants Qi and ßi. The terms involving ai result in frequency shifts of the primary
modes. The terms involving Cï are responsible for the additional frequencies shown
in Figure 7. The values of C2 and C3 may be determined by the amplitudes of the
forced 61 and 81 modes. Figure 8 shows the result when we numerically integrate the
Hamtonian system (30) with ai = cPi = 0, C1 = .35, C2 = 1.0, C3 = .8 and C4 = 2.1,
and use the same initial conditions as Figure 6. Pictorially, the agreement with the
contour dynamics results (Figure 6) is excellent, and their Fourier transforms (the
two curves in Figure 7) are nearly identical.
6 Conclusions and suggestions for additional work
We have shown that for a certain amplitude range the nonlnear dynamics of
the twocontour eddy can be very closely approximated by a four degree of freedom
Hamitonian system. Effectively, we have been able to reduce the number of degrees
of freedom from 250 (for the contour dynamics algorithm) to 4.
The next logical step is to investigate whether or not the Hamitonian system
(30) contains chaotic dynamics. The investigation into this question has just begun,
and we present only some preliminary results. It has not known even if the system
(30) is integrable or not, but there is strong evidence that it is not. As before, the
system admits an invariant (possibly related to conservation of angular momentum),
I = J22 + 2J41 + 3J61 + 4JS1 (31)
however, it would appear that even the case C3 = C4 = 0 is not integrable. To
demonstrate this last statement, in Figure 9 we show two surfaces of section for
246
0.06 0.030
0.05 0.025
0.04 0.020
:=' o. 03 0.015
0.02 0.010
"
0.01 0.005
jl
0.00 0.000
0 200 400 0 200 400
t t
Figure 8: Mode amplitudes versus time for the Hamiltonian system (30) with ai =
lPi = 0, Ci .. .35, C2 = 1.0, Ca = .8 and C4 = 2.1 (compare with Figure 6)
slightly different initial amplitudes. The first case appears to be integrable, while the
second does not.
One interesting aspect of the nonlnear theory of Section 5 is that it involves
modes al of which have much higher amplitudes in the inner contour. In fact, a
barotropic eddy consisting of but a single contour probably obeys similar dynamics.
If in fact the Hamtonian system (30) exhbits chaotic behaYÌor in the parameter
range of interest, then the ingredients for chaotic vortex motion are simpler than
were thought at the beginning of this paper.
The nonlnear ansatz presented in Section 5 could be applied to a general
nonlinear interaction between a wave and its spatial harmonics in any Hamitonian
system. For this particular case, however, the hole in the theory which needs to be
:fed in is a calculation of the coeffcients in the Hamiltonian from first principles.
This could be acheved by completing two steps. First, the equations of motion for
the contours ((1) and (2)) must be written as a Hamiltonian system. As this report
was being written, Phi Morrison completed this step for the case when r( 8) is sin
247
0,05
, f\ 1" /" 1d \, /\ \ /:
, Ì' ' ..,
,'",: ii' \ '.:,:.' ~
~"~ ;'/
, '
0,06
(b)(~)
, ,
.', ."0,05
/ /'\
0,04
0,04,
O,OJ
:.......... .
." ....;.:........ .
I'll
0,03 " ,:....,. " .
..........' .".,'..
~".:;.:".". ,". ".. . . . .
'.' . . ....~ .r:..~:.;....::...:~....:....
. '.' . .. ;.:".":: "."." :.
....:~~\.~.,..:.. ',:.'",' ,'. ~'. ~ :.i.:;'!?h?:./.." ,'.
.: :. '., .... '.
0,02
0,02
0,01 0,01
0,00 0,00
J 2 1 �� 2 J J 2 1 ��
9..191(l
Figure 9: Two surfaces of section J41 versus 841 strobed by 822 = 0 for the Hamitonian
system (30). (a) C1 = A, C2 = 1 (al other constants zero), A22(0) = .1, A41(0) = .05,
AS1(0) = .05. (b) As in (a) but with AS1(0) = .07.
gle vaued (this is always the case for the eddies of Section 5). Second, r(8) must
be expanded in normal modes and through knowledge of the Hamiltonian structure,
actionangle vaables extracted. This step is in process, but it must be that the
nonlinear theory presented in Section 5 will emerge as some part of the Hamtonian
formalsm. A Future project suggested by the Hamtonian nature of contour dynam
ics is the investigation of negative energy states and resonances, and their possible
connection to fiamentation.
7 Acknowledgements
I would like to thank Glenn Flierl for a number of useful discussions in regard
to this project, and Steven Meacham for giving out copies of his contour dynamcs
code so generously. Phi Morrison became interested in the project once Hamitonian
dynamcs were mentioned, and I am grateful for the time he took to teach me about
248
the subject. Thanks to al the GFD "dynamos", I had a great summer both in Walsh
and on the soft bal field.
8 References
. Aref, H. 1984 Stirring by chaotic advection. J. Fluid Meck. 143. 121.
. Babiano, A., Basdevant, C., Legras, B., Sadourny, R. 1986 Vorticity and pas
sive scalar dynamics of twodimensional turbulence. J. Fluid Meck. 183 379
390.
. Carton, X. and McWilams, J. 1989 Barotropic and baroclnic instabilities of
axsymmetric vortices in a quasigeostrophic modeL. Mesoscale/Synoptic Coher
ent Structures in Geophysical Turbulence Elsevier 225248.
. Childress, S. 1984 A vortextube model of eddies in the inertial range. Geophys.
Astrophys. Fluid Dyn. 29. 2964.
. Dritschel, D. 1988 Nonlinear stability bounds for inviscid two dimensional flows,
... l. Fluid Mech. 191. 575581.
. Flierl, G. 1988 On the instability of geostrophic vortices. J. Fluid Mech. 197.
349388.
. Kida, S. 1981 Motion of an ellptic vortex in a uniform shear flow J. Phys.
Soc. lpn. 5035173520.
. Kirchoff, G. 1876 Mechanik, Leipzig.
.. Morrison, P. 1982 in Mathematical Methods in Hydrodynamics and Integrability
in Dynamical Systems, AlP Conf. Proc. 88 13.
. Polvani, L. and Wisdom, J. 1990 Chaotic Lagrangian trajectories around an
ellptical patch embedded in a constant and uniform background shear flow.
Phys. Fluids A 2 123126.
. Kloosterziel, R. and Van Heijst, G. 1989 On tripolar vortices. Mesoscale
Synoptic Coherent Structures in Geophysical Turbulence Elsevier 609625.
249
The Rise and Fall
of Buoyant Plumes
C. P. Cauleld
D. A. M. T. P.
University of Cambridge
Abstract
We generalize the theory of Morton, Taylor, and Tuer (1956) to plumes of nonzero
initial mass flux. We classify plumes with respect to the balance between their initial
mass and momentum fluxes. We explain the efect fit descrbed by Morton (1959) of
a reduction in plume height in a stratified environment with increased momentum flux
in terms of unstratified behaviour. The Morton efect does not occur for al initial mass
fluxes. However, for any initial mass flux, there exists an intial momentum flux with
minimum height of rise. The behaviour of a plume rising from a source of fluid that
exhbits nonmonotonic density vaation with ming is investigated as a model for volcanc
eruptions, å.d a, condition for collapse is found.
i Introduction
Localised sources of fluid with density dierent from the denity of the ambient occur in
a wide range of geophysical, astrophysical and industrial contexs. Flows generated from
such sources are modelled using the entraient assumption, fit discussed by Taylor
(1945) (see Tuner (1986) for a comprehensive review). The infow velocity of ambient
fluid into the fluid rising from the source is assumed to be proportional to some charac
teristic vertical velocity of the source fluid. The constant of proportionalty is assumed
to be the same at al heights above the intial release height. This steady state model
assumes that the turbulence and velocity structures are preserved at al heights. Though
this model is very simplified, since it assimilates al the efects of turbulence into one con
stant of proportionalty (see List (1982) and Tuer (1986) for discussion), the description
and predictions of the model appear to be vad in many situations, and the results of
interest are relatively robust with respect to vaations in the exact vaue of this constant
of prportionalty.
Using this model the evolution of the behaviour of the fluid from a source can be
modelled by considering the conservtion of three quantities, namely mass flux, momentum
flux, and buoyancy flux, Le. the flux of density deficiency of the input fluid. Following the
seminal paper of Morton, Taylor, and Tuer (1956), we defie the haldth of the plume
arising from a source of buoyancy at a cerain height as b and its reduced gravity g',
,g' = 9 (Po.  PP ) ,Prej
250
where Po. is the density of the ambient fluid, P. is the denty of the source, and Pre! is a
characteristic density of the system. We alo identif a characterstic verical velocity w
, at each height, and assume that
Uin!'otD = aw ,
where 0: is known as the entrainment constant. The equation descrbing the evolution of
mass flux is d 2
dz (p7rb w) = 2¡nbow . (1)
Tils equation reflects the fact that the rate of increae of mass flux with height is balanced
by the entrainment of ambient fluid. The rate of increa of momentum flux is balanced
by the gravitational acceleration of the source fluid due to denity dierences, i.e.
d
dz (p7rb2w2) = pg'7rb2 . (2)
The rate of change' of buoyancy flux with height is balanced by the entraient of fluid
with density characteristic of that height, a balance descrbed by
..dd (7rb2w(po.(O) p,,)l = 7rN2b2w,Pre! Z (3)
where N2 is the Brut:"Vaisala frequency, defed by
2 _ g dN = dzPo.,
pre!
and Po.(O) denotes the ambient density at z = O. If we mae the fuher assumption that
the Boussinesq approximation applies (i.e. Po.  PI' .c Pre!) then the equations reduce to
tQ = 2aM1/2 ,
dz
tM = FQdz M 'd 2
F = QNdz '
(4)
(5)
(6)
where
Q = b2w ,
M = b2w2 ,
F = gib2w ,
(volume flux)
(momentum flux)
(buoyacy flux)
and we se that
b Q
= Ml/2 ;
MW' ,
Q
, F
and 9 
Q.
251
In the absence of stratification, Morton (1959) demonstrated, that since F is constant,
( 4) and (5) may be combined as
M3/2 d d2QM = QQ ,Fdz dz
which implies that
:;M5/2 +c = Q2 , (7)
where
2 SQ .. d/2e = Qo  5FiVO .
Thus (46) reduce to F = Fo ( constant), (7), and
.!Q = (20Q4 F)I/S(Q2 _ e)I/S .
dz
(S)
(9)
In the case of initial mass flux and momentum flux zeo, e = 0, and (S) reduces to
~ Q = (20Q4 F)I/SQiis . (10)
A similarty solution may then be found for b, w and g' (se 'ler (1973) for details), of
the form .
6b = SQZ ,
5 (9 )1/3 'w = . QF zI/3 ,
6Q 10n 1/3
, 5F (9 F) 5/3g=Q Z.6Q 10
(11)
(12)
(13)
These initial conditions characterze a wel known and exremely common phenomenon in
geophysics, namely the buoyat (pure) plume. We note, that in an untratified environ
ment, buoyant pure plumes rise without lit, with thei mas flux and radus increasing
with height as described by the similarty solution. However, in the geophysical contex,
ambient stratification is usualy present. Ths ha alo bee consdered using this model
(see Morton, Taylor and 'ler (1956) and Morton (1959)), but cae must be taken since
energy may be radiated away in the form of inter waves when an intruion occu in a
stratified environment (see List (19S2)). In thi ca, N is no longer zeo, but we can stil
reduce the governng equations to two by noting that (5) and (6) combine:
dM2 1 d 2
dz = 2QF =  N2 dz F ,
L
r
that is
2 F2M + N2 = e. . (14)
252
IT the ambient stratification is staticay stable, (since buoyacy fluX deceases with
height) (from (6) and see fig. '1), there will be a paricuar height at which the plume
fluid density equal that of the ambient. At such a height, frm (14), the plume will have
maxmum momentum flux. This is progressively,(and quite rapidly) lost, until at some
later height, referred to as the fial height of rise, the upwad velocity goes to zero (see
fig. 2). The fial height of rise of a pure plume predcted by this model has been shown
to apply very well to both experimental and geophysca plumes (see Tuer (1973)), and
indeed geophysicaly it appears that the pure buoyat plume is in some sene selected as
a natural structure. Morton (1959) found that incring the intial momentum flux from
zero actualy decreased the fial height of rise for point source plumes of zero intial mas
flux and fite buoyancy. Only plumes with ver lage momentum flux rose higher than a
pure plume, which has zero intial momentum flux, an apparently paradoxical result. Ths
report considers the effects of deviations from thi clsica siple model to more realstic
source conditions.
In Section' 2, we generalze this theory to arbitrar intial conditions of mass flux
and momentum flux in the untratifed envoironmen.' We clsi vaous plumes by the
intial balance betwee momentum flux and ma flux quatifed by the constant c in (7),
adopting a system of LaneSer et al (1989). We fid that aU plumes ultimately asymptote
to a solution with c = 0, that we ca the pur plume solution. However, the efective origi,
defied as the point at which the asymptotic pur plume sity solution would have
zero radius, is determed by the constant c.
In Section 3, we generalze the model to stratifed envionments. Although no sim
laty solutions exst, pure plumes (i.e. plumes with zeo intial mass flux and momentum
flux but fite buoyancy flux) have a we defed balce betwee momentum and mass
flux at al heights. Alo, for any intial mas flux Q, ther is a paricuar associated mo
mentum flux, M = M( Q), that causes a m;n;mum fi height of rise for a plume with
fied intial buoyacy flux. For smal intial ma flux, th m;n;mum has an associated
intial momentum flux greater than that of a pur plwie, and this exlai the Morton
efect. For sufciently lage intial mass flux, th m;n;mum ha a momentum flux less
than the pure plume. In this cae, al jetli source penetrate fuher than a pure plume,
and the Morton efect does not occu. Ths is liy to be important when a wel developed
plume passes through a region of vang stratifcation.
In Section 4, we consider the behaviour of a plume of fluid that exbits nonmonotonic
density vaation durg ming. Such plumes may be consider model for the behaviour
of several geophysical phenomena, e.g. volcac erption clouds, ra clouds etc. In the
cae where an intialy dense fluid becomes buoyat afer a ceai amount of mixig, we
derive a condition on the intial momentum flux such that the intial denity dierence is
so large that the plume collapses at some height, which is cacuated for several situations.
Section 5 presents conclusions, and detai fuher work.
253
2 U nstratifled Theory
The governng equations are (7) and (9) above. We nondienionalze on initial mass and
buoyancy fluxes, nondimensional quantities havig astersks. Thus
Q = QoQ* ,
(20a4F)i/S' *z = Q~ z ,
_Q2.c Oc .
Dropping the asterisks, (9) reduce to
~Q = (Q2 _ c)I/S .dz '
Note that, the plume with Mo = Qo = 0 ha c = O. Now, modifyng the concepts of
LaneSerf et al. (1989) we classify the plumes arsin from ��l sources of buoyancy with
initial mass flux and momentum flux in the followig way:
i: if c = 0 the plume is a pure plume;
ii: if c : 0 then the plume has exces momentum flux, and is a jet;
iii: if c :: 0 then the plume has momentum flux defciency, has too large initial radus,
and is a distributed source.
By defit��on ..00 : c :: 1, thus from (15) we se that mass flux always increases, and
eventualy, the, behaviour must asymptote to
(15)
~Q l" Q2/sdz '
the solution for a pure plume.
The radus b evolves according to
d 6 Fc
dzb = Ša  2Ms/2 (16)
Thus the rate of spread for a pure plume is 6:, and increa for a jet, up to the well
known maxmum of 2a initialy, when Qo = O. For a distributed source, the spread is less
than that of a pure plume, and indee if
",5/2
Q2 :: 4a.lo0 SF
the fluid actualy initialy contracts. Care must be taken in such caes (see Fischer et al.
(1979) and List (1982) for a fuer discusion of the jetlie behaviour), since the actual
vaue of the entrainment constant vaes, and the assumptions of silarty in turbulent
and velocity profies may become more questionable, as does the assertion that mig
takes place al the way across a Wide distributed source, right from the outset. However,
254
a reduction in the intial entrainment in the cae of a distributed sourCe will increase the
effect of the contraction described above. These efects are shown on fig. 3, where the
evolution of radius for thr~ dierent vaues of c (0.999, 0.0, and 0.999) are plotted. Since
each source has the same Qo = 1, the vaation in c is reected by a vaation in bo, with
the jet narower, and the distributed source wider. The nondiensionalzation is such
that the pure plume solution has bo = 1 and angle of sprea 3/5. We se that the jet
radius increases faster from a smaler bo, and evetuay crsses the pure plume'solution,
while the distributed source stars from a larger bo, Contras and eventualy crosses the
pure plume solution alo. Eventualy, however, we see that al three asptoticaly spread
linearly with slope 3/5, i.e. lie pure plumes.
This approach to plume behaviour ca al be se in fig. 4, where the evolution of Q
with height for these three different plumes is plotted. Note the lager increase in the jet's
mass flux since its momentum flux reuies a lager mas flux to balance it. The revere
is true for the distributed source.
The approach to a pure plume solution may be studied asymptoticaly. However, due
to the nature of the problem, a numerca approac ha be used. Since eventualy the
evolution of a sOUrce with arbitrar intial conditions is approxiated by the evolution
of a pure plume, we search for the "equivat pure plume solution" of any plume. In
dimensional term, if we defie the point where Qo = Mo = 0 (i.e. the point source) as Zo,
then the pure plume siÌnlarty solution for the mas flux Q is, from (1113), ,
6a ( 9 )1/3
Q =  aF (z  ZO)5/3 ~5 10 (17)
or nondimensionaly,
(3 )5/3Q = s( z  zo) . (18)
For the pUre plume, which follows this evolution frm z = 0, where Q = 1, we see
the efective origin is 5/3. For al source conditions we ca identif an efecive origin
similarly since eventualy the mass flux evolves as above. Calcutions of efective origin for
the three vaues of c are shown in fig. 5, with the asated equivaent pure plume solution.
Note that the distributed source approaces its equivaent pure plwre from outside, with
larger radi, while the jet approaces from inde, with smaer ra. Also, the efective
origin of the distributed source is ahead of that for the pure plume, which is in tur ahead
of that for the jet. The vaation of efective origi with c is shown in fig. 6.
m sumar, in an untratified envionment, source with arbitrar intial conditions
ultimately converge to a pure plume. However, the raus of an originaly jetlie plume is
larger than that of a pure plume, which in tur is larger tha that of a distributed source,
after an initial adjustment where the situation is reed. Ths is a trivial consequence of
the positioning of the effective origin. The implications of this in the stratified envionment
are investigated in the next section.
"l
j,
~ri
;
255
3 Stratified Theory
To recap, the equations that govern the behaviour in the stratifed environment are
d (F')1/4
dz Q = 2a c.  N2
d
dzF = _QN2 ,
(19)
(20)
and 2 F'c. = M + N2 ' (14)
where c. is a constant.
We nondimensionalze these equations on the ambient stratification N2 and the intial
buoyancy flux n. As before, stared quantities ar dienionless.
F = FoF* ,
M = 21/2 FoN1 M* ,
Q = 2S/8 a1/2 ~/4 NS/4Q* ,
z = 2S/8a1/2 F:/4N3/4z* ,
and the equations reduce to (once again dropping aserks),
~Q = (c _ !.F2)1/4dz · 2 '
d
dz F = Q ,
2 1 2c. = Mo + 2Fo .
The point source pure plume solution ha c. = 1/2 in this nondiensonalzation.
Since it only reaches a fite height (see fig. 2) a siarty solution analogous to that
described in Section 2 does not exst. Neverheless, there is a clealy defied evolution of
mass flux and momentum flux. We generale the approac of Morton (1959) to alow for
the possibilty of intial nonzero momentum flux. We defe two parameters, trM and trq
as follows, with nondimensional quantities beig asterske:
(21)
(22)
(23)
l( M'trM = N2W +.F
2M*2
~ 2M.2 + F*2
NS/2Q2
trq 
 NS/2Q2 + 2S/4Fol/2.F
== (1 + g'.2)1
(24)
(25)
(26)
(27)
256
These parameters are, respectively, a measure of the importance of momentum flux and
mass flux compared 'to the natural momentum and mass :fux scaes detered by the
initial buoyancy flux and stratification. The evolution of trq and trM for a point source
pure plume is shown in figs. 7 and 8. Intialy trM and trq ar zeo, and then they increase
to a maxmum (which is 1) when the plume cees to be buoyant. Subsequently, they
decrease, trM to zero, when w goes to zero (c!. fig. 2), and trq to a fite vaue, since the
model break down, and b + 00.
Combining these data, we obtain a locu in trMtq spac for a pure plume solution of
the equations, which is plotted in fig. 9. From fig. 7, we know that every t1q is associated
with a specifc height. Therefore, ea point on the 10cu may be considered to specfy
a height, h. For given intial mass :fux, a plume frm a source with momentum flux and
buoyancy flux calcuated from trM and trq at that point on the 10cu subsequently rises
like a point source pure plume that had stared frm z = h.
Thus in this contex, we defie pure plumes as plumes arsing from sources that lie on
this locu, for arbitrar intial mass :fuxes. For a given mass :fux, al points to the right
of this cue correspond to source with too much momentum :fux, whose plumes shal
be descrbed as jetl, analogously to before. Simarly, al po~ts to the lef correspond to
sources with too little momentum :fux, whose plumes sha be caed d��tributet "Duree".
Morton (1959) considered the behviour of plumes with trq = 0 and trM :F o. He
found that the height of rise actualy decea for trM ~ 0, reth;ng a m;n;mum at
around trM .. 0.7 and reacg heights higher tha the pur plume for trM ~ 0.99 (se
fig. to).' We investigate this countertuitive behviour, which we shal reer to as the
Morton efect, f��r general intial Qo. For trq = 0.12 (se fi. 11), there is an essentialy
simar progrsion. ,In this cae we se that the efec is geered into the ditributed
source region of parameter spac. Distributed source go higher than the pure plume
solution (which is marked with acrss) and ther is a m;n;mum rise height in the jet
region. However, o~ increasing trq fuher to 0.27, (se fig. 12), the plume solution rise
to a height greater than al distributed source with fite intial area, in the jet region the
height of rise monotonicay increas with trM, and there now is a m;n;mum rise height in
the distributed source region.
Fig. 13 shows the locu of mimum rise height for al possible trM and trq, and the
region where the generalzed Morton efec (i.e. jets go lower, ditributed source go higher
than the pure plume) takes place. This efect may be exlaied as follows. H the evolution
is largely unafected by the stratification, the untratifed behaviour domiates until such
time as g' /b becomes smal enough relative to N2 to alow stratifed efects to be signcat.
Thus if g' /b remai large compared to N2 for a sucient height, jets wi crss the pure
plumes in radius, and asymptote to a pure plume with efecive origi behd that of a
pure plume with the same initial mass :fux. H the stratifcation becmes important afer
this crossing event, the jet will rise to a 10wer height, provided that its momentum flux is
not too much larger than that of a pure plume. (But we remember from figs. 1 and 2 that
after g' goes negative, M and w rapidly decy to zeo.) Simarly a distributed source will
cross to a radus less than that of a pure plume, and provided that its momentum flux is
not too much smaler than that of' a. pure plume when g' goes through zeo, it will rise
to a greater height. There is thus' a balce between the competing efects of increasing
'
257
momentum flux "pushng" the efective origi fuher bac when stratmcation is "weak",
and yet, when stratification is strong, "pushig" weay buoyat or even dene fluid higher.
We thus obtai a minimun rise height where the reative efect of reg efective origin
is strongest.
H the stratification is very strong, any source with nonzo intial mass flux will have
a plume whose height of rise is substantialy detered by the intial momentum flux of
the source. (In this case, one is high up the tTMcq, cue.) Solutions of this type ca
occur when a pure plume solution, rising through a region of low stratification passes into
a region of high stratification. H the tranition occu at a height where g' /b is relatively
smal, the change in N2 may trigger a signcat diplaent from the pure plume locu,
and hence quite large vaation (from the siple pur plume model prediction averaging
the stratification in some way) in the fial height of rie. Th is the subject of fuher
investigation.
A fuher effect, that must be remember at al times, is that a nonzero intial mass
flux may change the vaue of tTq signficantly. In the clsica Morton, Taylor, 'ler model
of point source pure plumes, Qo = 0 and Fo is fite (which without loss of generalty we
take to be 0(1)), imd so, at z = 0, g', W l 00 (see fi. 2). Formaly, we have the scalngfor E ~ 1, '
g~ ,w  0 (~), bo  oCE) =* Fo ,Mo  0(1), but Qo  oCE) ,
and
,
~ ~N2.
We alo see that the initial intniion height is inte, and the Boussinesq approxiation
break down. On ,the other hand, if Qo is fite, g~ is al fite, and decreasing as Q
increases, thus increasing Qo decreases intial intruon height. In fig. 14 we show that this
efect, (equivaent to movig vericaly in UMuq spac in fig. 15) ca quite dramaticaly
reduce fial height of rise.
Thus, in certain situations, it appear to be important to take into account fite intial
mass flux from a source, especialy when the intial denity dierence between source fluid
and and the ambient is smal.
258
4 Nonmonotonic ~ensity Variation during Mixng
In several geophysical situations of interest, fluids exbit nonmonotonic vaation of den
sity with mixing. In a plinianstyle volcac erption, the efux frm the volcano is hot
but dense, due to the paricuate suspension withi the cloud. As the ver hot erupted
material entrains and heats the ambient, the bul denity deceaes, and eventualy the
colum may become buoyant. In this ca we have a denity that evolves with mixig as
shown in fig. 15. A dierent nonmonotonic denty vaation (se fig. 16) arses In moist
convective clouds (see Squie and Tuer (1962) for a fuer descrption).
We can model the essential fluid mechanca efecs of both of these situations using
a simplified modeL. We assume that the denity of the fluid ca be descrbed by
( QO)2 (po  Pezt)
P = Put + ..  Q (.. _ 1)2 .
Thus, when Qo/Q = .., P = Put, the exremal denty, and when Qo = Q, P = po, the
input density. .\ may be considered to be a mea or the stnictur of the nonlnearty of
the fluid mixing. II the lit Qo/Q = 0, the fluid is so wel mied that it must tend to
the density of the ambient. Thus
(28)
Å2(po  Pezt)
Pø = Pezt + (Å _ 1)2 . (29)
If we now' conc~ntrate on the situation shown in fig. 15, we se that
Put = Pmin ,
and the following regimes exist:
i: Å = 0 ~Po. = Pmin;
��: O.c Å .c 1/2 ~ Pmin .c Pø .c po;
��i: Å = 1/2 ~ Pm in .c Po .c po;
iv: .. + 1 ~ Po. + 00.
We see that
g' = .. ((po  pmin)) (2ÅQo _ Qã)Prel (Å  1)2 2'
and the other two governg equations are
(30)
~Q = 2oiMl/2dz ' (4)
.¡~$
,::;F
f
and
d g' Q2dz M = M . (30)
As in the unstratified case, the system may be integrted to reduce the order. We arve
at
~Q = (40oi4 Fo ) 1/5 (.\Q2 _ Q Q _ )1/5,dz (2Å 1) 0 em , (31)
259
where
Cm = ).Q2 _ QoQ _ 4a(2)" ~~)MS/2 , (32)
and
(2)"  1)/ Fo = (2)" 1)/(g~Qo) :: 0
at al times, by the defition of ). above. Once agai Cm is a constant.
Now let us make the following nondimensionaation
Q = Qo).Q* (=* Q*(O) = ).) j Cm = Qic* .
Then
C* = Q2 _ Q _ 4a)"(2)"  1)M5/2m '5FoQ~
We now obt~n a condition to determne when w = O. ,Clearly, from intial conditions
(33)
C .. ).().  1) , (34)
Q increases initialy,and ).().  1) goes through a minimum when). = 1/2. Thus if Cm ..
1/4, M will be positive always. Also, since Q is intialy increaing, if)" :: 1/2, Q*CQ*1)
increases also, and thus M ca never be ze in thi cae either. This is to be exected,
since). :: 1/2 corresponds to fluid that remJl;ml buoyat at al times. However, if)" .. 1/2,
increasing Q actualy decreases Q*CQ* 1), and thus if Cm :: 1/4, we wiUalways have
at some height, (which may be cacuated)
C _Q*2 Q*m  ,
i.e. M = O. Thus the condition for stopping of the plUme is
). ((1).) 4a(2.\ _1)ll/2) !+ 5FoQo .. 4 . (35)
We see that this occurs for smal initial momentum flux, and large intial denity anomaly
or mass flux, al of which requie that large amounts of mig must take place before the
plume can become buoyant. The height of rise for sever d��erent vaues of ). are plotted
in fig. 17. We note that the cures go off to inty when Cm = 1/4, and that the solution
is only defied for Cm :: ).  ).2. We must remember that afer these heights, the simple
model of entrainment, with simarty of velocity and tubulence structures, break down.
The process of mixing is liely to be exremely complicated about such a height. However,
jump conditions on mass flux and radius may be applied acs such a region between two
areas of approximately plumelie behaviour. Ths is the subjec of fuher investigation, as
is the behaviour of a plume of fluid that stars off buoyat, and then becomes dene. Ths
is analogous to the behaviour in a stratifed environment, and prelnar results suggest
that in this situation, for given initial mass flux, there alo exsts a well defied minimum
height of rise with ,associated initial momentum flux.
260
5 Conclusions and Future Directions
In this report we have descrbed a generalation of the classical Morton, Taylor, Tuer
model to nonzero initial mass flux, and investigated the efect of this generalzation on
the fial height of rise in a stratified environment. The relatively large vaations pre
dicted occur if the intial mass and momentum fluxes are fite when the stratification
becomes important. This model is likely to be applicable to the behaviour of plumes that
pass through a region of vang N2. As an example volcac erption clouds can' rise to
heights of tens of kiometres. Since the Bnit Vaiala fruency approxiately doubles
at about tOla, i.e. at the tropopause, (the boundar betwee the troposphere and the
stratosphere), where the plume would be liy to have low g' /b and lage M, Q, per
turbations from a pure plume balance may have signcat efect on the fial height of
rise, according to this modeL. The more realstic complicating fators of nonmonotonic
mixing behaviour should be included, especaly the interion of nonmonotonicity with
an ambient stratification. The litations, and assumptions of the model should alo be
investigated more deeply, and to this end, exerients should be conducted to test its
predictions. '
6 Acknowledgements
This work waS suggested by, and done with Andy Woods, who brought new meanng to
the word collaboration. His unailing enthusiasm, penetratin iiight and avaabilty for
discussion (paricularly while drvig) were greatly apprecate. I would lie to than the
Steering Commt,tee, and in paricuar Rick Salon, for orgasig such a stimulating and
enjoyable su.er, Jack Whtehead and W. H. O. I. for givig us acces to their laboratory
for some prelimiar' exerients, and Bob Frel, for al hi vaed help. Finaly, I would
like to ackowledge the support of H. S. E. Shefeld, U. K. and the contributions towads
my travel expenses of D. A. M. T. P. and Church College, Univerity of Cambridge.
7 References
Fischer, H. B., List, E. J., Koh, R. C. Y., Imberger, J. & Brooks, N. H. 1979 Mizing in
Inland and C00.tal Water8. Acaemic. 483pp.
LaneSerf, G. F., Linden P. F., & Hilel, M. 1989 Force, Angled Plumes. private commu
nication.
List, E. J., 1982 Tubulent Jets and Plumes. Ann. Rev. Fluid Meek. 14, 189212.
Morton, B. R. 1959 Forced Plumes. J. Fluid Meek. S, 151163.
Morton, B. R., Taylor G. i. & Tuer, J. S. 1956 Tubulent Gravitational convection from
maintained and instantaneous source. Proc. R. Soc. Lond. A 234, 123.
Squires, P. & 'Ier, J. S. 1962 An entraing jet model for cuulonibus updraughts.
Tellw 14, 422434.
Taylor, G. i. 1945 Dynamcs of a mass of hot ga rising in ai. U. S. Atomic Energy
Commis"ion MDDC 919. LADC 276.
Tuer, J. S. 1973 Buoyancy EJjectJ in Fluid.. Cambridge Univerity Press, 367 pp.
Tuer, J. S. 1986 Tubulent entrainment: the development of the entraient assumption,
and its application to geophysical flows. J. Fluid Meek. 173,431471.
261
Q, M, (lnrj F in IJ Stratified environment b, W, and 0.5g' in \J Slrotifieû "rivironmeni3.0r 3.0
~ ,.
. 2.5N ;N 2.5a:
"
";
~N 0 ,.0 Ni. ~
" "
.0
.0Z 2,0 z 2.0
..
..
c: l:
'"
..N
~ N
II
~
IIN
1.5
.: 1.5
:ë :ë0\
ëii
0\
1ii i
Õ 1.0 Õ 1.0c0 c
ëii
0
c
.¡¡
il C
E
il
E
"õ "õi:0 0.5 g 0.5z z
0.5 0.0 0.5 1.0 1.5
Point Source Plum'e (F 0= 1, Mo=Oo=O)
riG I
2.0 1.0 2.0 3.0
Point Source Plume (Fo=l, Mo=Qo=O)
0.0
1.0
F'&. 2.
3.0 1.0
Moss Flux against Height for various cRadius against Height for various c
0.5 1.0 1.5 2.0 2.5
Nondimensionalized Radius
.3.0
0.0
0.0
~2.5 ,.
N
.i 0.8
'"
"; ";
,. ,.
..
..000 0
~ 2.0 J : c=0.999 ~
..
..
c: l:0 P : c=O
~ 0.6N
~ ~
II II
N D,S. : c=0.999 N
1.5
:ë:ë0\ C'
.v .vI i 0.4
Õ Õ
c c0 1.0 0
ëii
.¡¡
c Cil II
E E
"õ "õ
c c 0.20 0z 0.5 z
0.0
0.0 0.5 1.0 1,5 2.0
Nondimensianalized Mass Flux (initially i:
FIef. 3
FIG. L¡
262
Radius açainst Heiçht ror Distributed Source (c.999)
6
5
n
..
~ "o
..
~
o
ß
"N
~
'~
:i
õ
.~
c
Ë',;
c
aZ
1
Virtuol Or;g;n (1,42)
2
o
Inilial Rodius (2,67)
2 3 4
Nondimensionolized Radius
1.4
1.5
~ 1.6
It
..
.
..a
~ 1.7i.
.,
ö
aN
���� leB
N
..
c:
'õi
1.9'¡:
0
õ
~
2.05
2.1
Rodius 09ainst Height for Plume (cwO)
l:to
'��
:i
ie
? 4
g
.0o 3N
r
ã
ca'~
"
Ë
'6
"
aZ
Virtual Origin (1.67)
5
2
o 2 3 4
Nondimensionalized Radiui
Flc,. ~
Variation of virtual origin with c
Radius ogoinst Heig"t (or Jet (c.999)
~
Nn
~
~
ooe
:.
l:
'"
'��
:i
ã
"
a'~
"
Ë;;
"az
3
No:'dimensionolized Radius
Jets Distributed Sources
2.20.0 0.2
1/(2c) (c
0.4 0.6 0.8 1.0
defined as 1 8aMo5/2/(5 F ��02))
F J& 6
263
.. ,.
N 2,5 . 2.5NOJ OJ
~ ~
.. ,.Na Na
u. u.
.. ..
'" CDz 2,0 z 2.0
., ..
ö ö
'" '"N N
" '
II II
N
1.5
N
' 1.5
'
~
.r
.c 0'
C1 'w
:r :r
0 1.0 Õ 1.0c c0 0
.Vi ïiic c
C1 Ql
E E
'0 '0c 0,5 ca 0 0.5z z
3,0 r
Variation of 0'0 with Height
0.0
0,0 0,2 0.4 0.6 0.8
ao = (Qo2/(Qo2+25/4aN5/2FoJ/2))
t/6. l
1.0
1.0
Variation of ao with aM for a Plume Solution
0.8
,.
..N
..ia
u.N '
..
'"
k 0.6ö
..
..
'"N
+
NaC
;: 0.4
a
EL
o
ti
0.2
0.0
0,0 0.2 0.4 0.6 0.8
aM (N2M20/(N2M2o+F20))
F 1&. 9
1.C
264
3.0
Variation of aM with Height
0.0
0.0 0.2 0.4 0,6 0.8
aM = (N2M20/(N2M2o+F20))
FI &: g
1,
. 3.0
Voriation of Final Height of Rise with a M
2.9
';
Nii
..
~2.8
a
u.
..
CDZ
..
"'ö 2.7
N
'
II
N
:: 2.6
.r
0'
'w
:i
o 2.5
c
o
ëñ
c
Q)
,�� 2.4
'0
c
oz
2.3
2.2
0.0 0.2 0.4 0,6 0.8
aM N2Mo2/(N2Mo2;Fo2))
Fi��: If)
1.0
2,6
Var;atic" cf Final Heigh, of Rise with a M
~
.,'" 2,5
'
~N
o
LL
'
"'
i:
ö
~ 2,"
~
II
N
£
,,?
OJ
:i 2,,3
0
c0
ïii
c
OJ
,~
'0
c 2,20
z
OM for a Plume
2,1
0.0 0,2 0.4 0.6 0.8
OM ( N2Mo2/(N2Mo2+Fo2))
Fll". II
1.0
1.0
Height of Rise relative to Plume Solution
0.8 Locus of Plume balance
0.6
0.2 1.0
0' M
F/&: i"l
265
1.65
Variation of Final Height ai Rise witn a M
..
~N 1.64
~
..Noi.
..
coZ
.
ö
'& 1.63
II
N
£
Ol
Q;
;¡ 1.62Õ
c0
ïii
c
Q)
E
'õ
c 1.610z
0'11 for a Plume
OQ :: ~'2~
1.60
0.0 0.1 0,2 0,3 0.4
O'M (= N2Mo2/(N2Mo2+Fo2))
1=1(,. 12.
Height of rise ogainst Q for, various initial conditio:'
2.0
1.8
I: ,~= 0
1.: Mo:()'15
3: Mo ::O.~ 'l
it; M.ø :0. l
..
~N 1.6
'
..No
~ 1.4
'"z
.
c:
~ 1.2
'
II
N
' 1.0
.i
Ol
'ij
:i 0.8
Õ
c
o
.~ 0.6
Q)
E
'õ
�� 0.4
z
0.2
0.0
a 2 4 6 8
o ( = (25a4Nl0Fo6)1/80')
Fi fr. It¡
Variation of Density with Mixing. for À= 1 14î,OI ~
0,8
0,6
0.4
0,2
c:
'Ë
~ 0,0Q
I0
..
0,2
0.4
0,6
0,8
1.0
0,0
Pmin
0.2 0.4
00/0
F/6. IS
0.6 0.8 1.0
LO
Variation of Density with Mix;rig. for ).= 1 14
0,8
0.6 Po
buoyant
0.4
0,2
"
o
E
~
'0 0.0
i
o
..
0.2
0.4
0.6
0.8
1.0
0.0 0.2 0.4 0,6 0,8
��ö/O
FJ6. 16
Height of Rise in Nonlinear regime for various ÌI
1.4
1.2
1.0
N
oc
.~ 0.8
c
Ql
E
"õc
g 0.6
0.4
0.2
0.0
0.00 0.05 0.10 0,20 0.25
c
FI6. Ir
266
Transport of a Chemical
in Stellar Radiative Zones
Brian Chaboyer
'
Abstract In this report, we examine under what conditions it is appropriate to treat
the transport of a chemical in a stellar radiative zone due to a large scale velocity field as a
pure diffusion process. We obtain an expression for this diffusion coeffcient in terms of this
velocity field and the turbulent diffusivities. This diffusive transport may be slower than
the transport due to advection by the velocity field. We show that our results are consistent
with observations of Li in the surface of stars, and may explain the work of Charbonneau,
Michaud and Profftt (1989) who found that some process was inhibiting the advection of
Li by EddingtonSweet circulation.
1 Introduction
In the convectively stable regions of a star, chemicals can be transported by large spatial
scale (slow time scale) circulations and by molecular diffusion. Large scale meridional circu
lations, such as EddingtonSweet circulation, are induced in a star by thermal instabilities
(Eddington 1925, Sweet 1950) and may be the dominant cause of the transport of chemicals
within a star, as the molecular diffusion coeffcients are very smal.
In attempting to model the evolution of a rotating star, the transport of chemicals due to
the large scale motions is sometimes treated as diffusion process with a turbulent diffusion
coeffcient determined from the advection velocity, (see, for example Endal and Sofia 1978,
Pinsonneault et al. 1989, Charbonneau and Michaud 1990) even though the equation which
describes the transport of a chemical (see equation 1) includes an advection term. In this
paper, we outline under what conditions it is appropriate to treat the vertical transport of a
chemical as a simple diffusion process, and find an expression for that diffusion coeffcient.
Another motivation for this work are the results of Charbonneau, Michaud and Profftt
(1989) who examined the depletion of Li in giant stars due to the advection of chemicals by
EddingtonSweet circulation on the main sequence (MS). The effects of turbulent diffusion
were neglected. For the youngest cluster they studied, no Li depletion is observed, contrary
to what would be expected from the EddingtonSweet circulation. In order to account for
this fact, Charbonneau et al. stated: 'one must investigate mechanisms that could have
reduced the ezpected transport through meridional circulation'. An obvious candidate for
267
such a mechanism is horizontal turbulent diffusion, which hinders the meridional advection.
Such turbulent diffusion may also be able to account for observations of Li in field dwarfs
by Boesgaard and Tripicco (1986b).
In section 2 of this report, we show under what conditions it is vald to treat the transport
of a chemical as a diffusion process. Our approach is very similar to that used by G.I. Taylor
(1953) who examined the dispersion of a chemical in a pipe with a shear flow. Section 3
compares our work with that of Charbonneau et al. and uses observations of giant stars in
open clusters, and of field dwarfs to estimate the average diffusion coeffcient which appears
in our equations. Our conclusions are presented in section 4.
2 Transport of a Chemical
The transport of a chemical with concentration c by both advection and diffusion is given
by
:t(PC) + v. (pc��) = V. (pD. Vc) (1)
where p = density, D = turbulent diffusion tensor, and �� is the velocity. The microscopic
diffusivities are very smal, and so do not enter into our equations. We are assuming that
the turbulent diffusion is due to smal scale, turbulent motion which is excited by the strong
differential rotation within a star which is induced by the large scale laminar velocity field ��.
We are using a tensor for the turbulent diffusion in order to take into account the possibility
that the smal scale motions are anisotropic. We wi assume that p and D are functions
of the radius only (in spherical polar coordinates). This assumption is vald for stars (such
as the Sun) in which the rotation does not introduce a significant departure from spherical
symmetry. We shal expand take the radial velocity component in spherical functions
U,. = L Un(r)Pn(cos 9) (2)
n
where Pn( cos 9) are the Legendre polynomials of order n, with n nonzero. The classical
EddingtonSweet circulation (Eddington 1925, Sweet 1950) is described by a single P2( cos 9).
We wi assume axal symmetry, so there wil be no dependency on the longitudinal direction.
It is convenient to express the concentration as
C = co(r) + bc(r,9) (3)
where the horizontal average of bc (denoted by : bc ) ) is zero and the horizontal average
of a function f is defined as:
1 r
: f )= '210 f sin 9 d9. (4)
268
Equation (1) may be expanded asô ô ô
P ôtCO + P ôtDC + pU,. ôr Co + pil. VDC
V. (pD. V (co + DC)) (5)
where we have assumed that p does not vary in time, so that
V . (pci) = pu,. ~~ + pil. V DC. (6)
Taking the horizontal average in spherical polar coordinates of equation (5) we obtain
ô 1 ô ( 2 ) 1 ô r 2 ôcoj
p ôt Co + r2 ôr r p .( Dc U,.:; = r2 ôr ir pDv ôr (7)
where we have separated the diffusion tensor into vertical (Dv) and horizontal (DH) com
ponents.
In order to calculate the advective flux .( Dc U,. :;, we need to determine DC. Assuming
that ÔCo I I ( )ôr :;:; VDC 8
then the term pil. VDC may be ignored in equation (5). Essentialy, we are assuming that
the concentration of a chemical varies much more strongly in the radial direction than in
the horizontal. This wil be true if the horizontal diffusion coeffcient is much greater than
the vertical diffusion coeffcient (see equation (10) and discussion thereafter). Subtracting
equation (7) from (5) yields
ô ôCo
p ôtDC + pu,. ôr
.. ~ r Dv pr2 ôDcj
r2 ôr L ôr
i ô r. ÔDcj
+ r2 sin IJ ôlJ isin IJDHP ôlJ . (9)
We now assume that
.fDH :;:; Dv ir (10)
where lH (lv) is the distance over which Dc changes in the horizontal (vertical) direction,
so that the first term on the right hand side of equation (10) may be ignored. As DH and
Dv are turbulent diffusion coeffcients, equation (10) requires that the smal scale motions
are much more vigorous in the horizontal than in the vertical. This will be easily satisfied
within the radiative regions of a star, in which the vertical velocities are inhibited by the
gravitational force (Zahn 1983). In addition, we wil assume that we are in a steady state,
which is established when
! .( DH
t  r2 . (11)
269
Under the above two assumptions, and replacing Ur by expression (2), equation (9)
becomes
r r2 8col 1 8 r. 8DCI
r lDH Un(r) 8r Pn(cosO) = sin080isinO 80 .
The Legendre polynomials are solutions of the equation
(12)
 n(n + I)Pn(cos D) = si~ 0 :0 ¡sin 0 :OPn(COS 0)1 (13)
thus, the solution of equation (12) is
DC=¿ r2Un(r) 8copn(cosO)
n n(n + 1)DH 8r (14)
where we have imposed the condition': Dc ).= O.
We may now calculate the average advected flux
.: Dc Ur). _ fo'7 Dc Ur sin 0 dO
¿_ r2U~(r) 8co r IPn(cosO)12
sinO dO
n 2n(n + 1)DH 8r 10
1/2 r2U~(r) 8co
r n(n + 1)(n + 1/2) DH 8r . (15)
Thus, subject to the conditions given in equations (8), (10) and (11), the transport of a
chemical within the radiative region of a star can be described as a pure diffusion process:
8 1 8 ¡ 2 8col
PCo =  r pDt8t r2 8r 8r (16)
with
Dt = Dv + D. (17)
where
D _ 1/2 r2U~(r)
·  r n(n + 1)(n + 1/2) DH .
In particular, if we consider EddingtonSweet circulation (n = 2 only), we see that
(18)
D  D r2U;(r)t  V + 30DH . (19)
3 Comparison to Li Observations
In order to complete this work, it is necessary to obtain expressions for the turbulent dif
fusion coeffcients in the vertical and horizontal directions. Unfortunately, this is a diffcult
270
problem, for which we have not been able to obtain a solution based on the principles of
fluid dynamics. The only path left to us to estimate the diffusion coeffcients is to consider
observations of the surface abundances of stars.
Lithium is an important probe of stellar transport processes as it is destroyed when
T :; 2.6 x 106 K and this temperature is generaly located in the radiative region of a star.
Thus, a star which has an initialy homogeneous chemical content, wil arrive on the zero
age main sequence (ZAMS) with a radial Li profie which is essentialy a step function and
defines a Li front. Below some critical radius (TLi) there wil be no Li, above TLi there wil
be the initial, so caled cosmic Li abundance. This critical radius occurs in the radiative
zone of most stars. If no chemical transport processes occur within the radiative zone of a
star, then the amount of and location of Li in the star would remain constant at its ZAMS
value.
Let us examine the effects of a large scale meridional circulation. Advection by Eddington
Sweet circulation, if it occurres, distorts the Li front. At the poles (where u" is positive), the
Li front wi be moved upward. Near the equator (where u" is negative) Li wil be carried
below TLi and so wil be burned. Thus, EddingtonSweet circulation wil cause a continual
decrease in the total amount of Li present in the star. However, as the star evolves on the
MS, the radius at which Li is destroyed wil move downward (in mass fraction) (Charbon
neau, Michaud and Profftt 1989, hereafter CMP), with a velocity of ULi. Thus, only when
lu,,1 :; IULil wil Li be progressively destroyed on the MS due to advection by Eddington
Sweet circulation. The velocity of EddingtonSweet circulation depends criticaly on the
rotation rate, U"ot; U" (X U~ot. Assuming solidbody rotation, CMP (who studied stars with
1.2 :: M/Me :: 2.0) found that no MS Li depletion occurres when U"ot :: 20 km/s. When
U"ot ~ 35 km/s, essentialy the ful amount of Li depletion due to advection by Eddington
Sweet circulation wi occur. For the mass range of interest here, rotational velocities are
typicaly 100 km/s, and so the ful amount of Li depletion is expected for most stars.
If the EddingtonSweet circulation penetrates the surface convection zone (as appears
likely and was implicitly assumed by CMP), then the ful amount of Li depletion wil occur
rather quickly. From Figure 3 in CMP, we see that a 1.5 Me with U"ot = 50 km/s, wil
have no Li at its surface after just 0.286 Gyr. However, Balachandran (1990), who observed
199 F dwarfs, found numerous stars with v sin i ~ 50 km/s in which the Li abundances
were near cosmic. This is a sign that the destruction of Li on the surface of a star due
to EddingtonSweet circulation is being inhibited. CMP suggest that a boundary layer
may form between the radiative zone and the surface convection zone, which prevents the
EddingtonSweet circulation from reaching the surface, so that no Li depletion wil occur
on the MS. The formation of such a boundary layer is unlikely, as there is no physical
reason for its existence. If it did form, this would imply that no MS Li depletion would be
observed. This is contrary to many observations of MS F stars (e.g. Balachandran 1990
and Boesgaard and Tripicco 1986b) which show significant Li depletion.
After a star has exhausted the H in its core, it evolves off the MS onto the giant branch.
During the rapid postMS phase of evolution, a very deep convective zone wil develop.
271
This wil mix Li depleted matter with Li rich matter, causing a diution of the surface Li
abundance, which is easily observable. If EddingtonSweet circulation has caused Li to
be destroyed on the MS, then the observed abundances in giant stars wil be lower than
that due to dilution during the postMS evolution. CMP compared observations of Li in
giants in three clusters (M67 age ~ 5 Gyr, turnoff mass ~ 1.3 Me, NGC 752 age ~ 2.2
Gyr, turnoff mass ~ 1.6 Me and NGC 7789 age ~ 1.6 Gyr, turnoff mass ~ 1.8 Me) to
that which would would be expected from standard stellar evolution with the advection of
chemicals due to EddingtonSweet circulation (diffusion was ignored). A major diffculty
with this work is that the MS rotation velocity (and its evolution) for a particular star is
not known  there is a spread in rotation velocities for stars with the same mass. Thus,
one must examine a large enough sample of stars in order to use statistical arguments.
CMP found that Li observations of M67 and NGC 752 were consistent with theoretical
values computed including the effects of advection. However, in NGC 7789, there appeared
to be no Li depletion in giant stars as would be expected from advection by Eddington
Sweet circulation. Furthermore, observations by Pilachowski (1986) found normal (comic)
Li abundances in stars near the turn off in NGC 7789 .
We may take this as a sign that horizontal diffusion is inhibiting the effects of the
advection. This wil occur when the time scale of advection is less than the time scale of
the diffusion, i.e. we are in a steady state, which requires that equation (11) be satisfied.
The time scale of diffusion is given by tdi = L2 / nt, where L is the distance over which Li
must travel in order to reach the surface, whie the time scale of advection by Eddington
Sweet circulation is tadvec = L/UES, where UES is the EddingtonSweet velocity. Hence, we
require that nt i LUES, which, from equation (19) implies that the horizontal diffusion
coeffcient must satisfy
n 1 UEsR2 1 R2H :; 30 L  30 tadvec' (20)
By assuming that Li is being diffused in the radiative zones of F stars, we are able to
estimate a time scale for the diffusion for stars with 1.6 ~ M / Me ~ 1.8. Clearly, the effects
of diffusion must show up somewhere between 1.6 Gyr and 2.2 Gyr. For our rough estimate
of the total diffusion coeffcient, we shal take the time scale of diffusion to be 2 Gyr. Note
that by postulating that diffusion is the principal means of transport of Li on the MS, we
require that the depletion of Li be observable on the MS in stars which are older than 2
Gyr. Further evidence that the time scale for diffusion is about 2 Gyr may be found in
Boesgaard and Tripicco (1986b) who observed Li in 75 field dwarfs. They found that 62%
of stars younger than 2 Gyr had the cosmic abundance of Li, whie 64% of stars older than 2
Gyr were Li depleted, with log(c/eo) ~ 2. This is in agreement with Pilachowski's (1986)
observations of no Li depletion at the MS turn off in NGC 7789. In addition, Hobbs and
Pilachowski (1986) found that stars near the turnoff in NGC 752 (age ~ 1.7 Gyr) had not
suffered any depletion in their surface Li abundance.
In order to ilustrate the expected behaviour from diffusion, we wil solve the simplified
272
C dC
= 0 , for al time
dx
C = 0, for al time
r= r
Li
r=R
*
Figure 1: Initial conditions on the Li concentration
equation (c.f. equation (16)) via
..
r
8 828tC = D 8r2c (21)
where we have ignored the effects of sphericity and of a variable diffusion coeffcient. We
have the initial condition that the Li concentration is c = Co when rLi ~ r ~ R.. The
boundary conditions are that Li gets destroyed at r = rLi, so that c = 0 when r : rLi and
that there is no loss of Li at the surface of the star, so 8cj8r = 0 at r = R.. A ilustration
of these conditions is shown in Figure 1. Although rLi is a function of time, we wil assume
it to be a constant. This is a fairly reasonably assumption, as for a 1.5 Mø star, L wil
change by less than 3% in 500 Myr.
The solution of equation (21) may be found by fourier decomposing c:
c = Co r an(t) sin ((; + mr) (~._r~~J J
which leads to 8an 7(2 D ( )2
8 =  (R )2 2n + 1 an't 4 .rLi
The solution for cis
(22)
(23)
4 ,,1 r 7(2 2 Dt J' ((7( ) ( r  rLi )J
c = Co 7( ~ (2n + 1) exp l4(2n + 1) (R. _ rLi)2 sm "2 + n7( R.  rLi . (24)
Observers determine the abundance of a chemical on the surface of a star on a decimal
logarithmic scale. After a short time, the abundance at the surface wil be given by the
slowest decaying mode, n = 0 and so the surface abundance wil be given by
log(cjco) ~ log ¡: exp (  :2 Zt) J
273
(25)
o0.5
..
10
i
~U

b.
.3
1.5
2.5
o 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
age (Gyr)
Figure 2: Expected Li depletion due to the transport by turbulent diffusion
where L  R.  TLi. When:i = 2, log( 4e:i /,rr) ~ 0.76, which is the level at which
observers consider Li depleted. Thus, we may associate the time scale of diffusion (2 Gyr)
with
7r2 D2 = "4 L2tD' (26)
From the Yale evolution code (Prather 1976, Seidel, Demarque, and Weinberg 1987), a
1.5 Mø star with the Hyades metalcity wil have L ~ 4 X 108 m and so we may estimate
D for a 1.5 Mø star as
D ~ 6.5 X 107 m2 jyr rv 2 m2 js. (27)
A plot of the surface Li abundance (as given by equation (21), with the diffusion coeffcient
given above) as a function of time is shown in figure 2.
It addition, it is also possible to obtain a lower limit to the value of DH from the fact
that the time scale of diffusion must be greater than the time scale of advection. From
equations (20) and figure 3 in CMP we see that for a 1.5 Mø star with Urot = 50 kmjs
DH ~ 2 m2js. (28)
274
4 Conclusion
In this paper, we have shown under what conditions a pure diffusion equation (as opposed
to an equation which included an advection term) may be used to model the transport
of chemicals due to large spatial scale, slow time scale motions, such as those induced
by thermal instabilties in the radiative zone of a rotating star. Thus, we have given
a theoretical justification for the work of such people as Pinsonneault et al. (1989) and
Charbonneau and Michaud (1990) who use a pure diffusion equatio~ to model the transport
of chemicals by EddingtonSweet circulation in a stellar evolution code. However, we are
unable to obtain an expression for the turbulent diffusivities which occur in our diffusion
equation.
In addition, we have shown that including the effects of turbulent diffusion may explain
the results of CMP, who found that Li was transported on a slower time scale than would
be expected from EddingtonSweet circulation. As horizontal diffusion inhibits the effect of
vertical advection, we can explain this result by assuming that the transport of Li was due
to turbulent diffusion, as given by equation (16). This alows us to an averaged diffusion
coeffcient (c.f. equation (21)) to be D = 2 m2/s for early F stars. This requires that
the horizontaly turbulent diffusion coeffcient be greater than 2 m2 Is. These value are
consistent with the observations of Boesgaard and Tripicco (1986b) who found Li depletion
occurred in dwarf stars older than 2 Gyr, and observations by Pilachowski (1986) who found
no Li depletion in MS turn off stars in NGC 7789 (age.~ 1.6 Gyr). The Li gap present
in the Hyades cluster (Boesgaard and Tripicco 1986a), which occurres for stars slightly
less massive than those considered here, clearly shows that the diffusion coeffcient is very
sensitive to stellar parameters.
Acknowledgements
I would like to thank my supervisor JeanPaul Zahn who made this project possible.
He conceived this project and guided me thrQughout the summer. He helped to make this
an interesting and enjoyable summer. I would also like to thank Andy Woods and Rick
Salon with whom I had numerous enlghtening discussions. Finaly, I would like to thank
al the fellows and staff of GFD who made this a very enjoyable summer.
References
Balachandran, S. 1990, Ap. J., 354, 310.
Boesgaard, A.M. and Tnpicco, M.J. 1986a, Ap. J. (Letters), 302, L49.
Boesgaard, A.M. and Tripicco, M.J. 1986b, Ap. J., 303, 724.
Charbonneau, P. and Michaud, G. 1990, Ap. J., 352, 681.
275
Charbonneau, P., Michaud, G. and Profftt, C. R. 1989, Ap. J., 347, 821.
Eddington, A.S. 1925, Observatory, 48, 78.
Endal, A.S. and Sofia, S. 1978, Ap.J., 220, 279.
Hobbs, L.M and Pilachowski, C. 1986, Ap.J. (Letters), 309, L17.
Landau, L.D. and Lifshitz, E.M., 1987, Fluid Mechanics, 2nd English Edition (Oxford:
Pergamon Press).
Pilachowski, C. 1986, Ap.J., 300, 289.
Pinsonneault, M.H., Kawaler, S.D., Sofia, S. and Demarque, P. 1989, Ap.J., 338, 424.
Prather, M. 1976, Ph.D. thesis, Yale University.
Seidel, E., Demarque, P. and Weinberg, D. 1987, Ap.J. Supp., 63, 917.
Sweet, P.A. 1950, M.N.R.A.S., 110, 548.
Taylor, G.I. 1953, Proc. Roy. Soc. London, A, 219, 186.
Zahn, J.P. 1983, Astrophys. Processes in Upper Main Sequence Stars (Gen��eve: PubL.
Obseratoire Gen��ve), 253.
276
MAGNETIC FLUX TUBES AND CONVECTION
Richard Kerswell
August, 1990
~
Abstract
Threedimensional cellular convection concentrates magnetic flux into ropes and
sheets when the magnetic Reynolds number is large. We examine the equilibrium
axsymmetric flux ropes sustained by cellular convection. The case of a compressible,
electricaly conducting fluid is studied and boundary layer analysis is found to yield
a selfconsistent solution for an externally driven convection field. Both kinematic
and dynamic regimes may be examined and a scalng for the maxmum value of the
amplified field may be deduced.
1 Introduction
Magnetoconvection is a complex process common in the astrophysical context. Here
a particular aspect is explored, motivated by observations of the solar surface. On the
sun, photospheric granules seem to concentrate the magnetic flux into regions with
intense local magnetic field. These flux tubes tend to form at granular boundaries
where local downflows exist. This expulsion of magnetic field lines from convective
eddies is a well known consequence of large magnetic Reynolds number flow (Parker
1963, Weiss 1966 & 1967, Clark & Johnson 1967, Busse 1975). When the total flux
is smal, concentration is limited only by magnetic diffusion. As the field strength
ens, further effects become dynamicaly important and both induction equation and
the equation of motion must be solved together. Galoway, Proctor & Weiss 1978 (1J
construct a model of this process in the case of an incompressible fluid and via bound
ary layer analysis are able to examine the transition from kinematic to the dynamic
regime. Here the sole opposition produced by the field buildup in the rope is the
Lorentz force. In an idealsed example driven by horizontal temperature gradients,
they observe a maxmum field strength in the rope proportional to the square root
of the ratio of the viscous to magnetic diffusivity. This study endeavours to extend
their model in order to capture more of the "physics" of the actual process. The
Sun is a complex mixture of partially or fully ionised gases and hence as such rep
resents a highly compressible fluid. In addition it is clear that the energy equation,
and hence the temperature field, requires a more careful treatment. Although the
precise structure of sunspots is unknown, it is clear that they consist of one or many
flux tubes erupting through the photosphere together. The observed cooling within
a sunspot is suggestive of a more generic cooling inside flux tubes. This is confirmed
by compressible magnetoconvection simulations conducted for example by Weiss and
coworkers.
As a consequence of including these aspects of the observed process, additional
277
forces play a role in the dynamic regime. The magnetic pressure, produced by the
field lines, causes a localsed drop in the fluid pressure. As a result the density tends
to decrease and the fluid experiences an increased buoyancymagnetic buoyancy. This
is found to reduce the magnetic field in the rope. In contrast and as a consequence of
downwellngs in the tube, the temperature is found to decrease in the presence of the
field and as a result amplifies the flux concentration. The crucial coupling between
magnetic and temperature fields occurs via a magneticfielddependent thermal dif
fusivity. This produces cooling within the rope and forces the presence of a thermal
boundary layer enclosing the flux tube The net effect is to produce an enriched self
consistency relation (cf ¡1 D for the axial velocity as well as a backreaction equation for
the axal temperature field. The model assumes scalings which ensure that the mod
ifications to temperature, pressure and density in the tube are al small. While this
undoubtedly oversimplifies the structure ( particularly in pressure and density ) it is
a reasonable starting point for assessing the effects of compressibilty in an analytical
modeL. That such perturbations can be dynamically important is a consequence of the
axis being a singularity in the equations of motion. Normally the regular solution is
chosen, however the presence of a boundary layer around the axs allows the possibility
of matching onto the singular solution. It wil be shown later that this can alow small
perturbations in the tube to produce 0(1) effects.
It should be noted that in pursuit of this selfconsistent model, the main assumption
made is that the tube is only slightly evacuated by the magnetic field. The model
suggests that an equilbrated state is reached long before the alfv��n speed within the
tube reaches the sound speed. This is contrary to the findings of numerical simulations
and observations of the above speeds on the solar surface. However so little is known
about the magnetic field strengths below the photosphere that such a scenario may
indeed exist deeper within the sun. On a similar note, the Boussinesq approximation
is used when it is only sensibly applicable in the solar interior. The flux tubes are
envisaged as extending at least 3 photospheric depths into the convection zone.
2 The Sun
The basic convective unit on the solar surface, a granule, has a length scale of ~ 103 km.
It's aspect ratio is unknown but commonly supposed to be 0(1). The photosphere
has a depth of 500km and hence granular convection can be taken as entering the
convection zone which makes up the outer third of the sun and extends ~ 2 x 105km
radially. Supergranules are coherent structures commonly comprising of ~ 30 gran
ules. Their scale is 3 x 104km and are thought to exist mainly in the convective zone.
Sunspots have dimensions of supergranules and are thought to extend at least 104km
deep. On the solar surface the temperature is 6000J( giving an acoustic speed of
~ 12kms1. In the flux tubes, field magnitudes are 1000  2000G producing surface
alfv��n speeds of order ~ 5kms1 and an Alfv��n Mach number ~ 0.5. Typical fluid
velocities are observed as ~ 2kms1 and the acceleration due to gravity at the sur
face is 274ms2. For calculation of dimensionless parameters the following scales are
adopted: L ~ 2000km , T ~ 6000J( & v ~ 2kms1. This gives estimates for the
278
Peclet and Magnetic Reynolds numbers as follows:
Rm = Lv = 2 X 109 T3/2 Lv ~ 106 (1)
17
Taking K. ~ 105 Pe = Lv ~ 104 (2)
K.
Hence 1 ~ Pe ~ Rm The thermal and magnetic diffusivities are shown below
(Priest 1982 p 312 ¡2J) al figures are subject to debate (main source Priest 1982).
3 Problem Formulation
3.1 Exterior Flow and Scalings
A cylindrical convection cell of aspect ratio 1 is considered and a steady flow sought in
which the magnetic and velocity fields are assumed purely meridional and azimuthal
dependence is ignored:
q= (qr(r,z),O,qz(r,z))
The magnetic Reynolds number Rm is taken as large and as a result the field is
assumed concentrated in a centi:al rope of radius R~1/2.( The magnetic field is also
flung outwards to the cell perimeter forming a true boundary layer. However in terms
of magnetic field intensity this layer is dominated by the central rope and may be
ignored.) The region outside the rope is labelled the exterior and is considered free
of flux. Implicit in the formation of the flux tube is the presence of convection eddies
sweeping the field into the axs. These are envisioned as toroidal with downflow at the
centre. The external boundary conditions are assumed such as to produce this flow
and no attempt is made to solve the exterior problem. Convection is assumed highly
turbulent and this is modelled using a turbulent viscosity. Inertial terms are neglected
in preference to the viscous terms. The Energy equation is considered in its simplified
form of the steady temperature equation. The effcient turbulent mixing present in the
exterior alows temperature advection to be replaced approximately by an adhanced
thermal diffusivity Le. ��.VT = K.m V2T becomes 0 ~ (K.m + K.e)V2T where K.m is the
molecular thermal diffusivity and Ke ~ Km represents an eddy diffusivity. The fluid is
taken as a perfect gas whose state of ionisation remains constant.
The relevant equations then become for the exterior:
o  V P + pff + PVT(V2Ü + 1/3VV. ��)
P = p~T
o V.(p��)
o V2T
(3)
( 4)
(5)
(6)
The exterior flow is scaled by balancing pressure and viscosity:
pc2 vc
L '" PVT L2
giving an estimate for the velocity
c2L
vc '" 
VT
279
10
10 SOLAR OIFFSI\'mES
\ E,R,Pncsl Il)~~ )
~ ,i
m~ Thcnn31 Oiifusiv;iy
Z
10
\\
o
10
~lagntic
Dirfuiv;iy
2Mm Dc h Mm
20Mm
CONV CE w1ih fLUX ROPE II 'TERnJ LA VB
L
ILLlP
280
The magnetic Reynolds number is then Rm = Lvc/r¡. If the initial constant magnetic
field permeating the cell is Bo, conservation of flux requires the field in the rope to
have size O(RmBo). The alfv��n speed is then Va = Rm~. Thereforeý¡ip
Va L Bo
Vc r¡ Vi
3.2 The Magnetic Flux Tube
Assuming the exterior flow field is given, the Induction equation is now solved exactly.
V x (�� x B) = r¡V2jj
Define X such that
'V x (0, x(r,z),O) = B = (_XZ ,0, Xr) r r
N ondimensionalizing
��.Vx = RmD2X
with D2 a 1 a a2= r ar ( : ar ) + a z2
as V2( ~Ø) = �� x (V X (~Ø)
r r
(7)
(8)
(9)
Introduce the boundary layer variable ~ = R;(2r
_ a 1 ax
u.'Vx ~ ~a~(~ a~)
Assume an asymptotic form for ��
��'" (~rg(z),O,f(Z))
Then 1 X~
2g~Xe + fXz ~ ~(T)e
It is helpful to adopt the VonMises coordinates of incompressible boundary layer
theory. Here the temperature and magnetic field equations are linear for the given
velocity field and the utilty of these coordinates is to remove the variable coeffcients.
Let ø = 1/2ç2F(z) be a modified streamfunction and then choosing gF + f~~ = 0
leads to
x;: = 2(FI J)ØX,p,p
This has solution
x = Xo(l e1/2eh(Z))
with cP = 1/2e F(z)
F(z) e r g/fdz
h(z) F(z)2 ¡z FI fdz
(10)
(11)
(12)
(13)
Thus the magnetic field has Gaussian structure.
281
3.3 Temperature
Intense magnetic field within the tube can be expected to reduce turbulence and in
these circumstances a diffusiondominated temperature field is unrealistic. We are
forced to incorporate the effects of advection near the flux tube due to the highly
anisotropic velocity field there. However rather than return to the exact temperature
equation with only molecular thermal diffusivity, we imagine that the diffusivity is stil
enhanced by isotropic turbulent mixing. This allows us to speculate that the thermal
diffusivity depends in some inverse way on the field strength which provides more
intense coupling between the fields. Due to the strongly unidirectional field in the
rope we should expect the thermal diffusivity to be anisotropic, however for simplicity
this is ignored and K. is written as K.(B).
��."VT = \7.(K.(B)"VT)
The exterior temperature field is unaware of the flux tube and may be expanded
asymptotically for r + 0 as follows:
Text = To(z) + r2Ti(z) + . . .
and the interior perturbation field Tint(~' z), which does feel the field, is superimposed
on this. Defining the Peclet number as Pe = Lva/ K.oo, the above equation becomes,
after dropping small terms:
Pe(f(z) dTo + fez) 8Tintdz 8z
+1/2~g(z) 8~tJ
(d2To T) 8/' I (dTo 8Tint)K. dz2 + 4 i + 8z E dz +
1 8 ( 8Tint )
+Rm ~ 8~ ~/'a (14)
Where /' is now nondimensionalzed by its value at infinity. The exterior temperature
field does not see a spatialy dependent thermal diffusivity and must therefore satisfy
the reduced equation:
dTo d2To
Pe fez) dz = dz2 + 4Ti
It is the interior perturbation field which adapts to the variable diffusivity.
r( dTo 8TintPei 1  K.)f(z) dz + f(z)~+
+1/2~g(z) 8~tJ
8K. I (dTo 8Tint) R ~ i.(C 8Tint)
8z E dz + 8z + m ~ 8~ ."K. 8~
(15)
Here the ordering of parameters 1 ~ Pe ~ Rm, adopted above, is motivated by
solar observations and the equation then naturally describes two regimes. Within the
interior, K. is different from 1 and hence the dominant balance is between first and last
terms. In contrast, outside the flux tube K. = 1, and the second term replaces the first
in importance. To allow progress, we postulate a simple structure for /'(B).
or rather
1  K.
1  K.
aBß
a(Xõ)ß ei/2ßeh(z)
(16)
(17)
282
with XO = Xô/ Rm and Q dimensionless although proportional to Bg. This choice
conveniently alows immediate integration.
( *h)ß( Pe )fdTo ~(1 _ 1/2ße2h(z)) = c åTintQ Xo Rm dz hß .. K, ~
using the boundary condition ~ = 0 ~ a~t' = O. Notice the LHS is +ve, therefore
Tint increases as it leaves the tubeas desired. Also note that as ~ + 00
T.' '" ~ Pe *ßhß1 jdTo 1 cmt ß Rm Xo dz n..
Such logarithmic behaviour is impossible to match directly with the exterior where
the perturbation field must decay to zero. An intermediary layer is necessary and
naturally appears from within the equations:
P f( )åTint ~ R ~~(cåTint)e z åz m ~ å~ .. å~
Outside the rope K, = 1 a constant and (::) ~ 1. Resealng 11
defining y = r~1/2 dq/ j(q) gives = (E£ )1/2~ andRm
åTint = .!~(ii åTint)
åy 11 åii åii
( Recall j(z) ( 0 and so z E ¡I, OJ maps onto y E ¡+oo, 001). A solution is needed
of this equation which can match onto a logarithmic singularity as 11 + 0 and which
decays to zero as 11 + 00. Explicitly a solution T(ii, y) is required such that
T(ii,y) '" A(y)lnii as 11 + 0
T(ii, y) + 0 as 11 + 00
(18)
(19)
The inner condition represents a distribution of sinks along the y axs. Hence the
thermal layer is the adjustment region in which fluid flowing downwards accommodates
the inner cool tube. The wellknown fundamental solution
T(ii,y) = .!e1i2/4Y
y
represents the effect of a i5fn source of heat placed at 11 = O. Hence the required
solution is a convolution over the source distribution with the fundamental solution:
T = _~ jY A(w) e1i2/4(yw)dw2 _= y  w
Notice the lower limit is w = 00 which corresponds to z = 0 Le. the top of the cell.
For convergence we assume A( w) + 0 as w + 00 which is clear physicaly and can
be seen to hold later.
We proceed to match these layers to leading order. Cal T inside the thermal
boundary layer Tint to distinquish from that within the flux rope Tint. For leading
order matching
lim Tint = lim Tint
e= 110
283
=? A(y) = ~ Pe Xõßhß1 jdToß RTf dz
1a P, e *ßjY hß1(w)dTo 1i2/4(yw)d
Xo e w2 ß Rm 00 y  w dw
(20)
=? fpartint (21)
The superscript part (particular integral) is used to indicate a solution forced to exist
by the dynamics as opposed to cj (complementary function) which exists as a con
sequence of boundary conditions. In the case of the temperature field, there exist the
complementary solutions F(z) and it is this degree of freedom which accommodates
partboundary layer matching. Above we have deduced the form Tint , however full match
ing must take into account the complementary functions generated. In the exterior
the perturbative temperature is zero and hence Tint = Tl:irt via matching at v + 00.
Turning attention to the v + 0 matching we have in the Interior:
~~(ç âTint)ç âç K. âç
Tpartint
(1 _ K.)jdTo Pedz Rm
a Pe *ßhß1jdTo 1 c
'" Xo  n",ß Rm dz
F(z)
(22)
as ç + 00 (23)
(24)Tefint
In the Middle layer:
!~(v âTint) = âTintV âv âv ây
Asymptoticaly as v + 0 , Tint can be written most generally as
Tint'" a(y)lnv + ß(y) + ��(y)v2Inv + 6(y)v2 +...
In terms of ç = (:.: )1/2v
Pe /
Tint'" a(y)ln(Rm)12Ç+ß(y)+...
'" a(y) lnç + a(y) In( :: )1/2 + ß(y) + ...
The first 2 terms dominate and must match onto Tint such that
( ) _ a Pe *ßhß1jdToa y  Xo ß Rm dz
As a consequence Tt!I = a:(y)ln(J::)1/2 and a temperature change is induced in the
inner boundary layer of size In( :.: )1/2 larger than the original temperature perturba
tion. Explicitly in the rope Tln~rt = o(Tt!I). The complete temperature field can be
written
Tint'" a(y)¡n( :: )1/2 + In ç)
as ç + 00 with the first term dominating. The selfconsistent relation for the tem
perature field can now be derived. Writing 1'( z) as the axal temperature, the back
reaction is Tt:! = l'  To i.e.
(25)
(26)
1/2 
T rT a Pe i Pe *ßhß1j( )dT .Lo =  n  Xo z ß Rm Rm dz
284
Notice the backreaction is negative indicating tube cooling and that the order of the
backreaction is O(a:': In(:.:)i/2), where O(a) = D."'/"'oo.
3.4 The Equations of Motion
We now turn attention to the equations of motion. Ignoring the inertial terms reduces
the problem to a linear one and alows decomposition of the velocity field into its bask
and perturbative components. We nondimensionalze P with Pooc2, B2 / JLP with v~,
external velocity field Uo by Vc and interior solution ui by Va.2  2Lc \7*(P* + (va?B* )VVa C 2 9 L L ( * ':) LVa B* t"*B* t"*2 (VcVT  i  P z +  . v + v uo + UiVa V V VaV
+1/3\7*(VcVT\7*.~ + V'*.uii (27)
Vall
Here * represents a nondimensional quantity and is hereafter dropped. The dimen
sionless parameter Q = !l(!!) is introduced,
v Vc
QRm( ~)2\7(P + ta)2B2)Va C 2 gL ': QRm?:pz + QRmB.V'BVa
t"2(VcVT  i / 't(VcVTt"  't i( )v Uo + Ui + 1 3 v  v . Uo + v. Ui 28VaV VaV
This represents only two equations due to the absence of angular dependence and
swirl components in u and E. The radial component is as follows:
QRm( ~)2i.(P +Va âT
Va 2 B2 
(~) 2) ~ QRmB.\7Br +
': tt"2(VCVT   J 1/3't(vcVT't  't  J1
T. v Uo + Ui + v v.Uo+ v.Uivav va (29)
The Alfv��n speed is that in the tube and hence Bz=O(l). As a consequence of
V'.E = 0, Br = 0(R;,i/2) and then the scale of terms in the above equation reads
O(QR~,(2) + O(QR~(2) + O(Rm) ~ 0
Clearly the first term must vanish
i.(P + (Va)2 B2 J = 0â~ c 2
(Va )2 B2=: P ~, z) = Pext ( T, z)  (  c 2
P _ P. _ ~(Va)2 *2h2 ç2h(z)
 ext 2 C Xo e
we use the ideal gas equation of state P = pT to produce
p. 1 e2 h(z)=:p= ext __(Va)2(X~h)2 e
Text + Tint 2 c Text + Tint
285
Using the earlier assumption that Tint ~ Text and neglecting second order changes in
the density alows the two distinct density changing processes to be separated:
P. T _ç2 h(z)ext
( int) 1/2(va)2( "'h)2eP ~ Pext      XoText Text C Text
The second term represents the buoyancy due to the temperatureyerturbation and the
third term is the magnetic buoyancy. The vorticity equation ( 9.\lx the momentum
eq.) reads as follows
gL ap a(X, !D2X) 1 2 VcVT
o ~ QRm2"(a )  QRm a( ) + D (rwo + rWilVa r r, z r VaV
Substitution of the density expression into this gives
1 2 ( VcVT 1
D rwo + rWi
r vav ( 1/2Q R gL _ apext + Rm Pext aTintm V~ ar Text ~1/2 ) a(, 1 D2 )
_( Va )2X~2 h3 Rm ~eç2h(z) + Q Rm X, ~ X (30)c Text a(r, z)
This divides naturally as follows:
Exterior : 
1 D2( VcVT 1
 rwo
r vav = QRm gL(apext)v~ ar
= QR gL (R~2 Pext aTintm V~ Text a~
_( Va )2X~2h3 R~2 ~eç2h(Z))C Text
a(X 1 D2X)
+QR ' j:m a( r, Z)
(31)
Interior Correction:  !D2(rwi1
r
(32)
Upon simplification the interior correction equation reads
a 1 a
a~(~ a~(~Wl)) QR~(2x~2(2h2h' + gc~ rh3 l~eeh(z)ext
+QR gL Pext aTintm V~ Text a~ (33)
which can be integrated twice and with the boundary condition WI = 0 at ~ = 0 yields
~Wl = iQR~(2x~2(A(Z)e + 1  eeh(z)l(h' + ~~ 2~xt)
gL Pext ¡Ç
+QRm 2r (Tint((, z)d(va ext 0 (34)
286
3.5 Vorticity Matching
The perturbation vorticity can be divided into two parts which are clearly distinct in
structure and generating process. The first arises from Lorentz and magnetic buoy
ancy forces and as a consequence sees only the flux tube. In contrast the vorticity due
to perturbative cooling sees both magnetic and therma.llayers. To clarify notation w
is used to indicate vorticity in the flux rope, w in the thermal layer a.nd w loosely to
describe vorticity in the exterior. (Notice the subscript i is dropped as only perturba
tion vorticity is considered.) Consider the Lorentz and magnetic buoyancy generated
vorticity.
rw = !Qx~2(A(z)e + i _ e~2h(z))(h' + g; ~)2 c 2Text
This generates a velocity field u~art via
aupart aupart aupart
 (r Z) Zrw=r  ~ra z ar ar
The complementary velocity field corresponds to some arbitary function of z to be
later specified by matching requirements.
u~f = F(z)
Outside the flux rope, the vorticity experiences no forcing and hence satisfies the
homogeneous equation
D2(rw)=0
Near the axs r ~ 0, this vorticity can be expanded generaly by
rw '" a(z)+ß(z)r21nr+'l(z)r2+... (35)
'" a(z)+ß(z)(tInR;//2+Re Inç)+'l(z)Re +... (36)Rm m m
We match vorticity fields at the flux rope boundary to leading order
lim rw = lim rw
~oo rO
which produces
a(z)
I RI/2
ß(z) n R:
!Q ..2(h' gL ~)
2 Xo + c2 2Text
!QA( ) ..2(h' gL ~)
2 z Xo + c2 2Text
(37)
(38)
All the expansion functions a, ß,. . . can be taken as being the same order in Rm as
they are implicitly coupled in the expansion. If a = O(Q) then this implies A(z) =
O( ln~:1/2). The A(z)Ç2 term is therefore sub dominant to 1 within the rope and the
vorticity can be taken as radialy independent to leading order.
_ 1 ..2' gL h
rw", 2QXo (h + 22T. )C ext as r~O
287
This drives a motion just outside the flux tube of
part 1Q *2(h' gL h )1Ûz ~  xo + i n r2 c 2Text
The complementary function û;f = F(z) is subdominant to the driven flow as r + 0
and hence can be neglected during the axial velocity matching. Continuity demands
lim (u~art + u~f) = 21 QXô2(h' + g;2T.h )(lnr = In~ In R;,1/2)e..oo c ext
When the rope vorticity is expanded asymptotically as ~ + 00 only integral powers
of Rm can arise. Certainly no In R;,1/2 terms are available and so the driven velocity
field can only accommodate the In ~ term. Hence we must have
1. cf = cf = ~Q *2(h' + 9L ~) I R1/2m Uz Uz Xo 2 2T. n me..oo 2 C ext
This represents the backreaction at the axis due to Lorentz and magnetic buoyancy
forces. The complementary solution has an order O(ln R;;1/2) larger than the driven
flow. This boundary layer magnification mechanism is crucial to the existence of the
selfconsistent model presented here. Such magnification allows backreaction forcing
to be treated as perturbative, but ensures that the resulting flows in the rope are 0(1)
and hence capable of equilbrating the system.
For the temperatureforced vorticity two layers must be fitted to each other and
to the exterior. In the Inner flux layer, the vorticity is forced by a functional of the
temperature along the axs.
T 9 L Rm Pext ¡v  .rw = Q v~ Pe T(z) 10 (Tint((, z)d(
a *ß Pe lY(Z) hß1(w) dT _(2/4(yw)d
Xo  e w2ß Rm 00 y  w dw
Vorticity matching at the inner edge of the boundary layer is automaticaly accom
plished by ensuring correct temperature matching treated earlier. Matching axal
velocities as /I ~ 0 trivialy produces
T Q9L Pext ie T (( )d(rw = iT. (int, Z
va ext 0
At the edge of the flux tube, in the limit as ~ + 00
~ Pe Xôßhß1 fez) dT (In( Pe )1/2+ ln~j
ß Rm dz Rm
Q9L Pext(z) a Pe *ßhß1f( )dT(~C2(1 (Pe )1/2 I C)j
v~ T( z) ß Rm Xo z dz 2" n Rm + n..
Q9L Pext(z) a Pe *ßhß1f( )dT(_~C2(1 (Pe )1/2 1 C)j
v~ T( z) ß Rm Xo z dz 4" n Rm + n..
i'T ( z)
In the Middle thermal layer
Tint ""
rwT ""
upart ""z
cf
""Uz
and Tint( (, z)
cf _ cfUz  Uz
288
(39)
(40)
(41)
(42)
(43)
( 44)
We look for boundary layer magnification at the external barrier of the thermal bound
ary layer.
As v  00
rwT " _~~QgL Pext X.ß f'XJ (d( ry(z) hß1(w) dT e(2/4(yw)dw2 ß v~ T(zr 0 10 100 Y  w dw
" ~QgL Pext .ß ¡z _hß1()dT dß v~ T(z)Xo 10 z dž z
(45)
~ rwT ( 46)
In the exterior, as before, the unforced vorticity has the general asymptotic expansion,
as r  0
rwT " a(z) + ß(z)r21n r + ��(z)r2 + ...
Matching rwT and rwT in the respective limits reveals that
( ) = _~QgL Pext .ß ¡z hß1()dTda z ß v~ T(z)XO 10 z dž z
The resulting axal velocity has asymptotic form û~art " a( z) In r and dominates the
complementary field û;f for r  0 . Matching to this leads to
u~art " a(z)lnv
u;f = a( z) In Pe1/2 as v  00
(47)
(48)
As a result the backreaction flow at the axis due to the temperature cooling is:
cf _ cf _ _~QgL I P 1/2 Pext .ß rz hß1()dTdUz  Uz  ß V~ n e T( z) Xo 10 z dž z
This velocity is negative and thus the temperature effect is amplifying on the convective
flow. Within the dynamic regime the axial velocity backreaction is O( 1) and if this is
not to disrupt the asymptotic form of the flow field within the tube, the backreaction
must possess the same limiting form i.e.
Ul = (~rgi(z), 0, !1(Z))
2
This requires the condition
~ 8p ~ 1
p8~
within the tube via the continuity equation \7 .(p��) = 0, and motivates our restriction
that density changes in the tube are small.
3.6 SelfConsistency Equations
The selfconsistency relations are
(J  fo)vc = (_~QgL i P 1/2 Pext .ß ¡z hß1()dTdß V~ n e T(z)Xo 10 z dž z
1 .2 1/2 i gL h
+2QXo In Rm (h + 2T. ))vaC ext
= ~ Pe I Pe 1/2 .ßhß1!C )dT
ß Rm n Rm Xo z dz
(49)
TTo (50)
289
Recal
e f g/fdzh( z) = 2 f F / I dz
And from continuity
g(z) = (pJ)'
P
h). 0 =? jZ p(z)dz ( 0
Then we solve
r_~QgL 1 P 1/2 Pext *ß rz hß1()dTdl ß V~ n e T( z) xo 10 z dz z
1Q *21 R1/2(h' gL h ))Va
+ Xo n m + 2 2 c 2Text Vc
~ Pe 1 Pe 1/2 *ß hß1 I( ) dT,
ß Rm n Rm Xo z dz
over z E ¡1,O) with boundary conditions h(1) = 0 and h(O) ( 00.
2 f~ Pext(Z)dz (h _ ho) =
Pext( z)
(51)
T  To = (52)
3.7 Scalings
We now list al the scalings implicitly taken in the model
. For density changes in the tube to be smallv '( ~)2 ~ 1
c
. Velocity backreactions to be O( 1)
1. Temperature
O(a Va In Pel/2Qg~) :: 1Vc va
2. Magnetic buoyancy
0(VaQlnRl/29l):: 1Vc m c2
3. Lorentz
0(:aQlnR~2):: 1
c
and for the driving effects to be smal
1. O(aQ,*) ~ 1Va
2. O(Q) ~ 1
3. O(Q~) ~ 1
. With regards Temperature back reaction
O( a In( Pe )1/2 Pe ) :: 1Rm Rm
290
. The exterior flow should be O( 1)
o ( Q Rm 9 L .! Va) ~ 1
v2 vT Va c
. The form taken for the thermal diffusivity 1  K. = oBß imposes the restriction
0=0(1)
A typical set of parameters which allow all effects to be felt by the system except
the temperature backreaction is
9L "" 1 (53)
c2
""
Q
1 Vc
(54)~
1 R1/2 Vn m a
( Va )2
In Pe1/2 (55)"" 0""
ln R~r(2c
v o ln Pe1/2 (56)
vT
~
Rm
Vc
o(ln Pe1/2) (57)
Va
Within this model, as ��0 = 0(1), the temperature backreaction is always negligible.
4 An Illustrative Example
In order to analyse the SelfConsistency equations ( eqns 51 & 52 ), the functional
forms of the leading fields Pext, To & fo are required. Idealy we would like to consider
asymptotic forms of realzable exterior flows, however this is another problem in itself
and wil not be attempted. Rather, we take the simplest forms possible and just
demand that they satisfy the appropriate limiting exterior equations.
On :0+ n2 v p  pz v U
o = V' .(p��)
o V'2T
(58)
(59)
(60)
These lead to the restriction that
d d2 fo
dz (ToPexi) + Pext = dz2
The simplest doublezeroed function is fo = z( z + 1) ( the axal velocity must vanish
at top & bottom of the cell ) and an associated linear famiy of fields.
1To = C  '2z
(2  lA)z
Pext = A + Jand
291
In what follows we have takenA=6 and C=l so that both To and Pext increase with
depth.
1To = 1  z
2
Pext 6  z
(61)
(62)&
We concede that, with the model as it stands, the temperature backreaction is negli
gible and so T(z) ~ To(z) is taken. Equation 51 becomes:
1!Z(12zz22 h _
6z 6z
&
to h( 1)
dh hd
+
dz 2
z(z + 1))
= ~abß ¡z hß1(q)dq + eb21  lz (dh + ~)2 10 6  z dz 2  z
= 0
o at z = 0
(63)
(64)
(65)
subject
Here
L .ß
a = a In Pe1/2'!LXo
v v; ß
e = ! In R1/2X.2 RmL2 m 0 cVJ.P
d = gL
c2
b = Bo
This parameterization is chosen to isolate the initial flux Bo. By varying this and only
this parameter, we can examine the equilibrium state reached in both kinematic and
dynamic regions. The value of d is largely unimportant in the solution and so is set
to 1. The solution for h can then be written as
h = h(a1/ßb ~)
, a2/ß
and hence the natural dimensionless parameter to adjust is Ji ß' and the plot to
examine is a1/ßbh vs a1/ßb. However in keeping with the original paper by Galloway,
Proctor & Weiss, we plot bh vs b which represents the magnetic field at the axs ( at
some z valuewe take z=1/2 ), divided by Rm verses the initial uniform field. We
consider various sizes of the control parameter JIß which alows the relative effects
of Temperature and Lorentz backreactions to be studied, smal values implying smal
Lorentz forces.
4.1 Results
The first 3 plots show the effect of varying ß on the solution. All have the same
qualtative form: a linear kinematic growth of magnetic field followed by a maximum
as the dynamic regime is reached. The effect of the Temperature term governed by
a is to amplify the flux tube compression and hence increase the axs field. The 4th
292
plot shows the field on the axs as a function of z. Note that the field is strongest near
the top of the cell, modellng the field entering the Photosphere and decreases to zero
at the base. That the maxmum is slightly displaced into the cell is a consequence
of the magnetic buoyancy. Plot 5 reiterates the similarity between various powers
of ß ). 1. The last two plots show the effect of gradually letting the temperature
backreaction dominate. We expected to find qualitatively the same results as before
but with perhaps increased maxima, and indeed this is what is observed for e ~ 1 ( ,
The bottom curve, e = 6, is that in plot 2 with a = 1 ). However for e : 1 there
are runaway solutions i.e. the solution does not equilbrate and the model appears to
break down. In this case no solution exists for an initial field exceeding some critical
finite value.
4.2 Scaling Results
From the plots, we see that a maximum magnetic field strength is achieved for a finite
value of the initial field. Typically
B = O(RmBo)
vRmB=O(LmQ)
The scalng of Q depends on the dominant backreaction. If the Lorentz force provides
the balance
=? Q '" 1/
I Rl/2v n m
1/1'/P )
In R~2
which agrees with Galoway, Proctor & Weiss 1978 (1). If the temperature backreac
tion dominates then
(Rm
=? Baxis = 0 L
BT, = 0 (Rm Vaaxis L c 1/1'/P c2 )a In Pe1/2 gL
Note that
Brxis = 0 (va In R~2 c2
Baxis c a In Pe1/2 gL
and on the sun this seems 0(1). Using the first estimate of order with the following
parameters:
Baxis = 2000Gauss = 0.2Tesla
J. = 1.257 X 106Hm1
1/ = 109/T3/2m2s1
L = 106m
T = 104J(
293
p = 3 x iO4kgm3
gives
4R2
~ Re = m
l081n Rm
Taking Rm ~ 106 implies Re ~ 5000.
5 Comments
. We feel that the energy equation requires a stil more careful treatment. Mod
ellng Eddy diffusivities involves entropy gradients as opposed to the rather sim
plistic approach adopted here of using only temperature gradients. Introducing
entropy wil complicate the system by coupling the energy equation and equation
of state. The effect is to be investigated.
. The stabilty of the flux rope equilibrium state is an obvious point of interest and
one which we would like to study. However a lack of an exterior convective flow
solution to base the analysis upon severely limits the utilty of any such effort.
Moreover, interaction with this flow would appear to be an essential point of this
stabilty problem.
. The model, as it stands, can accommodate only smal perturbations of pressure
within the flux tube. We would like to extend this to finite amplitude modifica
tions as seems more appropriate in the solar context.
. Plot 7 appears to show that the model merely breaks down for suffciently smal e.
However one can't help speculate whether this might indicate a critical bound
on the initial flux for such a convective state to exist, which might provide a
selection mechanism for the size of flux tubes. It is not out of the question that
such a breakdown mechanism could be associated with fibrillation of flux.
294
PLOT 1 : Effect of reducing the Temperatue Backraction d=.1 ,e=6 beta=3/2
0.05
0.01
Il: I
0.045
0.04
N 0.035


I
~ 0.03
en
.
lo
~ 0.025
S
~
e 0.02
e=
= 0.015
,,~~ ~~~~~~~~.. .. .... .

'.
..
.

'.
 
...... 1.=0.'
0.,.0,0 I
0.005
0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
BO the initial field
PLOT2: Effect of reducing the Temperae Backracon d=1,e=6 beta=2
0.04
0.035
0.01
/,/"""" .",."......."...,.,"',.",.".,."""" """ ",.,
Q='
0.03
s:

ii 0.025
N
en
.=
U 0.02
oS
~
~ 0.015
CQ
ct::. 0.1
((",o.m
0.005
o
o 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
BO the iniual field
295
0.035
0.03
0.025
~

i
II
N
en
.a
~
S
~ 0.015
e
~
=
0.01
0.005
0.18
0.16
0.14
0.12
0.1
N

.c
0.08
0.06
0.04
0.02
0
1
PLOT3: Effect ofreducing the Temperatur Backfeaction d=1,e=6 beta=3
Q== I
Q:O.I
4:0.01
o
o 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50.05
BO the initial field
PLOT4: A Typical solution for field along the axis
0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
z a= 1 ,d= 1 ,e=6, beta=2,80=2
296
I.~ o.ij~
,. I
~;¡ I
:i I
= O'lr
!
I
O,05~
oV
�� 0,2 0,4 0.6 0,8
ilO5~
O,()~
S
ii
.. ,
~
O.o ~
'~
,.
.æ
;¡
;; 0.0 ~
=
i
i
!
0,01 ~
i.~
¡
1.~
Ii
~
11
~
i
,. O'SIu
oS
;¡
:i '6~
=
0.4
"f
01
��
PLOTS' v.ious oeii "=I.~=I.==b
l=¡ i
J
I
..,.....'.~i,.,.'" " .." ".,. '.." " ¡S'j
.,' ',"""""" " ~..,
lI:4
I
I
i
i
!
0,05 U,I U,15 O,~ O,~5 0,3 0,35 O,.l 0,45 0,5
80 the iniiial field
(U5. PLOT6: Increasing the Effeci of Temper:ire i=l.d=l.~eia=~
HHH"i J
i
I
1
j
i
i'..", I
'.'.".'...., i
.....,........... ie=%.
.. .. I¡e.:!
~ '! 6
O.2~
1.2 l..l 1.6 u
80 The Iniiial Field
PL01i: Increasing the Effec of Tempemue .=I.J=I.~eia=Z
i
i: e.C.!
I
, ~ =o¡
000__00...0.0...0...0. " '..,.,. "ë~1.
0,2 o,~ 0,6 O,S 1.6 Ul 2L2 l..l
80 The Initial Field
297
6 Acknowledgements
I am indebted to Steve Childress for suggesting and supervising this work. His obvious
interest in science and physical insight have been inspirational and I leave here with
his words ringing in my ears: "Now Rich, just think about the physics and the rest
will take care of itself...." Hey, thanks, Steve. J must also thank Nigel Weiss, Mike
Proctor and particularly Bob Rosner for invaluable advice and encouragement. Thanks
also to Ed Spiegel, Phil Morrison and Andrew Soward for sharing their thoughts and
opinions and to Andy Woods for always being ready to argue the physics and drive
his car very fast. Of course, this summer has been much more than just this project
and for this I must thank al the fellows and faculty. Rick Salmon deserves a special
mention for making the summer run so smoothly and Ed Spiegel and JeanPaul Zahn
for introducing a bunch of applied mathematicians to astrophysics.
6.1 References
1. Galloway, D.J., Proctor, M.R.E. & Weiss, N.O. 1978 Magnetic flux ropes
and convection J. Fluid Mech. 87 243261
2. Priest, E.R. 1982 Solar Magnetohydrodynamics
3. Clark, A. & Johnson, A.C. 1967 Magnetic field accumulation in supergran
ules Solar Phys.2 433440
4. Parker, E.N. 1963 Kinematic hydrodynamic theory and its application to
the low solar photosphere Astrophys. J. 138 552575
5. Busse, F.R. 1975 Nonlinear interaction of magnetic field and convection
J.Fluid Mech.71 193206
6. Weiss, N.O. 1966 The expulsion of magnetic flux by eddies. Proc.Roy.Soc.
A 293 310328
7. Weiss, N .0. 1967 Magnetic fields and convection. In Problems of Stellar
Convection (ed E.A.Spiegel & J .P.Zahn), pp176187 Springer.
298
NATURY DRI DISPERION IN mTE POROUS LAYE
Stefan J. Unz
FR.l1. Theoretlsche Physik
Universltät des Saa/andes
D6600 Saaricken
Geany
ABSTRCT:
We consider convective motion in a tilted porous layer which is drven by a
vertical temJ?erature grdient and an additional temperature gradient between
the boundaries of the layer. We find an exact stationary flow solution
in that system. We discuss Taylor dispersion of passive paicles in this flow
and emphasize its geophysical implications. The existence of a zero wave
number instabilty in this system is shown. We discuss the stabilty of
the no flow solution in the system without the additional temperature
gradient between the boundares. For the vertical layer we derive nonlinear
evolution equations determining the behavior above the convecive threshold.
i INODUCTION
Convection in layers of fluids enclosed between impermeable boundaes
seems to be one of the most fascinating aspects in fluid dynamics (cf. the book
Platten and Legros (1984) for a recent review). The pardigm of these systems,
the RayleighBenard system (a horionta fluid layer subjec to a vertical tem
perature gradient) and its geophysical modification ( porous layer filed with
flUid) have be discussed in the past in grat detal (see e.g. Chandrasekhar
(1961) and for the porous medium case Beck(1972)). However little attention
has been given to the situation in which the layer is inclined . In the case of
porous media Caltagirone and Bories (1985) and Riley and Winters (1990) dis
cussed fixed temperature bounda conditions while Sen et al (1988) studied
fixed heat flux boundary conditions acting on the tilted layer. This report
forms a generalization and extension of their work.
Our work is motivated by the following geophysical situation: Suppose
there is an, inclined sandstone layer between layers of shale. The grvita:
tional field acts vertically downwards while a geothermic temperature gradient
acts upwards. If the thermal difusivity in shale and the porous medium filled
with fluid would be the same the isotherms would be horizontal. However.
299
in practice the thermal diffusivities can difer. Continuity of the normal
component of the heat flux requirs a declination of the isotherms at the
boundaries between shale and sandstone which is analogous to æi additional
temperature gradient between the boundaries. To model this situation we
study convection in a tilted porous layer which is subject to two drving
forces: (j a lateral temperature grdient along the boundaries and (jj a
temperature difference between the boundaes.
Our report is organized as follows: In secion 2 we descrbe our model
system. Section 3 contains the basic hydrodynamic equations in nondimensio
nalized form. In Section 4 we present the most simple solutions of the system
and in particular we discuss the structure of the non trivial flow solution.
A study of the "mean" motion of injeced passive particles in this flow (Taylor
dispersion) and its geophysical signifcance wil be given in Section S. In
Section 6 we find an infinite wave number instabilty of the flow solution. Up
to that point the full system With both grdients Is discussed. Section 7
contains some linear and nonlinear results if no additional forcing between
the boundaries is present. In Secion 8 we discuss the flow solution in the
geophysical problem mentioned above. Secion 9 summarzes our findings.
2 THE SYSTEM
We consider a porous layer of thckness d enclosed between two parallel
boundaries. Gravitational acceleration acts vertcally. The porous layer is tilted
about the horizontal axis With tilting angle cpo The layer is subjec to a vertIcal
temperature gradient ToG which vares linearly with z. To is the reference
temperature at the origin of the frame of coordinates (cf. Fig.1 ). In addition
there is fixed temperature difference AT between the boundaries.
" ~~
, i,')
L \ ILV
~o
i
l
~', ~ 'l
¡v\
'l
:lo\l~
~o
L i
L 1,)
\. ,lLv
"" ( 0
x
Fig.! The geometry.
300
We assume that the lateral extension is very large in comparson to its
thickness. By the way one should not be worred about the fact that for large
enough negative z the temperature may become negative. This is only an
arifact of our modeL. In practice this problem does not arse since the layer
is long but not infinite. We also require mass conservation in the layer.
In principle there are three parameters which can be vared independently:
The gradient along the boundaes, the temperature difference between the
boundaries, and the tilting angle.
3 BASIC EQUATIONS AN BOUNARY CONDITIONS
,
Following D.O. Joseph (1972) we use the generaized Darcy equation to de
scribe the evolution of the seepage velocity u of the fluid inside of the porous
layer given by
(polEl ÒtU = Vp + PI  (iiK)u . (1)
where LI is the viscosity of the fluid, E the porosity ( i.e. the ratio of void
volume to the total volume of the porous medium), p the pressure field,
g= gez the downwards acting gravity acceleration, K the penneabilty of the
porous medium, P the mass density field of the fluid and Po=p(x=O,z=OL
Note that Darcy's equation is linear in u and is valid only if u is small.
The evolution equation for the temperaure field reads
(Òt + u.V)T = x V2T, (2)
where x is the thennal diffusivity in the porous medium Uoseph 1972). We
assume the density varies linearly with temperature
P = Po L 1  ex (T  To) J (3)
where ex is the thermal expansion coefficient, To the temperature at x=O and
z=O. and Po = p(T=To). We also assume incompressibilty
V.u = 0 . (4)
We have used the OberbeckBoussinesq approximatlon,that all thennal
and transport quantities are assumed to be constant and density varations
enter only via bouyancy. We nondimensionallze as follows: length with d,
time with d2/x, velocity with x/d, temperature with x\l/exgdK and p/Po with
301
x2/d2 to arve at the foJlowing set of nondimensionalized equations
" "
(��t + EO) a = EV P E(p/Po) g ez (5)
V.a=O (6)
(�� t + a. V ) T = V2 T (7)
"pI Po = 1  (01 g) ( T  To)' (8)
Here p corresponds to the nondimensionalized ratio (p/Po), o=\ld2IxK is the
Darcy,Prandtl number which differs by d2/K from the bulk Prandtl number,
and g = gd31x2 is the nondimensionalized grvity acceleration. ez denotes
the unit vecor in z directon. We impose are fixed temperatures, i.e.
T=To(t+Gz) at the lower and T=To(t+Gz)+åT at the apper boundaries. ToG
denotes the temperature gradient along the boundaries. There is no flow
through the boundaes; a consequence of this and the constrnt of incom
pre,ssiblity is that there is no mean flow through the cell. Since the friction
in the Darcy equation is proportional to u we have no boundary condition on
the a component along the boundaes.
For convenience in the present stady we use a tilted coordinate system
(Ç,,,) (compare Fig,H. In that system z=Çsincp+'Tcoscp and ez=sincpeç+coscpe",
Thus we can write the bounda conditions as
TeÇ, ,,=0) = To (t + G ç sin ip) (9)
T(Ç,,,=!) = To ( 1 + GÇsinip) + ToGcosip + åT (10)
w(,,=O) = w(,,=!) = 0 . (11)
Writing V = (��ç,��"L. a=a(Ç,,,) =(u,w), and T=T(C;ii) the governing equations
of our system in the tilted frame read
(Òt + Eo)a = EV~  E(g oCT  To)Jcsinipeç + cosiperi) (12)(��t + U, V) T = "12 T (13)
V.a = 0 . (14)
302
These equations are the staring point of the following work. Let us note
several special cases which have been discussed already by other authors:
1 )No tilting, cp=O: Our system reduces to the RayleighBenard problem in a
porous medium with an effective temperature diference AT eff =  (AT + ToG)
between the boundaries. This case was already studied by Lapwood(948),
2) No lateral grdient, I.e. G=O, but AT:; 0 and cp arbitra: This corresponds
to a tilted RayleighBenard system which has been discussed just recently by
Caltagirone and Bories (1985) and Riley and Winters(t990).
3)Vertical layer without lateral grdient, I.e. G=O. In that case GUl (1969)
proved that the basic flow solution cannot be destabilzed by any AT.
To our knowledge the system we consider has not been studied before.
The DarcyPradtl number: If the porous medium consists of closely packed
spherelike objects with a mean diameter do. e.g. sand. then Kozeny's fonnula
(Joseph 1972) gives an empirical relation between porosity E and penneabUlty K
K = E 3 ( 150 (1  E)2 J  i d ~ . OS)
Typical values of E for porous media ar E=O.35 implying that K is about
6.7 104 d~ .Thus the DarcyPrandtl number is given by
o = 1500 0bulk (d/ do)2 . (6)
The bulk Prandtl number of the fluid. 0balk' is typically of the order 1. Since
we use the seepage concept for the structure of the velocity field eq., do
must be very small in comparson to d, for sand do is about 102 cm. The
layer widths d we focus on are typically of the order i cm or much bigger.
Thus 0:; 1.5 106 0bul k and the limit �� 0+ 00 seems to be a reasonable appro
ximation. Beyond this we note that for fixed do the thicker the layer the
larger the DarcyPrndtl number.
4 BASIC SOLUTIONS
We now derive simple stationary solutions of the equations (12) and (13).
4.1 No Flow Solution
The purely conductive solution has a stationar horizontally unifonn
temperature field and no fluid motion, i.e.
uml = ��tUml = Òt Tmi = 0 (17)
303
everyhere in the layer. Here the index ml refers to the motionless state. The
temperature field equation reduces to
2 2(dç + ��ii)Tml = o. (18)
Using the boundar conditions we easily find the temperature profie in the
layer
Tml(Ç;ri) = To + ToG ç sinlf + (ToG COSIf + ~T)ri . (19 )
Inserting this profie into the velocity field eq. again a condition
(To Gcos if + Ll T) sin if = ToGcosa: sina: . (20)
Hence the condu.ctive solution can only exist if either (a) there is no tilting,
i.e. 1f=0 or If=it, one reovers the Lapwood (1948) result, or (b) there is no
boundary forcing, i.e. AT=O and for fixed z the temperature variations at the
plates are the same. Thus as long as there is only a lateral temperature
gradient at the boundaries the conductive solution is the "trivial" solution
of our system.
4..2 Stationar Undiona Flow Soluton
Let us next try to find a stationar flow solution of the field equations
(12) and (13) supposing that there is only a flow solution in ç direcion. i. e.
ÒtUs = ÒtTs = 0 and Us = us(e,.,) = (us,OL (21)
Incompessibility implies that Us can depend on ri only. Thus we have to solve
��lluS = sina: ��iiTs  cosa: ��çTs
2 2
us��çTs = (��ç + ��ii )Ts '
(22)
( 23)
where the index s refers to the stationar flow solution. To find this solution
we write the temperature field as
T s( e,ii) = T ml (e,ii) + f(ri) (24)
where f( ri) is the" disturbance" of the temperature field caused by the nonzero
velocity Us ('1). Insertion of this ansatz into eqs. (21) and (22) leads immediately
to the following relations between the velocity field and f(ri):
30g
Us = (ToGstnep)1 ��~f (25)
��"us = tlT stnep + stnep ��" f. (26)
Both can be combined to an inhomogeneous thrd order equations for feii)
which takes the form
3
��" f  a��"f = b, (27)
where a=ToGsin2ep and b= ToG tlT sin2ep. The general solution of (27) is
feii) = f1exp(ri) + f2explyT)) + (b/ahi + cIa.
fi. f2, and c are determined by the thre bounda conditions
, 1
fhi=O) = fhi=U = f d" ��~ f = 0
o
co'rresponding to no temperature fluctuations at the boundaries and no lateral
mean flow. After some algebra one finds
fhi) = (t/2MT (exp(y/2)exp(y/2) J1 (exp( y(,,1/2) )exp( y(,,l/2))J
+ åT (T) 1/2) (28)
where the abbreviation y2 = ToGsin2ep was introduced. Note that depending on
the sign of G, y is real or imaginar. Combining this solution with (25) and (26)
one obtains the desired flow solution as
use,,) = (1/2) åT sinep H(y,,,) (29)
Ts(Ç,,,) = To + ToG(Çsinep + "cosep) + U/2)åT(1 + H(y,,,)) (30)
where H(y,T)) = sinh( y(T)1/2))/ sinh( y/2) if G ~ 0
H(y,,,) = sin(lyf(,,~1/2))/sin(lyll2) if G ~ O.
Let us note the following propertes of H(y,T))
1) H(y~O,,,) = 2hi 1/2)
2) H(y,,,=O) = 1
3) H(y,,,=1) = 1
4) H(y,,,=1/2) = 0
5) antisymmetry of H about midplane: H(y,1,,) =  H(y,,,) .
305
Since the basic flow is stationar Us and T s do not depend on the Darcy
Prandtl number. In the limit å T70 we recover the no flow sol ution (17) and (19L.
Thus the flow solution grows continuously from the no flow solution
when åT is increased from zero and grows linearly with åT. In the limit Y70
(G~O) we recover the solution of Caltagirone and Bories (1985L.
An interesting property of our solution is that varation across the layer
is detennined by the sign of G, the flow diection in the upper/lower half
of the layer is detennined by the sign of åT: There is upflow In the part
bordering on the hotter boundary and downfow in the other par.
In Fig.2 we sketch the velocity profies for different G and åT. Positive
G lead to a suppression of flow near to the midplane in comparison to the
case G=O. As G increases the flow becomes more concentrated near to the
boundaes £bundar layer flow). In contrat if y is negative the flow
becomes enhanced near to the middle of the layer. For negative G the flow
solution shows anomalies whenever 1"(1 Is an even' multiple of 1t. Then Us
diverges inside of the layer. Ths seems to indicate that our flow solution is
unphysical near to these points and that the flow solution is unstable below
the first divergence. We attribute this phenomenon to a breakdown of the
validity of the Darcy's equation which holds only for small velocity fields.
ib::O
lUr.,,) OJ~~ k (Ai)
i
t4 0
~ AT~O
\
,  ,
A 0 .. 0 .l 0
;
6~o 6:0 i (; '0
'. "\
4:1') 0 Al ') 0 .41:) 0
I l 
I
6~O
"
~
¡,
~
r
Fig.2 H (y, '!) sgn(å T) as function of TJ for several combinations of G and å T.
306
5 TAYLOR DISPERION IN TITE LAYE
In a seminal paper G.I: Taylor (1953) discussed the the motion of passive
particles injected in flows. In paricular he considered the long time behavior
of dye in a Poiseutle flow and showed that due to the combined action
of molecular diffusion and shear a paricle distribution initially
concentrated at a fixed position diffuses about the center of mass as a
Gaussian and is advected with the mean flow.
Following the approach of G.I. Taylor. we now derive the dispersion associa
ted with this flow. Note however the differences from Taylor's case:
our flow is generated by the bounda conditions on the temperature field
ditions on the temperature field and there is no mean flow.
The diffusion equations for the tracer paricles in our flow solution reads
ÒtC + ushi) ��cc = 09'2 c , (31)
where 0 is the molecular difusivity (for reasons of nondimensionalization
scaled with )(). Let us assume that varations of c in time occur slower than
variations of c caused by diffusion and advecion. as wil be the case for enough
downstream from the initial state. Then
ushi)��cc = D��~c . ( 32)
Splitting c into a mean concentration c (0 which is nothing but the average
orc over the layer width and a fluctuating part c'(C;ii), i.e.
o(C;rl) = c (0 + C'(C,li) (33)
and noting that one can neglect C varations of the fluctuations c' if these
are small in compason to the C varations of mean concentration. one arrves
at
��~ c' = D1 us(Yl) ��Cë. (34)
With these approximations, in the spirt of G.!. Taylor, we have transformed
the fully nonlinear adveciondiffusion problem to a linear inhomogeneous
differential equation for the concentration fluctuations which can be solved
analytically. Inserting of our flow profie and using that
��~ H(Y.TJ) = y2 H(Y.TJ) (35)
yields
307
��~ (c'  A(Ç)y2 H(y,ri) J = 0 (36)
where ACO=CåT12D)��çc sinlt is not ri dependent. This differential equation
can be readily solved to give
c' = ACÇ)y2 (HCy,ri)  (��"/HCy,ri)) ri + constJ
'I ri=O (37)
where we have imposed impermeable boundary conditions Cno transport of
tracer partcles through the boundaries): ��r¡c'=O at ri=O and ri=1.
The constant in c' is in general undetermined and will drop out in the fol lowing
calculation.
The mean concentration flux in ç diection is given byi i i
Jc = f cUs dri = ~COus dri + f c' Us dri. (38)o 0 0
Since our flow has no mean flow, the first term on the r.h.s. vanishes.Inser
ting c' into the second term on the r.h.s. yields
Jc = U/40) åT2 y2 sin21p F(y)��ç~
where FCy) is given by
, FCy) = f dri (H2Cy,ri)  (��riHCy;ii) ~=O HCy,ri)ri J
and has to be calculated for both types of the flow profie.
The mean concentration obeys a diffusion law in the long time limit. since
2ÒtdO =  ��çJc = DT ��ç cCO . (39)
From (39) we can read off immediately the value of the Taylor dispersion
coefficient which looks fonnally like a diffusion constant, but is in fact
caused by advection of the tracer particles. One finds
0T =  B(Ip,åT) y2 FCy) (40)
where
BCIp,åT) = C1I40) åT2 sin21p
and FCy), = (2YSinh2Cy/2)J1 (2y + ycoshy  3sinhy) if G=O
FC y) = (21yl sin2 (Iyl/2) J 1 (21yl + Iylcos Iyl  3sinlyl) if G oc O.
308
l
~F
II 1.49
C) 1.2Ii
\DV 1.0
~P 0.8
:"
i 0.6p
0.4
0.2
0.0
0.0 0.4 0.8
G=O
1.2 1.6
~
Fig.3 Variation of DT/DT(G=O,Ip=¡¡/2) with tilting angle Ip.the curves are
for y=3,y=0. and y=31.
10.3
'ô
II
102
..
'l
~
.. 101
p
10��
, 101
a 100 200
.3oe
XL
Fig.4 Varation of DT/DT( G=O) as function of y2"'G. Here lj=¡¡13.
309
Thus the Taylor coefficient consists of a part determined by tilting and ~ T,
i.e. B(ip,~T) and a part which describes changes of DT due to the temperature
variation at the boundaries. One reads off immediately that tilting and tem
perature gradient across the layer are necessar for a nonvanishing of DT.
Beyond this DT is invariant with respect to .6 T ~ .6 T, depends only
quadratically on ~T and is symmetric about ip=ir/2.In the limit y~O (G~O)
both branches approach the value
DT = (1/1200) ~ T2 sin2 ip. (41)
Here the maximum value of DT is reached for the vertical layer.
As in Taylor's case DT is proportional to the inverse of the molecular diffu
sivity. 'Let us now study the ip and G dependence of DT. In Fig.3 we present the
the variation of DT normalized by DT(O) which represents the value of DT with
G=O and :p=ir/2. Of course DT/DT( 0) drops to zero quadratically with ip if
ip ~O. For nonzero G the reduced Taylor coefficient is similar as shown in
Fig.3. The main difference is that negative G enhances its magnitude whereas
positive G lowers its magnitude. This behavior can be understood phYSically
by'remembering that the r.m.s. flow is bigger (smaller) in comparison to
G=O if G,O (G;,OL The bigger the mean flow in a layer half the more effective
is dispersion. This behavior can also be seen from Fig.4 where we present
DT/DT(O) as function of y2. For positive y2 the Taylor dispersion
coefficient decreases to zero in the limit y~oo, for negative y2 it grows
rapidly and diverges at y2=4'l2. This divergence is caused by the singular
behavior of the flow field for that y value.As we shall see in the next section
'this y value is the lower bound of stabilty of our flow solution (29).
Geophysicai implications: To estimate the order of magnitude of the
!.aylor dispersion coefficient we return to dimensional quantities
D=D~ and DT= DT~. For G = 0, the Taylor dispersion coefficient with dim
ensions is
!~ 
~.
¡,f.
'h
OT = (1/120))(2 0 1 åT2 sin2 ip (42)
Since in general ÄT is of the order 1 or smaller the Taylor dispersion coefficient
is typically
ÕT '" O( ~ (x/Õ ) ) (43)
Here x/D is an inverse Lewis number being typically bigger than one.
We deduce from (43) that the Taylor dispersion coefficient is of the order
of the thermal diffusivity of the transporting_fluid. This seems to be the
main result of our theory. Since in general 0 is much smaller than the
310
thennal diffusivity x (typical values D=105cm2/sec, x=103cm2/sec ) it
follows that the Taylor dispersion coefficient is much bigger than the
molecular diffusivity. Our theory can also be applied in the following ways:
1) The molecular diffusivity of a given passive trcer can be detennined by
experiment using the above theory.
2) Given the molecular diffusivity one can estimate the amount of material
which would be transported by Taylor dispersion through the layer. Assuming
that at time t=O all particles are concentrated at ç=O the solution of the
"diffusion" equation (39) is a Gaussian with respect to ç of the fonn
c(Ç,t) ~ (4'1DT t)1/2 exp(ç2/4DTt)
Here we have already used that DTlD D. _
The width of the Gaussian is given by 1=2( DTt)1/2. Thus assuming a åT of
order 1 , )(=103 cm2/sec, and D=105cm2/sec one can estimate the spread
of the particles due to disp.ersion: Supposing lp='l/2 after lh 1=3.3cm,
after ld 1=16.6cm, and after lOa I=lkm.
The final question is the nondimensionalized value of the temperature differ
ence åTdim between the boundares corresponding to åT. Since we scaled
temperature by x\J/exgdK the value of åTdim depends on fluid properties and
layer width. For water (x ~ \l ~ 103cm2/sec, ex~3 1041/grd) in sandstone
( K~2, 105 cm2) one finds that åTdim= U/6dJåT (grdcm). Thus the thicker
the layer the smaller the external temperature diference necessary to obtain
a fixed åT.
A final comment to our theory: We have assumed that the porous medium
is homogeneous and isotropic and ignored changes of D due the structure
of the porous medium.
6 BOUN FOR THE STABIUTY OF TH FLOW SOLUTION
If the layer is horizontal, lp = O. the results of Lapwood (1948) can be reco
vered: There is an stationary instabilty at Cs( k)=  (b. T +ToG)= ( 'i
2+ k2)2/k2
where the critical wave numbed which minimalizes the absol ute value of Cs)
is kc='I and the critical temperature gradient there is Ccrit=4'l2. In that case
there is no instabilty present if k=O. These results are concerned with the
stability of the conductive solution.
The situation is different if the layer is tilted. Then the linearzed equa
tions for disturbances u  (us ,0) and 6 = T  T s of the basic flow solution read
"
(dt +eo)\7':iy = eo( sinlpd1i9  COSlpaçe)
\ (44)
(dt  \72)6 =  (aii T s ��ç'l  ��çT s ��l'T'l + us��lç9)' (45)
311
Here we have introduced the stream function 'l which is related to the velocity
field by 11 (us,O) = (��'Y'l,��ç'l) if the flow is twodimensional and incom
pressible. The general solution of the above stability problem is difficult,
in particular since via T s and Us these equations are explicitely 'Y depen
dent and for arbitrary tilting angles ep there 1s a crosscoupling of
derivatives. However we can find an upper bound for the instability of the
basic flow sol ution analytically . To do this we consider the k = 0 limit of
the above equations at the stabilty threshold Cth = ToGth which are given
by
'
2
(Òt + ��O)Ò'Y'l =  Eosinep ��'Y8
2
(Òt  ��'Y ) e = Cth sinep ��ii'l.
(46)
(47)
Exact eigenfunctions of these equations are
'l('Y.t) = a(t + bet) sinÀ'Y + eft) cosÀ'Y (48)
e('Y,t) = c(t) + d(tl sinÀ'Y + f(t) cosÀ'Y (49)
where the boundar conditions 'l = e = 0 at 'Y = 0 and 1 imply that À = 2 'I n, e = a.
and f=c. Here n is positive and integer. Note that À='ln with odd n do not
fullfil the boundary conditions. This is significantly different from the
nontilted case. The time dependence of a, b. c. and d at Cth is exp( At) with
Re(A)= 0 and in general Im(A)= w.
Inserting these expressions into the k=O, equations leads to a characteristic
equation
w2 + (4'12n2+oeHw + 4'12n2oe + oeCthsin2ep = O. (50)
Equating imaginary and real pars implies that w=O. Thus the instabilty is
stationary and there is no oscilatory instabilty for k=O. From the real parts
one finds that stabilty thresholds are located at
Cth =  4'12n2 sin2 ep (50
for n= 1,2.... . The stabilty threshold with the smallest absolute value is the
one with n = 1. Thus a lower bound for the the instabilty of Us is given by
Cs(k=O) = 4'12 sin2ep. (52)
Whenever Co: Cs(k=O) our flow solution is unstable at least against infinite
312
wave length perturbations. Note that the stability threshold is independent
of the applied 11 T. This is caused by the fact that all terms containing
I1T enter via ç derivatives which drop out if k=O instabilities are discussed.
Thus the lateral gradient is the only destabilizing force. The zero wave
number instability exists whenever the tilting angle is nonzero.
From (52) it follows that in the limit If + 0 this instability vanishes.
Its minimal value is reached if the layer is verticaL. Gil (1969) has shown
that if G=O the basic flow solution cannot be destabilzed by any 11 T
if the layer is verticaL. We have shown with (52),that an additional lateral
gradient can destabilze this solution. We present Cs(k=O) reduced by its
maximal value as function of the tilting angle If in Fig.5.
cl w= 5
::
~
'õ 10
"JJ
~
~
u 15
20
o
o
//
baste s.ht~
¡tmdo.bUe ~i'isl , ,
I, l:: 0 ,pii ku\:v.:,,( s /. ,~. /, )
0.5 1 1.5 2 2.5 3
Ti1tig ai (pIm")
Fig.5 Variation of Cs(k=0)/4n:2 with 'P
Up to now we have not discussed the arbitrary wave number case. A search
for exact analytical eigenfunctions would be difficult. Thus numerical
investigations based on a Galerkin method have to be performed. This is
planned in the future.
7. TH LAYE WITH AT = 0
We discuss in this section the stabilty of the the no flow solution with
I1T=O and develop nonlinear evolution equations. We suppose that the Darcy
Prandtl number is infinite.
313
7.1 Stabtlty of the conductve state for a vertca layer
Here we want to calculate how the stabilty threshold bifurcates out of
the k= 0 instability. Thus we have to study the stationary case only. For the
vertical layer ( ip=ii/2) the equations linearized about the conductive state
read after insertion of a lateral Fourier ansatz '" exp( ikQ at threshold
(��2k2)1f=�� e11 11
(��~  k2) e =  C s ��;¡ 1f ,
(53)
(54)
or up to the second order in k
(��4  (2k2 + C )��2 J 1f = 0ii s 11 (55)
where the boundary conditions 'l=0=0 have to be fullfiled. Since these equa
tions are invariant with respect to k ~  k the following perturbation ansatz
of 1f and Cs holds
1f = 1f + 'l? k2 + Q(k4)o .. (56)
Cs = Co + C2 k2 + Q(k4) (57)
The order k=O solution was found already in Secion 6. In order k2 we have to
solve the equation? 2 2(��;+ 4'I )1f2= (C2+2)��ii1fo
= 4ii2 (C2 + 2)(a cos2iiri  bSin2iiriJ (58)
where Co =  4ii2 and 'lo in fonn of eq.(48) was used. The inhomogeneous
solution of 'l2 is given by
"
,
,h.
:l
'l2,inh= Ki ii (1cos2iiii) + K2 11 sin2ii;¡
with Ki = (b/4iiHC2 +2) and K2=  (a/4iiHC2 +2), A solvabilty condition follows
from the boundar condition on e. One finds
i�� ii 1f 2 I = (C2 + 2 )(  a/ 4ii) = 0 . (59)
o
Thus Cz =  2 and the stability threshold up to order k2 reads
Cs(k) =  4ii2 (i + k2 /2ii2 + 0(k4)J. (60)
314
From eq. (60) we infer that Co is in fact at least for small k the critical value
of the stability threshold. Beyond we note that '12 is zero.
7.2. Stabilty thshold for the tilted layer
In the tilted case things are somewhat different. The equations linearzed
around the conductive state read
(��~  k2)'I = ik costp 9  sintp ��'l 9
(��~  k2) e = ikC cos tp 'I  C sin tp ��ll'l
(61)
(62)
which are in general no longer invarant with respect to k ~  k. Thus the
the perturbation ansatz in that case must also incorprate terms proportional
to k
'I = If 0 + 1ft k + QCk2) , 9 = 90 + 9t k + Qtk2)
Cs = Co + Ci k + Q(k2).
Performing a calculation similar to the one in Section 7.1. one finds? .,
ci = 32'7'" cot tp / sin2 tp. (63)
Thus there are two stabilty thresholds bifurcating out of Co given by
Cs =  4;r2 sin 2 tp (i :r 2t12'!t cot tp k J (64)
where the one with the plus sign is the minimal one for positive k. (64) tells
us in the tilted layer in general the k=O instabilty is not crticaL. The relevant
first correction in k is linear in k and always destabUising with the exception
of the vertical layer. Thus one has to expect a behavior as sketched in Fig.6.
\
\
\
\
\\
"
.. ..
,
i
/i
i/ I/
EY
:'¡'
:t.
¡o
~
.'i\
 llllec(
 "i.rtiLa.e
~ it
Fig.6 Stabilty thresholds of the tilted and the vertical layer as function of
k.
315
7.3. nonlinea amplitude equaons for the vertca layer
Here we want to derive two coupled amplitude equations governing the
convective state above threshold. In the case of the vertical layer we know
from Section 7.1. that the k=O instability is crticaL. Thus an approach similar
to the one performed by Chapman and Proctor (1980) can be applied. The
basic difference between their work ( done for RayleighBenard convection
with fixed heat flux at the boundaries) and our work consists in the d��fferent
types of scaling.
To study long wave number instabilties one scales length in ç directions
with L where L is assumed to be the lateral extension of the layer. L has
to be very large in comparison to d. Defining £=d/L the resealing is ç+£1 ç
and ��ç + £ ��ç. Wave numbers are resealed according to k + £ k. Time wil be
æaled with £2 . The time scaling is motivated by the scale on which growth
temperature fluctuations can take place. Beyond this we scale 'I and 6 with £.
Motivated by the stability analysis of Section 7.1 we set
C = Co + L.£2 (65)
where we assume that Ll = 0(1. Thus LI measures for a,given £ the magnitude
of the the actual temperature gradient C = To G.
Then the resealed equations read
(£2��¿ + ��~)'I =  ��ii6 (66)
(£2 ��¿ + ��~  £2 Òt) 6 = ( Co + \J£2)��ii 'I  £2 (��ll 'I ��Çe  ��ç'l ��ll e). (67)
Note that the nonlinearities (which appear in the temperature field equation
only) are proportional to e2 according to our scaling. Since (66) and (67) are
invariant with respect to £ +  £, an appropriate ansatz for 'I and 6 in terms
of E is
('1,6) = ('10,60) + £2('12,62) + 0(£4). (67)
The zeroth order in E was already calculated in Secion 7.1. The only difference
is the interpretation of the amplitudes, which are now in general ç and t de
pendent. We find
'10 (C,t) = A(C,t) ( 1 cos 2nii) + B(C,t) sin 2nii (68)
99 (Ç,t) = 2n B(C,t)(1  cos 2nii)  2nAsin 2nlJ. (69)
Here again Co = 4ii2. Our intention in the following is the derivation of
evolution equations for the amplitudes A and B. To do this we have to solve
the order £2 which takes the form
316
(��~ + 4n2 ��ll ) If 2 =  4n2 (A��çA + B ��çB) + 2n ( Ò~ B  Òt B) +
+ (4n2 (A��çBB��çA) + 2n¡.lA  4n Ò~A + 2n��tA)sin 2nii
+ ( 4n2 (A��çA + B��çB) + 2niiB  4n ��¿B + 2n��tB )cos 2n'l (70)
Solving the inhomogeneous contribution of 1f2 and using the boundary cond
itions 1f2(i=1)  1f2(1l=0) = 0 and ��ll 1f2(1i=1)  ��iiIf2h1=0) = 0, which follow from
the velocity field and the temperature boundary condition respectively, leads
finally to the following coupled amplitude equations
2
, ÒtB = (4/3) ��ç B  (¡./3) B  2n( A��çA + B��çB)
?
c\A = 2 ��ç A  iiA  2n (A��çB  B ��çA).
(71
(72)
Let us discuss some properties of these equations. 1) Setting B=O the
nonlinearity in the A equation vanishes. The contrary is not true.
2) Th~ equations are invariant with respect to ç ~  ç , B~  B , A ~  A, with
respect to ç ~  ç , B ~  B, A ~ A, and with respect to ç ~ ç , B ~ B ,A ~  A.
They reflect three invariances (i ç ~  C i II ~ II , If ~  'l , e ~  e , (ii) ç ~  ç i
II ~ (ti) . 'l ~ If , e ~  e , and (tit ç ~ C , II ~ (tii) i If ~  'l , e ~ e
of the basic equations (66) and (67), respectively.
3)The linear stabilty analysis of the conductive state A=B=O yields that the
amplitudes A and B become unstable at different control parameter values:
A at a squared reduced wave number q 2= ¡./4o and B at q 2= ¡Ll2 . Going back
to the nonresealed variables this implies that A becomes unstable at
at CA(k) = 4n2U+k2I2r(2) and B at CB(k) = 4rc2(t+k2/rc2), Thus for Co( CA
the amplitude A becomes unstable first and drves via nonlinearities the
amplitude B to finite values.
Work on the nonlinear solutions of these equations is in progress.
8. ANOGY TO A GEOPHYSICAL PROBLE.
Let us now turn to the geophysical problem mentioned in the intro
duction which is motivated by considerations of DaVis et aL. (1985):
a tllted waterfiled sandstone layer embedded in shale under the influence of
a vertical temperature gradient. Now we show how the difference in the
thermal diffusivities of shale. )(s' and fluidfiled sandstone.x, are in the case
of stationar flow analogous to an external temperature difference between
the boundar~s sandstoneshale. In that case the temperature gradient along
the boundaries is ��çT = ToGsinip while the one normal to the boundaries is
��ll T = �� ToG cos cp. Here € = )(s/x. These boundary conditions come from the
fact that the temperature and the normal component of the heat fluxes have
to be equal at the boundary standstoneshale.
317
Using these boundary conditions and
following stationary flow solution
eqs. (11)  (14) one finds the
Us hi) = ( E  1) cot II E (y ,.11) (73)
where
E(y;r¡) = y sinh(y(rlt/2))/cosh(y/2)
E (y i TJ) =  Iyl sin(1 y I( TJ  1/2) ) / cos(lyl/2)
if G~O
(74)
if G..O,
and y2 = ToGsin2.ç. We remark that this flow solution vanishes. i.e. us=O. if
either, the thennal diffusivities )( and )(s of the fluidfilled sandstone and the
shale are equal or the porous layer is verticaL.
Comparing this stationary flow solution with the one found in eq. (29)
one finds as condition for equivalence of both flows
AT tan II sin i. =  21yl (e  1) tan ( lyl/2) if G..O (75)
AT tan II sin\Ç = 2 y (e 1 )tanh (y/2) if G;.O . (76)
Via (75) and (76) we can map the results for the geophysical problem to the
system with externally applied AT whenever the flow is stationar and the
layer is tilted (op nonzero and not equal to 'I/2)' This holds in particular for
our theory of Taylor dispersion. In nature e is typically between .8 and i.~
(see e.g. Davis et aJ.198S). Thus the sign of the equivalent AT depends on the
direction of the geothennal grdient and the value of e1.
9. CONCLUSIONS AN PERPECTIS.
To summarize our findings:
1) For the system descrbed in Section 2 we calculated a nonlinear flow
solution and gave an upper bound for the stabilty of this solution.
2) We studied Taylor dispersion in that flow and found that the dispersion
coefficient is of the order of the thermal diffusivity of the fluid filled porous
medium.
3) We discussed for the case of zero AT the stabilty of the conductive
solution and found that the vertical layer has a crtical zero wave number
instability. This is no longer the case for the inclined layer. Beyond this
nonlinear evol ution equations for the vertical layer were given.
To the future work:
1) We have' also derived nonlinear evolution equations for the tilted case.
Then a different scaling is used. since the nonlinear basic equations are no
longer invariant with respec to k~ k. These results wil be discussed in
318
detail elsewhere. In particular the problem of the distinguished limit for
'f ~;r/2 is of interest.
2) The stability analysis of the flow solution for the case of nonzero åT has
to be cared out.
3) An extension our theory of Taylor dispersion to the case of bulk fluids
(without the porous medium) has to be discussed (Unz and Woods, in prepara
tion),
ACKNOWLGMES:
I would like to thank Andy Woods for introducing me into this subject,
his contimous interest. advice. and encouragement. It was a pleasure to work
with him.
Gratefully acknowledged are also interesting discussions with Joe Keller, Ed
SpiegeL. and George Veronis. Special thanks also to Rick Salmon for perfect
organization of the GFD summer progrm. It was a great summer in every
way.
RECES:
j.1. Beck (1972), Phys. Fluids 15.1377.
j.P.Caltagirone and S.A.Bories (1985). JFM 155,267.
S. Chandrasekhar (1981). Hydrodynamic and Hydromagnetic Stabilty.
C. Chapman and M.R.E. Proctor (1980). JFM 199.760.
S.H,Davis, S. Rosenblat. J.R. Wood. and T.A. Hewitt (1985). Am.J.Sc. 285.207.
A.E. Gil (1969). JFM 35.545.
D.O. Joseph (972),Stabilty of Fluid Motions II.Ch. X. SpringerVerlag,Berlin.
E.R. Lapwood (1948) . Proc. Camb. Phil. Soc. 44,508.
J.K. Platten and j.C. Legros (1984). Convecion in Fluids.Springer Verlag.
D.S.Riley and K.H. Winters (1990), JFM 215.309.
M.Sen. P. Vasseur, and L. Robilard (988),Phys. Pluids 31,3480.
G.I. Taylor (1953), Proc.Roy.Soc.London 219.186.
319
Behavior of a Fifth Order System of ODE's with
Intermittency
by
N. Platt
Abstract
A fifth order system of ordinary differential equations is studied as
control parameters are varied. The system displays an intermittency
akin to type II intermittency. The intermittent behavior of one of the
variables is characterized by random switching on and off of the ac
tivity. Fixed points of the system are found and their linear stabilty
is studied both numerically and analytically. Different characteris
tics of the chaotic attractor are presented through time series, phase
plots and Poincare sections. Numerical study of the statistics of the
intermittency include the average length of the laminar phase near
the intermittency threshold, histograms of the length of the laminar
phases near the onset of the intermittency and the lacunarity of the
autocorrelation function.
1 Introduction
In this presentation we are interested in a dynamical system which ex
hibits a peculiar behavior characterized by some variables intermittently
turning themselves off and staying inactive for a considerable amount of time
and later having a burst of activity. In general, intermittency in a dynamical
system is characterized by a random burst of activity in a system already
undergoing some periodic oscila.tions. Figure i shows a typical time signal
obtained in a dynamical system with intermittency. This is to be contrasted
with Figure 2 showing a time signal obtained in a dynamical system under
consideration.
¡~
,~
¡,
l,
320
2 Governing Equations
We follow the presentation given by SpiegeP to obtain a dynamical system
with intermittency shown in Figure 2. Consider a mechanical system with 2
degrees of freedom driven by a force derived from a timedependent potential
and subject to a drag force. Then equations governing the motion are
av
(1 )x   f,VXax
av
(2)y   wyay
where f" V are constants. The time dependence of V enters through a time
dependent parameter z, V = V(x,y,z). We take the following differential
equation for z
z = f,(z + a(x2 + y2  1)) (3)
where a is a constant. We take a special case of the generic potential given
by Thom2 and obtain the following parabolic umbilic1 1V = _(x4 + y4) + yx2 _ z(x2 + y2)4 (4)
Thus, we arrive at the fifth order system of differential equations
x P (5)
p _x3  2xy + zx  wp (6)
y q (7)3 2
(8)q y  x + zy  wq
z f,(z + a(x2 + y2  1)) (9)
Here, a, v, f, are constants and we restrict them to be positive. If we set
x = 0, p = 0, then the third order system in (q,y,z) separates out. This
system is equivalent to a Lorenz system3 under a suitable' transformation of
variables4. If E = 0 then the z equation simplifies to z = const and thus we
ha.ve a Hamiltonia.n system with 2 degrees of freedom where H is given by
IH = V + 2(p2 + q2) (10)
321
3 Fixed Points
In preparation for the study of the linear stability of the system, we look
for the fixed points of the dynamical system. They satisfy
p 0 (11)
q 0 (12)
x(x2  2y + z)  0 (13)
_y3 _ x2 + zy
 0 (14)
z + a( x2 + y2  1)  0 (15)
Two cases arise:
Case 1: x =1 0
Then the equations of the fixed points can be written as
x l:vz  2y
a(x2 + y2  1)
o
(16)
(17)
(18)
z
(I + 2a)y3 + 3ay2 + (2 + a)y  a
These equations were solved numerically using Mathematica for, various val
ues of the parameter a ~ 0 and four real fixed points IIV were found with
fixed points I, IV and II, III related by equation 16.
Case 2: x = 0
In this case we obtain the following fixed points
~ a(x=O,y= V~,z= 1+) Fixed Point V
~ a(x=O,y=V~,z= l+a)
(x = 0, y = 0, z = a)
Fixed Point VI
Fixed Point VII
322
4 Linear Stability Analysis
To study linear stabilty of the fixed points we obtain the Jacobian of the
system
0 3x2 2y + Z 0 2x 2a€X
1 €V 0 0 0
\1xF = 0 2x 0 3y2 + Z 2a€y (19)
0 0 1 €V 0
0 X 0 Y €
and examine its eigenvalues at various fixed points Xo.
If x = 0 then the eigenvalues À satisfy
(À(€V + À)  Zo + 2yo)((€ + À)( 3y; +zo  À(À + €v))) = 0 (20)
Substituting for the fixed point VII, we obtain
(€ + À)(À(€V + À)  a)2 = 0 (21)
Thus,
Ài  €
€V  ý€2V2 + 4a
2
€V + ý€2V2 + 4a
2
Hence, for a :: 0 fixed point VII is always unstable. Similarly, it can be
shown that fixed point VI is always unstable if a :: O.
Stabilty of the rest of the fixed points was determined numerically by
Mathematica for various a :: 0, v :: 0, € :: 0 and is summarized below:
. Fixed points II, III are always unstable
(22)
(23)À2,3 
À2,3  (24)
· There exists €i, €2 with €2 ~ €i such that
Fixed Points I, IV unstable €i Fixed Points I, IV stable
Fixed Pt V unstable
€2
Fixed Pt V' stable
323
,
Numerical experiments on the dynamical system with initial conditions
x = 0.01 and the rest of the variables set to zero produce the following
diagram:
f1
Fixed Pt I Attracts
I
L Fixed Pt V Attracts
Intermittency
Thus, the threshold of the intermittency in the dynamical system is controlled
by the stability of the fixed point V and the parameter 'f when the fixed point
V is losing stability is going to be denoted by fc.
5 Intermittent Regime
For further studies we fix a = 6.5 and v = 4.125 and study the resulting
system in more detail near the intermittency threshold fc.
Numerically, we obtain the following values for the fixed points and crit
ical values of f:
f1 = 0.825
f2 = fc = 0.441279
(x = 0.82,y = 0.6.5, z = 0.62)
(x = 0.76,y = 0.45,z = 1.47)
(x = 0.76, y = 0.45, z = 1.47)
(x = 0.82, y = 0.65, z = 0.62)
(x = 0, y = 0.93, z = 0.87)
(x = 0, y = 0.93, z = 0.87)
(x = O,y = O,z = 6.5)
Fixed Point I
Fixed Point II
Fixed Point III
Fixed Point IV
Fixed Point V
Fix~d Point VI
Fixed Point VII
Table 1 lists the dimensions of the unstable manifolds of various fixed
points at the intermittency threshold fc = 0.441279.
324
Fixed Point Dimension Eigenvalue
l,iV,V 2D complex conjugate
II, III ID real
VI 3D complex conjugate + real
VII 2D both real
Table 1: Dimensions of the unstable manifolds at tc
Since the threshold of the intermittency is determined by the fixed point V
becoming unstable, and its occurs as a pair of complex conjugate eigenvalues
cross the imaginary axis, we conclude that this dynamical system displays
intermittency similar to type II intermittency. The main distinction is that
in a standard definition of type II intermittency5 the limit cycle, not a fixed
point, becomes unstable as a pair of complex conjugate Floquet multipliers
leave the unit circle in the complex plane (This is equivalent to a couple of
complex conjugate eigenvalues moving across the imaginary axis).
Figures 24 show the time series of x, y, z and Figure 5 shows the broad
band power spectrum of the signal x. The time series of x is characterized by
the intermittent switching on and off of the signaL. Here, x spends approxi
mately 2/3 of its time in the laminar phase. From Figures 3 and 4 we conclude
that in the y  z plane the dynamical system exhibits a Lorenztype chaotic
behavior, spending most of the time circling around either fixed point V or
VI with random jumps from one to another. Unlike a pure Lorenz system,
we note that fixed point VI has an extra unstable manifold when compared
to the fixed point V with a positive real eigenvalue. Hence, the dynamical
system spends most of its time in the vicinity of the fixed point V.
Figure 6 shows the x  p phase space plot. This plot is especially interest
ing since it describes the behavior of the system during intermittent bursts of
activity (in the laminar phase, x = p = 0). Figure 7 depicts the y  z phase
space plot. It shows two characteristic Lorenz type ears. Unlike the Lorenz
attractor, one of the ears is traversed by the solution much longer than the
other (see explanation in the previous paragraph). In addition, there are
some extra curves of motion imposed on top of the Lorenz attractor. We
believe that they correspond to the intermittent excursions of the dynamical
system during bursts of activity in the x  p plane. Figure 8 shows the phase
325
space plot in the x  z plane. Here, the laminar phase corresponds to the line
x = 0, and additional curves capture the dynamics of the chaotic phases.
Let us briefly describe the dynamics in the x  p plane. Treating z and y
as free parameters, we find the following governing equations:
p
P
_x3 + (zo  2yo)x  wp
(25)
(26)
x
Stability of the fixed point (x = p = 0) is determined by the eigenvalues
w:l Vf2v2 + 4(zo  2yo)À12 = .
. 2 (27)
Thus, (x = p = 0) is stable if and only if Zo  2yo :: o. Close to the fixed
point V, ZO  2yo is negative. Thus, fixed point (x = p = 0) is stable. As
the trajectory spirals out of the fixed point V along its 2D unstable manifold
it reaches a point where Zo  2yo becomes positive and thus (x = p = 0)
is unstable. As the solution traverses the spiral, the farther away from the
fixed point V the trajectory is located, then the longer Zo  2yo remains
positive. Eventually, the solution either jumps to another ear of the Lorenz
attractor and the process starts all over again, except much faster, or there
is an intermittent burst of activity in the x  p plane. After that burst of
activity, the trajectory is reinjected back close to the fixed point V (the least
unstable fixed point), and the whole process starts all over again. Thus, the
length of the laminar phase depends on how close to the fixed point V the
solution reinjects itself and the number of times the solution jumps from one
ear of the Lorenz attractor to another without significantly departing from
the fixed point (x = p = 0) in the x  p plane.
Figure 9 shows a natural loglog plot of the average length of the lami
nar phase vs distance from the critical value of the parameter f. The most
surprising feature of this plot is that it seems that the average length of the
laminar phase does not approach infinity as f  fc 5,6. One possible expla
nation for that phenomena is that as f  fc the total length of the laminar
phase diverges. Hence, it is possible that there is a range in the parameter
space close to fc where the average length of the laminar phase stays almost
constant, but the length of the chaotic phase is rapidly decreasing. The slope
of the straight line in Figure 9 is approximately 0.64.
326
Figure 10 depicts a projection of the Poincare section of the attractor onto
the x  p plane at f = 0.44. Here, the cutting plane is y = 0.93 and iJ ~ O.
This Poincare section describes the behavior of the attractor during chaotic
phases (x = p = 0 in the laminar phase). Figure 11 depicts a projection of
the Poincare section of the attractor onto the y  z plane. Here, the cutting
plane is x = 0.05 and :i ~ O. This Poincare section also depicts the behavior
of the system during chaotic phases. In Figure 12, we show the Poincare
section projection onto the x  p plane. Here, the cutting plane is z = 0.87
and Z ~ o. The laminar phase corresponds to the dots located at the origin.
Figures 1316 show additional Poincare sections.
6 Lacunarity
We consider a statistical moment, C, on a fractal set that depends on a
separation scale I 7. A selfsimilar fractal set may be expected to satisfy the
scaling law
C(l) = O'iC(pl) (28)
where p, 0' are real numbers. One of the particular solutions is
Co(l) = Aid (29)
where A is a constant, d = In 0' / In p. The general solution is
C(l) = IdX(ln I)P (30)
where P = In p and X is a periodic function of period 1.
We apply the above to the autocorrelation function of the time signal x2
given by
11T
C(l) = lim T X2(t)X2(t + I)dtT +00 0
Figure 17 shows a InIn plot of the a.utocorrelation functi0n at f = 0.441279
and Figure 18 is a blowup of the same. The indicated line has a slope
d '" 1.1. Also, periodic oscilations are clearly present in the plot, but their
period seems to decrea.se. Figures 19 and 20 show InIn plots of the autocor
relation function as parameter f is decreasing. The slope d is increasing as
parameter f is moving away from the critical value fc.
(31)
327
7 Histograms of the length of the laminar
phase
Figure 21 show InIn plot of the histogram of the length of the laminar
phases vs. number of phases at the critical value € = 0.441279. Smoothing
average have been applied. The slope of a straight line is approximately
3/2. Figures 2224 depict histograms of the lengths of the laminar phases
at various values of the parameter €, as indicated. It is interesting to note
that the slope is decreasing as € is moving away from the critical value.
8 Conclusions
In this report a fifth order system of differential equations with intermit
tency has been investigated. The nature of the intermittency (type II) was
identified. All fixed points of the system were found and linear stability anal
ysis performed. Phase space plots and Poincare sections have been employed
in an attempt to describe the physical nature of the intermittency and the
structure of the chaotic attractor. An interesting feature of the dynamical
system relating to the average length of the laminar phase was discovered.
It seems that the average length of the laminar phase does nut approach
infinity as € t €c (or at least it has an almost constant plateau for some
range of parameter € near €c). Some analysis of the autocorrelation function
of the signal and histograms of the lengths of the laminar phases has been
performed.
Future work should resolve the point raised about the average length of
the laminar phase near the threshold of the intermittency. Also, additional
work is required for a closer examination of the lacunarity of the autocorrela
tion function. Another avenue worth investigating is to look for a statistical
model describing the intermittency.
Acknowledgement: I would like to thank Edward Spiegel for his guid
ance and help in dealing with this problem. Most of the numerically intense
computations were performed at the Center for Fluid Mechanics, Turbulence
and Computation of Brown University, Providence, RI.
328
References
(1) Spiegel, E.S., A class of ordinary differential equations with strange at
tractors, Annals of the New York Academy of Sciences, 357, p. 305312,
( 1980 )
(2) Thorn, R., Structural stability and morphogenisis, W.A. Bemjamin, Lon
don, (1975)
(3) Lorenz, E.N., Deterministic nonperiodic flow, J. Atmos. Sci., 29, p. 53
63, (1963)
(4) Marzec, C.J. and E.A. Spiegel, Ordinary differential equations with
strange attractors, SIAM J. App. Math., 38, p. 387421, (1980)
(5) Berge, P., Pomeau, Y., C. Vidal, Order within Chaos, John Wiley &
Sons, Inc., New York, (1986) ,
(6) Argoul, F. and A. Arneodo, A threedimensional dissipative map mod
eling typeII intermittency, submitted for publication.
(7) Smith, L.A, Fournier, J.D. and E.S. Spiegel, Lacunarity and intermit
tency in fluid turbulence, Physics Letters, 114A, No. 89, p. 465468,
(1986)
329
'CI
o,Gi
0,030
1.560 USE4 2.4 2. 2.E4
Figure 1: Time Series of a System with
Intermittency
+2.0
". 0,0
1.0
2.0
2.0=.4.
i
'I
2,OSE' 2,IOE4 2.ISE4 2,20E4
Time
figure :3: Time Series or y at E(;
I,
I
��
I
+2.0
+1.0
)( 0.0
2,0
2.0E+4 2.ISOE+4 2.2O+42,OSO+4 2.10E+4
rim
Figure 2: Time Series of x at €c
+10,0
N
,S,o
,10,0
:!.OE': 2,05E4 2,IOE4 2,ISE' 2,20E4
Time
figurcl: Tille Series of :; at (c
330
+2.
x
'õ
E2
¡;
CL 0.0
a.
en
~0CL
'"
.3
2,0
44.0
2.
0,0 0,10 0.2 0.3 0,40 o.s
4.0
2.0
1.0 0.0
X
+1,0 +2.0
Freqnc I Hz
Figure 5: Power Spectrum of x at €c Figure 6: xp Phase Space Plot at €c
+10,0
+10,0
N
+5,0
10,0
.2,0
1.0 0,0 +1.0 +2.0
y
x
Figure 7: yz Phase Space Plot at Cc Figure 8: izPhase Space Plot. at Cc
331
S.O
4,0
.c
v
coo
'
3,0
2.0
12.SO 10,0 7.5 S.o 2.
log IE,Ec
Figure 9: In c n :; vs In IE  Eel
+10.0
N
. ...." '' " ""'~":'::"'"'~L,~i~.~',:",..,~;,i"_",,:_,,,
.:¥~(::~~:, .
5,0
,10,0
,2,0 ,1.0 0,0
y
+1.0
Figure 11: Poi ncare Section at (:
0.4L Cutting plane i: = 0,05
..2.0
0,0
+1.0
.o.5
0,0
Q.
Ð.5
1,0
1.SO
.2.0
.o,SO
.0,25
Q. 0,0
Ð,25
.(,5O
,1.0
d;rkf/
1.0 0.0
X
+1.0 +2.0
Figure 10: Poincare Section at E
0.44. Cutting plane y = 0.93
_....'.: "" .
, "':"'~:?:;~ii~tçik':~"'r \ :.,'
'.,7... .
. ~~ . :
, '
.(,SO 0,0
x
.0,50 +1.0
Figure 12: Poincare Section at ('
,OA4, Cutt.ing plane z = 0.87
332
+10.0
6.
+5.0
..0
N 0,0
N
2.0
5,0
10,0
2.0
1,0 0,0 +1.0 +2. +3,0 0,0
1.5 1.0
0.5 0,0 .0,50 +1.0Q
p
Figure 13: Poincare Section at t _
0.44. Cutting plane x = 0.05
Figure 14: Poincare Section at t _
0.44. Cutting plane y = 0.93
/
+1.0
.0.5
';'.;.~";" i~rMh:.. .0,25 ,,/,'~F~~Y:?~~":':' '
~~;,¡1~~
':~,/:'(~:~
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..
+l,5O
." r.... .
;".:' '.
) 0,0
o 0,0
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0,25
.1.0
050
0,250 0,0
p
+i,25O +i,50 .(,5O
,1.0
050 0,0
y
+l,SQ +1.0
Figure 1::: Poincare Section at t
OA4, Cutting plane z = 0.87
Figure 16: Poincare Section at E
0.44. Cutting plane z = 0.87
333
E4,441279 E,4412790,0
+1.0
\,0
0,0
0.0
Log(Q
+1.0 ...0
o
õi
~
2.0
~o
'
,2,0
,3,0
~
4,0
2.s 0,0 +i. +~.o +1.s 4,0
4,0 ,2,0
LogQ
Figure 17: In( C(ln vs In(l) at € _
0.441279 Figure 18: Blow up of Figure 17
eO,42 eo.3
+1.0 +1.0
0,0 0,0
;J.
"
~.~
~ ~
f;~
0
.3 8'
'~t:~
1.0~1.0
4,0
..,0 .!.o 0,0
Log(l)
.2.0 ..,0 4,04,0
.~_O 0,0
.2.0 ..,0
Log(l)
Figure 19: In(C(l)) vs In(l) a.t E = 0.42 Figure 20: In(C(l)) vs In(l) a.t E = 0,:35
334
eO,441279 eO,446. 6.
6.0
4.0 4.0
l ..De
" Žz
coco
S0'
2. 2.
0.02. 3.0 4.0 5,0 0,02.
Log Leng Lo Lengh
Figure 21: Histogram of n vs length at
f = 0.441279
Figure 22: Histogram of n vs length at
f :: 0.44
eO,43
eO,436.0 6,0
4,0 4,0
..
..
~,
D
D�� E
"z zco
co
:~~
0
S t
'
2.0 2,0
. f~;
,: .
0,0
2.0 3,0 4,0
Log Length
5,0 6,0 0,02,0 3,0 4,0
Log Lenglh
Figure 23: Histogram of n vs length a.t
t = 0.4:3.5
Figure 24: Histogram of 12 vs length a.t
f = 0.43
335
DIFFUSION IN POISEUILLE FLOW
Wendell T. Welch
Department of Applied Mathematics
University of Washington
Seattle, W A 98195
August, 1990
Abstract
Diffusion of a passive additive in Poiseuie flow in a pipe is studied. Exact and
asymptotic equations are derived for e, the average concentration over a crosssection.
The asymptotic equations contain higher derivatives than the simple diffusion equation
derived by Taylor. These higher order terms may explain the asymmetry in the
concentration profie observed experimentaly. The asymptotic equations are solved
for e and the results compared with Taylor's solution.
1. INTRODUCTION
Using physical reasoning, Taylor (1953,1954a) derived and solved the following dif
fusion equation for c( æ, t), the crosssectionaly averaged concentration of a passive
additive diffusing in a lamnar pipe flow:
¡Bt +uaz  a1a~1 c(æ,t) = q(æ,t). (1)
Here, u is the average fluid velocity in the ædirection, a1 is a coeffcient and q is
a source term. Keller (1989) has rederived this result by a more systematic method
which yields a generalzation of (1) containing third derivatives. We shal present this
derivation, evaluate the coeffcients which occur in it, and solve it.
The basic mechanism is the combination of radial diffusion with advection due to
a parabolic velocity profie, which spreads out the additive longitudinally. (Longitu
dinal diffusion is also present, but its effect is very slow.) Taylor assumed that the
radial diffusion happens so quickly that the concentration across each crosssection
could be considered uniform at any given instant, and thus he obtained analyticaly
a concentration profie which is symmetric about the centroid of the :fuid motion (a
point travellng with the mean speed of the fluid). Such a Gaussian profie, however,
is only observed experimentaly after hundreds of pipe diameters down the pipe; at
earlier times the profie is distinctly asymmetric. Thus Taylor's results seem to apply
only at large times.
336
By assuming that the time scales of radial diffusion and longitudinal advection
overlap, we have relaxed Taylor's major assumption and derived an asymptotic equa
tion governing ë. The distinctive feature of this equation is that it contains derivative
terms of higher order than in (1); for example, to order €3 (where € is an aspect ratio
of the flow) the asymptotic equation is:
(Bt + uå:i  C1å;  C2å;Bt  C3å~J ë(z, t) = a(z, t). (2)
The coeffcients C1, C2 and 03 depend on the variation in fluid velocity across the
pipe. We have solved (2) for various cases which incorporate some or al of these
higher derivative terms and have obtained an asymmetric concentration profile. This
profie should agree with experiments at times much earlier than Taylor's solution.
2. FUNDAMENTAL EQUATION
Let us consider a pipe paralel to the xaxs, with a crosssection D of any shape in
the y, z plane. Let u(y, z, t) be the zcomponent of the fluid velocity within the pipe,
and let c(z, y, z, t) be the concentration of a passive additive in the fluid. (Note: we
have assumed that the fluid is incompressible by requiring u to be independent of æ.)
Suppose that c satisfies the following diffusion equation and boundary condition:
(Bt + uå:i  å:ik:iå:i  åykyåy  åzkzåz) c = q, (y, z) E D (3)
(nykyåy + nzkzåz) c = 0, (y, z) E åD ( 4)
where ny and nz are the y and z components of the outward normal to D. In (3) q is
the sum of a source distribution q. and an initial concentration co:
q = q. (z, y, z, t) + h' ( t) Co ( z, y, z) . (5)
where q. is nonzero only for t ~ 0+. We also assume that c = 0 at t = 0. Here
k:i, ky, and kz are the diffusion coeffcients in the three directions and are functions
of (z, y, z, t); n:i and ny are the components of the unit outward normals to åD. We
denote by f ( z, t) the average over a cross section of any function f ( z, y, z, t), and by
f' = f  f, the deviation from the mean. We also write f' = Pf, where P is a
projection operator.
We would first like to obtain an equation for ë, and thus we divide each of u, k:i, c,
and q in (3) into an average and a variable part. Thus we can rewrite (3) as:
(L + V  !:T) (ë + c') = q + q' (6)
where L, V, and the transverse diffusion operator !:T are defined by:
L = Bt + uå:i  å:ik:rå:i, V = u'å:i  å:ik~å:r, !:T = åykyåy + åzkzåz' (7)
337
Taking the crosssectional average of (6) and simplifying yields:
Le + V c' = q (8)
where the term ÅT(e + c') drops out because of the boundary condition (4). We can
then subtract (8) from (6) to give the following equation for c':
( ÅT + L + PV) c' = q'  Ve. (9)
(The operator P always operates on everything to its right.) Solving (9) for c' and
substituting into (8) finaly gives an equation for e:
(L + V(ÅT  L  pvt1 VJ e = q+ V(ÅT  L  PV)l q'. (10)
Note that (10) is an exact equation for c. To use it, we make the approximation
(verified through experiments) that L+PV is smal compared to ÅT. More specificaly,
for a circular pipe of radius a, let l:i be a characteristic length of pipe travelled by the
fluid. We are interested in times such that:
at = l:i ~ 1
and we choose length and time scales such that a = 0(1) and u = 0(1) (or equiva
lently, a = O(t)). Therefore, a:i = O(t) and ay, az = 0(1). If we also assume that
k:i, ky, and kz are al 0(1) and that al perturbation terms are 0(6) where 6 oe t, we
see that:
(11)
L = O(t) + 0(t2), V = 0(t6) + 0(t26) oe O(t) + 0(t2), ÅT = 0(1). (12)
Thus we can rewrite part of the coeffcient ofe in (10) and expand it in a binomial
series:
V(ÅT  L  PVt1 = V ((1 LÅT1  PVÅT1) ÅTr1
= V ÅT1 t 1 + (LÅT1 + PV ÅT1) + (LÅT1 + PV ÅT1)2 +... J (13)
Substituting this expansion into (10) yields our basic asymptotic PDE. If we assume
that k~ = q' = 0 and keep terms only to O( t3), we obtain:
(a + ua:i  a:ik:ia:i + U'a:iÅT1u'a:i
+U'a:iÅT1(a + ua:i)ÅT1U'a:i + U'a:iÅT1 PU'a:iÅT1U'a:ij e = 7j (14)
338
We can simplify this equation further by recalng that u (and thus u') is indepen
dent of z and by assuming that u' and k:i are independent of t and z, respectively, to
give:
(Ôt + ua:i  (k:i  U'ßT1U') a~
+(u' ß 1 ß lu')a2a + (u' ß IUß IU' + U' ß 1 Pu' ß lu')a3J ë = qT T:it T T T T :i
(15)
Because u is not affected by ßT1, we can pul it outside the first coeffcient for a~.
Then, introducing a moving coordinate system, defined by e == z  ut, we can write:
(at  (k:i  U'ßT1U') al + ( u'ßT1ßT1U') alÔt + ( u'ßT1 PU'ßT1U') a:J ë(e, t) = q
(16)
This coordinate system moves with the mean speed of the fluid, i.e. the speed in the
æ direction averaged over a cross section.
Equation (16) is our basic asymptotic equation forëin pipe flow. From c(æ, y, z, 0) =
0, we obtain the initial condition:
ë(e,O) =0. (17)
Then we can solve (16) for ë and substitute back into (9) to obtain c', again assuming
L + PV ~ ßT and expanding as before.
Let us now apply (16) to the specific case of PoiseuIe flow in a circular pipe of
radius a. Thus u(r) = Uma:i (1 r2/a2) and we can calculate al the coeffcients in
(16) to obtain:
¡ ( 22) 42 43 Ja _ k a Uma:i 2 a Uma:i 2 a Uma:i 3 _ _ _t :i + 192kl' ae + 2880k: aeat + 23, 040k: ae c  q (18)
Here we have assumed that:
ky = kz = kl" kl' independent of z. (19)
As an example, let us calculate the coeffcient of the second derivative in (16). We
proceed from right to left, first calculating u':
( r2) 1 i211 ¡a U ( r2 )u' = u  U = Uma:i 1     u( r )rdrdfJ = ~ 1  2a2 7ra 9=0 1,=0 2 a2 (20)
Now let ß'i/u' = w(y, z, t). Thus w is that solution of
ßTW = u' (21)
339
which satisfies:
8w
 = 0 at r = a, and w = O.8r (22)
The first condition of (22) follows from the boundary condition (4); we require
each of the coeffcients in (16) to satisfy (4) so that c wil. The second condition is
due to the fact that, by definition, u' = 0; thus, since w is related to u', it must also
satisfy w = o.
Using (19), we can rewrite (21) in cylindrical coordinates as:
k,. (82w + ~ 8w) = u'8r2 r 8r (23)
Integrating twice yields:
w(r) = ~,.1 fi" ~ (£Ø eu'(e)de) d1J J + K (24)
where the constant K is determined by imposing the second condition of (22). Car
rying out this integration yields:
() Umøz + Umøz 2wr= r16a2 k,. 8k,.
The coeffcient of 81 in (16) is thus:
umøza2
24k,. (25)
   2 1Ø  u2 a2
kz  u'w = kz  2' u'(r)w(r)rdr = kz + m;zka 0 19 ,. (26)
To simplify (18), let us introduce a new space variable T = (kz + ø:;ik" ) t. Then
we can rewrite (18) in the form:
e  cee + ßceer + bceee = qkz+a (27)
where we define:
2 2a umøza=
 192k,. ,
4 2ß  a umøz
= 2880k2'
,.
alu~a.lI
c = 23,040k~v__ 22k+~z 192k,. (28)
340
3. SOLUTION METHOD
We wi seek a solution of (27) with q = 0 and with the initial condition
1
c(e,O) = Co(e) = c5(e)
7la (29)
This corresponds to a unit amount of the additive distributed uniformly in a cross
section of infinitesimal thickness at x = t = O.
To solve this problem we seek c in the form:
c = 1: j( k )eikeO'(k)1" dk (30)
Then j(k) is calculated from (29) via an inverse Fourier Transform as follows:
_ 1 ¡OO _ ike _ 1
j(k)  2 c(e,O)e de  2 2 2.7l 00 7l a (31)
4. TAYLOR'S CASE
The simplest form of (27) occurs when ß = c5 = 0, when it becomes the simple diffusion
equation G.I.Taylor derived in 1953 (ë, = Cu). Its solution, with initial condition (29),
is
_ l_c
c( e, r) = 27l3/2a2rI/2 e 4T
By assuming k;i = 0, we can write (32) as Taylor did:
_ 1_~
c( e, t) = 27l3/2a2tI/20.I/2 e 4at
(32)
(33)
where 0. = a;:it.. is the effective dispersion coeffcient of the additive in laminar pipe
flow.
Physicaly, this corresponds to a Gaussian profile of c vs x, moving with the mean
speed of the fluid and spreading out as t increases. The symmetry of this profile
about the centroid is of special interest. This is due to Taylor's assumption that
L + PV is negligible in comparison to ßTI in equation (10), i.e. that radial diffusion
is much faster than either advection or longitudinal diffusion. He assumed that radial
diffusion would act quickly to smooth out any differential concentration within a
crosssection before advection had a chance to act, or equivalently, that diffusion and
advection occur on such disparate time scales that they can be considered to act
independently. Thus, in this idealzed case, we can think of the additive as diffusing
341
symmetricaly about the stationary point z = 0 with effective dispersion coeffcient a,
and simultaneously (but independently) being advected by the fluid at the mean flow
velocity. This last statement assumes that we can consider all the additive as moving
with the average speed of the fluid. This is because we are dealng only with the
average concentration over a crosssection, and since the radial diffusion is assumed to
be instantaneous, we are effectively reducing the pipe down to a line with infinitesimal
crosssectional area, in which all of the fluid moves with the average speed of the flow.
Such a Gaussian profile is indeed observed experimentaly, but only after hundreds
of pipe diameters have been traversed by the fluid. The asymmetry of the concentration
observed at earlier times can not be explained by formulas (32) and (33); it is precisely
for this reason that we have considered the more general cases which follow, in which
we include the third derivatives in (27).
5. AIRY FUNCTION SOLUTION
Next we solve (27) with ß = 0 and 5 i O. Thus we must solve the following PDE:
Cr ëU + 5ëeu = O. (34)
We note that this case is not motivated by a particular physical meaning, since
in general O(ß) = 0(5) and thus ß can not be considered negligible. However, the
solution to (34) can be obtained exactly in terms of Airy functions, and thus we pursue
this case for the qualtative insight it may yield regarding the solution to the more
general equation (27).
Substituting (30) into (34) gives the dispersion relation:
u(k) = k2  i5k3 (35)
Thus we can rewrite (30) in the form:
ë = ~ 100 ei(1eHi1e2'T+61e''Tldk
27r2a2 00 (36)
where we have also used the initial condition (31). This can be transformed to an
Airy integral by letting k' = k  ~ to yield
ë = ~eU6H 27262 'T)1��O ei(1e( H ,16'T )+6'T1e')dk27r2a2 00
2~e('T2;:~~") /00 cos (k (e + ..r) + 5rk3) dk27r2a2 10 35 (37)
342
where v == ~. (The sine term cancels out because it is an odd function of k.) Thus,
using the definition of an Airy function, we have:
(t )  2~ (T2+v~,,1 7r A' ¡e + isrJC i", r  e 216 i 1 i27r2a2 (3c5rF (3c5rF (38)
This solution has been computed numericaly and plotted together with Taylor's
solution on the graphs which follow. These graphs çorrespond to different values
of time, where time has been measured in terms of the number of pipe diameters
traversed by the mean flow. In each graph, the horizontal axs indicates e in pipe
diameters, i.e. the number of pipe diameters away from the centroid, so that the
centroid is always at O. A vertical line has been drawn to indicate the starting point
of the additive (the plane :i = 0).
In this computation, the values used for the various parameters were taken from
an experiment Taylor performed using KMn04 in water:
 5 2/a = .0252 cm, kri = k" = .7 x 10 cm sec, Umari = .527 cm/ sec (39)
We can notice several features of these graphs. (All except the last two have been
plotted on the same scale to faciltate comparison.) First, the concentration in the Airy
function solution reaches a maxmum to the left of the centroid; this agrees with the
asymmetry we expect from observational data. Second, the maxmum concentration
is slightly lower for the Airy case than for Taylor's. Both of these effects are due to
the fact that we have assumed that L + PV in (10) is smal but not negligible in
comparison to LlT1. We expect radial diffusion to occur more slowly than in Taylor's
case, so that advection is felt before radial diffusion has smoothed out al differential
concentration over a crosssection. Thus, in this case, radial diffusion and advection
are not independent, and we do not expect a symmetric concentration profie about
the centroid. (Note that longitudinal diffusion occurs only in the coeffcient of eee
and has negligible effect, since usualy kri ~ a:;;i~",) We also notice that the profies
do approach a Gaussian, but only after several hundred pipe diameters have been
traversed by the fluid. This can be seen from the last two graphs at 500 and 1000
pipe diameters, which have been plotted on their own scales to put in perspective
the difference between the Airy and Gaussian solutions. At 500 diameters, one can
stil see a difference between the two curves, but at 1000 diameters they are virtualy
identical on this scale.
A few comments on the computation are in order. Ai(7J) was computed for each
7J = (:;T~;3 by either an ascending series for "smal" values of 7J, or by an asymptotic
expansion for 7J ). 3.7. Since an Airy function oscilates above and below zero for
negative values of its argument, the profie of e was ended as soon as it became zero.
The physicaly meaningless negative concentration given by the solution is possible
343
concentralion (averaged over cross see lion)
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344
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345
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because our original PDE (27) is only approximate (only accurate to order E3),and
thus its solution wil not necessarily retain al the physical aspects of the original
problem.
6. GENERAL CASE
Lastly, we consider the ful equatian
~  cee + ßeUT + 5cue = 0 (40)
(We do not pursue the case ~  cu + ßeUT = 0 , because it seems to be no easier to
solve than (40) and does not correspond to a particular physical situation.)
Substituting (30) into (40) gives the dispersion relation:
(k) = P  i5k3U 1 _ ßk2 (41)
Now let us rewrite our general solution (30) in the form:
c = 1: f( k )eT'P(1c)dk (42)
Here
ip = u(k) _ ike = k2  i5k3 _ ikvT 1  ßk2 (43)
6.1 Method Of Steepest Descents
We wi first use the method of Steepest Descents to approximate this integral for
large T. Thus, we wi look for stationary points k. of ip(k) at which the integrand of
(42) is a maxmum. We wil then evaluate this integral along a contour on which theimaginary part of ip(k) is constant.
To find the stationary points ofip(k), we set ip'(k.) = 0 and
solve for k.. This
yields the following quartic equation for k.:
(iß5  iß2V)k: + (2ißv  3i5)k: + 2k.  iv = O. (44)
Notice that, for a given physical situation, these values of k. vary only with Vj they
do not depend on T. To simplify matters, we introduce a new variable z, defined by
z = ikz, and rewrite (44) as a polynomial with al real coeffcients:
(ß5  ß2V)Z4 + (35  2ßV)Z2 + 2z  v = O. (45)
347
Thus in general there wil be four complex stationary points for each value of v.
These wil be maxma or minima of Re( If( k)) depending on the contour of constant
Im( If) along which we choose to integrate; that is, each k. wil in general be a saddle
point of Re( If( k)).
We could use several methods to solve (45) for z. Although we could use a formula
to solve the quartic analyticaly, this would be complicated, since so doing requires
decisionmaking based on the associated cubic and quadratic equations, and this would
have to be done for each value of v. To get around this problem, we look for ranges
of v within which we can solve the 'quartic easily (without such manual decisions). In
these ranges, then, the four z wil have distinctive characteristics (i.e. al complex, or
two complex and two purely imaginary below the real axs, etc). We then develop a
sold tion for (40) in a piecewise manner, treating each range of v separately.
Once we have solved (45) for the stationary points, either numericaly or analyti
caly, we must investigate the contours of constant Im( If) which go through the~e k.
in the complex k plane. Thus, we must find al values of k for which Imtlf(kH =
Imilf(k.H. Using (43) we have:
(P  i6k3 . ) _ (k:  i6k: .k ,)1 _ ßk2  ikv  1 _ ßk: i.v (46) ,
or, letting k = a + ib:
ß6a6  6a3 + 2ab + 36ab2 + 2ß6a3b2 + ß6ab4
av=
(1  ßa2 + ßb2)2 + 4ß2a2b2 (k:i6k: .k )1 _ ßk~ i.v, (47)
In general there will be two perpendicular contours going through each k.;we,
must pick the one to which we can deform the original contour (the real kaxs) of
the solution (42)., This wil also turn out to be the contour along which Re(lf(k)) has
its.niiiinum vâ.ue at k. (and hence the integrand of (42) its maxmum value at k.).
Furth��r, we may be required to include the effects of more than one stationary pointif two faÌ on the same contour. "
,I~ we know the stationary points and proper contours for a certain range of v, then
we'can expand If(k) about k. and approximate the concentration in that range by the
following formula from the method of Steepest Descents:
1
c( e T) ,. eTIP(1e.)
, ,. 1r3/2a2 (2Tlf''(k.))1/2
(48)
348
6.2 Res1Ùts
The zeroes of equation (44) have been calculated for a range of v and are plotted
on the complex k plane on the graph which follows. (This range of v corresponds to
100 : e : +100 pipe diameters at a time when the mean How has traveled 100
diameters from the z = 0 plane.) Ai;rows on these graphs indicate the direction of
increasing v or e.
There are two pairs of zeroes. Stationary points (i.e. zeroes) 1 and 2 are always
complex and have the same imaginãry part but opposite real parts. For v : 6/ß, they
have positive imaginary parts, and for v :; 6/ ß, negative imaginary parts.
Stationary points 3 and 4 are more complicated. For v ~ 6/ ß, both are complex
and again have the same (negative) imaginary part and opposite real parts. Simiarly,
for v :p26/ß, both are complex with the same (positive) imaginary part and opposite
real parts. But for the middle range, 6/ ß ~ v ~ 26/ ß, stationary points 3 and 4
are purely imaginary. For 6/ ß ~ v : 0 both are negative; for 0 : v : 6/ ß one is
positive and one negative; and for 6/ ß : v ~ 26/ ß both are positive.
The remaining task to calculate an approximate solution for this general case is
to investigate the contours of constant Im(cp(k)) for the different stationary points in
each range of v. One should then determine, for each range of v, those k. which are
minima of Re(cp(k)) and thus have contours which are deformable from the original
contour (the real k axs). Stationary points 1 and 2 wil in general have different
contours from those of points 3 and 4. Thus they should be investigated separately.
Then, with equation (48), a piecewise approximate solution to (40) can be obtained,
using appropriate stationary points and contours.
6.3 Numerical Integration
The integral in (42) with cp( k) given by (43) has been integrated numericaly for various
values of T, and this result is shown together with the Airy and Gaussian solutions on
the graphs which follow. (These graphs are displayed in the same way as the previous
set; al are on the same scale except the last one at 500 diameters, which has been
shown on its own horizontal scale.)
To compute this solution, a cautious adaptive Romberg extrapolation method (also
known as CADRE) was used. As the integrand of (42) wil oscilate very quickly for
certain values of v, i.e. those which yield a large magnitude of Im(cp(k)), there can be
errors due to cancellation of large terms, and thus these graphs should be regarded as
premliminaryonly. (Such errors wil 'be most significant at smal values of time, when
the exponential damping of the integrand due to T * Re(cp(k)) is also smal.)
349
Taking this into account, one can stil derive some qualtative insight from the
graphs. First, al three curves agree better as T increases, which is expected, since the
third order terms of (40) decrease in importance as t of (11) decreases (i.e. as time
and T increase).
Second, the numericaly integrated curve is indeed asymmetric, although this is
diffcult to discern from the graphs. For example, the maxmum concentration occurs
at approximately 3, 7, and 8 pipe diameters when the mean flow has traversed 25,
100, and 200 pipe diameters, resp~ctively. This asymmetry decreases as T increases,
again as expected.
Lastly, the numericaly integrated curve lies below both the other solutions for al
T. By conservation of mass of the additive, this means that the additive must spread
out more (i.e. faster) longitudinaly than either the Gaussian or the Airy function
theory predicts.
For smal T, then, the Airy function solution has two features which are absent
from the Gaussian profie: it lies under the Gaussian and it has an asymmetry about
the centroid. These features yield some qualtative insight on the exact profie which
the Tarlor theory cannot predict.
7. FURTHER WORK
There are various extensions of this work which suggest themselves. One could per
form finite differences on the truncated PDE (40) to obtain a numerical solution. A
comparison with experimental data could be performed as well.
In addition to Poiseuile flow, there are other geometries and flows which would
be interesting to investigate. Turbulent flow in a pipe is a natural extension of this
work and was discussed by Taylor (1954b). Laminar or turbulent flow in a layer, in
which the velocity u could, in general, have components in al three spatial directions,
is also of interest and has applications to estruaries. This work could also be applied
to a spherical geometry, with flow radialy outward and/or along spherical shells, to
model diffusion of chemicals within and out of a star (Chaboyer, 1990).
Finaly, to obtain an ezact solution to a problem with simultaneous advection
and molecular diffusion (as opposed to the approximate solutions found above), a
more simple model could be developed. The solution to such a problem might then
iluminate the physics observed in the more complicated geometries and flows.
350
Zeroes of Quartic  diams = 100
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ACKNOWLEDGEMENTS
I would like to thank Joe Keller for his marvelous insight, instruction and patience; I
thought and learned a lot, Joe! I also appreciate al that Rick Salon did, especialy
behind the scenes, to organize this year's program and keep it running so smoothly.
Thanks to al the professors and fellows for an intellectualy stimulating and wonderful
summer!
REFERENCES
1. Taylor, G.I., (1953), Dispersion of Soluble Matter in Solvent Flowing Slowly
Through a Tube, Proceedings of the Royal Society of London, A219, p.186.
2. Taylor, G.I., (1954a), Conditions Under Which Dispersion of a Solute in a
Stream of Solvent Can Be Used to Measure Molecular Diffusion, Proceedings
of the Royal Society of London, A223, p.446.
3. Keller, J.B., (1989), Diffusion in a Shear Flow (preprint).
4. Taylor, G.I., (1954b), The Dispersion of Matter in Turbulent Flow, Proceedings
of the Royal Society of London, A225, pp.473477.
5. Chaboyer, B., (1990), Transport of a Chemical in Stellar Radiative Zones, Geo
physical Fluid Dynamics Summer Program: Fellows' Project Reports, Woods
Hole Oceanographic Institution.
6. Abramowitz, M. & Stegun, I., (1970), Handbook of Mathematical Functions,
9th ed., Dover Publications, New York, pp.446448.
354
Classification of Similarity Solutions of the
TwoDimensional Convection Equations
Eric C. Won
o Introduction and Overview
This project applied a method developed by Sophus Lie for systematicaly solving differential
equations. E. Galois' work on solutÌons to polynomial equations (which started the theory
of discrete groups) served as Lie's motivation. Well after mathematicians discovered the
quadratic (x = (b:l ýb2  4ac)j2aJ, cubic, and quartic formulas, Abel proved that there
is no general quintic formula. Galois' work, however, showed how to associate with any
polynomial (of any order) a mathematical object which is now caled the Galois group of
the polynomial. The Galois group determines whether or not a polynomial can be solved
"by radicals," i. e. certain polynomials of fifth degree and higher can be solved exactly. In
addition to giving a "solvabilty" criterion, the structure of the Galois group of a solvable
polynomial also tells one how to construct those roots. Galois theory however, is accessible
only after one is quite familiar with the theory of discrete groups and the theory of algebraic
fields. '
Lie's work has led to a theory which is to differential equations as Galois' theory is to
polynomials: to each (ordinary or partial) differential equation, one can associate a Lie
group which shows one how to, in the case of an ODE (sometimes) reduce the order of the
equation, and in the case of PDE's:
1. Transform a given solution to another solution, and
2. Find simiarity solutions.
We are primarily concerned with simiarity solutions to PDE's. Thus, confronted with
a system of PDE's, one might naÏevely use point 2 of the theory to find several similarity
solutions and then use point 1 of the theory to (hopefully) discover new similarity solutions.
This effort would certainly lead one to some similarity solutions of the system of PDE's,
but if one has not solved al possible similarity ,equations, how does one know whether or
not one has found all simiarity solutions? One might also wonder, since point 1 alows one
to transform a given solution into another solution, whether there is a minimal or optimal
subset of similarity solutions on which one can then use point 1 to generate all possible
similarity forms? It turns out that the structure of the Lie group alows one to find this
optimal subset which greatly reduces one's work. Thus, the answer to the first question,
roughly, is that if one properly analyses the structure of the Lie group of a system of PDE's,
then one can find an optimal subset of simiarity solutions on which one can apply point 1
to find all others.
Lie's theory, which is a theory of continuous groups, involves the base manifold of depen
dent and independent variables and prolongations of the base tangent in the jet manifold of
the differential equation when applied to solving partial differential equations. The theory is
355
powerful in the sense that the determination of the Lie group is essentialy algorithmic and
one who wishes to apply the theory to find simiarity solutions to a system of PDE's need
not master advanced topics in differential geometry. Writers of several symbolic manipula
tion programs (MACSYMA, Maple, and REDUCE) have taken advantage of the theory's
algorithmic aspects and now have packages avaiable which begin the analysis. These pack
ages obviate the necessity for one to understand the subtleties of the theory. Rick's first
lecture gives a brief summary of the theory upon which the packages are based for whose
who do wish to have a sense of what the "black box" does. Part of the algorithm the pack
ages perform resembles the multiplication of arbitrary polynomials, which is a programming
problem assigned in undergraduate computer science courses.
Nevertheless, let us briefly review a few basics of the theory of Lie groups and Lie
algebras and at the same time give a brief overview of the method before we begin the
discussion of similarity solutions to the twodimensional convection equations. A simple
example of a Lie group is the xt plane with vector addition. It is a Lie group essentialy
since one can move up, down, left, and right by any amount one wishes and stil remain
within the space, and one can also not move at al (+ 0). It is nice to first consider the
plane with addition as a Lie group since it is easy to see its two subgroups, the xaxs
and the taxs, which are Lie groups as wellon the xaxs one can move up and down
by any amount which includes 0, and likewise for the taxs. Note that if one alows other
operations, such as rotation or scalar multiplication in addition to vector addition, the Lie
group will have a richer subgroup structure. For the moment, though, let us continue to
restrict our attention to the xt plane with vector addition.
The transformations described by Lie groups are al finite. It turns out that it is often
quite useful to consider the infinitesimal transformations related to a Lie group's finite trans
formations. It is not surprising that whenever derivatives are involved, it is often useful to
consider these related infinitesimal transformations in addition to (and sometimes instead
of) the finite transformations of the Lie group. The infinitesimal transformations for the
Lie group we are considering are az and a. The mathematical rules this collection of in
finitesimal transformations obey are no longer those of a group. Instead, these infinitesimal
transformations obey the rules of an algebra. It should come as no surprise, then, that the
infinitesimal transformations related to the finite transformations of a given Lie group are
caled a Lie algebra. The infinitesimal transformations which comprise the Lie algebra are
caled generators. Thus the generators for the Li,e algebra associated to the Lie group we are
considering, the xt plane with addition, are az and a. It is important to remember that
if one knows a Lie group one can determine its Lie algebra, and viceversa. If one extends
the operations alowed by our Lie group to include rotations and scalar multiplication, the
generators of the Lie algebra wil include ( taz  xaJ and ( xaz, tatJ respectively.
The Lie group associated with a given system of PDE's is caled the symmetry group of
the system, its related Lie algebra is caled the symmetry algebra, and the generators of the
symmetry algebra are caled symmetry generators. An arbitrary combination of syinmetry
generators will be caled a generator.
It turns out that the theory's algorithm does not directly specify the symmetry group.
Instead, given a system of PDE's, the algorithm shows one how to compute the symmetry
generators. Thus, we are first given the symmetry algebra, from which we can deduce
356
the symmetry group. Let us suppose that a given system of PDE's has the two symmetry
generators Vi and V2' Each symmetry generator gives rise to a way of transforming solutions
to solutions, say ti and t2. Thus if U is a solution to the system, then so are tiu and t2u.
So for example, if the system were a single PDE for u(:z,t) and Vi = 8;i and V2 = 8t, then
tiu = u(:zs,t) and t2u = u(:z,ts) where s is an arbitrary real number. An important fact
of the theory is that each possible generator gives rise to a similarity solution. So in general
one would expect that Vi gives simiarity solution Ui, V2 gives U2, and the combination
Vi + CV2 gives U*, where c is an arbitrary constant. Thus since Ui, U2 and U* are simiarity
solutions,
tiui, tiui, tiu*, t2ui, t2U2, t2u* (1)
are simiarity solutions as well. The solutions may not al be distinct, though.
The crucial structural property of Lie algebras for the purposes of this project is that
they are closed under Lie dragging, i. e. a given generator must carryjadvectjLie drag any
other generator into some (possibly trivial) combination of generators. Lie dragging, or
equivalently the adjoint ofvj by Vi is defined by
SAd(vi, Vj) = Vj + S(Vi, Vj) + '2(Vi, (Vi, Vj)) + . .. (2)
where h,) is the commutator or Lie bracket, and s is the arbitrary adjoint parameter. Note
that the adjoint is linear in its second argument, but not in its first. Now typicaly the Lie
dragging or the adjoint of Vj by Vi is Ad(vi, Vj) = vil �� = 1,2, i. e. Lie dragging or the
adjoint of one by the other has no effect. In this case, the solutions (1) are al distinct, and
one would need to explicitly solve for Ui, U2, and U*. If, however,
Ad(v2, Vi) = Vi + SV2 (3)
then one does not need to solve for U* since U* = t2Ul.' This is shown in
Ad(v2, Vi + CV2) = (vi + SV2) + CV2
since one can pick s = c which leaves Vi only. Thus if (3) holds, then (1) becomes
tiUi, tiu2, tit2ui, t2ui, t2U2, (t2U2)
where the group property that the product of two elements of the group is also within the
group has been used. So in this case, one had only to solve two PDE's to find Ui and U2
to obtain al similarity solutions (by using point 1 of the theory on Ui and U2)' Adjoint
values like (3), therefore, mean that the optimal subset wil be smaler than the ful general
set. Indeed in this example with (3) holding, optimal subset is f Vi, V2J instead of the most
general f Vi, V2, Vi + cv*J. The adjoint table which displays al adjoint pairs, manifests
most clearly this structure. Thus, the structure evinced by the adjoint table helps one find
the optimal subset of generators. More details can be found in Rick's second lecture. The
appendi includes an explicit example. We wil cal this procedure of using the adjoint table
to find the optimal subset of generators whose similarity solutions can be used determine
al other similarity solutions the adjoint reduction and wil cal a generator belonging to the
optimal subset an optimal generator.
357
1 Main Results
The equations of twodimensional convection are
\72Wt + J(W, \72W)
Pt + J (W , p)
pz + Pr\72\72W
\72p
(4)
(5)
where W is the stream function and p is the density. We consider an infinite fluid, and there
fore wil not be concerned with boundary conditions. In addition to the general convection
problem, (4) and (5) above, we also examine the high Prandtl number case:
o
Pt + J(W,p)
 pz + \72\72w
o
(6)
(7)
and the inviscid case:
\72Wt + J(W, \72'1)
Pt + J (W , p)
pz
O.
(8)
(9)
We used the SPDE package which was written for the REDUCE symbolic manipula
tion program to begin the computation of the symmetry generators for each case. After
completing the calculation by hand, we found the general system, (4) and (5), has five sym
metry generators and both special cases have seven each. The surprising result, however, is
that the five symmetry generators of (4) and (5) are included in each special case, i. e. five
of the seven symmetry generators of the high Prandtl number case and five of the seven
symmetry generators of the inviscid case are the five symmetry generators of the general
system. Furthermore, the remaining two symmetry generators in each special case are the
same up to a scalar which multiplies the ôp part of each symmetry generator.
Table 1 shows our findings for al three cases where 0:, ß, and 1 are arbitrary functions
of t. The top five symmetry generators are the five which are common to al cases. The
bottom two, which are enclosed by brackets, "t" and "��," are the additional symmetry
generators for the special cases. Note that the scalars A and B in the ôp term are the
only differences between the high Prandtl number case where A = 1 and B = 1, and the
inviscid case where A = 2, and B = 1. The right hand column of Table 1 shows the way to
transform solutions to solutions, which is caled, the symmetry group action on solutions.
The next step is to perform the adjoint reduction. For the general case, the optimal
subset consists of only two optimal generators: Ôt + CÔp and' CÔp + (ß(t )ôy + æß( t )Ô'I J +
b( t )ôz  y'l( t )ô'I1. Their similarity forms and corresponding similarity differential equation
sets are given in Table 2. The appendix gives the details of the computation for the first
generator Ôt + CÔp.
The adjoint reduction, similarity form, and similarity differential equation for the special
cases are given in Tables 4 and 5 in the same format. The similarity forms' are closely
related. The last similarity differential equation set in the inviscid is not included since it
is complicated. All systems in Tables' 4 and 5 are nonlinear (except the one not included).
S. Childress reports some colleagues are currently investigating Table 4 case B.1. with c=2
numerically with a supercomputer.
358
sym. generators group action on solutions
8t
8p
a(t)8i
ß(t)8y + xß(t)8i
y(t)8z  yy(t)8i
i tBt  'l8i  Ap8p
i x8z + y8y + 2'l8i + Bp8p
'l = F(x,y,t  s)
'l=F
W=F+a(t)
W=F(x,yß,t)+xß
W = F(x y,y,t) yy
W = ).iF(x,y,).it)
W ' ).2F().ix,).iy,t)
p=G(x,y,ts)
p=G+s
p=G
p = G(x,y  ß,t)
P = G(x y,y,t)
p=).AG(x,y,).it) ��
p = ).BG().ix,).iy,t) ��
Table 1. Symmetry generators and corresponding group action on solutions. a, ß, and
yare arbitrary functions of t, and), is a real constant. Pr ¿: 1 case: A = 1, B = 1.
Inviscid case: A = 2, and B = 1.
f V2'lt + J('l, V2W) = pz + V2V2W
1. Pt + J('l,p) = V2p
v = ci8t + c28p + a(t)8i + (ß(t)8y + xß(t)8i) + (y(t)8z  yy(t)8i)
general case
A.I. Ci =1 O. Adjoint reduction gives 8t + c8p
f W = f(x,y)
1.p =g(x,y)+ct
f J(f, V2 f) = gz + PrV2V2 f
1. c + J(f,g) = V2g
A.2. Ci = O. Adjoint reduction gives c8p + (ß(t)8y + xß(t)8i) + (y(t)8z  yy(t)8i)
J W = l (~x2  ~y2) + f(E" t) where x y
1 p = c(~ + ~) + g(E" y) E, = y(t)  ß(t)
J "'feet  "'4iE,feee + kfee + iJ = 11  ~ + Pr",2 feeee
1 gt  4iE,ge + c(;..fe + øE,) = "'gee
where
1 1
"'(t) = ß2 + y2
d
4i(t) = dt In(ßy)
ß yO(t) =   
y ßA d ß
4i( t) =  In dt y
Table 2. Similarity forms and differential equations for the general case.
359
to =  Pz + \72 \72 'lPr ~ 1 case: o = t + J ('l , p)
v = Ci 8t + C28p + a(t)8.¡ + (ß(t)8y + xß(t)8.¡) + (¡(t)8z  yy(t)8.¡)
+ C6 (t8t  'l8.¡  p8p) + C7 (x8z + y8y + 2'l8.¡  p8p)
A. The cases where C6 = C7 = 0 are exactly as in the general case.
B.1. C6 f. O. Adjoint reduction gives V6 + CV7
t'l =t2cif(= i)
t" , t"
_ i (Z Y)P  t"+l 9 ¡;, ¡;
f 0 = ge + \72 \72 f
1. 0 = ~(c + 1)g  c(çge + r¡g,,) + feg"  f"ge
2. v6, alone (c = 0)
J 'l =' f(:,y)
1 p = g(Zt'y)
f 0 = gz + \72 \72 f
1. 0 = g+ J(f,g)
C.1. C6 = 0, C7 f. O. Adjoint reduction gives c8t + V7
f 'l = e2ct f(xect, yect)
1. p = ect g( xect , yect)
L
~.
J~\
f 0 = ge + \72 \72 f
1. 0 = c(g + çge + r¡g,,) + J(f,g)
2. V7 alone
J 'l = y2 f ( ;, t)
1 p = ;g(:;,t)
f 0 = ge + 4(1  3ç)fee  4ç(1 + 4ç2)feee + (1  e)feeee
1. 0 = gt  feg  2fge
Table 3. Similarity forms and similarity differential equations for Pr ~ 1.
360
inviscid case f \72 \) t + J (\), \72 \)) =  Pz1. Pt + J (\) , p) , 0
v = Ci at + C2 ap + a(t)8'1 + (ß(t)ay + xß(t)a'1) + (,(t)az  Yi'(t)a'1)
+ Ca (tat  \)a'1  2pap) + C7 (xaz + yay + 2\)8'1 + pap)
A. The cases where C6 = C7 = 0 are exactly as in the general case.
B.1. Ca i= O. Adjoint reduction gives Va + CV7
r\) =t2cil(.. JL)
t" , t"
_ t 2 (Z y)p  gte' te
f _\72 I  c(ç\72 Ie + r¡\72 ITI) + J(I, \72 I) = ge
1. (c  2)g  c(çge + r¡gTl) + J(I, g) = 0
2. Va' alone (c = 0)
J \) = f(:,y)
1 p  g(z,y)
 t2
f \72 I  J (I, \72 I) = gz
1. 2gJ(I,g) =0
C.1. Ca = 0, C7 i= O. Adjoint reduction gives Cat + V7
f \) = e2ct I(xect,yect)
1. p = ect g( xect , yect)
f c(ç\72 Ie + r¡\721T1)  J(I, \72 I) = ge
1. c(g  çge  r¡gTl) + J(I, g) = 0
2. V7 alone
f \) = xyf(!., t)
1. p = VX ;(;, t)
Table 4. Similarity forms and similarity differential equations for inviscid case.
361
2 Shear Flow, Waves,.and Instabilities
In this section we briefly solve special cases of one of the simiarity differential equation
sets of the general case and interpret the solution. We examine the system A.2. of Table 2
which is rewritten for convenience below:
"'feet  "'aif.leee + klee + iJ
9t  aif.ge + c (;7 Ie + øf. )
ge c 2
   + Pr", leeee
, , (10)
= "'9ee (11)
where "', 8, ai, and ø are also given in Table 2 A.2. We first neglect diffusion by setting
the right most terms to zero and also set c = O. Recalng the definition of ai, we see (11)
becomes
d
9t  dt In(ß,)f.9e = O.
The method of characteristics suggests the new variable
r¡ = ß,f. = ß(t)x  ,(t)y (12)
which is like a transformation to Lagrangian coordinates. If we now use r¡ instead of f. as a
simiarty variable so that the similarity forms are
W  ~(~X2_;y2)+/(r¡,t)
p = c(~+ ;) +9(r¡,y)
and insert these into (4) and (5), then the similarity differential equation set becomes
a(ÆI'1'1 + 8) ß9'1  !: + PrÆ2/'1'1'1 ' (13), ,
ø
9t + c(r¡ ß, + 2/'1)  Æ9'1'1 (14)
where Æ = ß2 + ,2. Again neglecting diffusion, we now consider two special cases of this
system.
Setting c = 0 for the first case, we have
8d"'I7171 + 8) ß9'1
9t 0
which we can directly integrate. The result is
w = 21_ ((x2 _ y2)(ß,) + xy(ß2 ~ ,2)) _ G(ß(t)x _,(t)y) ¡t ß'" '"
H(ß(t)x  ,(t)y) A(t)(ß(t)x  ,(t)y)+  + '" '"
P G'(ß(t)x  ,(t)y)
362
where G, H, and A are arbitrary functions of one argument and we have neglected an
arbitrary function of time in the stream function q; which gives no physics. We now set the
arbitrary functions A = G = H = 1 = 0 which gives
q;  azy
p _ G'(zeiit)
which describe a shear :How.
If we now restrict ß and 1 to be constants, (13) and (14) become
'itf'1
c
 ßg""1 (15)
(16)gt + 2cf"  0
If we now differentiate (15) w.r.t. t and (16) w.r.t. 17, we have
fee,. = À 2 f"", where À2 = 2ßcit (17)
which we can integrate to
q; A(ß(t)z 1(t)y)e;\t + B(ß(t)z 1(t)y)e;\t + n(t)(ß(t)z 1(t)y) (18)
P  ~ (A'(ß(t):z 1(t)y)e;\t  B'(ß(t):z 1(t)y)e;\t + jt n(t')) + ~y (19)
where again we have neglected an arbitrary function of t in q;. If we adjust the arbitrary
functions A, B, and D properly, they wi be zero when t =, O. Thus
2c
py = .ß (20)
Now if (20) ��( 0 then as one moves upwards, density decreases. One would expect this to
be a stable configuration. (20) ��( 0 by (17) means the system (18) and (19) wi exhbit
waves. (20) :: 0 simiarly shows possibilties of instabilty.
Appendix The Essentials of Adjoint Reduction by
Example
Here we show how to proceed with part of the adjoint reduction of the general generator of
the twodimensional convection system. In particular we show how to go from the general
generator
v = Ciât + c2âp + e(t)â'l + (r¡(z)ây + :z~(z)â'lJ + (((z)â:i  y((z)â'l) (21)
where e, 17, and ( are arbitrary functions of t, to Ôt + câp via the adjoint table, given for al
cases studied in Table A. We wil refer to the first five lines of its top two tables.
363
V.3
Vi 8t 8p ç(t)8.¡
8i 8t 8p e&8c ç 8.¡
, 8 8t 8p ç 8.¡p
a( t)8.¡ 8t  sëx8.¡ 8p ç 8.¡
ß(t)8y + zß(t)8.¡ 8t  s(ß8y + zß8.¡ ) 8p ç 8.¡
¡(t)8z  yy(t)8.¡ 8i  s( y8z  Y'r8.¡ ) 8p ç 8.¡
t t8t  'i8.¡  Ap8p e&8t eA&8 e&(1+t8c)ç8.¡ )p
t z8z + y8y + 2'i8.¡ + Bp8p 8t B&8 e2&ç8.¡ )e p
Vi r¡(t)8y + z1j(t)8.¡
V.3
((t)8z  y((t)8.¡ ,
8i
8p
a(t)8.¡
ß(t)8y + zß(t)8.¡
¡(t)8z  yy(t)8.¡
t t8t  'i 8.¡  Ap8 p
t z8z + y8y + 2'i8.¡ + Bp8p
e&8c r¡8y + ze&8c 1j8.¡
r¡8y + z1jl)
r¡8y + z1j8.¡
r¡8y + z1jl)
r¡8y + z1j8.¡ + s( r¡". )8.¡
e&t8c r¡8y + ze&8c (t1j)l)
, e& (r¡8y + z1j8.¡)
e&8c (8z  ye&8c (8.¡
(8z  y(8.¡
(8z  y(8.¡
(8z  y(8.¡  s(ß()8.¡
(8z  y(8.¡
e&t8c (8z  ye&8c (t()8.¡ )
e&((8z  y(8.¡) )
Vi
Vj
t8t  'i8.¡  Ap8p == V6 z8z + y8y + 2'i8q, + Bp8p = Vr
8t
8p
a(t)8q,
ß(t)8y + zß(t)8q,
¡(t)8z  yy(t)8q,
t8t  'i 8q,  Ap8p
z8z + y8y + 2'i8q, + Bp8p
V6 + s8t
V6  s8p
V6  Sea + tëx)8q,
V6  s(tß8y + (tß)z8q, J
V6  s(ty8y + (t~)z8q,J
V6
V6
Vr
Vr  s8p
Vr + s2a8q,
Vr + s(ß8y + zß8q, )
Vr + s( ¡8z  yy8.¡ )
Vr
Vr
Table A. Adjoint table.. General case: omit third table and lines enclosed by tbrackets ��.
Pr ~ 1 case: A = 1, B = 1. Inviscid case: A = 2, B = 1.
364
After first recalng the definition of the adjoint of two generators (2), one should then
study the relevant portion of Table A payig particular attention to where the adjoint
parameter appears. Havig famiarzed ourselves with the adjoint table, we reason as
follows. Firstly, assume ci =f o. Let us drag v by the arbitrary generator ß(t)8y + zß(t)8il
where ß is an arbitrar function of t.
. iAd(ß8y + zß8il, ~v)
= (a  s(ß8y + z,a8il)) + c8p + e8il + (n8y + z7j8il) + (((8z  y(8il)  sUi()) (22)
= 8t + c8p + (e ~ sCå()) + (n8y + z7j8il  s(ß8y + z,a8il)) + ((8z  y(8il), (23)
where in (22) we have used the linearty of Ad(.,.) in its second argument, have let c = C2/Ci,
and also have absorbed Ci into e, n, and (. Now in (23) let us set the arbitrary function
ß = Y = ¡t n (so /3 = n). If we now set the adjoint parameter s = 1 and cal ¿ = e  (Ý(),
then v dragged by Y 8y + z Y 8il, is '
v = 8t + c8p + l8il + (8z  y(8il.
. tIf we now drag v by Z8z  yZ8il where Z = J ( we have
v = a + c8p + ¿8il.
And if we drag v by B(t)8il where B(t) = ¡t a And adjust s accordingly, we have our
result. Note that the results are unchanged if either e, n, or ( were O. The other optimal
generator for the general twodimensional convection equations is obtaied in a simar
manner, but with Ci = 0 in (21). Finaly, let us remark that it is helpful to have computed
the commutator table before computing the adjoint table.
References'
Olver, Peter J. Applications of Lie Groups to Differential Equations. SprigerVerlag GTM
107, 1986.
Salon, Rick. "A Fluid Mechancist's Introduction to Lie Symmetry Groups and Partial
Differential Equations." GFD Special Seminar, 1990. '
,Schwarz, Fritz. "Symmetries of Differential Equations: From Sophus Lie to Computer
Algebra," SIAM Rev. 30 (1988) p. 450.
Acknowlegements
Thanks to Rick Salon for his encouragement and assistance throughout this project. I
thank him for teaching me the theory and technique. His intuition amazed me many times,
and his facilty with calculation impressed me throughout our collaboration. Thanks also to
Steve Chidress both for suggesting this problem and for intermittent discussions along the
way. He also seemed to "know" what was interesting. Thanks to the other GFD Fellows
especialy Stefan Linz and to the other staff members for making this a truly firstclass
summer. And finaly, thanks to the EastmanKodak Company for generously supporting
me through this early part of my graduate studies, including this summer.
365
"H
1;
lt
~:
DOCUMNT LmRAY
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50272101
REPORT DOCUMENTATION 11. REPORT NO.PAGE WHOI.91.03
4. Title and Subtitle
1990 Summer Study Program in Geophysical Fluid Dynamics
STELLAR FLUID DYNAMICS
2. 3. Recipient's Ace_Ion No.
5. Report Date
October 1990
6.
7. Author(s)
Rick Salmon, Director, Edited by Barba Ewing DeRemer
9. Performing Organization Name and Addresa
8. Perfrming Organization Rept. No.
WHOI9103
10. ProjecaskIork Unit No.
Woods Hole Oceaogrphic Institution
Woods Hole, Masachusett 02543
11. Cotract(C) or Grant(G) No.
(C) OCE 8901012
(G)
12. Sponsoring Organization Name and Address
National Science Foundation
13. Type of Report & Period Covered
Technica Report
14.
15. Supplementary Notes
This report should be cited as: Woos Hole Oceaog. Inst Tech. Rept, WHOI9103.
16. Abstract (Limit: 200 words)
The 1990 program in Geophysical Fluid Dynamics had as its speial topic "Stellar Fluid Dynamics". Introductory lectues by
Edward Spiegel and JeaPaul Za paved the way for more speialze seminas on solar oscilations, neutron stas, stellar winds,
solar convection, and flows with strong magnetic fields. As usual, the lectues raged far beyond the speial topic of the summer,
with GFD fillng its tritional role as a clearng house for information between the varous fields that shar an interest in
rotating, differentiallyheated flows. Under the supervision of the staf members, our nine student fellows completed original
reseach projects. Their report appe in the 1990 volume, along with the lectue notes of Spiegel and Za, and summares of the
other lectures.
17. Document Analysis a. Descriptora
stellar fluid dynamics
geophysical fluid dynaics
sola magnetohydrynaics (M)
b. Identifiers/OpenEnded Terms
c. COSATI Field/Group
18. Availabilty Statement 19. Seurit Class (This Report) 21. No. of Pages
369
22. Price
Approved for public release; distrbution unlimited.
20. Seurity Class (This Page)
(See ANSIZ39.18) Sse Instructions on Reiiert OPTIONAL FORM 272 (477)
(Formerly NTIS35)
Department of Commerce
~)