Problems of Stellar Convection: Proceedings of the Colloquium Nr. 38 of the International...

Lecture Notes in Physics Edited by'J. Ehlers, M0nchen, K. Hepp, ZLirich, R. Kippenhahn, MLinchen, H. A. WeidenmiJIler, Heidelberg, and 1 Zittartz, K61n Managing Editor: W. Beiglb6ck, Heidelberg

71

Problems of Stellar Convection Proceedings of the Colloquium Nr. 38 of the International Astronomical Union, Held in Nice, August 16-20, 1976

ml

Edited by E. A. Spiegel and J. P. Zahn

Springer-Verlag Berlin Heidelberg New York 1977

Editors

Edward A. Spiegel Astronomy Department Columbia University New York, New York 10027 /USA

Jean-Paul Zahn Observatoire de Nice Le Mont Gros 0 6 3 0 0 Nice/France

ISBN 3-540-08532-? Springer-Verlag Berlin Heidelberg New York ISBN 0-387-08532-7 Springer-Verlag New York Heidelberg Berlin

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PREFACE

This volume constitutes the proceedings of Colloquium N ° 38 of the

International Astronomical Union, held at the Nice Observatory during the week of

August 16-20, 1976.

The scientific organizing committee was composed of L. Biermann,

F.H. Busse, P. Ledoux, B° Paczynski, E.A. Spiegel (chairman), R. Van der Borght,

N.O. Weiss and J.P. Zahn, They decided to adopt a format of general reviews followed

by discussion and informal contributions, much more in the spirit of a workshop than

in that of a classical colloquium. For this reason, the number of participants was

limited to about fifty~ but particular care was taken to represent a wide range of

interests and ages. It was also agreed that papers submitted for publication in the

proceedings, other than the invited reviews, should be refereed.

The colloquium was funded by the Centre National de la Recherche

Scientifique, whose Directeur Scientifique J. Delhaye was of great help, by the Comit~

National Fran~ais d'Astronomle, the City of Nice and the Nice Observatory. The Inter-

national Astronomical Union provided travel grants for young astronomers. Most parti-

Cipants were accomodated at the Centre Artistique de Rencontres Internationales, thanks

to p. Oliver and Ch. de Saran.

The local organization lay in the competent hands of D. Benotto

and R. Petrini. The social events were highlighted by a visit of the Music Chagall

under the guidance of its curator P. Provoyeur~ and followed by a concert given by the

Trio de Freville whose violonist, M.E. Mclntyre, was also an active participant of the

colloquium. R. Zahn took care of the ladies' prograrmne.

These proceedings were put together by D. Benotto and R. Petrini,

and D,O. Cough carefully checked them in their final form.

To all those quoted above, to the many others who also contributed

to the success of the meeting and to the editors of the Springer Verlag, we express

our warm thanks.

Jean-Paul Zahn

CONTENTS

Intr___oductory Remarks

E.A. SPIEGEL ..................................................................

I. M____ixing-Length Theory

- "Historical Reminiscences of the Origins of Stellar Convection Theory",

L. BIERMANN ................................................................... 4

- "The Current State of Mixing-Length Theory",

D. GOUGH ...................................................................... ~5

- "On Taking Mixing-Length Theory Seriously",

9.~M~H and E.A. SPIEGF~ ..................................................... 57

- " Observations Bearing on the Theory of Stellar Convection",

E. BOHM-VITENSE ............................................................... 63

II.___ Linear Theory

- "Dynamical Instabilities in Stars",

P. LEDOUX ..................................................................... 87

lll____~.Observational Aspects

- "Observations Bearing on Convection",

K.H. BDHM .................................................................... 103

- "Evolution Pattern of the Exploding Granules" ,

O. NAMBA and R. VAN RIJSBERGEN ............................................... 119

- "Granulation Observations",

A. NESIS ..................................................................... 126

- "Some Aspects of Convection in Meteorology",

R.S. LINDZEN ................................................................. 128

IV_~_NNumerical Solutions

- "Nu merzcal Methods in Convection Theory",

N.O. WEISS ................................................................... 142 - ,i C °

Ompresslble Convection",

E. GRAHAM ........................................ ............................ 151

VI

~t Rotation and Magnetic Fields

- "Convection in Rotating Stars",

F.H. BUSSE ................................................................ 156

- "Magnetic Fields and Convection",

~ . ~ ~ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

- "Axisyn~etric Convection with a Magnetic Field"~

D.J. GALLOWAY .............................................................. 188

- " Convective Dynamos",

S. CHILDRESS .............................................................. 195

VI. Penetration

- "Penetrative Convection in Stars",

J.p, ~ ................................................................. 225

- "The BDundaries of a Convective Zone",

A. MAEDER ................................................................. 235

- "Convective Overshooting in the Solar Photosphere;

a Model Granular Velocity Field",

A. NORDLUND ................................................................ 237

VII. Special To~ics

- "Thermosolutal Convection",

H.E. HUPPERT .............................................................. 239

- "The URCA Convection",

G. SHAVIV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255

- "Photoconvec tlon",

E . A . ~ ............................................................... 267

- " Convection in the Helium Flash",

A.J. WICKETT ............................................................... 284

VIII. Waves

- "Wave Transport in Stratified, Rotating Fluids",

M.E. Mc INTYRE ...........................................................

- "Wave Generation and Pulsation in Stars with Convective Zones",

W~ UNNO ..................................................................

290

315

VII

IX. Turbulence

- "Fully Developed Turbulence, Intermittency and Magnetic Fields",

U. FRISCH .... • ............................................................... 325

- "Turbulence : Determinism and Chaos",

Y. POMEAU .................................................................. 337

X. Appendix

- "Stellar Convection",

D.O. GOUGH ................................................................. 349

INTRODUCTORY REMARKS

E.A. Spiegel

Astronomy Department

Columbia University

New York, N.Y. 10027 U.S.A.

As president of the organizing committee of this meeting I was granted the honor of

opening the conference. But despite appearances I was only a figurehead that Jean-Paul

Zahn somehow decided to set up, Whatever hismotivatlon was, his execution was excellent

and my first remark must be an expression of my admiration for the marvelous job that

he and his associates have done in providing all the spiritual, intellectual, and

material advantages that we found waiting for us in Nice. Let me assure you though,

that a token president is not without uses and I wish I had known this before accepting

the job. I spent the days before and during the conference running routine errands,

carrying luggage, and being reprimanded for some of the minor things that inevitably

must go wrong in many large gatherings, I was even scolded because the name of some-

one who had not said he was coming was omitted from the list of participants. And when

the time was running short during the meeting, I was obliged, in a statesmanlike gesture,

to cUt my scheduled one-hour talk to eight minutes. But rank has its privileges and mine

was to be informed of the guiding principles behind the organization of the conference.

~ermit me now to share these with you.

It happened that the first day of the conference coincided with that of a large po-

litlcal convention (in another place,happily), and that suggested a convenient meta-

phor for describing the divergence of viewpoints among the participants. Let us there-

fore discuss the politics of stellar convection theory.

At the extreme right of the convective political spectrum are those who want to

write down the full equations and solve them. The ultra-conservatives, as I shall call

them, have virtue but no results that apply directly to stars.

At the other extreme of the convection spectrum are the radicals who want to write

down an algorithm for computing stellar structure that contains adjustable parameters

which can be fit to well known cases. In an extreme version of this we would write :

R = II where R is the radius and R± is an adjustable parameter. If we fit the parameter

to the sun we get R = 7 x 10 ]I cm and the resulting formula turns out to describe a

large number of stars tolerably well. I think it is fair to say that no one at the con-

ference was this radical, but it would be hard to deny that there have been things in

the literature that have these overtones.

But let me come to the political views represented by the actual participants. I

cannot be too specific since many participants have sometimes yielded to expediency and

shifted ground shamelessly. Nigel Weiss is a case in point. This paragon of the right

has recently (with Gough) written a paper on stellar mixing-length theory in what must

be the greatest fall from grace in recent memory. Having viewed this behaviour with

alarm let me point with pride to the spectrum of opinion represented here.(The infrared

has been filtered out.)

In these proceedings we have a coverage of this spectrum from mixing-length theory to

computations on the full equations ( for limited parameter ranges ). Naturally, this

represents more than most astrophysicists need to know about convection. Some will merely

read this introduction, expecting to find out where the best current approach is des-

cribed, hoping that this will be consistent with the constraint that results are to be

found in a finite time ........... say months, I have,of course, anticipated this need,

but am not sure I can meet it.

Douglas Gough and I spent the summer in a Cambridge drought trying to prepare a

statement that will answer such a specific question. Naturally, I didn't have time to

give the results of our lucubrations in my spoken introductory remarks. Nor did Gough

manage to fit them into his lecture on standard mlxing-length theory. That does not

mean that we could not put it all into one of our manuscripts. But which paper should it

be in? The solution is that we have prepared a joint appendix which I am told will appear

somewhere below.

Our conclusion is that a non-local mixing-length theory seems to be the best that

one can do at present. Unfortunately, this is not a precise a~tement and we simply

give an outline of how such an approach might be made and try to give an indication of

the physical assumptions needed. There are other ways to go about this, and our aim is

merely to suggest the level of sophistication in mixing-length theory that we think may

be warranted in stellar models.

I have indicated the spread in the approaches to convection discussed below as a

kind of abscissa. There is also an ordinate which represents a spectrum of complications

that arise in convection theory in specific kinds of stars on stages of evolution, or

refer to effects that are usually presented but ignored in first approximation. If we

must look mostly to the left to get usable results for stellar structure theory, it is

equally true that we usually turn to the right for guidance about how to handle these

special effects. For even if the solutions of the conservatives are not directly usable

for stars, they can be extended to include compressibility, rotation, magnetic fields,

compositional inhomogeneities, penetration, and, if we would just take the trouble, coup-

ling to pulsation. The hope is that what a special effect does to a conservative's solu-

tion it will probably do to a radical's model. This half-truth in practical terms means

that by seeing what rotation does to Boussinesq convection in two-dimensional or modal

convection, you may build enough intuition to make a cogent argument about what it does

to stellar convection. For example, when stellar model-builders want to decide what to

do about semiconvection, let them read Huppert's article on thermohaline convection. No

doubt many astrophysicists will not care for this general viewpoint unless it happens

to lead to answers according with what they need to coax their models into agreement

with observations. Eric Graham's discussion is a good case in point.

Graham has numerical solutions for fullyc~mpressible three-dlmensional convection

in a layer several pressure (and density) scale heights thick. Apart from a charming

tendency to swirl aboutj his flows look startlingly llke Boussinesq convection, and

he detects no sign of scale heights influencing his dynamics. Radicals will probably

ignore this result. What else can they do?

Lest I seem to give too much credit to the conservatives let me point out their

main fault : they rarely include effects in their calculations that are motivated by

purely astrophysical convection problems, but rather study traditional effects. If they

want to prove me wrong about this let one or more of them do a proper Boussinesq calcu-

lation of the URCA convection problem suramarlzed here by Giora Shaviv. This example does

not have the double entendre of something like rotation that interests meteorologists

also. So much for the ordinate.

In these proceedings we shall also leave the phase I have been describing to have a

look at recent trends in turbulence theory. Those who have followed this subject at all

know that it too has something of a political spectrum and some of the extreme conser-

vatives of turbulence report here on current approaches. Urlel Frisch will translate the

right wing's latest credo, fractal dimensions, into terms the leftist can understand,

and Yves Pomeau will tell us about aperiodic oscillations. These both refer to forms of

mathematics that may help us to see what turbulence is. Pomeau's talk is concerned with

systems of o.d.e.~s that give periodic solutions except in certain parameter ranges

where they go into aperiodic, almost random hehaviour, The suspicion has been around

for many years that this behaviour may have the mathematical ingredients that give tur-

bulence its stochastic features and, lately, attempts to formulate this idea precisely

have been mounted. But even if this does not turn out to work, it does not hurt to know

about aperiodic oscillators in other contexts. The funny behaviour of the solar cycle

during the reign of Louis XIV may have been a manifestation of such an aperiodic

oscillator of interest to this audience.

This has been a lengthy introduction yet it has not told you the full range of topics

to be covered. I hope that it gives you a flavour of what to expect in looking over the

proceedings. I am told that all the contributed papers have been refereed and so the pro-

lixity stops here. There is not even a concluding oration to be reported. Of course, I

happen to have a manuscript called "Convection in Stars III..." that might have served,

but that is destined for other things. However, a brief summary of developments before

this meeting is in Gough's report for IAU Commission Mestel and it is reprinted here

with bibliography. Its adequae~ as a summary may be a measure either of the rate of pro-

gress in this subject or of Gough's perspicacity.

HISTORICAL REMINISCENCES OF THE ORIGINS OF

STELLAR CONVECTION TKEORY

(1930- 1945)

Ludwig Biermann

Max-Planck-Institut fur Physik und Astrophysik

Munich, Germany

To set the stage for the report to follow +) , let me start with a

quotation from A.S. Eddington's classical work "The Internal Con-

stitution of the Stars", published exactly half a century ago.

Eddington wrote: "We shall not enter further into the historic problem

of convective equilibrium since modern researches show that the hypo-

thesis is untenable. In stellar conditions the main process of trans-

fer of heat is by radiation and other modes of transfer may be

neglected." (paragraph 69, p. 98).

Only 19 years earlier Robert Emden had outlined for the sun a very

different picture; in paragraph 4, chapter 18, of "Gaskugeln" - also

a classical treatise which influenced research well into the thir-

ties - he had stated with equal conviction that the energy radiated

into space from the photosphere could be brought there almost ex-

clusively only by convection, and that the granulation depicts the

cross section of the ascending hotter and the descending darker

currents. Emden's manuscrip£ had been read in the proof stage by his

brother-in-law Karl Schwarzschild, who advised Emden on a number of

points. It seems worth noting that Schwarzschild's somewhat earlier

work on the radiative equilibrium of the sun's atmosphere was discussed

in Emden's monograph (in chapter 16, par. 13) ++) .

+) The text includes a number of points, which actually came up in the later discussions during the colloquium or, in one case, during the IAU assembly in Grenoble. The author is indebted to many colleagues, in particular to D. Gough, L. Mestel, M. Schwarzschild and N. Weiss for important comments.

++) Whether Schwarzschild, who had proven that radiative transport prevailed in the sun's photospheric layer and had formulated the quantitative criterion for the stability of such layers, was in complete agreement with Emden, is not quite clear, though Emden's text does convey this impression.

Part of the change of the scientific argument during the intervening

time (1907-1926) was of course due to Jean's and L~demann and Newa]]'s

discovery that at the pressures and temperatures prevailing in the

stellar interiors, all the molecules and atoms would be broken up into

almost bare atomic nuclei and electrons and that, as a consequence,

radiative transport was recognized to be most efficient, in Eddington's

scheme of 1926 almost too efficient. In retrospect it looks however

as if Eddington, in laying the foundations of the theory of radiative

equilibrium in stars, did apparently not fully appreciate the power

of even very slow turbulent convection in transporting energy in

stellar interiors; that he neglected the possible influence of the

surface conditions was perhaps related to the fact that they were,

at the time when Eddington wrote his book, essentially inapplicable

to most stars. +)

This review will be concerned mainly with three developments, which

took place between 1930 and 1946: I) the application to stellar in-

teriors of the convective transport equation developed in hydro-

dynamics, which led to the proof that the adiabatic temperature

gradient is, in the case of thermal instability, a very good appro-

ximation there; 2) the influence of the surface boundary condition in

determining the extent of outer convection zones, which in certain

circumstances may comprise the whole star; 3) the introduction of the

scale height as a measure of the mixing length used in the transport

equation. Their connection with some further developments which may

come up during the present conference can be sketched only very

briefly.

To begin with item I, we note first that observations and measure-

("turbulent) mass ments of transport processes by non-stationary • "

motions in the earth' atmosphere and in the oceans had led, between

1915 and 1925, to a reasonably successful theoretical scheme. G.I.

Taylor (1915) and W. Schmidt (1917) noted++)that the quotient of the

+) Concerning both points it is instructive to reread his discussion of the point source model (ICS, § 91). It should be added, however, that Eddington (as far as the present author was able to find out) fully accepted the change of position which occurred during the period under review in this report. ++) Apparently, as a result of the war conditions, independently, as can be judged from their papers of 1915 and 1917.

flux of some quantity - like the momentum, the salinity or heat - and

the gradient of the content per cm 3 or per gram of the same quantity

led to consistent values of this coefficient, for which W. Schmidt

introduced the term "Austausch" (gr cm-lsec-1). In the special case

of heat flow ("Scheinleitung") the excess of the actual temperature

gradient over its adiabatic value, A?T, multiplied with the specific

heat for constant pressure, has to be used, because only small pressure

differences arise if the Mach number is <<I, as is usually true in the

earth~atmosphere. For interpreting the observed va~s of the "Austausch"

concepts developed by G.%. Taylor and by L. Prandtl n in particular the

"mixing length" were most useful.(These had been introduced first for

the case of dynamical instability.)These concepts relied to some ex-

tent of considerations analogous to those of kinetic theory. The

analogue of the molecule of kinetic theory is an element of the fluid,

which (having detached from its surroundings as a consequence of the

given instability due either to a super-adiabatic temperature gra-

dient or to the dynamical situation, for instance shear) moves as a

whole over some distance until it mixes again with the surrounding

fluid, in Prandtl's original thought the mixing length i was determined

by the geometry of the situation, e.g. the distance from the nearest

boundary or the diameter of the unstable region, and this was used

in the first application to stellar interiors. The somewhat later idea,

to link the mixing length with the scale height, will be discussed

below.

TO determine the velocity it was considered that pressure equilibrium

is only slightly disturbed, whereas - the flow of heat by conduction

or radiation being usually much slower - the temperature of an ele-

ment of the fluid is determined by the adiabatic gradient, such that

in case of thermal instability a rising element is less dense and

hotter than the surrounding fluid, and as a consequence is accelerated.

These considerations led t~ an expression for the velocity (v) of the

moving elements, which can be written in the form (g = acceleration

of gravity)

2 AT g£2 A?T v ~ gi • --~ T

For the convective transport of heat (H K = erg/cm2sec, Cp = specific

heat per gr for constant pressure, p = density) wewrite

H K ~Cp •piv • AVT

v 2

~pv • ~ R

(p - pressure =(R /D)pT; R - gas constant, ~ - mean molecular weight),

suppressing again a constant of order unity.

In meteorology the largest scale for which the transport equation has

been used with at least qualitative success (by A. Defant, see

W. Schmidt 1925), is the meridional transport of heat from the equa-

torial to the polar regions, which raises the average temperatures

of our own latitudes quite noticeably, the meridional heat flux ex-

ceeding the "solar constant" by roughly two powers of ten (as it must).

In this case the elements are low pressure systems which, due to their

rotation, have a certain stability; it happens, that the mixing length

and the pressure, but of course not the temperature and the density,

are comparable to those in the outermost layers of the sun's hydrogen

convection zone.

The application to stellar interiors was contained in a G6ttingen thesis

of 1932 (L. Biermann 1933), which owed a great deal to Prandtl's ad-

vice. For the convective zone around the centre, due to the conversion

of hydrogen into helium (supposed to be highly temperature sensitive

as is true for the C-N cycle), the mixing length was taken to be of

the order of 10.OO0 km, and the flux to be transported assumed to be

of the order of 1012erg/cm2sec, about 15 times that on the sun's sur-

face. For the third power of the velocity a combination of the equa-

tions given above leads to

3 HKg£ R* v p

which was found to be or order 1011 (cm/sec) 3. This then leads to

IAVTI ~ 10 - 8 or ~IO-51VTI

Though there is some incertainty about the coefficients of order

unity contained in the equations given above, it is clear that the

result last stated, that a relatively very small excess over the adia-

batic gradient is sufficient to carry all the flux, is quite insensi-

tige to errors made by the phenomenological theory used here. Two

years later Thomas G. Cowling, who used a slightly different approach

and formalism, recovered the same result. Since 1933/34 it has become

standard practice to use, in theoretical models of star's interiors,

the adiabatic temperature gradient whenever that required for radiative

transport exceeds the adiabatic one +) . In this form the stability cri-

terion was formulated first by Karl Schwarzschild for the solar atmo-

sphere (1906); in meteorology an equivalent criterion had been in use

already many years earlier.

Concerning item 2, we note first, that during the years 1934-38, an

attempt was made to explore the question whether partially or wholly

convective stellar models (not only such with a central convection zone)

lead to a more complex picture of the overall constitution of the stars

than the one given by Eddington (as was finally found to be the case).

Such models would result from a higher luminosity than the radiative

ones, for the same radius and opacity if the luminosity could be re-

garded as an (effectively) free parameter. Near the surface two cir-

cumstances, the second of which was not at once fully appreciated have

to be taken into account: the existence, in all stars with not too high

surface temperature, of convective zones due to the partial ionization

of hydrogen and helium, which had first been investigated by Uns~id in

1930, and (second) the necessity of using in the photosphere of every

star the radiative transport equation together with that of hydro-

static equilibrium, which means that the pressure must be of order

g/c, < being the opacity (gr-lcm 2) of the photosphereic layers - a

relation which of course could be written down with better accuracy.

The importance of this boundary condition for the case under discussion

(L. Biermann 1935) was emphasized by T.G. Cowling in 1936, but its

application was held back for some years by the (then) poor knowledge

of the value of the photospheric opacity for all stars later than

spectral class A.

+) ~The first such model being Cowiing's point source model of 1935.

Shortly before the discovery by Rupert Wildt that the negative hydrogen

ion is the main source of the photospheric opacity in such stars, an

attempt was made (L. Biermann 1938) to use approximate values for the

photospheric opacity based on work of Pannekoek which fortunately led

to approximately correct answers. It was found (a result which was

soon confirmed on the basis of Wildt's pioneering work of 1939, and

subsequent work of others) that in the surface of the sun and similar

stars (with photospheric pressures of the order of IO5dyn/cm 2, such

that the radiation pressure is only ~ 10-5p) convection was likely

to be an efficient mechanism of transport of heat not only in the

main parts of the hydrogen convection zone, but also in its outer

layers; as a consequence the adiabatic gradient should at least appro-

ximately be established in the hydrogen convection zone up to its

outer boundary in the middle or deep photosphere.

In order to i l l u s t r a t e the importance of this result le t us look at a diagram taken from the author's paper of 1938. This diagram, with the

logarithm of the total pressure (P = p+pR ) and of radiation pressure

(pR) as coordinates, shows besides the adiabats ( f u l l l ines +) ) the

lines of constant ratio PR to P (Eddington I-8, weak lines), further-

more the "dominant" ionization potential ~++) and lines for fixed

ratio of ~ to kT, of which the one for ~ = 0 corresponds to the limit

of degeneracy ("Entartungsgrenze"). Near the bottom are shown the

photospheric values of the pressure (for given effective temperature)

for three values of the surface gravity, including that for the sun.

The crossed line marked

log ad

indicates the zone of rapid increase (inwards) of the opacity and as a

consequence of the radiative temperature gradient, and the incipient

decrease of the adiabatic gradient (inwards) due to the additional

degrees of freedom (using the terminology of the kinetic theory of the

+) The branching above logT ~ 5 corresponds to then existing uncertainties regarding the chemical composition, especially the value of Z (in the terminology in use now).

++) For which the degree of ionization is ~I/2 according to Saha's formula.

J. !

! ..

...

!

i I

J ...

. !

! |

,i

!

~r

,q

~ i.

.i

~

"V

%

11

specific heat) resulting from the increasing degree of ionization. Above

= 50 eV, the ratio of the radiative gradient to the adiabatic gra-

dient

(d log PR/d log P)R

(d log PR/d log P)ad

is essentially given by the value of 1-B, such that for the special

solution corresponding to Eddington's model, for given mass and lumi-

nosity, and for larger values of I-8, the radiative gradient is smaller

than the adiabatic one and Schwarzschild's stability criterion is ful-

filled. For smaller values of 1-8 the radiative gradient increases,

such that the instability limit is reached soon and the adiabatic

gradient (d log PR/d log P ~ 8/5 for stars of about the sun's mass,

for which I-8 <<I) is the smaller one and convection prevails. +)

It is thus see~)that the surface boundary condition effectively results

in such a relation between PR and P that the total pressure increases

inwards at first considerably faster than the radiation pressure; in

the regions with temperatures around some 104 degrees, I-B has a mini-

mum and the radiative gradient is much larger than the adiabatic one,

the opacity being approximately proportional (I-8) -1 according to

Kramers' law (in that temperature region aotually still larger), where-

as the adiabatic gradient is approximately constant (~ 8/5). With the

increase of I-8 along the adiabates inwards, the stability limit is

gradually approached and in the case of the sun finally reached at

T ~ 106 (with modern values of the chemical composition Eddington's

model would show I-8 ~IO -3 for the sun). It is therefore clear that

in the sun and in similar stars the outer convection zone must comprise

a substantial number of scale heights until the radiative temperature

gradient decreases below the adiabatic one, such that the inner boun-

dary of the hydrogen convection zone should be at a temperature

T ~> 106 and a depth of ~ 1OO OOO km. This result has been confirmed

by the much more accurate computations of recent years. Dwarf stars

of later spectral class should have still deeper convection zones,

+)The presentation attempts to retrace the steps, by which the complete stellar models with deep outer convection zones and the fully convective models were actually found. For a more detailed review see L. Biermann 1945.

12

and it was proposed in 1938 that late type giants might even be fully

coDvective, a result which was recovered many years later by Hayashi.

In retrospect, it is easily seen why all these possibilities had not

been noticed earlier: that convective transport could be efficient up

to almost photospheric layers, was a rather remote possibility on the

background of the earlier theory - though not on that of Emden - and

a quantitative discussion of the power of convective transport was

hindered by the lack of reliable knowledge of the photospheric pressure

(which had been highly underestimated before 1938/39).

The largest uncertainty in transferring the mixing length theory to

astrophysics - our item 3 - is evidently connected with the value of

to be used, under the different circumstances. For the work done in

1938 described above the observed size of the solar granulation had

been taken as a guide; this choice +) whioh fortunately did not intro-

duce serious errors was until 1943 replaced by the answer which has

been the basis of all subsequent work, and which is to use the local

scale height, defined either by the density gradient or that of the

pressure, as measure of ~, such that £ is given:

I I ~Av-~J or Iv--f~j"

Since there should be a nondimensional factor of order unity, which re-

quires separate d£scussion, the two expressions are under most circum-

stances equivalent. The idea behind this choice is that in any case the

largest elements should travel farthest and reach the highest velocity

but that an element of the fluid, after having travelled over a density

scale height, should have changed ~ts shape to such an extent that it

is likely to break up into smaller fragments and to mix with the sur-

rounding fluid. This approximation is of course precisely in the spirit

of Prandtl's original ideas on the subject, and had most probably been

discussed with him. On a slightly different background, an equivalent

proposal had been made by E. 0pik already in 1938~ +)

+)An at least approximate determination of the size distribution function of the solar granulation became possible recently (J.W. Harvey, M. Schwarzschild, 1975), whereas for red giants the observational situation is still less clear (M. Schwarzschild, 1975).

++) The work reported here under items 1) and 2) had remained unknown to ~pik until his work of 1938 was completed, cf. his "Note added in proof" (~pik 1938). For a more recent review of the general problem see M. Schwarzschild, 1961).

13

To conclude these reminiscences I would first like to mention that the

results discussed under item 2), led for the sun to a proposal concern-

ing the old question, why a sunspot is dark, the answer being, that the

strong magnetic fields observed in the umbra should inhibit the convec-

tive transport in the layers underneath the spot+); it has been pursued

by a number of authors up to the present and may come up again at this

colloquium.

Of the various formalisms suggested to improve upon the one given above

for the heat transport, the scheme proposed by E. B~hm-Vitense 1953, 1970

1958 became the most widely used. Much more recently R. Ulrich has pro-

posed still further refinements, which are particularly useful at rela-

tively low densities. All attempts to determine the exact relationship

between the mixing length and the scale height (and/or other parameters)

have not been really satisfactory so far, though comparisons with obser-

vational data on the integral properties of stars (including their

chemical composition) and on the position of the instability strip in

the Herzsprung-Russell Diagram have been used with some success; only

for the sun its known age provides an independent parameter and thereby

a check that the mixing length theory leads to essentially reliable re-

sults (D.O. Gough, N.O. Weiss 1976). It seems that only a deductive

theory of stellar convection would offer the chance to go beyond the

present essentially phenomenological approach used hitherto; at least

one contribution at the present colloquium will, I understand, deal

with this problem.

+) L. Biermann 1941.

14

References

Biermann, L., 1933, Z. Astrophys. ~, 117,

Biermann, L., 1935, Astr. Nachr. 257, 269.

Biermann, L., 1938, Astr. Nachr. 264 , 395.

Biermann, L., 1941, Vierteljahresschrift Astron. Ges. 76, 194.

Biermann, L., 1943, Z. Astroph. 2_22, 244.

B~hm-Vitense, E., 1958, Z. Astrophys. 46, 108.

Cowling, T.G., 1935, Mon.Not.R.astr. Soc. 96, 18.

Cowling, T.G., 1936, Astron. Nachr. 258, 133.

Eddington, A.S., 1926, Internal Constitution of Stars

Emden, R., 1907, "Gaskugeln"

Gough, D., Weiss, N.O., 1976, Mon.Not.R.astr.Soc. 176, 589.

Harvey, J.W., Schwarzschild, M., 1975, Ap.J. 196, 221.

Opik, E.J., 1938, Publ. Obs. astr. Univ. Tartu 30, No. 3.

Prandtl, L., 1925, Zeitschr. f. angew. Math. u. Mech. ~, 136.

Prandtl, L., 1932, Beitr. Physik freier Atmosphere 19, 188.

Schmidt, W., 1917, "Der Massenaustausch bei der ungeordneten Str6mung

in freier Luft und selnen Folgen", Kalserl. Akad. d. Wiss., Wien.

Schmidt, W., 1925, "Der Massenaustausch in freier Luft und verwandte

Erscheinungen", Verlag von Henri Grand, Hamburg.

Schwarzschild, K., 1906, Nachr. KSnigl. Ges. d. Wiss., G~ttingen, No.1.

Schwarzschild, M., 1961, Ap. J. 134, I.

Schwarzschild, M., 1975, Ap. J. 195, 137.

Taylor, G.I., 1915, Phil. Trans. R. Soc. Lond. A, 215, 1.

Taylor, G.~., 1932, Proe. R. Soc. Lond. A, 135, 685.

Ulrich, R., 1970, Astroph. & Space Science ~, 71.

Ulrich, R., 1970, Astroph. & Space Science ~, 183.

UnsOld, A., 1930, Z. f. Astroph. !, 138.

Vitense, E., 1953, Z. Astrophys. 32, 135.

Wild, R., 1939, Astroph. J. 9_O0, 611.

THE CURRENT STATE OF STELLAR MIXING-LENGTH THEORY

Douglas Gough

Institute of Astronomy & Department of Applied Mathematics

and Theoretical Physics, University of Cambridge

SUMMARY

The basic assumptions of the mixing-length formalism are described, and the

theory is developed with a view to representing convection in stars. Directions

in which the results might be improved and extended are indicated.

I. INTRODUCTION

Aside from some recent pioneering work by Latour, Spiegel, Toomre and Zahn

(1976 a,b), the mlxing-length formalism in one or other of its guises remains

the sole method for computing the stratification of convection zones in stellar

models. Little attention is usually paid to assessing the accuracy of the models,

partly because there is a general feeling that mixing-length theory is so un-

certain that the task would be fruitless, and partly, perhaps, because of an

optimism that the theory will soon be superseded by something better. There

appears to be no better convection theory emerging that migh= be applicable to

stars in the foreseeable future, however; the mixing-length is likely to stay

with us for some time. It is perhaps time, therefore, to take stock of the

situation, and to ask whether the methods currently employed can be made more

reliable.

The first stage of any enquiry of this kind must be a definition of the

physical model upon which the theory is based. What started as little more than

an order-of-magnitude estimate of turbulent transport processes has subsequently

16

been taken rather literally in some contexts. It is therefore important to

appreciate what the assumptions are, and where the uncertainties lie. Only after

that can there be some hope of improving the representation of the physics. And

as a byproduct, one might see how best the theory might he extended to describe

more general situations than those for which it is customarily employed in stellar

physics.

But perhaps most important of all is to appreciate the degree of contact

with reality. Astronomical verification of the mixing-length prescription is at a

very primitive level, and permits only a poor assessment of the validity of the

functional forms of the formulae describing the convective transport processes.

2. THE IDEAS BEHIND MIXING-LENGTH THEORIES

The mixing-length idea was introduced independently by Taylor (1915), Schmidt

(19|~ and Prandtl (|925) to provide a means of understanding the transport of vor-

ticity, heat and momentum in turbulent fluid. By analogy with gas kinetic theory the

fluid is considered to be composed of turbulent 'eddies', 'parcels' or 'elements'

which advect properties such as heat, in the case of thermal convection, and

vorticity or momentum, in the case of shear turbulence. An element arises as a

result of instability, with about the same properties as its immediate environment.

It travels with a characteristic speed ~ through a mean-free-path or mixing

length { , and finally breaks up because it becomes unstable itself, and merges

with its new surroundings. This breakup into smaller scales of motion is

considered to be instantaneous. It is the mixing-length description of the

beginning of the turbulent cascade; velocity components of the consequent small

scale motion and the associated temperature fluctuations are assumed to be un-

correlated so there is no contribution from them to the overall transport of heat

and transverse momentum. From such a description it is a straightforward matter

to estimate the mean heat flux or shear stress in terms of ( , ~ and the

structure of the mean environment. To complete the theory a procedure for

obtaining ~ and ~ must be found.

In the case of shear flow Prandtl (1925) assumed the turbulence to be more

or less isotropic and so equated the velocity ~'- perpendicular to the mean motion

to the velocity fluctuation in the mean flow direction induced by the shear.

Prandtl assumed the turbulent elements to be momentum conserving and obtained an

expression for the shear stress in the form of a product of the mean velocity

gradient and a turbulent transport or exchange coefficient ~u~ (Austausch

coefficient) where ~ is density. Thus in his form of the theory, turbulent shear

stresses (Reynolds stresses) behave like viscous stresses with the Austausch

coefficient being a sort of turbulent viscosity, a concept that had been discussed

previously by Boussinesq (1877). The Reynolds stresses take on a somewhat

17

different form if it is assumed that turbulent elements conserve vorticity (Taylor

1915, 1932) and sometimes this yields better agreement with experiment (e.g.

Prandtlp 1952). In either case it is only the mixing length ~ that remains un-

determined.

In the case of free convection there is no externally imposed velocity scale

as in shear flow, and it is necessary to consider the dynamics of the turbulent

elements in greater detail. This can be done only after the mixing-length model

is mere preclsely defined. During its existence a turbulent element is accelerated

by the imbalance between buoyancy forces, pressure gradients and nonlinear

advection processes. In addition it can gain or lose mass by entrainment or

erosion. As a result of ignoring different combinations of these processes,

approximating the remaining ones in slightly different ways and making slightly

different assumptions about the geometry of the flow, different physical models

have emerged. They all predict similar heat transports when the mean atmospheric

structure is time independent, which is hardly surprising because the formulae

can be obtained from barely more than dimensional reasoning. As a consequence,

the differences between the physical assumptions are not usually emphasized in

the astrophysical literature, perhaps because it is difficult to differentiate

astronomically between rather gross variations in the functional form of the

turbulent heat flux.

It was pointed out by Prandtl (1926) in a discussion of turbulent shear flow

that~in the absence of a driving forcejturbulent drag would cause an element of

characteristic size ~ to lose its kinetic energy after travelling a distance of

about ~ . This is simply because turbulent drag at high Reynolds number is

proportional to the square of the velocity, and hence also to the kinetic energy.

Thus if the mixing length represents both the element size and the mean-free-path

it is inm~aterial whether one postulates unimpeded motion followed by instantaneous

annihilation, as would be natural by direct analogy with gas kinetic theory, or

continuous momentum exchange between the element and its surroundings. This led

to the first and perhaps the simplest description of the dynamics of thermal

convection : namely an exact balance between buoyancy force and turbulent drag

(Prandtl, 1932). Convective elements are assumed to achieve this balance

instantaneously, which implies that their inertia is unimportant. They move

through a distance ( comparable with their own diameter, conserving their heat,

and then instantaneously mix their excess heat with the new surroundings. These

ideas were applied to stellar convection by Biermann (1932, 1937, 1943) and

Siedentopf (1933 a,b, 1935).

The model can be made more consistent by assuming interchange of heat between

the element and its surroundings to be continuous toop as was emphasized by ~pik

(1950). Then heat and momentum exchange are treated similarly. Since there is

always an exact balance between buoyancy force and turbulent drag, and between the

18

rate of increase of a temperature fluctuation and the diminution of that

fluctuation by heat exchange, it doesn't matter where the element came from, end

the mixing-length description of annihilation of elements can be dispensed with

entirely. This is not the case, however, when the star is pulsating (Unno 1967,

Gough 1977).

The alternative approach of considering elements to he accelerated

adiabatically from rest by buoyancy alone was adopted in the later papers by

Biermann (e.g. 1948 a,b). Pressure forces and turbulent drag can be incorporated

approximately into the dynamics without changing the functional forms of the

equations used, though they introduce different factors of order unity. In more

recent work that includes radiative heat exchange (e.g. Vitense 1953, BShm-Vitense

1958) it is common to ignore turbulent exchange during the life of an element and

invoke instantaneous breakup to account for all the nonlinearities that occur in

the equations governing the turbulent fluctuations.

3. EQUATIONS OF MOTION

To simplify the presentation attention will be restricted to a plane parallel

fluid layer. It will be assumed that horizontal averages, which will be denoted by

overbars, are independent of time and that there is no mean mass transport through

the convection zone. The horizontally averaged momentum and total energy equations

can then be written

dz

. . . . . ~ . 1 ' (3.2)

where z is the vertical co-ordinate of a Cartesian system (x, ~, z ), ~, ~,

are gas plus radiation pressure, density and specific enthalpy; ~ = ( uL, ~r u~r )

is the fluid velocity, ~t is the z,r component of the viscous stress [ and ~z

is the vertical component of the radiative energy flux F~ which will he assumed,

again for simplicity, to be given by ~e diffusion approximation

= - K ~ Z T , (3.3)

where T is temperature and K = ~.c'Ty3Xj~ , ~ being the radiation density

constant, c the speed of light and ~ the Rosseland mean opacity. Perturbations

in the gravitational acceleration ~ = ( O, o,-~ ) have been ignored. Equation

19

(3.3) can be replaced by a more general though local representation of radiative

transfer, such as the Eddlngton approximation (Unno & Spiegel 1966), without

adding unduly to the complexity of the analysis; adding a nuclear source term to

equation (3.2) or casting the problem in spherical geometry introduces no new

conceptual difficulties.

Equations (3. i) - (3.3) must be supplemented with a continuity equation and

an equation of state. The system is then completed with formulae for the

convective fluxes. It is these that the mixing-length theory must provide.

In studying the dynamics of convection it is usual to separate all quantities

into mean (horizontally averaged) and Eulerlan fluctuating parts, as in

p = ?(=) + ] ~ ' ( = , ~ , ~ . ~ ) , (3.4)

where t is time, and to subtract the mean equations from the full equations of

motion from which they were derived to obtain equations for the fluctuations. It

is at this point that serious assumptions are first introduced. Though it is

rarely stated explicitly, in almost all attempts to model stellar convection the

Boussinesq (1903) approximation is used~ this can be justified only when the

scale ~ of the motion is much less than the pressure and density scale heights

of the layer (Spiegel & Veronis 1960, Malkus 1964). In this approximation the

viscous terms and the kinetic energy flux ~?~-~; in equations (3.1) & (3.2)

are neglected which renders (3.2) indistinguishable from the mean thermal energy

equation. The equations for the fluctuations, in this approximation, are

= -- _ ~- ~ -XT?'-~ T ~ , (3.5)

d.iv ~ = O~ (3.6)

20

f = - ~ - T ~ , (3.8)

where,with the exception of fl, all mean quantities are considered to be constant

over the scale ~ of the motion. In these equations c~ is the specific heat at

constant pressure, ~ : - ( ~ [ . f l l i . T ) ~ and

az ./~G az . (3.9)

Moreover, the convective momentum and heat fluxes in equations (3.1) and (3.2)

simplify to

(3. I0)

~-- fh~r ~ rE? ~ T i . (3.11)

Quantities such as X and ~ are considered to be functions of the thermodyn~nical

state variables f and T whose fluctuations are related by (3.8). The pressure

fluctuation appears only in the momentum equation, and has no thermodynamical

significance. Indeed it can be eliminated by taking the double curl of equation

(3.5), the vertical component of which, after use of (3.6), becomes

V -~ V O~- ~, a~,,U - V,~T " = O , (3.12)

where

u - ( u . , u : , u . ) - ~ . v ~ - L~.v .~ , (3.~3)

~. ~/~ , ~ ~/~ ~d V;~V ~-~. The viscous stress has been omitted from the momentum equation (3.5), and

hence from (3.12). This is justifiable in stars because the Reynolds number

characteristic of the largest convective eddies, which are the only motions

21

treated explicitly in stellar mlxlng-length theories, is large. The continuity

equation (3.6) indicates that dynamically the fluid behaves as though it were

incompressible; thus the occurrence of acoustic waves is prohibited. Density

fluctuations that do arise serve only to provide buoyancy. Equation (3.7) is the

fluctuating thermal energy equation, and not the equation for fluctuations in

total energy.

It should be emphasized that the Boussinesq approximation is not an essentlal

component of the assumptions of mixing-length theory. Other less restrictive

though more complicated approximations to the equations of motion could be used,

such as the anelastlc approximation (Ogura and Phillips 1962, Gough 1969) which

holds for low Mach number convection in deep layers of gas. Like the Boussinesq

approximation it filters out the possibility of acoustic waves, whilst retaining

some of the features of compressibillty. No attempt to develop a mixing-length

theory with consistent use of such an approximation seems to have been made.

4. LOCAL MIXING-LENGTH FORMALISMS FOR A STATIONARY ENVELOPE

In the Boussinesq approximation~mean thermodynamical state variables are

considered to be constant over the assumed scale ~ of the motion. Another

approximation, common to the formulation of most mixing-length theories used in

stellar structure computationsp is to treat the superadiabatic lapse rata ~ in

the thermal energy equation as though it were constant. Though in practice it is

found that this approximation is poor, because at the top of the hydrogen

ionization zone in particular ~ varies on a scale much shorter than { , it is

usually retained because it brings great simplicity to the mixing-length formulae.

It permits the heat flux and Reynolds stress to be expressed at any level in the

envelope solely in terms of the mean conditions at that level. For this reason

the resulting theories are called local.

Formulation under the assumption of balance baleen buoyancy and turbulent drag

Most mixlng-length descriptions are £ormulated in terms of rising and falling

fluid parcels having typical radius ~ ~ and which at any instant can be

characterized by a single vertical velocity ~ and temperature fluctuation T"

In the early discussions by Prandtl, Biermann and Siedentopf the parcels were

presumed to travel at their terminal veloclty, buoyant driving being balanced

exactly by turbulent drag. Thus in the vertical component of the momentum

equation (3.5) the time derivative was essentially ignored and the nonlinear

advection terms were replaced by a drag force of the form u~/~{) . If the

pressure fluctuation is ignored the coupling between vertical and horizontal

motion is removed, and a relation between the vertical speed and the temperature

fluctuation results!

22

From here on overbars are omitted from mean quantities where no ambiguity results.

Initially it was common to consider the fluid motion to be adiabatic. Equation

(3.~), with the right hand side set to zero and

integrated along the trajectory of the parcel.

distance the parcel moves~one obtains

~.VT' ignored, can then be

Taking ~ to be constant over the

(4.2)

where ~ is the vertical displacement of the parcel above its point of origin.

The constant of integration is zero since T' is assumed to be zero at ~ = O. A

typical parcel at any instant might have travelled say half the distance I , so a

typical temperature fluctuation 8 and velocity um can be obtained from (4.1) and

(4.2) by setting ~ - ~. Noting furthermore Lhat parcels with T'>O rise

and parcels with Tk O fall, the heat flux ~ can be estimated by replacing

with ~-~ . Then

- - - ¢

The Reynolds stress may be estimated in a similar way to be

(4.4)

The numerical factors in front of these formulae vary from paper to paper,

because the precise definition of ~ and in particular the relation between parcel

size and mean-free-path is not universal, and because factors of order unity can

be introduced to account for effects of pressure fluctuation or imperfect

correlation between ~ and T" .

Kinetic theory of .acceleratin~ fluid elements

The alternative approach is to imagine the fluid parcel to accelerate from

rest. It is usual then to ignore the nonlinear terms in the momentum equation.

The influence of pressure fluctuations can be estimated by working from equation

(3.12), and introducing typical horizontal and vertical waven~bers by setting

V, =-k and z. = -- ~= This is perhaps not quite as crude an

approximation as one might first imagine, because these relationships are

satisfied by the convective eddies of linear stability theory whose visual

23

appearance is not wholly dissimilar to the eddies of intensely turbulent

convection. The linearized form of (3.12) then becomes

where

- = o , (4.5)

----- I "*' k,,/k, (4.6)

The only difference between equation (4.5) and what one would have obtained from

the linearized vertical component of (3.5) with the pressure fluctuations ignored

is the factor ~ The pressure fluctuations divert vertical motion into

horizontal flow, thereby decreasing the efficiency with which the motion might

otherwise have released potential energy. The effect in this approximation is

simply to increase the apparent inertia of the vertically moving fluid, without

changing the functional form of the equation of motion. In some derivations

equation (4.5) is obtained directly from (3.5), the factor ~ being introduced by

analogy with the virtual inertia of a body moving in a potential flow.

When integrating equation (4.5) it is usual to regard the temperature

fluctuation as a function of the parcel displacement ~ , and approximate it by

the leading term in its Taylor expansion. Of course for adiabatic motion

equation (4.2) indicates that the leading term is the only term present. The

operator ~ in equation (4.5) can be replaced by ~/~ without further

assumption~ since in linear theory there is no distinction between Eulerian and

Lagrangian time derivatives of perturbation quantities. The equation can then be

integrated to yield

For adiabatic motion, (4.7) together with (4.2) complete the description of the

dynamics. If typical velocity and temperature fluctuations defined by setting

= -~ are used as before to estimate ~ and p~ , t h e same equations (4.3)

and (4.4) are obtained, aside from factors involving ~ . Note that pnessure

fluctuations could have been incorporated into the original formulation of the

theory by dividing the right hand side of (4.1) by ~ .

Heat exchange between fluid 2arcels~dtheoenVironment

Heat exchange between fluid parcels and their surroundings is most simply

accounted for by treating equation (3.7) in an analagous way to the momentum

equation. Retaining only the leading term in the Taylor expansion of ~(~) in

the linearized version of (3.7) and integrating along the trajectory yields

24

-r' i f - (4.8)

where k ~ = k~* ~ When ~ =~ this is precisely the same relation that

one would obtain by neglecting the time derivative in (3.7) and replacing

~,.VT' - ~ by the estimate ~T'/(~ t) for the turbulent

heat exchange. In deriving this equation the fluctuating part of (3.3) was used;

fluctuations in ~ do not arise because, consistency wi~h assuming ~ constant,

the gradient of ~ is small compared with ~VT'I . Whereas for adiabatic motion

the wavenumbers entered only in their ratio~ in the nonadiabatic theory their

magnitudes are also required for estimating ~u F; . Taking the mixing

length t o be a measure of the vertical extent of the eddy suggests

Proceeding as in the adiabatic case, but with (4.8) replacing (4.2), one is led to

Fo = ¢ ~-"~'l" s-' [-¢',*R's) ''~- I ] ' Kp, ~,~.~o)

[(, ~-~5) '~- ¢ (~r~ / '0 t ~ , (4.~1)

where

~(S/T)p t"" 5 ~ (:K/,,,~)"" (4.1~)

is the product of the Prandtl number and a Rayleigh number based on ~ , and

(4.13)

is a geometrical factor of order unity.

When ~S >>I the convective motion is almost adiabatic and

(4.14) I=o ~, ~ " z ~ S ' / " 1<18 ,

25

which are the same as (4.3) and (4.4) aside from the factors involving ~ . In

the solar convection zone this condition is satisfied everywhere except in

extremely thin layers at the top and bottom~ the latter and sometimes even the

former being too thin to be resolved in normal stellar structure computations.

When 7"5<~ I,

Fo = ~ ~-'/'~' S" ~ , (4.16)

--( 0 ?

which are relevant in substantial regions of some red giant envelopes.

Other formulations

In essence, the derivation presented above in terms of accelerated and

subsequently annihilated fluid parcels is the same as that of Vitense (1953;

Behm-Vitense 1958) whose prescription has been the most widely used for computing

stellar models. Her 1958 formulae for the heat flux imply (4.10) with ~ = 2 and

= J~/~ Vitense also studied the case when fluid parcels are optically

thin, end adjusted numerical constants so that the optically thick and optically

thin formulae gave the same result at unit optical thickness. A smoother and

probably more accurate transition between the two limits can be obtained by using

the Eddington approximation to radiative transfer (Unno & Spiegel 1966, Unno 1967,

Gough 1977).

The derivation in terms of continuous turbulent exchange of heat and

momentum has been adopted by Unno (1967) and is similar to an earlier approach of

~pik (1950) in terms of convective cells. Opik's formula for ~ is

mathematically somewhat more complicated than (4.10)D but it takes similar values

if ~ and ~ are chosen suitably (Gough & Weiss, 1976). The differences between

the values of ~ predicted by Vitense and Unno arise mainly from the different

assumptions about flow geometry and the slightly different constants of order

unity appearing in the approximations to the equations of motion~ rather than from

apparent variances in the physical models.

5. REMARKS ON THE ASSUMED STRUCTURE OF THE CONVECTIVE FLOW

In order to complete the prescription for ~ and ~ the parameter ~ ,

26

which depends on the geometry of the flow, and the mixing length ~ must be

determined. The latter is one of the most difficult parts of the theory. Here

the criteria that might be most important in affecting a choice of ~ are

discussed; the mixing length is postponed to a later section.

What one might consider the most natural choice of ~ depends very much on

the mixing-length model that has been adopted. Thinking in terms of approximately

spherical parcels of fluid rising and sinking through the background medium

suggests adopting the formula for the virtual inertia of a sphere. The particle

is thus considered in isolation, and ~ = 3/2. Different values would be

obtained if the parcel were thought to be aspherical.

An alternative approach is to ass~ne the return flow around a moving parcel

is confined to the immediate environment by the interaction with neighbouring

eddies, so that a relatively compact convective eddy or cell is formed. To

determine ~ the shape of the eddy must be specified. For want of a more reliable

procedure, the marginal or unstable modes of linear theory might be used to

describe the flow in convective cells. This has the computational advantages of

being simple to calculate for local mixing-length theory, and of providing a basis

on which to generalize to nonlocal theories or theories that one might hope to

apply to convection in more complicated situations. Of course it is not clear

what boundary conditions are the best to adopt, but that is unlikely to be

crucial; it is expedient to choose those that yield the simplest solutions. Thus

for the relatively simple theory discussed in the previous section one might set

(cf. Chandrasekhar 1961)

(5.I)

in an obvious n o t a t i o n , where ~ and ~ are s i n u s o i d a l i n z and depend p o s s i b l y

on t, and the planform f depends only on x and y and satisfies

v , " f - - k: f - , I . (5.3)

Equation (4.5) was derived from (3.12) with a flow such as this in mind.

If the mixing-length model is one in which a statistically steady eddy is

maintained, the continual turbulent interchange of momentum and heat may be

regarded as being due to an eddy viscosity and conductivity. The convective

cell is thus like the marginal mode of linear stability theory, for which ~ = 3.

If, on the other hand, the model is one in which the eddy grows and subsequently

breaks up, Spiegelts (1963) suggestion of choosing the most unstable mode is more

27

appropriate. The rationale for this is that it is the most rapidly growing mode

that would dominate the flow. Its shape is approximated by (Gough, 1977)

k~'/k: = ± {I • [ 3 ~ l ( h - ~ ' ~ 5 -t ~) ' /~ ] ' /~ ' } , (s.4)

given that its vertical extent 7T/kz = ~ is fixed. Thus ~ varies monotonically

from 2 to i as S varies from 0 to

Other possibilities come to mind, such as choosing the mode that maximizes

F c , though that results in a shape not very different from the one implied by

(5.4), with ~ varying between 5/2 and I. One might consider averaging over an

ensemble of different eddies, but then one is faced with the problem of choosing

the distribution function.

The flow geometry also enters directly into the mean equation for the hydro-

static support of a stellar envelope. In a plane parallel layer only vertical

transport of vertical momentum matters, but in a star vertical support is provided

partly by horizontal stress. If the star is spherically symmetrical, the Reynolds

stress tensor is axisymmetrical about the vertical and has only two distinct

eigenvalues. Thus in spherical polar co-ordinates the components depend on only

two quantities, which may be taken to he T~ and ~ . The equation for hydro-

static support, which is nontrivlal only in the radial direction, may be written

a (5.5) * = o ,

where r is the radial co-ordinate. If the turbulence is isotropic, ~ = 3 and

the Reynolds stress behaves like a pressure of magnitude F~ .

6. FURTHER REMARKS ON THE DYNAMICS OF CONVECTIVE EDDIES

When the growth and subsequent annihilation of convective elements or eddies

is taken into account, mean transports are usually estimated by using

characteristic values of the velocity and temperature fluctuations. These values

are normally taken to be the actual fluctuations at ~ =~ , when the element

has moved half its mean-free-path; Vitense took a space average over an element's

trajectory, which in local theory is equivalent. The approximation implies

certain assumptions about the creation of eddies, which become apparent as soon

as an explicit attempt to average over all turbulent elements is made. The

discussion that follows is based on the ideas behind Splegel's (1963) formulation

of the theory, though the details are somewhat different from that approach.

A specific mixing-length model

Consider, for example, the evaluation of the heat flux ~ ~ rcl,~--~. It

is convenient to have a specific model in mind, and to this end a flow field of

28

the kind (5.1) - (5.3) will be assumed. Thus the flow is represented by a

conglomerate of cells or eddies that form, grow and subsequently break up. Though

it is envisaged that there is a degree of randomness in the positions at which

the eddies form, each is presumed to stay in the same place during its existence.

An eddy centred at height :~o that originated at time to is thus described by

until the moment of annihilation, the functions W and ~ being determined by the

linearized forms of equations (3.7) and (3.12). If ~(2o,~°~ ~) is the

probability that the eddy has not yet been destroyed, the average of any function

over all eddies at height z. is obtained by weighting that function with

and integrating over all possible times of creation. Thus if eddies have mass ~,t

and are created at a rate ~ per unit volume per unit time, the heat flux at

2 f 7-° is

Fo = ~ c T ! W ® ~ a ~ o . (6.3)

Here all the eddies have been assumed to be centred at z~ where they contribute

maximally to F~ • In principal an average over eddies centred in the range

(Zo-~ , Zo ÷ ~ ) ought really to be taken. In local theories this does no

more than multiply the right hand side of (6.3) by a constant factor of order

~ity.

The growth Of convective eddies

The velocity and temperature fluctuation amplitudes depend on the initial

conditions of the eddy. One of the assumptions of the mixing-length approach is

that turbulent fluid elements originate with the same properties on average as

their ium~diate environment, though in any individual eddy there must be some

deviation from the mean state for otherwise that eddy would have no identity. It

must be assumed that the eddy grew from a non-zero perturbation, but provided the

initial amplitude is much smaller then the average value, the precise details of

the initial conditions are unimportant. In this discussion the conditions that

lead to pure exponential time dependence of ~/ and ~ will be chosen, merely to

simplify the mathematics. Then

29

W ; W . ~

(6.4)

where V~o and ~o are the initial values of k~J and ~ , and are related by

The growth rates 0~ are given by

(6.s)

o-~ ~ ± J v ~ . ~ - - ~ = ~ , ~ - ' s - ' ~ " [ + ( ~ t ~ ' 5 ) ' I ~ - l ] , (6.6)

where ff~ m ~ ~ ~/~ T and ~ ~ kLK/lf%. Note that if W, and ~o are of

the same sign the eddy grows, but if they are of reverse sign it decays.

It is presumed that in the initial perturbations, which arise out of the

smaller scale turbulence resulting from the breakup of both the major heat

carrying eddies and possibly also from larger eddies that make lesser contributions

to the heat flux, the velocity and temperature fluctuations are uncorrelated. Thus

only half the eddies have VJ o and ~ of the same sign and subsequently grow. The

other half make an insignificant contribution to the heat flux and are ignored.

Eddy creation rate and initial qonditions

The expression for the heat flux depends explicitly on the eddy creation

rate n and the amplitudes 9~/o , ~.. These are governed by the background

turbulence. In a statistically steady state, however. ~ can be evaluated from

the statement that the entire fluid volume (or some constant fraction of it) is

occupied by eddies. Thus

%- (6.7)

It is much more difficult to specify the initial amplitudes, and at this point it

will be observed only that if the flow is to be controlled for most of the time by

the linearimed dynamics leading to (6.4), and not by the eddy breakup process.

then ~/. and ~. should be small compared with the average amplitudes. Thus

~x~ (~) >> | , where ~- . ~'+ and "~ is the mean lifetime of an eddy. If r is

30

estimated by integrating ~/ in (6.4) and setting t h e resulting displacement

equal to ~ , this condition becomes

x ~ ~{/Wo >> i . (6.8)

Eddy annihilation hypothesis

Finally it is necessary to obtain ~ , which depends on the disintegration of

eddies. The most natural interpretation of the mixing-length annihilation

hypothesis is that a fluid element is considered to break up as it is displaced

through &~ with probability a~/~ : (W/~)a~ . In other words the element has a

mean-free-path { , and the probability of its annihilation is proportional to the

shear in the eddy and is not explicitly dependent on the details of its past

history. It follows that

.h_ t - - I ~-,(l='-I:.)

: } { ' + oc,, t. ( 6 . 9 )

The tffrbulent fluxes

It is now straightforward to evaluate

terms are ignored, equation (6.7) for the eddy creation rate yields

" , '<~ = o 7 / ( t , < >- - y) ,

where ~ is Euleris constant. Equation (6.3) then becomes

~f c t o" w . e . i- X-'c <:"(': "':') } = .t.(t. 'x-y) I eq,{Z~(t:4:o) 4t:o

¢c~ T t_'-~ "~

- - s-' r(, , - f s ' ' - ? ' K ] [I/).(t,,>,-~,).] ~ '~7 '

F~ If terms 0(~-) of the leading

(6.zo)

(6.11)

(6.12)

the factor 2 in the denominator arising because only half the eddies have positive

growth rates. This expression has the same form as (4.10), and can be made equal

to it by setting

31

~.. s' ~ ~, = e _~ 15. (6.13)

The stress ~ can be computed similarly, and yields

?~ = ([.x-~) ' (6.14)

which reduces to (4.11) when (6.13) is again used for ~ .

Remarks

It is now evident that the rough estimates (4.10) and (4.11) imply a value

for % , which approximates the degree of amplification of the velocity and

temperature fluctuations during the lifetime of a convective eddy. Of course the

precise value (6.13) must not be taken seriously, especially since it was a

obtained by exponentiating/poorly estimated quantity, though one may be tempted to

take comfort in the fact that it is at least reasonably consistent with (6.8). It

would be preferable if some independent method of determining the initial

conditions could be found. It is worth noting, however, that (6.12) and the

corresponding expression for ~ depend only logarithmically on A , so the method

of estimating the initial conditions need not be very accurate. The constraint

(6.5), which was applied only to minimize algebraic complexity, is not an

essential part of the formulation; other relations between Wo and ~o lead to

expressions for ~ similar to (6.12). These also have multiplicative factors

that contain the logarithm of the amplification ~ in the denominator.

The theory can also be formulated in terms of rising and falling fluid

parcels, with the only difference that the integrals in (6.3), (6.7) and (6.9) are

now considered to be evaluated along the parcel trajectories. In the local

approximation the two approaches are identical.

The discussion in this section has not led to any modification in the

mixing-length formulae, lts purpose has been to highlight the role of the eddy

creation process in determining ~ and ~k .

7. THE CHOICE OF MIXING LENGTH

Expressions (4.10) and (4.11) for F= and ~ are both increasing functions

of ~ . Since the philosophy of the mixing-length approach is to concentrate on

only one scale of motion at any level in the fluid, namely the scale that

contributes most to the fluxes~ the largest value of ~ compatible with the

dynamics must be chosen.

32

When modelling laboratory flows the choice of [ seems straightforward. The

largest eddy that one can imagine is determined by the geometry of the vessel

col%raining the fluid. Thus at any point it is usual to take

= ~ x , (7.1)

where ~ is the distance to the nearest boundary of the container and ~ is a

constant scaling factor of order u~ity which is determined by comparison of theory

with experiment.

Of course the dynamics of the flow could be such as to prevent this largest

conceivable eddy from growing, so that ~ is determined mainly by other factors.

Thus von K~rm~n (1930) suggested that for turbulent shear flow the mixing length

should be taken to be a multiple of a scale length of the mean shear. It seems

that for most laboratory applications this yields results that agree with

experiment less well than (7. i).

A stellar convection zone is not bounded by a rigid container, and must

therefore determine its own length scales. But just how ~ should be specified is

not clear. It is most common to follow yon K~rm~n's philosophy and choose

(7 .2 ) I[ - , ~ H ,

where H is a scale height of the mean stratification, though some stellar models

have been Computed using the lesser of (7.1) and (7.2) (IIofmeister & Weigert 1964;

B~hm & S=~ckl 1967). 0plk (1938) took H to be a scale height of density. This

choice has been favoured also by Biermann (1943), and by Schwarzschild (1961) who

argued that it is the distortion of expanding or contracting fluid elements as

they experience substantial changes in mean density that limits the size of an

eddy. This reasoning is not Boussinesq, and introduces some representation of

the effect of compressibility into the prescription. Vitense (1953) set H to a

pressure scale height. This has been widely used since, presumably for reasons

of computational convenience.

It is unfortunate that the attempt to incorporate compressibility into the

local theory results in a choice of ~ that does not reduce continuously in any

natural way to the kind of value that is favoured for laboratory applications.

Stellar convection zones are often many scale heights deep, which is currently

unattainable in the laboratory, but it would have been encouraging had some

experimental verification of the theory been feasible, even if it were in a

parameter range inappropriate to astrophysics. Only astronomical checks are

available at present, but these appear to provide little support for the details

of the theory.

33

8. cALIBRATION OF THE HEAT FLUX FORMULA

Stellar models are usually computed using equation (4. I0) for the convective

heat flux, with (7.2) defining the mixing length. The Reynolds stress T~ is

rarely included. The constant ~ is calibrated by comparison with observation

and then taken to be a universal constant, though some authors do not bother with

this nicety on the grounds that the mlxing-length formulae are too unreliable for

the calibration to be meaningful. The determination of ~ can be affected either

by constructing a solar model with the correct luminosity and effective temperature

or by comparing the slope of a theoretical lower main sequence with observation.

The two methods give results in reasonable agreement with one another when H is a

pressure scale height, though they are subject to considerable uncertainty. In

particular, uncertainties in the solar composition are reflected in the

calibration, as are uncertainties in the opacities, equation of state and nuclear

reaction rates. The results depend also on assumptions concerning the mixing of

material that has been processed in the core. In addition to these, additional

uncertainty is introduced by inaccuracies in the numerical techniques, whose

existence is indicated by discrepances between the results of different

invest igato r s.

It should be noticed that the conclusion =< -~ I does not satisfy the

condition ~ << I upon which the Boussinesq approximation depends. The theory

is therefore not internally consistent. However ~ is not much greater than

unity, and the effects of compressibility may be insufficient to modify the heat

flux severely. More serious is the local approximation that regards ? as being

approximately constant over the length scale ~ It is one of the aims of non-

local mixing-length theories to rectify this flaw.

The calibration of ~ rationalizes the astronomical data, but it does not

provide a test of the mixlng-length theory. The reason is partly that convective

envelopes of solar type stare are approximately adiabatically stratified every-

where except in thin transition regions above the hydrogen ionization zone. The

sole function of the convection theory in determining the gross structure of the

star, therefore, is merely to prescribe which adiabat characterizes the bulk of

the convection zone. This depends on the jump in temperature across the

transition region, but that hardly depends on the detailed functional form of the

expression for Fc • Indeed if ($.10) is replaced by (4.16), even though qL5 >> I

throughout almost the entire convection zone, the solar calibration requires

o( = 1.35 x 10 -3 when ~ and ~ take the values implied by BShm-Vitense's (1958)

formulae (Gough & Weiss 1976). The resulting solar model is barely distinguishable

from that computed in the usual way. In red giants there are regions where

q~5 < J , but the envelope models are insensitive to details of how the two

asymptotic limits (4.14) and (4.16) meet.

34

The calibration of ~ does not even determine both asymptotic limits, since

there is considerable uncertainty in the geometry of the convective motion. The

discussion in §5 suggests that ~'/~'does not vary greatly, but the possible range

for ~ is very wide. This influences only the limit (4.16)~ which hardly matters

for solar type stars. One might anticipate, however, that a calibration of q

would be possible by comparing theoretical red giants with observation.

An investigation of the sensitivity of red giant evolution to changes in the

constants in the mixing-length theory has been reported by Henyey, Vardya and

Bodenheimer (1965), and interpreted in terms of the asymptotic limits (4.14) and

(4.16) by Gough and Weiss (1976). Plausible variations in q do induce

noticeable changes in the evolutionary tracks on the H-R diagram, but it appears

that other uncertainties in both the theory and observation at present prohibit

calibration of q by this method.

A potentially more sensitive test for ~ might have been a measure of the

maximum depth of penetration of the convection zone. In some red giants the

convection zone extends deep enough to mix the products of the nuclear reactions

to the surface. The extent to which the convection zone has penetrated in such a

star could be determined in principle from observations of 017/O 18 ratios in red

giant atmospheres (Dearborn & Eggleton 1976). Computations by D.S.P. Dearborn

and myself, however, have revealed that such a test would probably not provide the

required information, for at its maximum depth the convection zone of a red giant

envelope is adiabatically stratified almost throughout, and the heat flux is

determined by (4.14). But this does not necessarily imply that there is nothing

to be learn~. Conditions may be sufficiently different in red giants that a

recalibration of (4.14) would yield a different result. This might shed some

light on the variation of ~ , whether density or pressure scale heights are

appropriate, or whether (7.2) is even a useful assumption.

9. THE REYNOLDS STRESS

It is common practice to ignore the turbulent stress ~e = I~£ in the mean

momentum equation (3.1). One reason, perhaps, is that to justify the Boussinesq

equations upon which mixing-length theory is based the convective velocities must

always be substantially less than the sound speed. This implies ~e << ~ '

However, it is the derivative of ~ that appears in (3.1), and if la?~/arl is

evaluated in a stellar model that has been computed without that term, it can be

the case that it exceeds Id~/=[rl by a considerable degree in the transition

region at the top of t3ae convection zone, even though ~ might be small compared

with ~ . Another reason for ignoring Te is because to do so removes

singularities from the equations of stellar structure and thus makes them much

easier to solve.

35

The equations, in the plane layer approximation, are

(9.1)

and

aT

The heat flux and Reynolds stress are given by (4.10) and (4.11) in terms of ~ .

Equations (9.1), (9.2) and the equation (3.9) defining ~ can be rewritten as a

system of first order equations for ~e, ~ and T thus:

(9.3)

- ( 9 . 4 )

aT = l

(9.5)

Here ~ and ~ are regarded as functions of Tt The usual stellar structure

computations are governed by the spherical analogues of (9.1) and (9.2) with T~

ignored, which is of one order lower than the system considered here. Solutions

of (9.3) - (9.5) are to be sought satisfying T~ = O at the edges of the

convection zones, where F/K - ~$/c~ also vanishes.

To analyse the nature of the singularities it is sufficient to consider the

structure of equation (9.3) near a boundary of a convection zone. Since S -~ 0

as the boundary is approached, the asymptotic forms of (4.10) and (4.11)

(9.6) Fo ~ A t'~ )

?~ ~ t 6 ~ --, ( 9 , 7 )

may be used, where A and B are nonvanishlng functions. If =he origin of z is

taken to be at the base of the convection zone, and all the coefficients in (9.3)

are expanded in a Taylor series about the origin, only the leading terms being

38

retained, the resulting equation is found to have the structure

am (9.8)

where M, N and Q are positive constants. In deriving this it was assumed that the

mixing length does not vanish at the boundary of the convection zone. The

singularity introduced by the Tt 'f~ term is best analyzed by setting ~4, z -- ~ and

writing (9.8), with the small last term neglected, as a linear system in terms of

a new independent variable ~ :

~ (9.9)

The coe f f i c i en t s ma t r i x of the r i g h t hand side has eigenvalues

" N "- J ± N ~ + ~ M ~ (9.1o)

which are real and of opposite sign, indicating a saddle point at the origin.

A similar analysis may be performed at an upper boundary. The resulting

equation for ~i has the same structure as (9.8), except that now the coefficients

M and N arenegative. Provided N ~" >/ ~ IMJ , both eigenvalues ~,~are real and

of the same sign, indicating that the origin is a nodal point, though if N ~ < EIMJ

the solution is a spiral that cannot satisfy the condition that ~ vanishes at the

boundary of the zone. This latter situation might arise if too large a mixing

length is chosen.

I~ is because the upper singularity is either a node or does not permit a

physically acceptable solution that inward n~nerical integrations from the

atmosphere of a star cannot be successful. Stellingwerf (1976) has pointed out

that an outward integration might workD and has presented a solution to a simple

model problem. Realistic stellar envelopes can be computed in this way only if

the convection zone is thin; otherwise a more stable nm~erical procedure must be

adopted.

Attempts to include T~ in realistic stellar envelopes have been made by

Henyey, Vardya and Bodenheimer (1965) and by Travis and Matsushlma (1971). In

both cases the structure equations were simplified in a manner tantamount to

ignoring the heft hand side of equation (9.3)~ thereby reducing the order of the

differential system and removing the singularities. Henyey et el. anticipated

that this approximation was not serious. Unpublished computations by Baker,

Gough and Stellingwerf of RE Lyrae envelopes with shallow convection zones using

37

the full system of equations revealed that at least in those stars the effect of

~ is not profound. Its inclusion smooths out the region near the top of the

convection zone~ so that d~/ar remains smaller in magnitude than ~/~ ,

and has little influence on the remainder of the convection zone.

iO. REFINEMENTS AND'GENERALIZATIONS

The discussion in ~6 demonstrated how ~ and y~ depend on the growth rate

of convective eddies. This dependence was emphasized by Spiegel (1963), who

also showed how the expressions are modified when viscosity is considered.

The averaging procedure used to derive (6.11) and (6.14) does not depend on

the precise nature of the turbulent flow. The description of the breaking up of

eddies is not refined enough to distinguish between the different circumstances

to which the theory might be applied. Detailed descriptions of the dynamics is

confined to eddy growth, and is contained in the expression for ~ . It is to

this that refinements and generalizations are most easily made.

Transport by small-scale turbulence

As an illustration, an attempt will be made to incorporate into the dynamics

the exchange of heat and momentum by smaller scale turbulence that was ignored in

6. It will be assumed that turbulence on a scale smaller than the heat

carrying eddies is isotropic, so the transport might be roughly represented in

terms of a scalar eddy diffusivity

= k" . (lO.1)

where ~' is a characteristic velocity and k' a characteristic wavenumber of the

background turbulence. This diffusity will be taken to be the same for both

momentum and heat. Its value is related to the velocity and length scales of

the major eddies, whose disruption seeds the small scale motion, and may be

rewritten

-- ~(_~.~)" '~k- ' -- ~ k " ( ~ - ' O '#"

( / r )

(lO.2)

where ~ is o f order unity and depends on the spectrum of the turbulence. It is

likely that E is only weakly dependent on the amplitude of the convection and

can probably be safely asstuned constant. This expression can now be incorporated

into the expression for the growth rate of a disturbance in a viscous conducting

38

fluid (e.g. Spiegel 1963):

, r = - I I, (lO. ~)

where ~ and ~ are the effective thermal diffusivity and kinematic viscosity:

= ~ ~ K/~c? , ~ = ~. (10.4)

Equations (10.2) - (10.4) define a growth rate ~ which can be substituted into

(6.11) and (6.14) to obtain equations for ~ and ?~ The prescription is

algebraically more complicated than the previous formulation which led to (4.10)

and (4.11), though its effect can be approximated by simply multiplying the value

of ~ obtained previously by the factor {I ~ ~e~ (~ -~"/~"

It is perhaps not surprising that the modifications to the results hardly change

the functional dependence of ~ and ~ on S, because the two extreme approaches

discussed in ~ 4 led to the same formulae. The new results may be no better than

(4.10) and (4.11), because the attempted improvement to the representation of the

physics may be insignificant compared with the errors that remain. It should be

noted, however~ that the modifications cannot simply be absorbed into the

definition of ~ .

The small scale turbulence not only influences the dynamics of the larger

eddies but also contributes directly to the fluxes. The heat flux can be

accounted for by replacing K by fc~ in the equation for the radiative flux.

The relevant Reynolds stress component must be augmented by ~f ~; which can

be written as [?~ , where ~ is yet another undetermined parameter of order

unity that depends on the spectrum of the turbulence.

Other refinements can be included, such as a representation of entrainment

and erosion of eddies, or the generation of waves. The former has been considered

by Ulrich (1970a), who used the meteorologists' model of convection based on

rising thermals. D.W. Moore and Spiegel (unpublished) considered the influence of

acoustic generation by convective eddies, and found that this noticeably reduces the

turbulent velocities when the Mach number is of order unity. Generation of gravity

waves with wavelengths comparable with Z , which occurs at the boundaries of

convection zones, probably requires a nonlocal theory for an adequate description.

Further refinements are discussed by Spiegel (1971).

Convection in s lqEly rotating stars

Aside from suggesting improvements to the standard theory, this approach can

be used to formulate mixing-length theories for more general circumstances.

Rotation or a magnetic field, for example, can easily be incorporated into the

39

stability analysis that determines ~ . If the convection zone is rotating, the

maximally contributing eddies are rolls aligned with the horizontal component of

the rotation rate /~ (e.g. Chandrasekhar 1961). Their growth rate is determined

by

(~÷~)~- p~ ~'(~-,)n ~} ÷ ~ = o, (lO5)

where 11 is the vertical component of J~ . Only if _(I is small might one reasonably

hope to obtain meaningful results by just using this growth rate in the normal

mlxing-length formulae, since the effect of the rotation on eddy disruption has

been ignored. In that event the solution to (10.5) can be approximated by

-, ~ -' ~ '/~ (10.6)

and ~ is given in terms of it by (6.11). Note that J~ measures the local

rotation in the vicinity of the eddy, and should therefore be interpreted not as

the angular velocity but as half the vorticity of the mean flow.

It is more difficult to calculate the Reynolds stress. The rotation

introduces a degree of order to the turbulence that destroys the axisymmetry of

the stress tensor and rotates its principal axes. Provided -(I ~ << ~ the

effect is small and for the purposes of computing the hydrostatic structure of

the star can no doubt be safely ignored. Equation (6.14) can be used for ~

with (r determined by (10.6). But this approximation is not good enough for the

horizontal components of the mean momentum equation, since the relatively small

off-diagonal terms in the stress tensor generated by the rotation are important

for determining the angular momentum transport by the turbulence. It is

straightforward to construct a Reynolds stress tensor from the eigenfunctions of

linear stability theory, but in the absence of experimental tests it would be

most unwise to rely on it.

.Influence of a magnetic field

A magnetic field ~ can be treated similarly, provided its turbulent

distortion may be considered random and does not lead to organized concentrations

such as sunspots. Once again the turbulent motion is most efficient as rolls,

aligned with the horizontal field, and the growth rate is determined by the

equation obtained from (10.5) or (10.6) by replacing 4.~-'(~-0Jl ~ by

ir~/{ ~ , where ~ is the vertical component of ~ . The caveats

concerning the Reynolds stress mentioned in connection with rotation apply here

too.

40

Nuclear react!one and composition sradients

The interaction between nuclear reactions and convection is of particular

interest when reaction timescales are comparable with ~-'. This can be the

case in late stages of stellar evolution. The fluctuations in energy generation

rate induced by the convection influence the eddy dynamics through modifications

in both the temperature and chemical composition of fluid elements. The

convection influences the nuclear reactions not only via ~ and ~e , but also

by transporting the products of the reactions,

The mixing-length theory can be generalized as before. Variations =~ in

the abundances x t of elements £ must now be taken into account when calculating

both the buoyancy and,of course, the energy generation rate £(T, T, ~ per unit

mass that must be introduced into equation (3.7). The amplitudes W and ~ are

now determined by

~ w ( 'ST" 0 (lO.7)

(lO. 8)

' defined in a way analogous to Vv' and @ in where X~. is the amplitude of =¢~ ,

(6.1) and (6.2), and

The summation convention is being used. The abundances are determined by

where ~ measures the rate of production of ~ . The linearized fluctuation

equation derived from (10.9) can be combined with (IO.7) and (10.8) to yield the

following equation for e- :

-' ~ = 0 (lO.lO)

where

41

R 5 ~- Cf'f=j ~.r,'r ' 1Z"r "- 5W .~,,,~

and ~j is defined by

(O '~ , j - l~,j - T P . , . j . j / I ) S j~ = ~ , , -

where ~j is the Kroneeker delta.

Overbars on mean quantities have been omitted as usual. Once again Y~ and i~

are given in terms of 0-by (6.11) and (6.14).In addition one can easily derive

for the flux of =c 6 , which is in the m direction,

(10.11)

chemical The first term represents the transport that arises solely because~elements are

created or destroyed at different rates in upward and downward moving fluid. The

second term represents turbulent mixing, though that too is influenced by the

reactions and is not a simple scalar diffusion.

In the special ease when there are no reactions 5~3 ~ 0-" ~U '

and

F~ = - (~-f)-' T~ ~-Z ' ( l o . 1 4 )

which leads to a simple diffusion equation governing the mean abundance ==~ .

Convection in pulsating envelopes

Application of the mlxing-length formalism to stellar pulsation is somewhat

more complicated than the examples considered above, because now the time

dependence of the coefficients in the fluctuation equations must be taken into

account. Additional assumptions must also be made. 0nly a few brief remarks are

made here, since detailed discussion of this problem is to be found in Unno's

42

contribution to this volume.

The case of radial pulsations is the simplest to discuss, provided attention

is restricted to fundamental and low overtone modes ~hat vary on a length scale

greater than ~ . If Lagrangian co-ordlnates defined in terms of the mean flow

are used to describe the pulsations, but locally defined Eulerian co-ordinates

for the convection, the equations governing the convective fluctuations are

rather similar to those used for a static atmosphere, though additional terms

must be added to account for the mean dilation and ~ must be modified because

the co-ordinate frame is no longer inertial. A mixing-length theory can therefore

be developed by following one of the procedures outlined in ~# 4-6.

Unno (1967) formulated a theory by generalizing the model that assumes

continuous turbulent exchange of momentum and heat between a convective element

and its surroundings. Though the general growth of convective fluctuations

during the lifetime of an eddy is ignored in this approach, acceleration of a

convective element and modulations in its temperature induced by the pulsations

are taken into account. An eddy is presumed to maintain its identity, deforming

instantaneously with the mean environment. The theory now requires one to

recognize that the lifetime of an eddy is finite, for a turbulent eddy retains

some memory of the conditions at the time of its formation. Thus much of the

apparent simplicity enjoyed by this model when applied to a stationary stellar

envelope is lost. Alternatively, the discussion of ~ 6 can be adapted for a

pulsating star by introducing the appropriate time dependence into the equations

of motion (Gough 1977).

These approaches each require an explicit statement about how the initial

state of a convective element depends on conditions at the time of creation.

Since the mixing length, which determines both the destruction rate and the

initial dimensions of elements, is assumed to depend only on the mean (horizontally

averaged) state and not on convective fluctuations, it is perhaps most natural,

and certainly simplest, to assume that it has the same functional form as for a

stationary envelope (and thus does not depend explicitly on time derivatives) and

to make a similar assumption about all other aspects of creation. It must be

realized that this is yet another unverified assumption of the theory. It may not

be a good approximation, for although it is the mean stratification of the

convection zone that controls which eddies grow most rapidly, the level of

turbulence at the instant of creation presumably does have some influence on the

perturbations out of which those eddies grew.

A similar objection may be levelled at the assumption that the mixing length,

when it determines eddy annlhilation, depends only on the mean environment.

If breakup is determined by shear within the eddy, perhaps the current eddy

dimensions provide a more appropriate length scale. These depend on the history

of the eddy and not just on instantaneous conditions. Likewise turbulent drag

43

and heat exchange depends on the current eddy size, and also on the intensity of

the small scale turbulence which may not vary in phase with the larger eddies.

The different versions of mixing-length theory yield different formulae for

F= and Tk when applied to radially pulsating stars. This emphasizes the

uncertainties in the assumptions. The differences offer some hope of choosing

between them by observation.

The range of possible asstEaptions widens further when nonradial pulsations

are considered. One prescription has been offered by Gabriel, Scuflaire, Noels

& Boury (1974) who generalized Unno's approach in a natural way. Amongst the

approximations is the neglect of anisotropy in the turbulent flow, which

circumvents the complicated problem of determining how the changing shear

associated with the pulsations modifies the convective velocity field. Anlsotropy

appears to have a more complicated influence on the pulsational perturbations of

the heat flux and Reynolds stress in this case than it does for radial pulsations,

so the ass~m~ption may be critical. It would be useful to know how sensitive

pulsations of stellar models are to changes in this and other assumptions in

order to assess where effort to improve the theory might most profitably be

dlrected.

One of the motivations for developing a convection theory in a time-

dependent envelope is to study the pulsations of the cooler Cepheid and RE Lyrae

variables. Unpublished computations by N.H. Baker and myself of the linear

stability of such stars to radial pulsations, using a generalization of the

formulation in ~ 6, indicate that the modulation of ~ generally has a

stabilizing influence on the pulsations, and is responsible for determining the

red edge of the instability strip.The phase of the modulation of T~ is such as

to &rive the pulsations in some regions of the convection zone and damp them in

others. The driving is greater in the cooler stars, and may be a significant

factor in the excitation of the long period variables.

Commen t s

These examples illustrate how the basic ideas of mixing-length theory might

be applied to a variety of situations. The generalizations all concentrate on

describing the dynamics of the major eddies prior to breakup, and ignore the

more difficult issues concerning creation and annihilation. To do more would

require a more sophisticated study of the mechanisms of turbulence.

In particular, there is no prescription for determining the mixing length.

One could choose the same value as one believes is applicable to ordinary

convection. In that case the theory predicts, for example, that a vertical

component of rotation or magnetic field reduces the heat flux. It appears how-

ever that there can be circumstances where rotation increases the heat

44

flux through a convecting fluid (Rossby 1969, Sommerville & Lipps 1973, Baker &

Spiegel 1975), which shows that the mixing-length prescription hasn't even

predicted the correct sign of the change. Perhaps the influence of a small

composition gradient is more reliably described, because the perturbation is via a

scalar rather than a vector field, and influences the dynamics only by modifying

buoyancy. However this could be the case only when that modification is small,

for we know from experimental studies of thermoha!ine convection that once

composition gradients are sufficient to change the stability characteristics of

the mean stratification the gross structure of the flow suffers a qualitative

change (Turner 1973). Thus the theory is not immediately adaptable to semi-

convection.

ii. NONLOCAL THEORIES

One of the obvious inaccuracies in the theory developed above comes from

assuming velocity and temperature fluctuations to depend on only local properties

of the environment. This would be justified if ~ were much less than all

relevant scale heights, but the stellar calibration suggests that this is not the

case. In particular ~ can vary on a scale much shorter than [ . Nonlocal

treatments take some account of the finite extent of convective eddies, and lead

to prescriptions for F, and ]~ that involve averages over distances of order ~.

Thus sharp gradients in ~ no longer lead to rapid variations in the convective

transports. Moreover, the treatments aim at representing overshoot into adjacent

stable regions.

There are two nonlocal properties of eddies that can be represented in a

straightforward way. One is that an eddy centred at •, samples ~ over the

range Cz.- ~£ , z.-e ~£); the other is that ~(z) and ~(z) are

determined not only by eddies with z,= z , but by all the eddies centred between

zo-~ and zo ~-~e. These can be taken into account within the framework of

the Boussinesq approximation, which entails ignoring the variation of all other

variables over the scale of an eddy.

Avera~in~ over eddies

The only place ~ enters into the formulae (6.11) and (6.14) for ~ and ]%

is in the growth rate o- . As was noticed by Spiegel (1963), the linearised

equations of motion used to determine the eddy growth are the Euler equations of

a variational equation for o- , whose solution is (6.6) with ~ replaced by

45

[ ~Cz') w'(z', z) a.z.'

where ~;(z, Zo) is t_he vertical component of the velocity of an eddy centred at

~o and the range of the integrals is the vertical extent of the eddy. With the

introduction of (6.1) this becomes

z - i {

(11.1)

Taking account of contributions to ~ and F~ from eddies centred at

different heights leads to similar averages: the assumptions (6.1) and (6.2) imply

that both ~T' and u; ~ have a m dependence quadratic in cos [~z- Zo)/~ ] .

Thus if Fc ° (z) and ~ (z) are defined as the right hand sides of (6.11)

and (6.14) with # replaced bye>, the nonlocal formulae for the heat flux and

Reynolds stress may be written

Fo = z I F~oC~O) ~o~[.~(~-~.)/~]az. z-~

(11.2)

(11.3)

Use of these expressions converts the ordinary differential equations of stellar

structure that obtain from local mixing-length theory into integro-differential

equations.

The fluid element approach

The extent of the region over which ~o and ~o are averaged in (11.2) and

(11.3) depends on the mixing length in its role of being a measure of the eddy

size. The description of mixing-length theory in terms of rising and falling

46

fluid elements, howeverp averages over a mean-free-path~ and so depends on ~ as a

measure of the annihilation rate. The analysis will now be repeated for this more

commonly used picture, to illustrate how uncertain the fine details of the theory

are. The arguments are similar to those used by Spiegel (1963).

It is a straightforward matter to repeat the analysis of ~ 6 with the fluid

particle picture in mind. The mathematical structure is almost the same, with the

principal difference that the integrals in the equations leading to (6.9) and (6.ii)

are now to be considered as llne integrals along fluid element trajectorles. It

is simplest to use the vertical displacement ~ of a parcel from its initial

position to define an independent .variable S according to

As = a~/£ . (n.4)

The bottom and top of the convection zone are assumed to be at 5-o and

Then the contribution to ~ from rising elements is approximately

I ,S $ - So

F~,( I ~ s) * I o - ' ~ T { " v .......... a S o , , ~ ¢c, ~s ( s - s . ) e - o

5 ~ S I *

(n.s)

and that from sinking elements is

~e-s

I I s' o-'~ T{" - ~ ....... (11.6) : l,~, - 7 - - ( ~ . - , ) e a~o .

In obtaining these equations the ~emperature fluctuation of an element was taken

to be

T ' = o - ' / t / - r t s.) .......... ~ . . ( ~ - ,

which was estimated by integrating W in (6.4) and using this and (6.5) to

eliminate ~- t o and @j in the expression for ~ . Once again ~" is assumed

to be defined in terms of an average (~> such as (ii.I) to take account of both

the finite size of fluid elements and the fact that they traverse a finite

distance through their environment. Note that the creation rate ~t has been

taken to be the same as in the local theory. It has been assumed that the

47

motion is not necessarily vertical, as does Spiegel, the cosine of the angle made

by the velocity with the vertical being denoted by ~ • Thus Fee =[~ is the

flux due to elements moving in directions between ~ and ~, ~p when ~ ~ O

The total flux is obtained by integrating over ~ and yields

i &,

O

(11.7)

where ~x is an exponential integral. The expression for ~ is similar. These

averages are rather different from (11.2) and (11.3), the main weight coming from

5 - S. I = 0.6 rather than being concentrated near zero. The value of

defining the initial conditions is once again undetermined. If it is fixed by

insisting that (11.7) approaches (4.10) in the limit e ~ O, one finds

~-P'- d >, "" e ~ 7 . ( n , 8 )

The averaging procedure lnboth this formulationandthe eddy approach is

rather crude, and depends in particular on an assumed structure ~r the velocity

and temperat~e fluctuations based on local theory. Other versions of the theory

that pay more explicit attention to the motion of elements have been formulated,

not~ly by Faulkner, Griffins and Hoyle (1965), Ulri~ (1970a). ~avlv &

Salpeter (1973) and Maeder (1975). Nordlund (1976) has recently studied a model

based on rising and sinking columns. The differences in outcome between the

various procedures ~pears to derive mainly from variances in the rather a~itrary

~oices of scal~g factors.

Spiesel's theory

A major drawback to the methods described so far is that they require one to

solve the equations of motion for the eddies. This becomes especially awkward

when the theory is generalized ~r ~plication to more complicated circ~nstances,

such as pulsating stars. It may be poss~le to alleviate the difficulties by

working within he framework suggested by Spiegel (1963) who started from an

element conservation equation ~ phase space. Spiegel considered a plane parallel

atmosphere and set

{ , (n.9)

48

where ~ is the element distribution function and v is the magnitude of the

velocity. The term ~(g~)/~L , which depends on the dynamics of elements

and which would normally appear on the left hand side of a conservation equation,

has been absorbed into the source function O_ ° This equation can be formally

solved for ~ in terms of (A , as is sometimes done in radiative transfer theory,

and the heat flux and Reynolds stress computed by averaging appropriate moments

of ~ over ~ . In particular, the heat flux is

Fo = I F L'j'a = (ll.lO

where ~' is the specific enthalpy fluctuation in an element. Rather than

discuss the element dynamics explicitly, Spiegel simply assumed that I~'I6~(so)

is independent of 8 and then chose it to make (II.i0) reduce to (4.10) in the

limit ~ ~ O. The result is

I S,

F= (s) = o F~o(So') E ~ ( I s o - s l ) &So , (n.n)

with I given by (6.13). This result differs from (11.7) because of the

assumption about the functional form of l&'l~.

AR~roximations

Since integral equations are not readily incorporated into most stellar

structure programmes it is tempting tO approximate the equations for ~ and ~

with differential equations, Spiegel's approach now exhibits the advantage that

one can immediately draw on the techniques of radiative transfer theory. In

particular, Eddington's first approximation provides simple equations relating

and ~ to 4~> that are no doubt accurate enough. To obtain the equation

for ~ , for example, moment equations are first constructed by multiplying

(11.9) by ~'~ and by ~ and integrating with respect to ~ , remembering

that ~'@ o when ~ ~ 0 . This gives

dh__ _ 3" -- O , &s

(Ii.12)

49

&S (11.13)

where

!

7 " I ~" I "a~' • - i

Eddington' s approximation is t o t a k e

where ~, and ~_ are independent of ff .

_~ a~F~ _

This implies }~ -- ~ 3" , and hence

F~ = - F=o , (11.14)

where ~=~ (cf. Travis & Matsushima 1973). The equation for ~ is similar.

But there remains the problem of finding an approximate equation determining <18> .

Guidance may be found by attempting to rederive an equation of the type (Ii.14)

directly from the integral relation (11.11).

The approximation (11.14) is equivalent to replacing the kernel ~ (so-s) =

E-~(I 5o-5 I) in (ii.ii) by the simpler function

"K°CSo-S) = ~ I , ~ ' I ' ( - b l s o - s l ) ( n . 1 5 )

with b - a. Equation (ii.I) might therefore be approximated in a similar manner.

But how dues one best choose b? Equation (11,II) may be rewritten

I , F0(~) = }('oCS.-s) ~'(s.)aso ~ -X.C~o s)]~(s°)aSo . ,mm

- FJ °~ cs) -r F~ c'l (s) , 01.1~)

w h e r e

F~o(S) , 0 .< s .< s, .-~ C~) =

0 • $ < 0 , S.> S,

(11.17)

50

The limits of integration have formally been written as ~ ~ , and are meant to

denote positions well into the bounding stable regions where ~ is small. Obvious

adjustments must be made when two convection zones are close together, or if the

domain of integration includes the central regions of the star.

It is clear that b is best chosen in such a way as to minimize the magnitude

of ~o). This problem is of a kind that has been encountered in radiative

transfer theory (Monaghan 1970) and statistical mechanics (e.g. Barker &

Henderson 1976) and its solution depends on the features of (Ii.Ii) one wishes to

represent most accurately. Here, an approximation will be sought that roughly

represents the solution when the scale of variation [r of <o is not a great

deal less than ~ ; to find a representation approximately valid for all scales

~r would entail an analysis of the equations that determine Fco . Thus ~(5o)

is replaced by its Taylor series about s, and the leading terms of the expansion

of Y$'~ so generated are made to vanish. The first two terms are automatically

zero, and the third vanishes provided 5 =~. This result differs somewhat

from the value obtained from the Eddington approximation, which is a representation

that appears to be good at both extremes of ~r , at least for radiative transfer.

If equation (ii.I) is treated similarly one obtains

6~ ~> _ ~#> = _~ , (ii.18)

where b, which is calculated as before but now with ~Cc5) = Icos~7[5 , is

given by

If equations (11.2) and (11.3) are used to determine ~ and p~ , this value

must also replace a in (11.14) and the analagous equation for ~.

The differential equations determining the mean structure of the star, with

this approximation to nonlocal mixing--length theory, is of order five higher than

when local theory is used. Computing time is therefore increased. However the

singular points at the edges of the convection zone discussed in ~ 9, and the

numerical difficulties associated with them, are no longer present. Equation

(II.14), its analogue for ?~ and equation (11.18) should be solved subject to

the boundary conditions ~ -+ O, T~ -~ O and ?> -*~ as 5 -~ ~:

51

Comments

The factor of about 5 by which the values of b obtained from the two kernels

differ emphasizes the different roles played by the mixing length. Fluid crossing

the midplane of a convective eddy of diameter ~ is likely to have risen vertically

by perhaps about half the radius, which is only about ~ the mean-free-path of a

fluid element. The mean-free-path and the element or eddy diameter have been

arbitrarily set equal, but perhaps a factor of order unity should have been

introduced between them. Consequently the coefficients in (11.14) and (11.18) are

parameters that, like the mixing length itself, are not determined by the theory

but await calibration by comparing theoretical models with observation.

The most obvious testing ground is the top of the solar convection zone,

where overshooting into the stable regions can be studied, However it is in the

regions of overshoot that new uncertainties seem to enter. The integral equation

(11.16) for ~ , for example, explicitly assumes that the stable regions provide

no source of convective elements: ~= O outside the convection zone. The

decceleration of elements in the stably stratified regions is represented by the

averaging of ~ , but this does not adequately account for the possible

oscillation of elements and the generation of waves. Negative values of ~ would

he required, and Spiegel (1963) has suggested replacing ~ in the formula (6.11)

for ~o by its real part, presumably to account for the damping of those waves.

However, that does not account for possible propagation of energy by the waves,

and subsequent dissipation far from the site of generation. A more careful

analysis of the coupling between the convection and the waves must be undertaken

before one can have confidence in the procedure.

Calibration of nonlocal theories is at present in an unsatisfactory state.

Attempts are made to construct model solar atmospheres and to compare overshoot

velocities or limb darkening with observation, adjusting parameters where

necessary (e.g. Ulrich 1970b; Travis & Matsushima 1973; Nordlund 1974), with some

diversity in the conclusions. Indeed Spruit (1974) has fitted the limb darkening

function using a local mixing-length theory. Moreover, though the models are

constructed with an averaged ~ , p is not always averaged and ~ is ignored

entirely. Ulrich (1976) has recently investigated the sensitivity of solar type

model atmospheres to variations in the parameter b in the kernel (Ii.15), with

< ~ , and to the addition of a multiple of a delta function to that kernel.

The refinements end generalizations discussed in ~IO can easily he

incorporated into these nonlocal procedures. Nonlocal effects of small scale

shear turbulence have been discussed by Kraichnen (1962), using rather different

arguments which suggest that at very high values of 5 there is a qualitative

change in the functional dependence of ~ on S . Scalo and Ulrich (1973) have

incorporated nuclear reactions into a nonlocal theory.

S2

12. COMPRESSIBLE CONVECTION

Compressibility plays two kinds of role. First it influences the structure

of the convective flow discussed above, and secondly it introduces new phenomena

that are not represented by the Bousslnesq approximation.

Various studies of the structure of the linear eigenfunctions of convective

motion in a compressible atmosphere have been made, though no attempt has been

made to incorporate the results into a mixing-length theory. This is not

surprising. One reason is that the mathematical difficulties are rather greater

than for Boussinesq theories, but a more fundamental reason is that it is not at

all clear how the mixing-length hypothesis should be interpreted in these

circumstances. It should be recalled, however, that the assumption ~ == H is

based on arguments concerned with the structure of eddies in a compressible

stratified medium, and that in some sense, therefore, compressibility is

acknowledged. New phenomena that must he considered include the pressure

fluctuations in the equation of state, which not only modify the structure of the

eigenfunctions but also must be included in the formula for the heat flux. Viscous

dissipation must be included in the mean energy equation. Unno assesses the

importance of such mechanisms in his contribution to this volume.

13. CONCLUDING REMARKS

The mixing-length formulae derived in ~ 4 are based on very rough order-of-

magnitude estimates. The physical argt~nents supporting them are based on

imprecisely defined models. Moreover the observational evidence for the validity

of the formula for the heat flux is very weak; the Reynolds stress is ignored in

almost all stellar structure computations.

Even if it could be ascertained that the Boussinesq formulation outlined in

this article is sound, there would still be the difficulty of extrapolating the

theory to stellar conditions where compressibility is important. The major point

at which compressible arguments are invoked is in the choice of the mixing length

It is commonly believed that effective heat carrying eddies cannot extend

over much more than a scale height H of density or pressure~ and accordingly

is taken to be of order H • The solar calibration of the heat flux is not

inconsistent with this assumption, though it is inconsistent with the conditions

under which the Boussinesq approximation is justified. However~numerical

computations of compressible convection that either solve the equations of fluid

motion directly in two dimensions (Graham 1975) or three (Graham, these proceedings)

or represent the solutions in the single-mode approximation (Toomre, Zahnj Latour

& Spiegel 1976b; Van der Borght 1975) predict large eddies extending over the

entire convection zone that show little tendency to break up into smaller scales.

But perhaps the computations do not mimic solar conditions well enough, since they

lack the thin transition zone at the top of the convective region in which the

53

temperature gradient is very strongly superadiabatic. The convection just beneath

the photosphere is observed to have a characteristic length scale comparable with

H and it is no~ unlikely that the vertical scale is similar. Whether in the

region below~ the dominant scale of motion is always of order H, or whether it is

quite differen~is hardly relevant for most purposes, because any plausible

formula for ~ implies that beneath the first scale height the temperature

gradient is very close to being adiabatic. Moreover the detailed structure of the

transition zone doesn't influence the interior significantly, so any formula for

Fc with an adjustable factor multiplying it can serve to construct models of the

sun and solar type stars that have the correct luminosity and radius. Of course

the motion in the transition zone is important for determining the photospheric

velocity field, but here mixlng-length theory is currently inadequate for making

reliable predictions.

One must not conclude from these remarks that a good convection theory is

unnecessary to stellar evolution theory for modelling solar type stars. On the

contrary, though it is only the integrated properties of the transition zone that

are required to determine the adiabat deep down, a theory is required for extra-

polating from models of the sun to other solar type stars. Andj of course, as

soon as one wishes to discuss the structure of a stellar a~nosphere, a knowledge

of the subphotospheric velocity field is essential.

The structure of convective envelopes of red giants is more sensitive to ~ ,

but calibration is difficult because there are other uncertainties in both theory

and observation. The degree of overshooting and consequent material mixing at

the edges of convective cores is also of interest, but difficult to assess

observationally.

It is common practice to argue that because the mixing-length hypothesis, in

whatever guise it is to be used, is so uncertain, it is hardly worth the trouble

to calculate its consequences accurately. Indeed it is sometimes the case that

so coarse a mesh is used for the numerical solution of the stellar structure

equations that the solutions are not resolved in the convection zone, and that

the differential equations are therefore not adequately represented by the finite

difference equations. It is also coEmon, once a formula for ~ has been decided

upon~ not to calibrate the mixing length, nor even to report precisely the formula

that was used for Fc • Though it may be true that in our present state of

knowledge there is little reason to prefer, say, a red giant model computed with

a mixing-length formula that has been carefully calibrated on the main sequence

to a model computed with a similar formula that has not, there would be greater

hope of improving our understanding of stellar convection and its influence on

stellar structure if investigations were more meticulously carried out and

reported. The prospects of an imminent supersession of mixing-length theory by a

theory that is demonstrably more reliable for describing stellar convection zones

54

is bleak. Therefore it seems worthwhile to invest some effort into trying to

improve the theory we already have. Modern sophisticated mixing-length theories

have achieved some measure of success in describing turbulent flows in the

laboratory (e.g. Launder & Spalding, 1972), so there is some hope that the effort

would not be in vain.

E.A. Spiegel and I have recently been attempting to consolidate the theory by

synthesizing the ideas that have been severally used in the past. The approach is

based on a two-fluid model, one so-called fluid being an assembly of thermals and

the other being the background environment. Entrainment, erosion and turbulent

exchange of energy and momentum are represented in the equations of motion, using

laboratory calibrations where possible. The goal is to derive a set of equations

determining the heat flux and the Reynolds stresses that would be applicable to a

sufficiently wide variety of circumstances for a meaningful calibration to be

possible. The success or failure will be reported elsewhere.

It has been the aim of this article to clarify the ideas and assumptions

behind the simple mixing-length theories used in astrophysics, and so provide a

basis for the necessary improvement and generalization to circumstances more

complicated than those for which the theory was originally formulated. Some

indication of how this might be achieved has been given. Other measures that may

have to be taken include abandoning the idea that the flow can always be

described adequately in terms of a single length scale @ This may be necessary

for a theory of semiconvection, for example. It must be realized, however, that

many attempts to improve or generalize the theory involve additional physical

mechanisms, and co~Isequently the introduction of new parameters that must he

determined by observation.

55

REFERENCE S

Baker, L. and Spiegel, E.A., 1975, J. Atmos. Sci., 32, 1909

Barker, J.A. and Henderson, D., 1976, Key. Mod. Phys., 48, 587

Biermann, L., 1932, Zs f. Ap, 5, 117

Biermann, L., 1937, A.__~N., 26__4, 361

Biermann, L., 1943, Zs f. Ap, 2__2, 244

Biermann, L., 1948a, Zs f. Ap, 2_5, 135

Biermann, L,, 1948b, FIAT Review of German Scienqe (Astr. Ap Costa.), 161

BShm, K.-H. & StSckl, E., 1967, Zs f. Ap, 6__6, 487

B~hm-Vitense, E., 1958, Zs f. Ap, 46, 108

Boussinesq, J., 1877, M~n. div. savants Acad. Sci. Inst. France, 2__3, No I

Boussinesq, J., 1903, Th~orie analytique de la chaleur, Tome II (Paris, Gauthier-

Villars)

Chandrasekhar, S., 1961, Hydrodynamic and hYdrom@gnetic stability, (Oxford Univ.

Press)

Dearborn, D.S.P. and Eggleton, P.P., 1976, QJRAS, 1__7, 448

Faulkner, J., Griffiths, K. and Hoyle, F., 1965, MNRAS, 12__99, 363

Gabriel, M., Scuflaire, R., Noels, A. and Boury, A., 1974, Bull. Acad. roy

Belgique, C]. Sci, 6__0, 866

Cough, D.O., 1969, J. Atmos. Sci., 2__6, 448

Cough, D.O., 1977, Ap J., in press

Cough, D.O. and Weiss, N.0., 1976, MNRAS, !76, 589

Graham, E., 1975, JFM, 70, 689

Henyey, L., Vardya, M.S. and Bodenheimer, P., 1965, Ap J., 142, 841

Hofmeister, E. & Weigert, A., 1964, Zs f. Ap, 59, 119

Kraichnan, R.H., 1962, Phys. Fluids, 5, 1374

Latour, J., Spiegel, E.A., Toomre, J. and Zahn, J.-P., 1976a, Ap J.., 207, 233

Launder, B.E. and Spalding, D.B., 1972, Mathematical models of turbulence

(Academic Press)

Maeder, A., 1975, Astr. and Ap, 40, 303

Malkus, W.V.R., 1964, Ceophys. Fluid D~amics (Woods Hole Oceanographic

Institution) _I, i

Monaghan, J.J., 1970, MNRAS, 148, 353

Nordlund, ~., 1974, Astr. and Ap, 32, 407

Nordlund, ~., 1976, Astr. and Ap, 5_~0, 23

Ogura, Y. and Phillips, N.A., 1962, J. Atmos. Sci., 1__9, 173

~pik, E.J., 1938, Publ. Obs. astr. Univ. Tartu, 3__0, No 3

0pik, E.J., 1950, MNRAS, IIO, 559

Prandtl, L., 1925, Zs f. angew. Math. Mech., 5, 136

Prandtl, L., 1926, Verhandl. II intern. Kongr. tech. Mech., ZSrich, p 62

56

Prandtl, L., 1932, B eitr. z. Phys. d. frelen Arm., 19, 188

Prandtl, L., 1952, Essentials of Fluid dynamics, (London, Blackie)

Rossby, }{.T., 1969, JFM, 36, 309

Scalo, J.M. and Ulrich, R.K., 1973, A p J., 183, 151

Schmidt, W. 1917, Sitzun~sbgr. ~Kais. Ak. d. Wiss. We!n), Abt !!a, 126, 757

Shaviv, G. and Salpeter, E.E., 1973, Ap J., 184, 191

Schwarzschild, M., 1961, A2 J., 134 , i

Siedentopf, H., 1933a, A.N., 247, 297

Siedentopf, H., 19335, A.N., 249, 53

Siedentopf, H., 1935, A.N., 255, 157

Sonm~erville, R.C. and Lipps, F.B., 1973, J. Atmos. Sci., 30, 590

Spiegel, E.A. 1963, Ap J., 138, 216.

Spiegel, E.A., 1971, Ann. Re v. A. AP, 9, 323

Spiegel, E.A. and Veronis, G., 1960, Ap J., 131, 442 (correction: 135, 655)

Spruit, H.C., 1974, Solar Phys., 34, 277

Stellingwerf, R.F., 1976, Ap J., 206, 543

Taylor, G.I., 1915, Phil. Trans., A, 215, I

Taylor, G.I., 1932, Proe. Roy. Soc., A, 135, 685

Toomre, J., Zahn, J.-P., Latour, J. and Spiegel, E.A., 1976b, Ap J., 207, 545

Travis, L.D. and Matsushima, S., 1971, Scientiflc Report No 024, (Penn. State

Univ., Astr. Dept)

T ravis, L.D. and Matsushima, S., 1973, Ap J., 180, 975

Turner, J.S., 1973, Buoyancy effects in fluids, (Cambridge Univ. Press)

Ulrich, R.K., 1970a, Ap Sp. Sci., --7, 71

Ulrich, R.K., 1970b, Ap Sp. Sci., 7, 183

Ulrich, R.K., 1976, Ap J., 207_, 564

Unno, W., 1967, Publ. Astr. Soc. Jap~j 19, 140

Unno, W. and Spiegel, E.A., 1966, Publ. As tr. Soc. Japan, 18, 85

Van der Borght, R., 1975, MNRAS, 173, 85

Vitense, E., 1953, Zs f. Ap, 32, 135

yon K~rm~n, T., 1930, Proc. III inter. Contr. App. Mech. ~ Stockholm~ _I, 85

ON TAKING MIXING-LENGTH THEORY SERIOUSLY

D. O. Gough

Institute of Astronomy and Department of Applied Mathematics and Theoretical Physics, University of Cambridge

and

E. A. Splegel

Astronomy Department, Columbia University New York 10027, U.S.A.

There has been progress in convection theory in the past decade, mainly in

the problem of mild convection. Yet, we are still not able to cope with vigorous

convection such as we face in the envelopes of late-type stars. Most astrophysic-

ists therefore use mixing-length theory and get on with calculating their models.

As this sltuatlonmay continue for a while, it may be a good thing to consider

what mlxing-length theory really is and to see whether it can be taken seriously as

a physical model for stellar convection.

Different authors mean different things when they speak of mixlng-length

theory. Here, we interpret the theory in terms of the specific model in which a

star is composed of a background fluid through which discrete, well-deflned parcels

of fluid move. These parcels may be thought of as quasiparticles whose number

density is sufficiently high that they constitute a second fluid permeating the

background fluid. The convective model is therefore a two-fluld model loosely

resembling the composite of radiation and matter familiar in astrophysics, except

that the quasipartlcle fluid is more complicated than the photon gas.

In applying this model we must write down equations of motion for the

quaeiparticles. We have to specify the nature of the quaslpartlcles, and most

people, with varying degrees of explicitness, treat them as idealizations of the

buoyant thermals described by meteorologists. Fortunately, there is by now some

guidance provided by laboratory data on the motion of isolated thermals in both

laminar and turbulent fluids. Turner (1963, 1973) has described these experiments

and has outlined the simple theory which has been evolved to describe them. In

particular, he assumes that the thermals are small compared with any scale heights

so that gradients across them, both inside and Just outside, may be neglected. Only

in their vertical motion do they sense the presence of the ambient temperature

gradient.

Turner's description allows for turbulent exchange of heat, momentum~ and mass

between a quasiparticle and the ambient medium. With some slight modifications of

his discussions we may derive the following set of equations governing the motion

of quasiparticles. We display these Just to give some idea of their form:

58

dm ~ lu-Ul - Elv~ , (i) d--~ =~ i

du d m ~ = g(m-m) + ~ (m U)

+ P0 zllU-U-I (~-u)-&~(u-U) , (2)

dtd-~h = w d~_p dz ~ EllU-UI +-- q](h-6) , (3)

dx (~)

Here m, x = (x,y,z), ~ = (u,v,w) and h are the mass, position, velocity and

specific enthalpy of a thermal, p, ~, 6 and ~ are the local means of density,

pressure, enthalpy and velocity of the ambient medium at x. E 1 and E are cross

sections for entrainment and erosion (both of the order of the geometrical cross

section of the thermal), X is the ambient turbulent velocity at x, m is

ambient mass displaced by thermal, ~ is the acceleration of gravity corrected for

the hydrodynamic mass of the thermal, q-i is a thermal decay time allowing for

radiative and turbulent exchanges, and ~-i is a similar viscous decay time.

Evidently these formulae must contain some fudge factors to be obtained by

comparison with measurements, be they experimental, meteorological, or astrophysical.

In astrophysical treatments of convection many of the effects modelled in these

equations have been included. Turbulent exchange of momentum between fluid elements

and the ambient medium was included in the early theories (e.g. Prandtl 1932,

Biermann 1932, Siedentopf 1933) and ~pik (1950) allowed for turbulent exchange of

heat. Ulrich (197Oa,b) has adopted the formulation of Morton, Taylor, and Turner

(1956) in his studies. However, when astrophysicists use these equations of motion

they generally replace them by algebraic equations; that is they essentially replace

d/dr by w/Z where % is a length to be specified. This gives rise to the usual

local mixing-length treatment. Sometimes, some or all of these algebraic equations

are averaged over height with some arbitrary weigh= function to produce a nonlocal

extension of the theory (e.g. Ulrich 1976).

Such reductions of the dynamical equations for thermals have not been favoured

in the meteorological literature. Certainly, they are not suitable for use by

anyone interested in studying the interaction of stellar pulsations with convection.

An alternative procedure, first attempted by Priestley (1953, 1954, 1959) for

hydrostatic convective layers, is to solve the differential equations and use them

together with some hypotheses about the distribution of initial conditions of

quasiparticlea to compute the heat flux. This has also been attempted for linear

pulsation theory (Gough 1977). But in both instances one has to build in some

information about the number density of each kind of quasipartiele at each height,

generally by specifying creation rates. This becomes quite an undertaking for the

nonlinear pulsation problem and even the formulation of the calculation has not

been agreed upon. The manner of incorporating the dynamical equations into the

convection theory thus poses a major difficulty in applying this kind of model. As

we have hinted, it requires a prescription of the number of quasipartlcles for each

value of the parameters, and this distribution must be specified in a way that is

compatible with the dynamics.

Formulated in this way, the model resembles kinetic theory and, in an

attempt to capitalize on this, a transport equation was written down as if the

quasiparticles satisfied Hamiltonian dynamics (Spiegel 1963). Deviations from this

ideal behaviour were compensated for by introducing a source term in the transport

equation. Amodlfieatlon was suggested by Castor (unpublished manuscript) who

renounced the simple form provided by Hamiltonian dynamics and wrote a continuity

equation for the one-particle distribution in the phase space of the quaslparticles.

The phase space was enlarged over the usual six dimensional ~-space of position

and velocity to include the temperature of a single quasiparticle as a phase

parameter. In doing this one loses the volume-preserving feature of the phase fluid,

which raises questions about the meaning of the approach, especially when one

attempts coordinate transformations. Yet it seems to us a useful thing to write a

continuity equation for the phase space density of quasiparticles and, for the

present, ignore some of the niceties. We modify Castor's choice and use specific

enthalpy (~ather than temperature) of the quasiparticle as a variable and add an

additional phase parameter~ the quasiparticlemass. We have then an eight-

dimensional phase space in which the density of representative points is

f(x,~,h,m;t). The continuity equation satisfied by f is:

~f + ~ • ~ • ~ (~f)+ ~ (mf)= ~(-~-) ~-~ ~ (xif) + ~ (uif) + ~ ~m coll

(5)

where dots denote differentiation with respect to time and a collision term has

been introduced. The collision term is supposed to express the turbulent

destruction and creation of quasiparticles; through this term we may represent our

crude understanding of turbulence. It seems inadvisable to use a form llke the

Boltzmann collision integral since the interactions are probably not dominated by

two-body collisions. Instead, it is perhaps best to include a loss term like -f/~

to represent the destruction of quasiparticles, where T is a time required for

the quasiparticle to travel its own diameter. This term then embodies a basic idea

of mixlng-length theory. But what about the creation term?

The generation of new quasiparticles is not really understood, and to quantify

it, a specific model is needed. Often one imagines that quasiparticles grow from

small fluctuations because of the instability mechanism. However, in a turbulent

medium the fluctuations are not small. In the quasiparticle picture we think of

the new quasiparticles as decay products of the old ones to represent the turbulent

80

cascade process. Their development through instability is already included in the

dynamical equations.

The problem, of course, is that we do not know much about the decay products

following the destruction of the quasiparticles, and this is the first clear

difficulty that must be faced in completing the theory. It is becoming increasingly

clear in turbulence theory that the turbulent spectrum is strongly influenced by

the number of decay products in the breakup of a quasiparticle, and possible models

have been discussed which may provide guidance (cf. Frisch 1977). We shall not offer

any preferences in the present discussion. Our aim instead is to bring out the

points at which physical assumptions are needed to make the mixing-length model

cogent.

Once a form for (Sf/St)coll is decided, the remaining difficulties are

computational. This is not to belittle them; they are fierce and a moderately

reasonable approximation scheme is not immediately apparent. The computational

methods depend on the way one uses equation (5), and that has to be discussed next.

We believe that it would be sensible to try to construct moment equations

from equation (5). For example, multiplication of equation (5) by m followed by

integration over d5~ = d3u dm dh gives

3P m [/ ~ 8f [ --+v.F =|,~-f, mdsn- j~fds~ • (6) 3t "m #x--, coll

where

f Pm = J mfd5~

is the mass density of the gas of quasiparticles and

(7)

r (8) F m = J m-ufd5f~

is the mass flux of the gas. The last term on the right of equation (6)

represents the mass exchanged with the background fluid by entrainment and erosion.

One may compute other moment equations, but we shall not do that here. We

should however mention that the number of moments goes up faster than in ordinary

kinetic theory or transfer theory. Quantities like fluxes of enthalpy and mechanical energy arise and there is the all-important turbulent stress tensor:

Ti j = f mui~jfd5 ~ (9)

Once a hierarchy of moment equations has been written down [a skeleton version

has recently been studied by Stellingwerf (private connnunication)] the problem of

closing it off must be faced. A possible approach, resembling the moment method,

is to decide on an approximation for f and use that guess, for that is all it is

at present, to get approximate expressions for the higher moments in terms of the

lower moments. Once this is done, a last problem of principle remains. One must

81

still decide how to describe that part of the fluid that does not move in

quasipartieles. Should this be thought of as a zero fluctuation condensate of the

quasiparticle gas? Or should one describe the background as an ordinary laminar

fluid acted on by the stresses and so forth generated by the quasiparticles? The

latter course seems decidedly preferable to us, especially for treating penetrative

convection, where most of the matter may be in the background fluid. If that is

accepted, the next course of action is to write the dynamical equations for the

background fluid including the mass, momentum, and energy sources indicated by the

moment equations of the quaslparticle gas. Then, in principle, one has a complete

set of equations for the dynamics of a star with turbulent convection, but for the

present without rotation or magnetic field.

Now we have tO come to the kay question: is this what has to be done or are

we to be saved from it by a 'real theory' starting from the full fluid equations?

We think that the inediate prospects for a sound fluid dynamical approach are not

bright. And even the approximations to such an approach as are on the horizon

promise to be far more demanding computationally than the scheme summarized here.

At present, untold computing hours are being lavished on stellar models using

a mlxlng-length theory whose reliability is untested off the main sequence. It

seems to us that if this situation is to continue it would be well to take the

mixlng-length theory seriously. In particular, one should be clear on the turbulence

model one is using and not simply alter the standard formulae according to whim,

as is often done in the literature. We are not saying that alternative general

structures to that given here may not be preferable. Nor are the procedures we

outline meant to be rigid. The present version of a mixing-length procedure is a

synthesis of ingredients existing in the literature and we have done no more than

put it together to show that a cogent discussion of mlxing-length theory is

possible. We have especially tried to show where the physics is missing and to

indicate a framework for including it. The resulting equations are in principle

capable of dealing with many of the problems of current interest, such as the

nonlinear interaction of pulsation and convection. Those coping with such questiOns

are all too familiar with many of the problems we have raised. But they, as we

ourselves, have sometimes dealt with these problems piecemeal and have not tried to

put them into context by working with a concrete general model. We are claiming

here that the specification of such a model is possible and desirable and that if

one can be constructed, stellar convection theory may begin to seem more rational.

We thank the SRC and the NSF for supporting our work on this subject.

62

REFERENCES

Biermann, L. 1932, Zs f. Ap, 5, 117

Frlsch, U. 1977, these proceedings

Gough, D. O. 1977, Ap J., ~!~, 196

Morton, B. R., Taylor, G. I. and Turner, J. S. 1956, P~0q. Roy. Soc., A23~, 1

Oplk, E. J. 1950, MNRAS, !~, 559

Prandtl, L. 1932, Beitr. z. Phys. d. Freien Arm., !9, 188

Priestley, C. H. B. 1953, Austr. J. Phxs., 6, 279

Priestley, C. H. B. 1954, Austr. J. Phys., Z, 202

Priestley, c. H. B. 1959~ Turbulent transfer in the lower atmosphere (University

of Chicago Press)

Siedentopf~ H.

Spiegel, E. A.

Turner, J. S.

Turner, J. S.

Ulrich, R. K.

Ulrich, R. K.

Ulrich, R. K.

1933, A toN', 247, 297

1963, Ap J., !~, 216

1963, J. Fluid Mech., !~, 1

1973, Buoyancy effects in fluids (Cambridge University Press)

1970a, Ap Sp. Sci., !, 71

1970b, Ap Sp. Sci., !, 183

1976, AP a . , ~ Z , 56~

OBSERVATIONS BEARING ON THE THEORY OF STELLAR CONVECTION II

Erika B~hm-Vitense

University of Washington, Seattle, Wa.,U.S.A.

and

University of C~ttingen

Germany

SUMMARY

It is shown that the best way to get information about efficiency of convective

energy transport in the hydrogen convection zones in stars other than the sun is pre-

sently contained in continuous energy distributions of A and F stars. Scans show that

convective energy transport must be much more efficient than thought hitherto. The scans

indicate that rapid rotation enhances convective energy transport.

Figure I lists all the observations, we could think of, which are influenced by the

outer convection zone and which can therefore give us information for our convection

theory.

I. STELLAR STRUCTURE AND EVOLUTION

Already Hoyle and Sehwarzsehild (1955) pointed out that with increasing efficiency

of convection the radius of a star shrinks.

If we describe the efficiency of convection by the one parameter £, the characteris-

tic length or the mixing length of the convective flow, and if we further assume that

the ratio £/H = ~ is depth independent, where H is the pressure scale height - an assump-

tion that may not be valid, see for instance B~hm (1958), Schwarzschild (1974) - then

this one parameter ~ can in principle be determined from the observed radius of the star.

Unfortunately, the radius depends also on the initial abundances Z, Y and CNO (Iben 1963)

as well as on the age of the star. From Iben's study we see that for given T a ~Iog L e

0.3 is obtained for A log ~ ~ 0.5, but one also finds A log L % 0.3 for a change in

abundances Y and Z by a factor of 2. A similar change in log L is obtained if opacities

are computed with different approximations. Different authors therefore obtain different

values of ~ from the radii of the sun and the stars, ranging from ~ = 0.4 to e = 2.

From the position of the main sequence we can therefore presently only conclude that

log e = 0 ~ 0.5.

l m

ixin

g t

o &

fr

cQ

de

ep

la

ers

tim

e sc

ale

o

f e

vo

luti

on

on

M.S

. ~

- (b

lue

stragElers]

Li,

Be,

B

qe-

C,

N,

O,

J H

e

wh

ite

dw

arf

abu

nd

ance

s

CONVECTION

stratification i~

deeper l

ayers

stratification

(K o

f a

dia

ba

t)

in

surf

ac

e

lay

ers

rad

ius

%-

shap

e o

f th

e K

- tu

rn o

ff

tra

ck

po

siti

on

of

red

g

ian

t b

ran

ch

~-

Tcenter

white dwarfs

(conli.g time) ~

F v

(0

4-

on

set

of

con

vec

tio

n ~

-

B-V

gap

on

M.S

red

b

ou

nd

ary

of

cep

hei

d

stri

p

Tce

nte

r ~f

th

e su

n

ne

utr

ino

s)

sola

r v

elo

cit

y

fie

ld

dam

pin

g of

sola

r o

scil

lati

on

s (G

old

reic

h

+ K

eele

y)

ce

nte

r to

ll

mb

v

ari

ati

on

o

f su

n

ve

loc

ity

~

--

.~

field

T

4--

chro

mo

sph

eric

heating

I em

issi

on

CoII,Mgll

absorption

He 10830

stellar wind

(Vro

t)

curve of growth

osc

illa

tio

ns

--P

~'~

of sun

Vturb

or grad W

solar intensity

fluctuations

+ F,

(T)

->line profiles

llne intensities

va

ria

bil

ity

o

f red

sup

erg

ian

ts

"~

(Se

hw

arz

sch

ild

)

FIGORE !

Ways in wh

ich the effects of t

he outer convection zo

nes ap

pear

in astronomical observations and which

therefor~ can

give us

information ab

out convection theory.

65

The same difficulties apply to studies of the evolutionary tracks which again

depend on Z, Y and = and also C, N, 0 abundances. In fact one would like to know

in order to find the other parameters°

An additional difficulty is encountered when studying H.Ro diagrams:In order to

compare theoretical and observed tracks we have to relate the theoretical parameters

T e and g to the observed ones: color and m or M . The relation between the color and v v

Te, g depends not only on Z, and the CNO abundances, but also on the influence of con-

vection on the observed energy distribution, i.e. the changes of the T (T) relation in

the surface layers. This could possibly be important on the whole evolutionary tracks

for stars of spectral types F and later.

There are several indications that the colors are indeed influenced by surface

convection :

(a) Theoretical color Computations for giants in radiative equilibrium giving

U - B as a function of Z for a given value of B -V show a maximum of U - B for

Z ~ 0o3 ° Z (B~hm-Vitense and Szkody 1974). This maximum is not well seen in the oh- ® servations (Wallerstein et al. 1966).

(b) For intermediate Z values we find a discrepancy between the observed and com-

puted M 4 = (B - V) - (V - r) index (Mannery et al. 1968) for giants in radiative equi-

librium (B~hm-Vitense and Szkody 1974).

(c) Canterna (1976) finds similar problems ~or his metallicity index C - M.

All these discrepancies show that for a given energy distribution in the red there

is less energy observed in the blue and violet region than predicted by the radiative

equilibrium models. Scaled solar Bilderberg models, i.e. models with a decreased tem-

perature gradient in the deeper layers, would ease the problem. It might be emphasized

however, that the discrepancies only exist for giants with 0.I Ze ~ Z < Ze,not for very

metal poor stars. It could,therefore, also be related to an error in the line blanke-

ting computation, for instance a wrong value for the microturbulence.

While the decreased ultraviolet flux is in itself an interesting problem and might

well tell us something about convective overshoot in stars with different Z, it also

tells us that we have to study and understand this convective overshoot before we

can deduce the "observed" evolutionary tracks in the L, Te diagram and proceed to de-

re,mine Z, Y, C, N, O, the age t, and finally e .

We then conclude that the stellar evolution computations require a knowledge of

rather than provide one.

66

II. SURFACE PHENOMENA

A. DIRECT MEASUREMENT OF VELOCITY FIELDS

The most direct observations of convection are the velocity fields which for stars

can only be observed by broadened line profiles. If the velocity field changes over one

mean free path of a photon it will lead to a broadening of the line absorption coeffi-

cient thereby increasing the line width and the equivalent width. If the velocity chan-

ges only over much larger scales it will lead to a broadening of the intensity profile

only and not change the equivalent widths of the lines. According to the influence on

the equivalent width we describe the two effects as micro or macro-turbulence (or possi-

bly rotation). However, we have to keep in mind that an increase in the equivalent width

could also be due to other effects than small scale velocity fields. Also there is no

reason why we should have only small scale or large scale turbulence, in fact we expect

a continuous turbulence spectrum of all scales. (For isotropictu~ulence see the dis-

cussions by H. and U. Friseh 1975, by G. Traving 1975 and by E. Sedlmayr 1975). There-

fore, we have to be careful with the interpretation of the socalled microturhulence.

If we believe that the microturbulence as determined from the equivalent widths by means

of curve of growth analysis really is a measure for the small scale velocity field,

which could be either due to turbulence or to laminar velocity gradients, then we find

the picture first compiled by Wright 1955, see Figure 2.

Generally the microturbulence increases for decreasing densities in stellar atmos-

pheres as is expected for convective velocities: Since the convective flux F c Q. V 3

p = density, V = convective velocity, we expect for a given F ~ F, where F is the total C

energy flux, that V = p-I/3. We do however not observe the decreas£of the microturbu-

fence for hot stars for which F << F is expected. c

A compilation of more recent data especially for giants has been given and dis-

cussed by Reimers (|976).

For supergiants Rosendhal (1970) and van Paradijs (1973) have more. recent studies,

qualitatively confirming the results shown by Wright (]955).

For main sequence stars Baschek and Reimers (1969) did a detailed investigation

especially in order to study possible differences between A m and normal A stars. Chaffee

(1970) extended the study to cooler stars. Andersen (1973) repeated the determination

with the new Fel oscillator strengths (Garz and Kock 1969). The result is shown in Fi-

gure 3. An increase in Vturb is found when going from late F to early F stars in quan-

titative agreement with convective velocities obtained for ~ = |. However, for earlier

stars the expected decrease in velocities is not found. For later stars an increase in

Vturb is suggested - though not observed for the sun - also in contradiction to expec-

tations from convection theory.

67

I / , - -

1I - -

111 - -

I V - -

V - -

X X~&A =x~

A ~

O OO

AA & •

X

O 0 O •

Oo

• & 4,

1

o@

o o

@

• •

o

@

I I I i I I ! I ! 1 I I 05 B0 B2 B5 A0 A5 F 0 F 5 GO G 5 K 0 K 5 M 0

Veloci~ (km./~0

• 1-2

• 2-3

o 4-5

A 5 - 7

• 7 - 1 0

A 10-15

15-20

X 20-30

X > 3 0

FIGURE 2:

Microturbulent velocities for stars of different spectral types and luminosity classes as given by Wright (1955).

68

70

At• A ~ •

Q &

i % o16

! ...... I ......... I ..... I I f

O O ~ OO

• • ® @_..

I 1 I ,,I,, I 1 t 0.7 0.8 0,9 :~.o

FIGURE 3 :

Microturbulent velocities for main sequence stars of different effective temperatures according to Andersen (1973). The open symbols refer to previous investigations by Baschek and Reimers (1969) and Chaffee (1970) using old oscillator strengths, the filled symbols to Anderson's determination with the Garz and Kock (1969) oscillator strengths. V, ~ refer to A m stars.

69

Another puzzle is provided by the measurements of Allen and G~eenstein (1960) and

Wallersteln (1962) showing that in Pop. II dwarfs Vturb ~ 0, a result which is certain-

ly not expected from convection theory , but these studies will have to be repeated with

new Fel oscillator strengths in order to be sure. Reimers (1976) attributes the increase

of Vturb for late type stars to possible measuring errors. Baschek and Reimers (1969)

suggest that for the A stars the high Vturb is caused by a large number of pulsation

modes similar to the ones studied recently by Lucy (1976) for e Cyg.

In short, measured values of Vturb sometimes do and sometimes do not agree even

qualitatively with exceptations from convection theory, indicating either that our

expectations are sometimes quite wrong, or more likely, that the measured microturbu-

lence has quite often nothing to do with convective velocities. How then do we know

when they do and when they do not ?

Even more difficult is the judgement of the observed depth dependence of the micro-

turbulence (Huang and Struve 1952, Rosendahl 1970). In general there does not seem to

be any observed contradiction to the assumption that for other stars the depth depen-

dence of Vturb is similar to the one observed for the sun.

B. INDIRECT MEASUREMENTS OF VELOCITY FIELDS BY MEANS OF ATMOSPHERES

AND CORONAE

(a) Chromospheric emission :

It is general belief that for the formation of classical solar type chromospheres

a velocity field is a necessary condition. We do not know whether it is also a suffi-

cient condition. The different strengths and the age dependence of the Call K 2

emissions for otherwise similar stars show the importance of a second parameter, pro-

bably the magnetic field. The absence of chromospheric emission may therefore not be

proof of the absence of a velocity field, only, if for a given spectral type we never

find chromospheric emission, I would believe this to indicate the absence of efficient

convection.

Chromospheres in cooler stars are seen by means of CaII K 2 and MgII h and k

emission or by the ]0830 line of HeI in absorption. O.C. Wilson (1976) has made ex-

tensive studies of the CaII K 2 emission in G and K stars. His results are shown in

Figure 4. In the same graph I have also plotted the bluest stars that have been ob-

served to show Mgll emission and I0830 HeI absorption according to Zirin (1975). There

appears to be a line in the HR diagram on the blue side of which the chromospheric ac-

tivity seems to stop. In the low luminosity part is not quite clear to me whether the

Call K 2 emission stops for somewhat more red stars than the Mgll emission. If so, it

~uld be an effect of the larger abundance and ionization energy of Mgll. If they

stop at the same time itshould indicate a cause different from ionization.

In the same graph I have also plotted the reddest Pop. I Cepheids according to

Sandage and Tammann (1974).

70

:E

- 6

- 4

- 2

0

2

4

GO ' I ' ' ' I ~ ' ' l l l l l ' I ' ' " ............... i ......... ' I

• • o " °" ° ~ ° . " "" ":" " -

• " " ° ".i ea=e/7 ; ° , . . . . . . - / -

o ° o . . , . . . = ~ .

/ . : . x o . . . . ~ :. . . . . . '~4,~n,:.~.:

/ - : ~ ~ . . . .

. , . - " X ~ . " , ' . : : . , . . . , , , ,

'2.00 0.40 0.80 1.20 1.60 B-V

+ S T A R S W i T H M g l l E M I S S I O N x B L U E S T S T A R S W I T H H e A B S O R P T I O N L I N E S

• R E D C E P H E I D S

FIGURE 4:

The color magnitude diagram for G and K stars with CalI K 2 emission (ooee) taken from Wilson (1976). We have added an additional point for Procyon (FOIV). We have also added ++ for the bluest stars observed to have MgII h and k emission and xx for the bluest stars showing He absorption lines (Zirin 1975). Also shown are the positions for Cephelds close to the red boundary of the instability strip. The straight line roughly marks the boundary for stars with or without observed signs of solar type chromospheres.

For the higher luminosities the red boundary of the instability strip appears to

agree roughly with the boundary line for Call and Mgll emission. There is of course

Call K 2 emission observed for some Cepheids and also for @ Cyg but this is supposedly

due to shockwaves created by pulsation. The agreement of these two boundary lines is

not surprising since we see no other reason fo= the breakdown of the pulsational in-

stability but the onset of efficient convection which reduces Frad, thereby reducing

the driving force.

Since the theoretical line for the onset of efficient convective energy transport

depends on ~ we can check which e should be chosen to make the theoretical and observed

boundary lines agree. Assuming that ~ is the same for all stars in this region - an

assumption which has been criticized by Schwarzsehild (1974) - we found agreement for

=0.5, 2, 3 or 5. This can be seen from Figure 5 (B~hm-Vitense and Nelson 1976).

(If e should not be same for all stars, then £ = R 2 appears to be also a possibility

except for la supergiants.)

As already noted earlier, the extension of the instability strip boundary reaches

the main sequence at about FO or B - V ~ 0.3. So F stars would be expected to have

efficient convection, while A stars would not, but they appear in the extension of

the instability strip, as mentioned earlier.

(h) Stellar rotation:

It has been suggested that stellar rotation will be braked by means of the stellar

wind and the magnetic field. (See for instance Kippenhahn ]972; further references are

given there.) Since stellar winds for later type stars are due to the presence of co-

ronae which are linked to stellar convection zones,the decrease of the rotational ve-

locities for the F stars may also mark the onset of efflc~ant convection. In Figure 6

(B~hm-Vitense and canterna 1974) we show the dependence oi the rotational velocities

on B - V for main sequence stars for different clusters. For field stars there is a

drop in v sin i for B - V ~ .25~ a second drop seems to appear for B - V ~ 0.4. r

For some of the clusters the drop at B - V = 0.40 is the more pronounced one. Apparant-

ly the final drop in v sin i does not occur where we expect convection to set in but r

only for cooler stars. It seems to occur at temperatures where the hydrogen and helium

convection zones merge.

C. TEMPERATURE INHOMOGENEITIES

Convective temperature inhomogeneities are expected to be largest for F stars.

We might look for evidence in the integrated light.

In ~igure 7 we compare continuum energy distribution of stars whose surface is

assumed to be half covered with an atmosphere with T e ~ ~ 100 ° and half with T e = 8340°.

The average would be 7500 °. The resulting energy distrib ion would appear as that of

an atmosphere with T = 7750 °. The inhomogeneous atmos ~ere~ therefore~ would re- e

72

I I ' H I I, I I ' I ~ i

0 • J- , H~z

5

c:) o,°~, //o "(~-)

..o/ I T ,I , I

7 0 0 0 6 0 0 0 5 0 0 0

,°, °°*' / /

• .00a L 0

| , ~ l . - - ~ ! 6 0 0 0 5000 ?000

FIGURE 5 :

Taken from B6hm-Vitense and Nelson (1976) this figure shows a comparison of theoretical and observed (---- or - - ) boundary lines for efficient

convection in the luminosity T e diagram (T e = T'-400 +_ 150~. Different values for the ratio of the mixing length £ to the pressure scale height H were

assumed, symbols are given in the graph. To obtain the points in Figure a we required that £ -- < 21 D for a con-

sistent theory, where D is the thickness of the unstable layer. The theoretical ( .... ) and observed boundary lines agree roughly for £ = H.

For the points in Figure b we assumed that ~ < DF, where D F is the extent of the zone where F c > 10 -2 - F, where F is the total flux and F c is the convective flux. No agreement between observed and theoretical boundary line can be found for any value of </H.

In Figure c we required i= D I at the boundary, where D 1 is the extent of layer for which V - V' > O. I, with the usual notation. No agreement between theoretical and observed boundary can be found.

In Figure d we required £ = D at the boundary, where D m is the extenc

of the layer for which V - V' = 0.~ (V - V')max, where (? - V')ma x is the maximum value of V - V' in the convection zone under consideration. No agreement can be found between observed and theoretical boundary lines.

For details of the derivation see B6hm-Vitense and Nelson (19761.

73

3 0 0

2 0 0

I00 ¢.l (g

3o °"

= 2 0 0 b~

I00

30 O"

2 0 0

I00

(0) o F IELD STARS

!%~', O e 0 0

"~ e • •O

(c) PRAESEPE

ee = e •

e ~ • e e ~ o

e • o e • •

• • •d l , e O N ,, ,* ~,

' (e ) PLEIADES "

• ; ; .

e

. o • •

o ~ ' = ' I " I t I I | ,

O 0 0.1 0 .2 0.5 0 .4 0 .5

J I ] I I

(b ) COMA BERENICES

e • • •

(d) HYADES-

• ..- .

"., ,,. ,- • e e • e o

• •

(f)

e

e

= PER

0

B-V

o• e

%

I ! [ ...... d ~ • I

O.I 0.2 0.3 0 .4 0.5 0.6

FIGURE 6 :

The rotational velocity for field stars and for stars in different

clusters. TWO discontinuities in the velocity distribution of field stars are

suggested, one at B - V ~ 0.25 and the other at B - V = 0.40. For the clusters

the decrease of vsin i at B - V % 0.40 is more pronounced. The figure is

based on measurements by (a) : Abt and Hunter 1962 and Slettiebak 1955; (b)

and (d): Kraft 1965; (e) : Dickens et al. 1968; (e) : Andersen et al. 1966;

(f) : Kraft 1967.

74

o

un

0~1 I

I0.0

10.5 -

II . 0 -

HOMOGENEOUS RADIATIVE T, .-' 7 6 0 0

I I I

T e ~ 8 0 0 0 (,% log Fv:+ 0 .12)

i I I I 3.6 3.7 3.8 3.9

log X

FIGURE 7 :

The continuum energy distribution of a star whose surface is half cover-

ed with an atmosphere with T e = 6800 ° and half with T e = 8340 ° is compared

with the continuum energy distribution of stars with T e = 8000 ° (shifted

by Alog F v = 0.12) and T e = 7600 ° . The energy distribution of the inhomo-

geneous star is indistinguishable from that of a homogeneous star with

T e ~ 7750 ° .

75

semble one with a temperature about 250 a higher than its actual T e

In Figure 8 (from B~hm-Vitense 1972) we compare line profiles of a homogeneous

star and an inhomogeneous one with AT = ~ ]000 °. The T e and the corresponding frac-

tions of the surface area on the inhomogeneous star were chosen in such a way that in

Hy the~lines agree for both the homogeneous and the inhomogeneous stars. The B - V

colors then turn out to agree also. One finds that, because of the larger contribution

from the hot component, the spectral lines for the inhomogeneoua star are generally

weaker by ~ ]0% than those for the homogeneous one. The effect is especially strong

for the Ca lines,which are reduced by about 30%. One is reminded of the A stars oc- m

curring in this temperature range which show weak Ca lines, however for the A m stars

the other metallic lines appear generally stronger.

D. THE INFLUENCE OF CONVECTIVE ENERGY TRANSPORT ON THE CONTINUOUS

ENERGY DISTRIBUTION

The major influence of convection on the stellar structure is due to the convec-

tive energy transport which reduces the temperature gradient. If ~is happens in visible

layers then we might be able to see changes in the continuum flux distribution. F stars

are especially promising since the instability sets in rather high regions of the at-

mosphere and we have a pronounced minimum in the continuous absorption coefficient K C

longward of X = 3647 ~, where we can see down into the atmosphere tot = 2.

In Figure 9 I have plotted temperature stratifications for various depth dependen-

ces of the radiative flux F r = F - F c. Also shown are the corresponding energy distri-

butions in the visual region for stars with T ~ 7900 °. e

The curve shifted upwards (separate scale) shows the depth dependence of the ra-

diative flux as computed with the mixing length theory of convection.

The model with the steepest decrease in F r is a scaled solar, the socalled Bilder-

berg model. In an earlier study we had found (Bbhm-Vitense 1970) that the UBV colors

of main sequence F stars could not be obtained for radiative equilibrium models but

that scaled Bilderherg models would do fine. The other models were computed in order

to see whether we could reproduce the observed energy distributions with less ex-

treme reductions of the radiative flux. Little change from radiative equilibrium F

is obtained for these models.

It is interesting to note the very sensitive dependence of the surface tempera-

tures on the depth dependence of the convective flux. A low surface temperature as

obtained for one of the models should be apparent in the line spectrum which should

possibly look generally like that of a metallic line star.

From mixing length theory we expect a rather abrupt onset of convection for

T e ~ 8000 a. When using scaled Bilderberg models for convective stars we find a color

change of A(B - V) = 0.07 (B~hm-Vitense 1971). A similar value was also obtained by

Matsushlma and Travis (1973) with their nonlocal theory of convection if they use

76

,,, .... ~ i ..... ~ ...... f, +, , , i ~ ~ " 7

,.o,~// ? .oo~// ? '~W/ ~ "°° ¢I I /,7--°? ..~,y-t,:oo~ it~oo t k oo.

, , , , , , ,4: : , t , , , , . , ,r!,:, ~ - , , , , . , , , . T . . , ~; , : 7

.<. ,&,oo~;;;~oooo ¢ A,oo-- ,X/,,o~t I x / , "" / .,[.__.J/,,oo, -Ti-"J,,oo, ~ ItD,oo,-; iJ/-,,oo, t

, I l l l ,~. I'~, 25, . . . . . , , , , , ,,,1" , f 9 ~,1", I , i l l , , , i l , , , ' l ..7,,o+jf ~,,ooo+;/~ ~ ~-, ,<>o_+'-, '~;t • ' 7 h , , - ,4/.-o • ,,~/~,,oo. t-J~,,°°, t__,Yk,,°°. :-__5/~,,°°. t " -,~--~,, ,. 9 < ~ : ~ ..... Ro:: ~ , ~ G , : - ,', , , :'9:1 O' .02 .04 ,06 .08 0 ,OZ .04 .06 .08 0 .02 .04 .06 .OB 0 .Oa .04 .06 .08 1.0

,,~,[~]

FIGURE 8 :

The line profiles of convective stars with AT e = ~ iooo ° are compared with those of a homogeneous (i.e. AT e = O) convective star and a star in radiative equilibrium, The inhomogenous star shows generally somewhat weaker lines especially of Calcium. The figure was taken from B6hm-Vitense 1970.

77

! o m *< t -

0 w !

0 K

16

/ °

_~~....__../ (o) • 6-

" I O ~ ! . 0 5 I O 4 , , - - P - " I . I I I I I i

5x lO "4 I0 " z 0.1 0 .67 3,5 21.1

-~ ,-~'

- ----"~-x~¢,,-- -

4 ,0" ,0~\ 10 (b) 2 I I I II I\ I m

3 ~lSJlO - 4 tO "2 0.1 0 .67 3.5 21.,

I.C ~ C _o ~

3.6 3.7 ~.e 3.9 log ),

FIGURE 9 :

In part (a) we give the temperature stratlfications for several models, in part (b) the corresponding depth dependences of the radiative flux and in part (c) the corresponding continuum energy distributions. The ooo in part (a) refer to the radiative equilibrium model for the stable layer and mixing length model for the convective layers. The corresponding depth dependence of the radiative flux is shown by the displaced curve (b) (with sparate scale). The long dashes refer to the scaled Bilderberg model. The other curves refer to trial models that were computed in order to see whether we could also obtain a reduction in the energy emitted Just longward of the Balmerjump with a smaller decrease of the radiative flux in layers ~ > I. This does not seem to be possible.

78

= | ( which however,leads to difficulties with the observed solar center to limb

variation). This abrupt color change should he observed as a gap in the observed B - V.

The presence of a gap at 0.2 < B - V < 0.3 for field stars was noticed by Mendoza

(1956). He also noticed that this gap is not present for Pleiades stars. Figure ]0

shows that this gap is indicated more or less pronounced in different clusters (B~hm-

Vitense and Can terna 1974), though its position changes slightly for different clusters.

We suspect that rotation my have influence on the onset of convection. If this inter-

pretation of the gap is correct, then field stars with B - V < 0.22 should be in ra-

diative equilibrium, those with B - V > 0.29 should be influenced by strong convec-

tion, leading to the color change.

Figure 11 shows the result of observations for field stars by Oke (1964), for

Hyades stars by eke and Conti (1965), and by Baschek and Oke (1965) for A m stars.

In the same Figure we compare these scans - corrected for the change of absolute

calibration (Oke and Schild 1970 and Hayes 1975) and for line absorption - with con-

tinuum energy distributions for radiative equilibrium models and for scaled Bilder-

berg models.

The right hand side of Figure 11 shows the result for the Hyades stars. Except

for the small deviations around 4000 ~ the scans show good agreement with radiative

equilibrium models (solid lines) for B - V < 0.2 and agreement with the scaled Bilder-

berg models (dashed curves)for B - V > 0.3 as we expected from the study of the colors.

The left hand side shows the results for the field stars, which display the gap

very clearly at B - V = 0.22. Unexpectedly we see the influence of convection already

for ~ Arl with B - V = 0.14.

Figure 12 shows the results for A stars. The cooler ones clearly fit convective m

models and not radiative ones,

We have made additional scanner observations of main sequence H~ades stars with

different rotational velocities (B~hm-Vitense and Johnson 1977). In Figure 13 we see

the results. Weather and instrumental problems reduced the accuracy of our Hyades

observations, but we can notice some interesting results: our bluest star is p Tau

with B - V = 0.24) i.e., at a B - V where the field stars show the gap. Unfortunately

we have only rather poor measurements for that star so our conclusions are somewhat

shaky but it seems this star shows effects of convective energy transport.

For 57 Tau we have plotted both Oke's and our 7all and spring measurements in

order to give an impression of the uncertainties in the observation which Conti and

eke estimate for their measurements to be of the order of om.02. For the Hyades our

uncertainties may be larger. We think that also for 57 Tau th~ ccnvective energy dis-

tribution fits better than the radiative one, though it is not quite conclusive.

For the field stars our scanner results are seen on the left hand side of Figure 13.

For stars with B - V ~ 0.3, all these stars clearly show the decrease in the UV as

given by scaled Bilderberg models in indicating very efficient convective energy

7 9

"°"o,° . t .... . 1

, , . , , . . . . . . . O if, , ., ' F ." .. • "..:" |2 FIELD . "" _ .

~.. , . . . . . . . ,.n 01-."41,:, ..J,M. LW.~.." ." [.,:.' m

| ~ i "1 f ~' . " " .|.

I- ": '" , ; o , L ~ . , , . ~ " : ' " " . " FIELD STARS J.M. ( 0 ) 1 l (b]" 02~ I I I IJ O ~ . | l ' I • l ,

• 0 0 .2 0 . 4 0.6 "" 0 0 .2 0 .4 0.6 8-V 8-V

I i -- x I I I .... J 0 . 8 1 , , ~' - , io I F I E L D . ' ~ -0.II" COMA ...C.

I '" :lit° el ';"II ~. . s.R " , ~ "." .

1.41____.I_-*,4.~ I " I I I 0 2 1 1 ... • , . , ( d 0 0,1 0 .2 0 .3 0 .4 " 0 0 .2 0.4 0,6

b -y B -V

"°i I , ,,,,,,,,,, , , , , . . . . . o J / , , , - - - } t = O~ HYADES . ~p,j,~ ? 0 P R A E S E P E =.:.j.",.~

':":": " "~!1 ' " "

o, , . : , . : . ' , f . , ' = o . ,~ , . , , , (e o , ," " , ('

0.2," 0 0.2 0 .4 0.6 " 0 0 .2 0 .4 0 .6 B-V B-V

= ".." .'.~ " % , 1 =, o~ .:, ".- . , , , ; . ' . . . : . ~ J ~ o., I -.-,: .-:. o°~r , " . , , - . , o~ . . . . ",, , •

' 0 0 .2 0 .4 0 ,6 0 0 ,2 0 .4 0 ,6 B - V B - V

FIGURE 10:

The two color diagrams for field stars and different star clusters. This

figure is taken from B6hm-Vitense and Canterna (1975) and is based on measurements by (a): Johnson and Morgan 1953; (b): Johnson et al. 1966; (c): Str~mgren and Perry 1965; (d): Johnson and Knuckles 1955; (e): Johnson |952;

(f): Johnson et al. 1962; (g): Johnson and Mitchell 1958; (h): Mitchell 1960. The gap for O.22 < B - V < 0.29 is very pronounced for field stars. It

is present in most of the clusters, though not visible for the Pleiades.

80

3.6

3.8

4.0

(.3 4 .2 +

~ 4.4 0

4.6 O4 I

4.8

5.0

5.2' 3 .6

- "e -

~ ' - [ ~ 8 7 0 0 q . h= A I , O . O I ~e( "r ~%= "~ . ~-..'~-~kq. ~ , ~-~ ~ )r "%~ ~ . K Teu. 7 7 . ~ O o o ~ i ' ~ , ° . . . . , ~ . - - : ~_ ~ . ~AT, 0,4 - ~ - . ~ , , o . ' ~ ~ ~ ' ~ ~8 ,ooo - ~ . ~ . _ - , ~ . ~ . . . . _

[ ~ 4 |QU t ~)~ ~-~_-~ -.~A,, 76 /1%°0, ~ ~A75.o ,6

_ "x. .XPsc, 7 o ..o~o ~ ~.o,0_

o. s . , . ~ ~ , .E ( o - s , , , . . . 7 , o o ° . . ~ l ' r ~ . . . ~Oo- FO, 0.33 ~ ~=-ru , v.~%_

" ' - AVERAGE" 7 6 Tou, 115 I I 1 I~ I I LINE 1 F 0 , 0 . 3 2 I

3.7 3.8 3.9 4.(J3.6 3.7 3.8 3.9 4.0 log k

FIGURE 11:

Shows scanner observations ( .... ) by Oke (1964) and by Baschek and Oke (1965) corrected for line blanketing and corrected for the new calibration by Oke and Schild (1970) and Hayes and Latham (1975). The points (-) and • demonstrate the difference obtained for the continuum with different measured line blanketing corrections. Also shown are the computed continuum energy distributions for radiative equilibrium models ( ) and for scaled Bilderberg models ( .... ). For the Hyades stars, shown in the right hand column, radiative equilibrium models can represent the observed distribution rather well if B - V < 0.22. For larger values of B - V a reduction of the flux for

~ 4000 ~ is ~pparent indicating a flat temperature gradient in the layers T ~ i. For the field stars shown on the left hand side the violet flux is reduced already for 8Ari with B - V = O.14 and I Ps¢ with B - V = O.19. The energy distributions can be represented quite well with scaled Bilderberg models, i.e. models with an unexpectedly large convective energy transport in layers with ~ ~ I.

The number given beside the star name gives the rotational velocity vsin i, the number beside the spectral type gives B - V.

81

4,0

4"11 4.2

4.3

4.4

° 0 ÷ 4

~ 4.6 _o ~. 4.7 N

I 4 . 8

~e ~dO0, 60 Leo

900 15 " 0.20

m , , , . , . o

f.. 7~0 63, Tou 0.30

4,9 ~ -OC--" -- -- 1 _-- -. 5.0 ~q .

rUMo 5.1 r . 0.35

5 2 / I ! 3.6 3.7

log k

| I" 3.8 3.9

FIGURE 12 :

Shows the A m star scanner observations by Baschek and Oke (1965), cor-

rected for line blanketing and the new calibration, (notation as in Figure 11). The bluest star, 60 Leo, can in the average be well represented by a

radiative equilibrium model. (A discontinuity might be suggested at a wavelength, where a discontinuity in the OI continuous ~ occurs). In 15 Vul with

B - V = o.20 some convective energy transport may be present. For 63 Tau and T UMa the flux reduction in the violet is even stronger than predicted by the scaled Bilderberg model, however, the line corrections may be some-

what uncertain. For A m stars convection appears to become important for about the same

B - V as for normal stars.

82

transport in the top layers of the convection zone. For the field stars in the gap

(0.22 < B - V < 0°29) we can almost match the observed energy distribution with ra-

diative equilibrium models except for the sharp downturn just longward of the Balmer-

jump. It seems they try to have convection like Hyades stars but do not quite make it.

I do not understand this difference between the Hyades and the field stars.

In the last Figure 14 we have plotted the T , derived for the different stars e

by these comparisons of scans and computed energy distributions as a function of their

B - V given in the literature. Also given are the spectral types and the VrSin i. The

filled symbols indicate stars matching convective models, the open ones radiative

equilibrium energy distributions. Stars for which the decision could not be clearly

made are given in brackets. They are mostly the field stars in the gap. If we leave

out these uncertain ones, then we see two sequences, one for radiative and one for

convective energy distributions. The stars with high VrSin i occur exclusively on the

convective branch. I would interpret Figure 14 as telling us that generally convection

will become efficient for B - V > 0.22, however fast rotation will cause an earlier

onset of efficient convection leading to a reddening of the star by A(B - V) = 0.07

as given by the scaled Bilderberg models.

Could the decrease of the flux longward of 3647 ~ be a direct result of rapid ro-

tation without involving convection? Collins's results (1965) show that such an effect

can only be expected if the star rotates close to the Roche Limit and if at the same

time we look almost equator on, i.e.~ sin i ~ ]. For A stars this should lead to

v sin i ~ 350 km/sec, which is much larger than the observed values. r

Also indicated in Figure ]4 are the values for the A m stars. After correcting

B - V for the additional line blanketing - Baschek determined A(B - V) ~ 0.05-0.07 -

they fall on the same two sequences defined by the normal stars. There does not seem to

be any difference with respectto convectiono I might mention that we c~zmot reproduce

Baschek's and Oke's scanner measurements for any of the A m stars which we measured,

namely ]5 Vul, T UMa, 60 Leo, even though we do reproduce the energy distributions of

normal stars~ except for some minor discrepancies for 45 Tau. We are inclined to con-

clude that the A stars are all variable on time scales of the order of decades, a m

timeseale that reminds one of the solar cycle. We are presently checking on the

variability. This result is only preliminary.

III. SUMMARY

We have pointed out that stellar evolution computations presentl F need a good con-

vection theory rather than give us relevant information. The measured micro-and macro-

turbulent velocities may tell us something about convection, but we do not really know

when. Temperature inhomogeneities are hard to measure. The continuum energy distribution

in the UV for stars with B - V ~ 0.30 clearly shows the effect of a reduced temperature

83

5.6

5.;

4.8

o ~

L~ 4.4

l

4.0

, , , , ~ ' , i , , , ~ , , I , - ~__~a3oo~ ~...~-.~. ~o.o

~ooo. 5"~ CAq, ,s0 ~ "

;;'._;o~%~oo -%-~,<25 ~ o ; ~"',.. ,.u.~,.5 ~ ~ • /'.~ - . ~ , u - " " - - ' l = ~ F 2 0 . 5 6 " ~ - ' ~ " ~ " ~ A , ¢ , - , : ;n -'~OO.W'x%&.~STTou 100

.~mC_ .e,O, ' " ~00_ "~ ,~ ~'~ . . . . . . © " - " ~ A9 -FO" 0 2 8

/~ ~ _ _ ~ . r t u y g , D . " " " ~'~,~..=G'0. ~ e , A7 0,2 ~, _ . ~ H D 2 4 3 5 7 , 5 0 . • ~ - ~ % . F 4 , O . 3 8 ~ ~'~ln."n"n"n"n"n"~L • ' ~ ~ m "~' '-~61~uUr -x=~,FI 0 3 4

• *" ~ * ' - - ~ [ 0 ~ -~FO 0 2 6

,, ro-~oOc - ' _~-- '~=-~¢,~ "~FO, 0 , 2 6 ~; ,~- " 2o "~ ~'f '50 . - ¢ - - ¢ .00¢

/ . . . . r _ FO, 0 . 5 0 ~ . . . . . . ~6000~4, F2 0 .37 .." ..~oo~--?~'-.~ .. oo~~71ooc , ,

i l i I i i i ¢ ~ I i I ~ ' ' , , , ~ ' ~ , 3.6 3.7 3.8 3.9 4 .0 " 3 6 3 7 3,8 3,9 4 .0 '3.6 3,7 3,8 3,9 4 , 0

~og X

FIGURE 13:

Shows the scanner observations by B~hm-Vitense and P. Johnson (1977). For 57 Tau we compare 2 sets of measurements of B6hm-Vitense and Johnson with those of Baschek and Oke to give an impression of the uncertainties. ~Indieates stars for which we have only 1 or 2 usable scans. Otherwise notation as in Figure 11.

The scans confirm our previous conclusions. The stars in the "gap" i.e. with 0.22 ~ B- V ~ 0.29 show evidence,of convection, the field stars as well as the Hyades stars. HD 27429, a rapidly rotating star, shows an especially large reduction of the blue and violet flux.

84

9000

8500

Te 8000

7500

7 0 0 0

0

OAI,118 I

• HYADES, CONV. 13 HYADES, RAD. 0 FIELD, RAD+ • FIELD, CONV.

I !

• A5, 76 B Ari

r + A7' 1600 + ( c } ~7,77 a.eAs,+-+P,,+)~

A6, 9 0 0 _ -A7, 70 +C

A7.5, 52D + ( r ) mmA8, t30

AT, 74 r'l(OA5 ' 175) (FO, 5 0 1 1 A 8 , 215

A9 ,92 rt ( (0 FO))

((FO,5OO})m A9, I00

lI FO, 60 FO

(FO0) FI,SC

FO, I I S I I ~ F 2 F 0 , 8 5 +/ c + I I F 2

• F 2 F 2 , 9 0 0

I I I O, I 0.2 0 3 0.4

B - V

150

FIGURE 14:

Gives the relation between T e and B - V as determined from the comparison of computed and observed energy distributions. Filled symbols indicate matches with scaled Bilderberg models, i.e. convective models, open symbols

matches with radiative equilibrium models. Dubious cases are given in brackets, they always appear twice, once matched with a convective model and once with

a purely radiative model. The spectral types and the rotational velocities VrSin i are given for each star. Leaving out the dubious cases we find that

for B - V ~ 0.30 all stars are convective. For smaller B - V we find two branches: One for radiative equilibrium stars and one for convective ones.

The high VrSin i occur all at the convective branch, indicating that rapid rotation appears to enforce convective energy transport in the late A stars

rather than to inhibit it as was previously suggested (B6hm-Vitense and Canter-

na 1975).

This conclusion rests on the interpretation of the somewhat uncertain energy distributions of p Tau, 69 Tau and 57 Tau.

+r refers to radiative equilibirium A m stars, +c to convective A m stars. After correcting B - V for the additional line blanketing they fall on the same two branches as the non A m stars.

85

gradient in layers ~ ~ 2/3 indicating an unexpectedly large convective energy flux in

these layers. Stars with B - V < 0°2 and VrSin i < |00 km/sec mostly appear to be in

radiative equilibrium. Stars with 0.10 < B - V < 0.30 may be convective, they are al-

ways convective if they have large v sin i. r

The point B - V = 0.30 also marks the extension of the red boundary line of the

instability strip to the main sequence. We take this as an additional evidence that

this boundary line actually marks the onset of convection in the HR diagram. With local

mixing length theory I/H = ] leads to agreement between the theoretical and observed

boundary lines for convection, neither larger nor smaller values for I/H will do.

The author's research described in this review was made possible by an NSF grant,

which is gratefully acknowledged. I am also grateful for a U.S. senior scientist award

from the "Alexander yon Humboldt Stiftung".

REFERENCES:

Abt, H. and Hunter, J.: 1962, Ap. J. 136, 381 Allen, L.H., Greenstein, J.L.: 1960, Ap. J. Suppl. ~, 139 Andersen, P.: 1973, P.A.S.P. 85, 666 Andersen, C.M., Stoeckley, R. and Kraft,R.P.: 1966, Ap. J. 143, 299 Baschek, B. and Oke, J.B.: 1965, Ap. J. 141, 1404 Baschek, B. and Reimers, D.: 1969, Astron. and Astrophys. ~, 240 B6hm, K.H.: 1958, Zs. f. Ap. 46, 245 B6hm-Vitense, E.: 1970a, Astr~n. and Astrophys, 8, 283 B6hm-Vitense, E.: 197Ob, Astron. and Astrophys. 8, 209 B6hm-Vitense, E. and Szkody, P.: 1974, Ap. J~ 19~, 607 B6hm-Vitense, E. and Canterna, R.: 1975, Ap. J, 194, 629 B6hm-Vitense, E. and Nelson, G.: 1976, Ap. J. in press B6hm-Vitense, E. and Johnson, P.: 1977, in preparation Canterna, R.: 1976, private communication Chaffee, R.H.: 1970, Astron. and Astrophys. 4, 291 Chandrasekhar. ~. :1961>,Hydrodynamic and Hydromagnetic Stability (Oxford,

Clarendon Press) p. 135 Collins, G.W.: 1965, Ap. J. 142, 265 Dickens, R.J., Kraft, R.P.~ and Krzeminski, W.: 1968, A . J. 73, 6 Frisch, H. and U.: 1975, Physique des Mouvements dans les Atmospheres

Stellaires (Centre National de la Recherche Scientifique, Paris 1976)

Garz, T. and Kock, M.: 1969, Astron. and Astrophys. ~, 274 Hayes, D.S. and Lantham, D.W.: 1975, Ap. J. 197, 593 Hoyle, F. and Schwarzschild, M.: 1955, Ap. J. Suppl. 13 Huan9, S. and Struve, O.: 1952, Ap. J..116, 410 Iben, I.: 1963, Ap. J. 138, 452 Johnson, H.L.: 1952, Ap. J. 11_~6, 640 Johnson, H.L. and Morgan,W.W.: 1953, Ap. J. 117, 313 Johnson, H.L. and Knuckles, C.F.: 1955, Ap. J. 122, 209 Johnson, H.L. and Mitchell, R.I.: 1958, Ap. J. 128, 31 Johnson, H.L., Mitchell, R.I. and Iriarte, B.: 1962, Ap. J. 136, 75 Johnson, H.L., Mitchell, R.I., Iriarte, B.;and Wisniewski, W.: 1966, Comm.

Lunar and Planet. Lab. No. 63 Kippenhahn, R.: 1972, In Stellar Chromospheres, Proceedings of ~ASA Collo-

quium.

86

Slettebak, A.: 1955, Ap. J. 121, 653 Str~mgren, B. and Perry, C.: 1965, unpublished report, Institute of Advanced

Study, Princeton, N.J. Traving, G.: 1975, Physique des Mouvements dans les Atmospheres Stellaires

(Centre National de la Recherche Scientifique, Paris 1976) Wallerstein, G.: 1962, Ap. J. Suppl. ~, 407 Wallerstein, G. and Helfer, H.L.: 1966 Ap. J. 71, 350 Wilson, 0.C.: 1976, Ap. J. 205, 823 Wright, K.O.: 1955, Transactions of the IAU, IX, 739 Zirin, H .: 1975, Reprint, Hale Observatories (Carnegie Institution of

Washington, California Institute of Technology, BBSO No. O150)

Kraft, R.P.: 1965, Ap. J. 142, 681 Kraft, R.P.: 1967, Ap. J. 148, 129 Lucy, L.B.: 1976, Ap. J. 206, 499 Mannery, E.J., Wallerstein, G. and Welch, G.A.: 1968, Ap. J. 73, 548 Matsushima, S. and Travis, L.D.: 1973, Ap. J. 181, 387 Mendoza, E.E.: 1956, Ap. J. 123, 54 Mitchell, R.I.: 1960, Ap. J. 132, 68 Oke, J.B.: 1964, Ap. J. 140, 689 Oke, J.B. and Conti, P.S. : 1966, Ap. J. 143, 134 Oke, J.B. and Schild, R.E.: 1970, Ap. J. 161, 1015 Paradijs, J. van: 1973, Astron. and Astrophys. 23, 369 Reimers, D.: 1976,Physique des Mouvements dans les Atmospheres Stellaires.

(Centre National de la Recherche Scientifique, Paris 1976) Rosendhal, J.: 1910, Ap. J. 16__~0, 627 Sandage, A. and Tammann, G.A.: 1971, Ap. J. 167, 293 Schwarzschild, M.: 1975, Ap. J. 195, 137 Sedlmayr, E.: 1975, Physique des Mouvements dans les Atmospheres Stellaires.

(Centre National de la Recherche Scientifique, Paris 1976)

DYNAMICAL INSTABILITIES IN STARS

P.LEDOUX

Institut d'Astrophysique

Unlversit~ de Liege

SUMMARY

The linear dynamical instability at the origin of convec-

tion in stars is reviewed and shown to depend essentially on the sign

of

A = ] d__~p ! dp p dr rlp dr

which is the usual argument of convection criteria. The case of two

or more superadiabatic regions separated by subadiabatic ones might

well deserve more detailed attention.

Once this instability is partially removed by the setting

in of convection its effects must be balanced by dis ~pation terms

if a stationary state is to result. This yields the value of a Ray-

lelgh number.

If energy generation is included in the non-c0nservative

terms, possibilities are somewhat enriched including a case of dyna-

mical instability in presence of A<0 (usually stable) but very small

in absolute value.

I. THE GENERAL PROBLEM

In the context of this conference we are not interested in

dynamical instability towards purely radial mQdes since convection

cannot manifest itself through these modes. We are thus left with the

problem of the response of the star to non radial perturbatius which

88

we shall assume very small to allow a linear treatment.

Of course, non linear effects may be of very great interest

and importance and are at least partially included in some of the ap-

proaches to steady convection as, for instance, in the mixing length

theory or various numerical attempts usually with other slm~lifylng

hypothesis and simple geometry (of. Spiegel ]971, 1972 Nordlund, 1976).

One of the most recent and most direct attaeksin general stellar cir-

cumstances is due to Deupree (1974, 1975 a-b, 1976). As far as I am

aware it has not revealed new instabilities such as may occur for

instance in metastable situations. It has not either restricted the

domain of significant dynamical instabilities but it has yielded

interesting information on the development of these instabilities

such for instance as a strong asymmetry between upward and downward

motions.

The study of non radial stellar oscillations does not go

far back. If we exclude Lord Kelvin's discussion of the homogeneous

incompressible sphere (Thomson, ]863) and some more or less timid

references by Moulton (|909) and Shapley (1914), the first significant

paper is that of Pekeris (|938) in which he solved the problem of the

non radial perturbation of the homogeneous compressible s~he~e.

Pekeris used the usual separation in time and spherical

coordinates of the Euclidian perturbations f~(pt,0', T',~') and of

the radial component of the displacement ~r

f'(r,O,~,t) - f'(r)pm(cosS)eim~e i~t -£~m~£

He showed thatp for each value of the degree ~ of the sphe~cal har-

monic~ apart from a positive spectrum with an accumulation point at

infinity corresponding to the pressure modes (or p modes, a~), there

89

existed a negative spectrum corresponding to gravity modes (or g

modes, a~) with an accumulation point at zero, all these modes being

(2~+I) degenerate with respect to m. An illustration of the distribu-

tion of the ~2 as function of ~ and the order of the modes can be

found for instance in Ledoux (1974).

Of course~ in this case, all the g modes (o~<0) are unstable

but I don't think that the connection with convection was pointed out.

Note that Pekerls choice of dependent variable (~= dlv ~) led him to

miss the so called fundamental mode (or f mode or Kelvin mode) which,

in this case, is exactly the same as the unique mode (for a given £7

of the incompressible sphere (~=O).

The next important paper is that of CowllnE (1941) in which

he tackled the case of the general polytrope of index n(p = Kpn+l/n).

In this case, the general problem is of the fourth order as it does

not split into ~wo second order differential equations which can he

solved successively as for the homogeneous model. However Cowling

noted that, except for the lowest modes and lowest values of £, the

perturbation of the gravitational field can 5e neglected without

serious effects. With this approximation, the problem reduces again

to the second order and, as Cowling showed, can be assimilated to a

St=rm-Liouville problem for large enough u 2 (high p modes) or small

enough I c21 (high g modest, the f mode for each ~ falling la between

the corresponding p and g spectra.

Furthermore if the generalized ratio of specific heats F 1

(or ~ in a pure gas) satisfies the inequalities

n+l r1> n (I)

all the o 2 are positive (stable g modes : g+) while, if 8

90

r < n+l i n (2)

they are all negative leading to instability (unstable g modes : g ).

But the criterion for convective instability in terms of

A = l dp l ap (3) 0 dr FlP dr

can be written very generally, even in the relativistic case (Thorne,

1966, Kovetz, 1967, Islam, 1970)

A >0 (4)

and becomes in a polytrope, with the usual notation,

A = 1O dEd--~0 (n -n_+i)111 >0 ( 5 )

or since (dO/dE) is negative

n+l (6) rl < n

2 which is identical to condition (2). Thus all the Og are negative

(dynamical instability towards non radial perturbations) provided the

criterion for convective instability (4) or (5) be satisfied every-

where in the star.

The negative g spectrum in the homogeneous model may be

interpreted in the same way since in that case

A I dp >0 Flp dr

gl

But apparently, it was only slowly (Ledoux, 1949), at least

in astrophysics, that the connection between dynamically unstable g

modes and the sign of A became to be recognized and that convection

became to be considered as the end effect of this instability.

In the general case, it is not difficult to have A appear

explicitly in the equations which can be written

~P _ O' ~r dp -- - -- + .... d i v ~r (7)

p p p dr

0~i ~ rip div ~r = ~2~r- grad (@' + ) + ~ A P .....

= r3 -I ! grad o ( ~ - i div ~)' - i_~a di~) i~ p o P

p' + rip ( °' + A~r) ~ ~ p(~- !div ~)' o is p

(s)

(9)

F 2 p iaCp p

A~' = 4~Gp' (11)

where

I kPikA - + -- ~ iVk e =£N E P

represents the total heat liberated per second by nuclear reactions

and viscosity.

One may note that A is related to the Brunt-V~is~l~ frequency

N by

N 2 ,, - gA

02

and to

S = r2 - i I d1~ I dT (12) r 2 p dr T dr

by

a = 4 - 3____/~ s + ! d£ (13) 8 ~ dr

if 8 is the ratio of the gas pressure ~o the total pressure.

The adiabatic approximation (right hand members neglected

in (8), (9), (I0)) is sufficient to discuss dynamical stability and

it should enable us to understand the correlation noted above between

the sign of A and the stability of the g modes.

If we neglect ~ we can for i~stance write a second order

differential equation for ~- ~r/r :

where

9 d~ 6 i do I d

dr 2 ( ) dr

÷ ¢ { 020 _ t ( t + 1) o 2 A8 + 6

rip r 2 0 2 r 2

d dr (PI~)

+ 3 i do I d 0r 2 r ~ ~r'r + ( ) dr (Plp)) ( ) r~p d-7 < ) } - o (14)

_0r2 )

For high p modes,o 2 large, the equation simplifies very

much and shows the acoustic character of these modes. However, they

are without interest in the present context since, in realistic stel-

93

far conditions, all the o 2 are always positive. But in the same way P

if one considers high g modes, I ~21 small so that terms proportional

to 02 can be neglected, the only term left over which contains 02

is the second one in the coefficient of ~ in (14), i.e.

~(~+l) A_~

r2 ~2

and the problem is essentially of the Sturm-Liouvllle type with a

parameter %~ 1/o 2 . It is well known in that case that if A keeps

the same sign throughout, the elgenvalue (l=I/=2)~fll be of the oppo-

site sign. In other words~if A is positive (convective instability),

at least the small u 2 will be negative. But it has been accepted g

generally that all the 02 will have this same sign, because it is g

difficult to see how the sign of 02 could change in going from small

to larger values if the sign of its coefficient is constant. Anyway

fairly recently Grisvard, Souffrin and Zerner (1972) using the second

order system

dv =(~( ~ + 1) Pr2 i p2tr l d--{ c 2 - r l ~ ~ w = aw (15)

dw (s2+ Ag) P d--r = ~ v = by (16)

equivalent to (14) managed to prove without any assumption as to the

order of magnitude of ~2 that the latter are all real positive if A

is everywhere negative. Thus a necessary condition for dynamical in-

stability is that A be positive at least in some part of the star.

In that case, the authors succeeded in establishing an upper limit for

the modulus of any negative c 2 g

94

I o~t < Max (Ag) (17)

where the maximum is taken on the region where A>O. They showed also

that 1o21 increases with £, i.e. when the horizontal wave length g

decreases.

However if the investigation of Grisvard et al is free of

any asymptotic restriction, it still treats the perturbation of the

gravitational potential 4' as negligible. Would the previous results

hold if 4' is not neglected, especially for the lowest modes and har-

monics? Lebovltz (1965 a-h, 1966) tackled this general problem on the

basis of the variational principle established earlier by Chandrasek-

her (1964)~ which expresses o 2 as the extremum of integrals exten-

ded to the whole configuration. This enabled him to show first that if

A<0 everywhere (convective stability) all the o 2 are positive which

is thus a sufficient condition for dynamical stability towards non

radial perturbations. It is also a necessary condition as he showed

later, since the existence of a region however small with A>O entails

negative eigenvalues. This implies also that one can find solution~

of the differential equations with appreciable amplitudes in that re-

gion only.

In this respect~the work of Ledoux and Smeyers (1966) was

more or less complementary since they pointed out indeed that when A

changes sign in O~r~R the asymptotic form (02 small, g modes) of equa-

tion (14) has s turning point in A = 0 and that the g spectrum splits

then into two : one of positive eigenvalues (a~ 20) corresponding to

modes oscillating in space with appreciable amplitudes in the convec-

tively stable region (A<O) and decaying exponentially in the unstable

region (A>O) while the other spectrum of negative eigenvalues (o~ <0)

95

corresponds to modes oscillating in the convectlvel F unstable

region (A>O) and decaying in the stable one.

Using equations (15) and (16), Scuflaire (1974, cf. also

Osakl 1975) showed that these properties of the unstable modes sub-

sist even for the first few modes~ i.e. they oscillate only in the

unstable region. However~ it may happen that stable modes oscillate

in a slightly superadlabatic region, as for instance in a condensed

model where the stable g modes characteristic of the central stable

core may continue to oscillate in an external convection zone.

2. MULTIPLE UNSTABLE ZONES

The unstable modes are of greatest interest here and an

interesting problem arises as to their behaviour if there are two

or more regions with A>O separated by stable zones (A<O). Tassoul

and Tassoul (1968) in a discussion of asymptotic g modes suggested

that there should be as many distinct unstable g spectra as there

are unstable regions.

However, this is not obvious since even in the simplest

case of two turning points (for instance two convectlvely unstable

regions separated by a stable one) the usual analysis of a single

turning point in terms of Bessel functions cannot be simply repea-

ted (Langer, 1959) at each of the two turning points. A solution

should rather be sought in terms of Weber functions allowing to cross

the two turning points at once.A considerable amount of literature

exists on the subject and a very recent paper by Olver (1975) opens

the way to straightforward applications. They may bring to our

attention, at least in special cases~ unstable solutions whlch~In

the above case, may have large and comparable amplitudes in the two

unstable regions with a relatively minor reduction of this amplitude

96

across the stable region especially if the latter is fairly narrow

with a density gradient only slightly subadlabatlc. Such modes could

be particularly efficient in mixing the whole star. This conjecture

is somewhat supported by some simplified or numerical investigations.

For instance, Goosens and Smeyers (1974) have found more or less

"accidental resonances" between some of the stable g modes of two

stable regions separated by an unstable one (Just the opposite of

the case considered above) giving rise to a stable mode with large

amplitudes in both stable regions decreasing only moderately in bet-

ween in the unstable region.

Other examples have been treated by students in Liege

which lead to slmilar conclusions. Consider for instance, the case

of a heterogeneous incompressible model composed of superposed layers

of different densities presenting two unstable discontinuities,

(Pin - Pext )<0~ separated by a stable one. The behavlour of the

elgenfunctlons associated with the two negative eigenvalues can

depend drastically on the closeness of these eigenvalues. In general~

i.e. as long as those two o 2 are not very close~ each of the elgen-

functions has a sln~le maximum at the unstable interface with which

it is associated. Kowever when the parameters of the discontinuities

are varied~ one of these solutions may acquire a secondary strong ma-

ximum at the other discontinuity, the minimum between the two remai-

ning apprecla~le ~r1~n t~n~s elgenvalue becomes very close to that cor-

responding to the other discontinuity.

3. THE EFFECTS OF THE DYNAMICAL MOTIONS AND THE REDUCTION OF THE

SUPERADIABATIC GRADIENT

Many of the examples where dynamically unstable modes have

been found are artlfleiel because the superadiabaticity (A) has been

fixed a priori at a much larger value than is ever llkely to occur

97

in stars. In such cases, the time-scales of tke growing motions

which are proportional to (A) -I/2 are rather short. Of course in the

end, these violent motions lead to the establishment of a convective

zone through which A is reduced to a very small value Just sufficient

to allow the residual energy no longer transferred by radiation to

be carried by convective currents. This implies important readjust-

ments in the internal structure of the star including transfer of

mass to deeper layers and various feed-back effects which may in-

crease considerably the extent of the convection zone with respect

to the initial superadlabatlc region. For instance, in going from

an homogeneous compressible sphere to a polytrope n - 3/2(FI=5/3)~

the energy released, if the mass and the radius are those of the sun,

is of the order of 0.i GM~ /R G = 5.1047 ergs in a short time of the

order of one hour. Of course this is an extreme case, but even if the

energy release is reduced by a factor I0 I0 and the duration increa-

sed by a few orders of magnitude, it would still remain a fairly

spectacular phenomenon which was considered at one time (Biermann~

1939; Schatzman, 1946) significant for the interDretatlon of novae.

In any case~ I suppose that there are no serious doubts

that the readjustments contemplated above would lead in the end to

the same model as the one that could be built a priori using the

usual method of having a convective adiabatic zone initiated at the

point where the radiative gradient becomes exactly equal to the adia-

batic one (A ~ 0).

In fact things are a little more subtle as the reduction

of the superadlabatlcity must proceed only so far that the subsisting

excess provides the necessary buoyancy force to balance the energy

dissipated by viscosity and conduction (radiative or otherwise). Of

course, when A and 1o21 are still large, the effects of these dissi-

98

pation terms could Be evaluated hy a pertuT~atlon metkod (similar

to that used for vibrational stability when ~2>O) yleldlug a damping

coefficient ~' correcting the adiabatic time dependence By a factor

-olt e

But one knows how cumbersome (cf. Ledoux 1974), the ex-

pression of ~' is. yo illustrate the situatlo~, it seems Better to

revert to a simple case as, for instance, the plane layer with cons-

tant coefficients. Let ~ and n represent respectively the thermome-

tric conductivity a=d the kinematic coefficient of viscosity. Assu-

ming a time depe~d~nce e st, one finds for the g modes, if A is posi-

tive and larger than K,~ or <+n

s = gA) z2- 2 (lg) k 2

where k~ and k z (k 2 = k~ + k ) are respectively the horizontal and

the vertical wave number. In this case, dlssipatio~ simply hinders

a little convection hut does not affect it very much.

On the other hand, when convection is established, the

effects of the dissipation forces are of the same order as those of

the residual buoyancy and the above formula is no longer significant.

In that case and if one includes the rate of energy Ke~eratlon c

(significant in the deep interior), one gets an equation equivalent

to that of Defouw (1970) with

~)g E L T = - _ = T ' L0 O

where v represents the sensitivity of ~ to T,~ =(dlog~)0/dlo~ T.

After separating a secular root s 3 = -n k 2, the dispersion relation

for g modes, gives solutions

99

1 ~ (~ - I) ) s = - y~ (~+ ~)k 2 -CT

P

2

+ i --~ (19) -'{~ ((~ - n)k2 CpT (v- i) 12 k zk~l gA} I/2

which for vanishing E reduces to the ordinary Raylelgh solution of

the B~nard problem

i + I s -- - y (n + ~)k 2 -{~ (~ 2 l/2

~) 2k4 kH + ~ gA} (20)

which yields back (18) if (k2/k2)gA>>~2k 4 and n2k 4

On the other hand, if we approach the marginal case

2 2 (st0), then (kH/k)gA must take the appropriate value to make the

square root exactly equal to the first term, i.e.

2

kH gA ~ nk 4 nk2 C~ (v-l) k 2

If e is negligible~this condition becomes

~Ad 4 = k6d 4

~n 2 k H

where the depth d of the layer has been introduced. If k z =~/d and

a ~ kHd , one gets finally

~Ad 4 = ( 2+ a2)3

<~ 2 a

= R

which is the usual value of the Rayleigh number. The energy genera-

tion could reduce somewhat the value of R. In this marginal case,

100

the tlme-scale- !/s Becomes infinite but a circulation sets in with

a finite turn-over time.

Thus convection~although originating essentially in a dy-

namical instability (conservative terms) with a short time scale,initi

ares fast motions to modify the medium itself in such a way as to

reduce gA fIBal~y to the same order as the dissipative (non conser-

vative) forces.

As Defouw (1970) pointed out, the energy generation term

has a destabilizing influence (cf.19) and if

CeT (v-l)> (n+~)k 2 P

it contributes directly to the instaDillty. If A>O, this effect

simply relnforc~ the Buoyancy.

On the other hand if A<O, and such that

2

k 2 p

the effect would correspond to a case of vibrational instability (or

overstability) of stable g modes. But if

2 kH gA<~ (< - -V 2 c T (v-l)

P

it seems that a growing non-oscillatory motion would arise which

might lead to some kind of convection, although A<O.

101

REFERENCES

BIERMANN, L. 1939, Z. Astrophys., 18, 344

CHANDRASEKHAR, S. 1964, Astrophys. J., !39, 664

COWLING, T.G. 1941, Mo~. Not. Roy. Astr. Soe., I01, 367

DEFOUW, R.J. 1970, Astrophys. J., !60, 659

DEUPREE, R.G. 1974, Astrophys.J., 194, 393

DEUPREE, R.G. 1975a, Astrophys.J., 198, 419

DEUPREE, R.G. 19755~ Astrophys.J., 201, 183

DEUPREE, R.G. 1976, Astrophys.J., 205, 286

GOOSENS, M. and SMEYERS, P. 1974, Astrophys. Space Sel., 26, 137

GRISVARD, P., SOUFFRIN, P. and ZERNER, M. 1972, Astron. and Astro- phys., 17, 309

ISLAM, J.N. 1970, Mon. Not. Roy. Astr. Sot°, 150, 237

KOVETZ, A. 1967~ Z. Astrophys., 66, 446

LANGER, R.E. 1959, Tra~s. Amer. Math. Soe., 90, 113

LEBOVITZ, N.R. 1965a, Astrophys. J., 142, 229

LEBOVITZ, N.R. 1965~, Astrophys. J., 142, 1257

LEBOVlTE, N.R, 1966, Astrophys. J., 146, 946

LEDOUX, P. 1949, Contribution ~ l'~tude de la s~ructure interne des

~toiles et de leur stabilitY, M~m. Soc. Roy. Scl. Liege

Coll. in -8 °, 4°s~r. T IX, Ch. III, sect. 4, 5, 6

LEDOUX, P. et SMEYERS, P. 1966, Compt. Rend. Acad. Sci. Paris, S~r.B., 262, 841

LEDOUX, P. 1974, in P.Ledoux et al (ed.) Stellar Instability and

Evolution, IAU Symposium n=59, Part Vl, p. 135

MOULTON, F.R. 1909, Astrophys. J., 29, 257 0

NORDLUND,A. 1976, Astron. and Astrophys., 50, 23

OLVER, F.W.J. 1975, Phil. Trans. Roy. Soc. London A, 278, 137

OSAKI, Y. 1975, Publ. Astron. Soc. Japan, 27, 237

PEKERIS, C.L. 1938, Astrophys. J., 88, 189

102

SCHATZMAN, E. 1946, Ann. Astrophys., 9, 199

SCUFLAIRE, R. 1974, Astro~. and Astrophys., 34, 449

SHAPLEY, H. 1914, Astrophys. J., 40, 448

SPIEGEL, E.A. 1971, Ann. Rev. Astron. Astrophys. ~, 323

SPIEGEL, E.A. 1972, Aun. Rev. Astron. Astrophys., IO, 261

TASSOUL M. a~d TASSOUL J.L. 1968, Ann. Astrophys., 31, 251

THOMSON, W. 1863, Phil. Traus. Roy. Soc. London, 153, 612

THORNE, K.S. 1966, AsCrophys. J., 144, 201

OBSERVATIONS BEARING ON CONVECTION

K.H. BbHM

University of Washington~ Seattle, WA., U.S.A.

SUMMARY

I. Solar observations contain a considerable amount of information on the hydro-

dynamics of stellar convection. We emphasize and discuss especially

(a) the existence of two very different cell sizes,

(b) the unexpectedly high cross-correlation between vertical velocities and tempera-

ture fluctuations in the granulation,

(c) the fast "downdrafts" in intergranular regions,

(d) the existence of cells much larger than the scale height,

(e) thes~mnge behavior of the temperature fluctuations in the supergranulation, and

(f) the importance of convective overshoot.

2. The Li-Be problem and its possible relevance as an indicator of convective

overshoot is briefly summarized.

3. Convection may have a stronger influence on the observable properties of He-

rich ("non-DA") white dwarfs than of most other stars. We discuss especially

(a) the persistence of outer convection zones through a very wide range of effective

temperatures,

(b) the occurence of convection in high layers of the atmosphere,

(c) the relatively high efficiency of convection in white dwarf atmospheres, and

(d) the relevance of convection to the cooling problem.

! • INTRODUCTION

The main ~£iculty of the present topic is due to the fact that almost all

spectroscopic and color observations of cooler stars are somewhat related to the

convection problem but that there are so few observations which seem to be really

crucial in this context. Almost all of the really decisive observations seem to refer

to the sun. Though in the stellar case the conclusions concerning convection can be

extremely interesting, the way to reach them is often indirect and some doubt is

usually possible.

Consequently we fee~ chat the topics of our discussion are quite obvious as long as

we look at evidence from the Sun, but that our selection of topics in the stellar

cases will probably be somewhat subjective.

104

I shall first report on the solar evidence. After that I shall go on to discuss

some observational evidence for convective mixing processes in stars.

Finallywe shall consider a class of stellar atmospheres whose average temperature

structure is strong, changed by the presence of convection in rather high atmospheric

layers.

II. OBSERVATIONS OF SOLAR CONVECTION

A. GENERAL REMARKS

As we all know there are at least two groups of phenomena which are thought

to be direct indications of the presence of convective motion (including overshoot)

namely granulation and supergranulatlon. In the case of granulation it is more or

less generally accepted that we see the direct effects of convective motion, in the

case of supergranulation some slight doubts may be possible. Nevertheless most

astronomers believe that supergranulation is the manifestitation of the penetration

of large cell convection.

In addition to these direct convective effects there are indirect effects which give

us some information about the structure of the convection zone. There are firstly the

observed eigenmodes of the so-called five-minute oscillation. I refer to the work

of Ando and Osaki (1975), Deubner (1975), the review given last year by. J.P. Zahn

as well as the very interesting comment by Mclntyre (1975). Secondly, there are

observed mixing effects in the sun which may or may not be due to convective mixing.

I refer specifically to the very low abundance of Li 7 and the apparent absence of Li 6.

I shall first discuss the direct evidence for convection from the observations of

granulations and supergranulation.

One of the most interesting observed features of solar convection is that there are

two scales of motion present near the surface of the convection zone, namely the

granular scale of about 2000 km (maximum of the A I power spectrum) and the super-

granular scale of about 32000 km. Both numbers correspond only to the relatively

flat peaks of broad power spectra. This statement is of course not new. However,

it is surprising how few theoretical attempts to understand this fact have been made.

However, there are some notable exceptions including especially the paper by Simon

and weiss (1968).

Let me now try to summarize briefly our knowledge of granulation and then of the

supergranulation.

B. GRANULATION

How do we describe the observational information on granulation ? There

are essentially two possibilities. Either we use a description based on the autocor-

relation function and power spectrum or we look at single granules and, maybe, derive

from them the properties of an average (or typical) granule.

105

For some time it seemed that a description using correlation functions and power

spectra would be the only useful method. However, more recently studies of properties

of individual granules have found great interest. The reason for this is of course that

a description is needed which permits a separate study of the granules on the one

hand side and the intergranular lanes on the other side.

In studying granulation we should like (ideally) to obtain a three-dimensional picture

of the velocity and the temperature fluctuations. This requires observations of very

high spatial resolution, observations in a number of different lines (which are formed

at different depths) and center-to-limb observations. One of the fundamental diffi-

culties is, as we all know, the separation of the granular and the oscillatory velocity

fields.

One way to do this is to identify the region in the k-~ plane which corresponds to

granular motion. A simpler method is to assume that all horizontal scales smaller

than'b 4000 km correspond to granulation whereas all motions with a larger horizontal

scale are due to oscillations (cf. Mattig, Mehltretter and Nesis ]969, Beckers

and Canfield 1976).

In addition, one finds that the observed velocities first decrease with increasing

height and then increase again (cf. Canfield and Mehltretter 1973, Mattig and Nesis

1974). This fact can also be used for a separation of the two velocity fields.

The component which decreases outward is usually identified with convection and

convective overshoot whereas the increasing (or constant) component corresponds

to the oscillations.

Even today the problem of correcting the observations for the effects of the

contrast transmission function of the telescope and of the atmospheric seeing seems

to be a very difficult one. This is true for the determination of the intensity

and the corresponding temperature fluctuations but even more so for the velocity

determination. Consequently we find a rather large scatter of the observational

results which is certainly to a considerable part due to seeing differences. Another

difficulty is due to the fact that different authors use different spatial domains

and use lines which are formed at different depths in the atmosphere. Consequently

it does not make much sense to take averages of measurement by different authors.

Rather, we shall take the detailed high resolution observations by Canfield and

Mehltretter (1973) as a typical example of modern results. These authors find a definite

decrease of the r.m.s, velocity outward before it starts to rise again. (This

result has been confirmed by Mattig and Nesis 1974, Mattig and Schlebbe ]974,

Musman 1974, and others).

The largest r.m.s, velocity is found for the line formed at the relatively

largest depth. The relatively faint Fel line 5178 shows a formation depth of about

40 Pan above the zero level of HSRA. Canfield and Mehltretter find a r.m.s, velocity

of .54 km/s for this line if corrected only for instrumental effects and of .73 km/s

if a reasonable correction for seeing effects is applied. By extrapolation the

authors find a r.m.s, velocity of .8km/s at the height of continuum formation.

106

It seems that these velocities are very roughly compatible with predictions of the

simple mixing length theory.However, this type of theory (without convective overshoot)

predicts a much steeper variation of the r.m.s, velocity than is observed.(See ~ig.l.)

Any theory which includes overshooting (even linear mode calculations, el. B~hm 1963 a)

can reproduce the observational results in a qualitative way.

0 .8

.~ 0.6 E

> 0 . 4

0 . 2

~l I i "" I ' I ' I I - -MIXING LENGTH WITHOUT

_'~ O V E R S H O O T

~ Canfield, ~\ X,~ Mehltretter ~\ ( 1 9 7 3 ~

\ \ " . ,~ ,~o . . ~ - \ \

\ "- \ , ~ ~" -. ~X"3130 km

1 5 6 0 k i n " I , I " ,L I I

0 2 0 O 4 0 O h (kin)

6 0 0

F I G U R E l

Comparison of the observed (vertical) velocity stratification (corrected for finite telescope aperture and for finite slit width hut not for atmospheric seeing~ Canfield and Mehltretter |973) with some simple theoretical results. The filled circles represent the observations by Canfieldand Mehltretter (1973), the (long) broken line is the corresponding interpolation curve. The solid line shows the results for the standard (local) mixing length theory with I = H. The vertical velocity distribution for two linear modes with horizontal wavelength calculated for a detailed model of the solar convection zone (B~hm 1963)is given by the (short) broken curves~ Note that the linear modes contain an arbitrary amplitude factor.

107

.-2 ~.oo c

,4 t,=

0 . 7 5 n~ LU

0 a. 0 . 5 0 LU >

0 . 2 5 ..J W n,-

%

' I ' I ' I ' I ' -

-- ~\\ / , I X5171

I '~

/ / \ \ " --_rx5,64 -1 ", \ \ t xs,za /

, I i I ,- "" 1" J I ~

2 0 4 0 6 0 8 0 I 0 0 k x 10 - 4 km -I

FIGURE 2

Comparison of the velocity power spectra for the lines Fel ~ 5171 and Fel /% 5164, 5178, 5180 and the power spectrum of the intensity fluctuations in the continuum. The diagram is based on the data given by Canfield and Mehltretter (]973). The maxima of the contribution functions for the lines 5178, 5164 and 5180A occur at a height between - 20 km ands70 km in the Harward-Smlthsonian Reference Atmosphere, the maximum for the line ~ 5171 lies at h% + 470 km i~ the HSRA. Note that the velocity power spectrum for the lines formed near the depth of formation of the continuum (~ 5178, A 5164, ~ 5180) looks very similar to the intensity fluctuation power spectrum for the continuum.

108

The continuum r.m.s, intensity fluctuations are~ 8.2% with a reasonable seeing

correction (2% T-fluctuation) which (if naively interpreted) is lower than the

mixing length predictions (~20%).

However, Canfield and Mehltretter believe that the higher value of ~ I ~ 12%

derived earlier by Mehltretter (19711 from his observations may also be correct.

The spatial power spectra of the intensity fluctuations show (after correction

for seeing effects) a rather flat maximum at a horizontal wave number between

3 and 5 x |0 - 3km -I (~ between 2000 and 1250 km).

Itshould be noted that a number of investigators find the peak of the power

spectrum at considerably longer wavelength (cf. Mattig and Nesis 1974). We assume

that this is a consequence of seeing effects.

Power spectra ~f the velocity fluctuations for lines which are formed relatively

deep in the photosphere seem to approach a power spectrum similar in shape to

the A I power spectrum (Canfield and Mehltretter 19731 indicating that

T-fluctuations and velocity field are really related below T 5000 ,~ ,4.(Fig.2).

Earlier, Frazier (1968) had found a comparably high correlation between continuum

intensity and the velocity in the Sill 6371 llne which is formed very deep. (The

peak of the contribution function is at • ~a ,65,)

It is really astonishing how high the v - ~ I correlation is. One has to

remember two things to appreciate this fact. Firstly, the velocity and the

continuum intensity which are beeing cross-correlated do not refer to the same

layers. Secondly, there is a possibility that for turbulent convection even at

one given point the cross-correlation could be considerably lower than 1

(cf. Spiegel 1966 a).

It is also interesting to note that there seems to be roughly a 30 second phase lag

in the sense that the maximum granular continuum follows the velocity (Frazier 1968,

Edmonds and Webb 1972, Musman 1974, Beckers and Canfield 19761.

We shall now proceed to the description of individual granules which, as mentioned

earlier, turns out to be useful and very interesting from a hydrodynamic point

of view. A very fundamental study of this type has been carried out by Beckers and

Morrison (!9701. They observed a large number of granules at ~ = .84, .70, .60

and determined the velocity field of an average granule from these observations.

By making use of the different positions on the solar disk (different~)these

authors derive the vertical as well as the horizontal velocity field for the average

granulum. Since not all granula have the same size and shape we can not expect

to get a really quantitative picture of the hydrodynamics of a granulum by this

procedure. Since in the reduction process the granules are positioned such that their

centers fall on the same point the upward velocities at the center of the granule are

enhanced in the averaging process whereas horizontal and downward velocities (occu-

ring in the outer parts of the granulum) are reduced. Nevertheless the "average"

granulum shows very clearly the basic structure of the velocity field in a granulum.

(Fig. 3).

109

| i I ,

Z ~ k m I 0 ~

200 m / ~

I I I I iooo looounl

J

FIGURE 3

The vertical cross-section of the velocity field of an "average" granule. (See text for a brief description of the averaging procedure). From Beckers and Morrison (1970). By permission of Reidel Publi. Co., Dordrecht.

Of course, we have to keep in mind that all modes of smaller scale are smoothed out by

atmospheric effects and by the averaging process.

The importance of such investigations lies in the detection of an ordered horizontal

outflow from the granulum. This is, of course, not too surprising, but it is certainly

an observational fact which is not emphasized in the mixing length theory.

The investigation of individual granules has lead to the discovery of another interes-

ting phenomenon namely the "exploding granules". (Carlier, Chauveau, Gugan and

RDsch 1968, Musman 1972, Beckers and Canfield 1975).

The phenomenon is shown (in a rather schematic way) in Fig. 4. Musman (|972) has

interpreted this sequence of events as being due to the interaction Of a thermal

with the stable layers above the convection zone. Ha has carried out a laboratory

experiment in order to confirm this point of view.

110

4 MINUTES B MINUTES LATER LATER

APPARENT EXPANSION VELOC ITY

I.Skm/s

o 0 SMALL UNUSUALLY

BRIGHT GRANULES

HAS EXPANDED; SHOWS

SLIGHT DARKENING AT

CENTER

FURTHER EXPANSION

WITH DARK REGION AT

CENTER

GRANULE (RING) FRAGMENTS AND FADES

FIGURE 4

Schematic representation of the development of an "exploding" granule (after Musman 1972, see text). Note that the bright granular region is drawn dark here.

Another important group of investigation is concerned with phenomena occuring in the

intergranular regions. It turns out that observers find a number of unexpected pheno-

mena and that these observations change our idea about solar convection considerably.

According to Deubner (]975) observations show that very large downward velocities can

be observed fairly often in the dark intergranular lanes. He quotes that observations

which have been made under excellent conditions show often downward velocities (uncor-

rected for seeing effects) of 2 km/s. He argues that these observations lead to correc-

ted downward velocities of about 4 km/s or possibly to even somewhat higher values.

These results indicate a very large asymmetry between upward and downward motion and

should be of great relevance to an understanding of the hydrodynamics of solar convec-

tion. It is not surprising that this effect is not visible in the results of Beckers

and Morrison who take average over many granules of different sizes.

111

It should be emphasized that it is not clear how large a fraction of the intergranular

lanes really show these great downward velocities. Deubner discusses these problems

in connection with the occurence of magnetic flux ropes in the photospheric network

as observed by Stenflo (1973) but he definitely considers the possibility that the

downward velocity is very high in all intergranular regions.

B. SUPERGRANLILATION

We all know that the supergranular velocity field pattern is strongly correla-

ted with the chromospheric network. The average diameter of supergranulation cells is

about 32000 km. Cells with diameters in the whole range from about l0000 km to 60000 km

are present. The typical horizontal (outflow) velocities are 0.3 - 0.4 km/s (Simon and

Leighton 1964, Deubner 1971). Vertical downward motions specifically in the magnetic

regions at the borders of the cell are in the range 0.] - 0.2 km/s (cf. Simon and

Leigthon 1964, Frazier 1970, Musman and Rust 1970, Musman 1971, Deubner 1971, Worden

1975). Supergranular motions have been detected also deep in the photosphere as

indicated by Deubner's (1971) measurement of CI 5380 line (X-v 7.7 eV ).

It is generally agreed that the main downdrafts occur in the rather localized

magnetic regions at the cell boundary.

Upward motions at the center of supergranulum of the order of 50 m/s may be present

but have not yet been confirmed. (Worden and Simon 1976).

In order to judge whether supergranulation is really a convective motion a study of

the corresponding temperature fluctuations is of great importance. The results of

such studies are somewhat confusing though a clarification may now be in sight.

Somewhat surprisingly one finds normally a slight temperature increase at the cell

boundaries (~ 2.5°K) whereas according to the usual picture of a simple convective

cell a temperature decrease would be expected (Beckers 1968, Frazier ]970). It is

usually assumed that these very small temperature increases are due to the increase

of the magnetic field and are not directly related to the convective flow (Liu 1974).

One might hope to find the expected velocity -- ~ T correlation more easily in deeper

layers. Recently Worden (1975) has studied the supergranulation structure at 1.64

which represents the deepest observable layer in the solar photosphere. Using the

HSRA (Gingerlch et al. 1971) we find that at this wavelength we look down to about

T qu 1.8 where the temperature is ~ 7000°K. Worden finds an 0.7~intensity increase

near the cell boundaries. This has to be compared to an 0.4% intensity increase in the

higher photospheric layers. These results can be translatedinto a temperature diffe-

rence ofabout 50 ° in the deep photospheric layers. This rather small difference indicates

that the supergranulation (if it is convection) contributes only very little to the

total convective energy flux in the visible layers. Nevertheless, the fact that such

large convection cells do existis very important from a theoretical point of view.

I do not think that this conclusion is very much influenced if supergranulation is an

overshoot phenomenon or if the supergranulation elemen~ are counter cells (of. Spiegel

1966 b).

112

Some of the i~ediate theoretical implications of the existence of convection cells

of supergranular size have been discussed by Simon and Weiss(1968 a, 1968 b).

They point out among other things that we should expect that the depth of cell is

not very different from about one fourth of i~s diameter. This leads to a cell

depth of about ]0000km which is clearly much larger than a scale height. Simon and

Weiss are to some extent guided by the results known for polytropic atmospheres in

which the scale height is of the same order of magnitude as the distance to the

surface of the convection zone. The possibility that cells of supergranular size will

transport a large fraction of the convective energy flux in deep layers is of great

importance. Simon and Weiss also predict giant cells which are comparable in size to

the total thickness of the solar convection zone and which in the meantime have also

been detected observationally.

We close this chapter by listing briefly some of the interesting and hydrodynamically

relevant properties of solar convection in table I.

TABLE I

SOME INTERESTING PROPERTIES

OF SOLAR CONVECTION

| . GENERAL

A. Existence of two completely different cell sizes (2000 km and 32000 km)

2. GRANULATION

A. Very high correlation between ~J and A T (more than 80%)

B. Flat maximum of ATpower spectra between ~ = 1200 and ~ = 2000 km

C. Very fast down-drafts (~u 4 km/s) in intergranular regions

D. "Cell" structure

E. Convective overshoot

3. SUPERGRANULATION

A. Cell sizes much larger than scale height

B. Concentration of magnetic fields at cell boundaries

C. Positive ~ T in high layers, negative ~ T in deep photospheric layers

of cell boundaries, ~ T unexpectedly small.

113

I I I CONVECTIVE MIXING AND THE Li-Be PROBLEM

When we discuss convection in stars it is quite clear that the observational

evidence concerning convection has to be rather indirect. Our conclusions are usually

based on a mixture of observations and theoretical predictions (which often use a very

crude form of theory), Sometimes we even forget that theoretical assumptions are used

and claim that the result is purely observational though it is not.

Problems of convective mixing definltely belong to this category. In many cases

we certainly see effects of mixing but we can not be absolutely certain that this

mixing is really due to convection. Even if it is convective mixing it can have

happened rather recently or during an earlier phase of evolution. Probably~ the

best known example of such situation is the lithium-beryllium problem. As is well

known, the solar atmosphere contains no or almost no Li 6 and in contradistinction to

other old material in the solar system (chondrites) very little Li 7. In order to

explain this it is usually assumed that surface material has been mixed into layers

of 2.5 x 106K where the lithium is destroyed by (p, ~ ) reactions. A progressive

decrease of the lithium content from G 2 V to K O V stars (Wallerstein, Herhig and Conti

1965, Zappala 1972) has been attributed to the effects of pre-main sequence convection.

(Bodenheimer |965, |966). In addition amain sequence depletion time scale of about

1.5. x 109 years has been found (Herbig 1965, Zappala 1972). It is not yet completely

clear whether very slow convective overshoot (cf. B~hm 1963, 3966, Weymann and Sears

1965, Spiegel |968, Strau~, Blake and Schramm 1976) in combination with the relevant

nuclear rates can produce this time scale. However, Strauss,Blake and Schramm (1976)

argue very strongly that convective overshoot can explain the observed lithium abun-

dances. They emplasize the fact that the enersetically possible convective overshoot

theoretically decreases with decreasing mass of the star and that this is just the

feature which is required in order to explain the observations. If this result should

be confirmed the investigations of the Li problem would give us an excellent opportunity

to study convective overshoot below outer stella r convection zones. It seems that the

lithium problem is more useful in the study of stellar convection than many other mixing

problems since the time scale is known well and since it refers to main sequence stars

which are better understood than the advanced phases of stellar evolution in which most

other mixing processes seem to occur. The fact that it is probably related to convective

overshoot makes it especially interesting.

114

IV INFLUENCE OF CONVECTION ON ATMOSPHERE STRUCTURE AND COOLING TIMES IN COOL

WHITE DWARFS.

After these brief remarks about the Li problem I finally would llke to talk

about a group of convection problems which are also closely related to the above ques-

tions and in which I have been strongly interested in recent years. This is the field

of convection in white dwarf stars. I personally feel that there may be more drastic

effects of convection on observable properties in the ease of white dwarfs than in most

other stars for the following reasons (of. Bohm ]968, 1970, Van Horn 1970, Wiekramasinghe

and Strittmatter 1970, B~hm and Cassinelli 197], Wagner 1972, Grenfell ]974, Fontaine

et al. 1974, Fontaine and Van Horn 1976, Muchmore and B~hm 1976).

]. Atmospheric convection persists over a wider effective temperature range

in white dwarfs than in any other stars. This is especially true for the roughly 30 %

of white dwarfs which are called non-DA's and which have very He-rlch atmospheres. In

these objects outer convection zones are present in the effective temperature range from

about 25OO0°K down the lowest effective temperatures.

2. Convective instability sets in very high in cool. white dwarf atmospheres.

In a number of models this occurs at optical depths higher than O.O1. Consequently the

atmospheric structure is strongly influenced by convection. This is especially true for

white dwarfs with) say, Tel f < 8000 ° in which the high atmospheric density leads to

relatively high effectivity of convection and consequently to a temperature stratifica-

tion which differs considerably from a radiative equilibrium stratification (cf. Wagner

1972, Grenfell }972). In extreme cases (non-DAs with 3000°K < Tel f ~ 5000 °K) the convec-

tive flux reaches values of about 60 % of the total flux at ~ ~ ].0 (B~hm, Carson)

Fontaine and Van Hors |976). So, we should be able to detect the presence of convection

without difficulty by studying the line spectra Of cool non-DA white dwarfs like

Van Maanen 2 and Ross 640. (However, it should Be emphasized convection in cool white

dwarfs has to be studied through its influence on the mean stratification of the atmos-

phere. The velocities in these high density atmospheres are much too small to he detec-

ted directly.~

3. The existence of two different types of white dwarfs atmospheres, one consis-

ting of almost pure helium (non-DA), the other one of almost pure hydrogen (DA) has lead

to the suggestion that convective mixing processes in combination with gravitational

separation and/or accretion may be important in determining the chemical composition of

white dwarf atmospheres (cf. Strittmatter and Wickramasinghe }97}, Shipman 1972, Baglin

and Vauclalr ]973). However, the subject is controversial and simple convective

115

mixing is insufficient to explain the non-DAs (Koester 1976). Nevertheless, some

interesting convective mixing effects are to be expected since it is clear that

white dwarfs have a very ~hin envelope of hydrogen or helium surrounding the main

body of the star which consists of C and O (el. Weidemann |975).

6.0- I--

_.o

5.5-

!

0 -5 -10

I I

. . . oooo °

,,, ! ! . . . . . . . .

/ . _ - - BOO0 °

. . - - - - . . . - I o o o =

. . . . - 6500 °

5 . 0

6.5-

-20

FIGURE 5

Temperature stratification in the outer layers of cool He-rich (non-DA) white dwarfs. The diagram is based on calculations by Muchmore and B~hm (1976). We have plotted the logarithm of the temperature as a function of the local degeneracy parameter = = - ~ /kT (with ~ = chemical potential). Tel f is the parameter. The solid parts of the curves correspond to convection zones, the broken lines to the radiatlve-conductive zones below. The diagram shows how the temperature in cool He white dwarfs approaches its asymptotic core value T c and how large a fraction of the temperature rise towards T c occurs in the (geometrically thin) convection zone. This gives some indication of the importance of convection for the determination of the T e and Tel f relation (see text).

116

4. In the case of white dwarfs the evolutionary time scale is strongly

influenced by the time it takes for energy flux to go through the outer non-

degenerate and partially degenerate layers. When convection breaks throngh this

"insulating" layer the cooling becomes considerably faster. Consequently, the

cooling time and the luminosity function of white dwarfs is influenced by the

presence of convection (cf. Bbhm 1968, Ostr~ker 1971, Lamb and Van Horn 1975).

Eventually we shall he able to derive information about convection from all these

different effects. Some indication of the importance of convection for the deter-

mination of the relation between core temperature and Tel f is given in Fig. 5.

Finally it should be emphasized that the comparison of theoretical predictions of

white dwarf convection with observations involves one complication which

is not present in most other stars. The calculation of the equation of state, the

adiabatic gradient and the specific heat is made rather difficult by the presence

of partial degeneracy, pressure ionization and electrostatic interactions between the

particles. (In non-DAs of Teff < 3800 ~ we face some fascinating problems because

drastic complications of the equation of state, including partial degeneracy, occur

already within the atmosphere, see B~hm, Carson, Fontaine and Van Horn ]g76).

However, we hope that the problems related to the equation of state will be overcome in

the foreseeable future.

ACKNOWLEDGEMENT S

This work has been in part supported by NSF grant AST 74 - 24343 A 0].

A part of this paper was written while the author held the Gauss Professor-

ship of the Gbttinger Akademie der Wissensehaften. I am grateful to the

Akademieand to Professor H.H. Voigt and the other astronomers of the Gbttingen

University Observatory for their kind hospitality.

t17

REFERENCES

ALTROCK, R.C, and MUSMAN, S. 2976, Ap J. 203, 533

ANDO, H. and 0SAKI, Y. ]975, P.A.S.J. 27, 58]

BAGLIN, A. and VAUCLAIR, G. 1973, Astron. Astrophys. 27, 307

BECKERS, J.M. 1968, Solar Phys. 5, ]08

BECKERS, J.M. and CANFIELD, R.C. 1976, in Physique des Mouvements dans les Atmospheres Stellaires (ed. R. CAYREL and M. STEINBERG) Editions du C.N.R.S., p. 207

BECKERS, J.M. and MOKRISON, R.A. 2970, Solar Phys. 14, 280

BODENHEIMER, P. 2965, Ap J. 142, 451

BODENHEIMER, P. 1966, Ap J. 24_~4, 103

B~HM, K.H. 1963, Ap J. 138, 297.

B~HM, K.H. 2966, Z.f. Naturforschung 2l_~a, ]107

BOHM, K.H. 1968, Astrophys. Space Sci. 2, 375

BOHM, K.H. 2970, Ap J. 162, 919

B~HM, K.H. and CASSINELLI, J.C. 197], Astron. Astrophys. 12, 21

BSHM, K.H., CARSON, T.R., FONTAINE, G. and Van HORN, H.M., 1976, in preparation

CANFIELD, R.C. and MEHLTRETTER, J.p. 1973, Solar Phys. 33, 33

CARLIER, A., CHAUVEAU, F., HUGON, H. and R~SCH, J. ]968, Comptes Rendus 266, 199

DEUBNER, F.L. 2971, Solar Phys. 17, 6

DEUBNER, F.L. I975, Astron. Astrophys. 44, 371

DEUBNER, F,L, 2976, Astron. Astrophys. 47, 475

EDMONDS, F.N. and WEBB, C.J. 2972, Solar Phys. 25, 44

FONTAINE, G. and Van HORN, H.M. 1976, Ap J. Suppl. 3]_, 467

FONTAINE, G., Van HORN, H.M., BOHM, K.H. and GRENFELL, T.C. ]974, Ap J. 193. 205

FRAZIER, E.N. 1968, Ap J. 152, 557

FRAZIER, E.N. 1970, Solar Phys. 24, 89

GINGERICH, O., NOYES, R.W., KALKOFEN, W. and CIINY, Y. 1971, Solar Phys. 28, 347

GRENFELL, T.C. 1972, Astron. Astrophys. 20, 293

GRENFELL, '£.C. 2974, Astron. Astrophys. 3]_ , 303

RERBIG, GoH. ]965, Ap J. ]4], 588

KOESTER, D. 2976, Konvektive Durchmischung und Accretion be] weissen Zwergen Habilitationsschrift Kiel

LAMB, D.Q. and Van HORN, H.M. 2975, Ap J. 200, 306

LIU, B.Y. 1974, Ap J. 189, 359

MATTIG, W., MEHLTRETTER, J.P. and NESIS, A. 1969, Solar Phys. 10, 254

MATTIG, W. and NESIS, A. 1974, Solar Phys. 36, 3

MATTIG, W. and SCHLEBBE, H. 1974, Solar Phys. 34, 299

McINTYRE, M.E. 1976 in Physique des Mouvements dans les Atmosphares Stellaires (ed. R. CAYREL and M. STEINBERG) Editions du C.N.R.S., p, 262

MEHLTRETTER, J.P. 2971, Solar Phys. 16s 253

118

MUCHMORE, D.0. and BDHM, K.H. 1977, Astron. Astrophys. 54, 499

MUSMAN, S. 1971, I.A.U Symposium n ° . 43, p. 289

MUSMAN, S. 1972, Solar Phys. 26, 290

MUSMAN, S. 1974, Solar Phys. 36, 313

MUSMAN, S. and RUST, D.M. 1970, Solar Phys. 13, 261

OSTRIKER,J.P. 1971 ~nn. Rev. Astron. Astrophys. 9

SHIPMAN, H.L. 1972, Ap J. | 7 7 , 723

SIMON, G.W. and LEIGHTON, R.B. 1964, Ap J. 140, |]20

SIMON, G.W. and WEISS, N.O. 1968 a, Z. Astrophysik 69, 435

SIMON, G.M. and WEISS, N.0. 1968b, in Structure and Development of Solar Active Regions (ed. K.O. KIEPENHEUER) D. Reidel, Dordrecht-Holland, P.]08

SPIEGEL, E.A. 1966 a, in Stellar Evolution (ed. R.F. STEIN and A.G.W. CAMERON) New-York: Plenum Press, p. 143

SPIEGEL, E.A. ]966 b~ I.A.U. Transactions ]2 B, 539

SPIEGEL, E.A. 1968, Highlights in Astronomy (ed. L. PEREK) Dordrecht: Reidel Publ. Co.

STENFLO, J.O. 1973~ Solar Phys. 32, 41

STRAUSS, J.M., BLAKE, J.B. and SCHRAMM, D.N. 1976, Ap J. 204, 481

STRITTMATTER, P.A. and WICKRAMASINGHE, D.T. 197], M.N.R.A.S. 152, 47

VAN HORN, H.M. 1970, Ap J. (letters) ]60, L 53

WALLERSTEIN, G.W., HERBIG, G.H. and CONTI, P.S. ]965, Ap. J. 141, 610

WEGNER, G. 1972, Ap J. 172, 451

WEIDEMANN, V. 1975, in Problems in Stellar Atmospheres and Envelopes, (ed. B. BASCHEK, W.H. KEGEL and G. TRAVING) Heidelberg - New-York: Springer Verlag, p. 173

WEYMANN, R. and Sears, R.L. ]965, Ap J. ]42, ]74

WICKRAMASINGHE, D.T. and STRITTMATTER, P.A. 1970, M.N.R.A.S. ]50, 435

WORDEN, S.P. 1975, Solar Phys. 45, 52]

WORDEN, P.S. and SIMON, G.W. 1976, Solar Phys. (in press) abstract: Solar Phys. 45, 547 (1975)

ZAPPALA, R.R. 1972, Ap J. 172, 57

EVOLUTION PATTERN OF THE EXPLODING GRANULES

O. Namba and R. van Rijsbergen

Astronomical Institute, Utrecht

SUMMARY

The evolution pattern of the so-called exploding granules has been studied on

the basis of a time sequence from Princeton Stratoscope pictures of the solar granu-

lation. Some preliminary results are presented (section3). A new interpretation of

the phenomenon is suggested (section 4).

1. INTRODUCTION

During the morphological study of the solar granulation (Namba and Diemel,1969)

we were puzzled by some large granules which have a round darkening in their centre

often with several dark canals radiating from it toward the boundary. In 1967 we began

a study of this remarkable class of granules, which are now referred to as "exploding

granules" though the term is somewhat misleading. The exploding granule phenomenon was

discovered by Roseh and his co-workers (Carlier et al., 1968) in moving pictures ob-

tained at the Pic du Midi Observatory, an example of which was shown during the 1967

I.A.U. Meeting at Prague, Musman (1972) has proposed the first theoretical model of

the phenomenon on the basis of his laboratory experiment and observations obtained at

the Sacramento Peak Observatory. In this paper we present some observational features

of exploding granules, which may be of some importance for the theory of non-stationary

convection.

2. OBSERVATIONAL MATERIAL

The observational material is a time sequence of high-definition photographs of

the granulation obtained on the Stratoscope flight of August 17, 1959. The duplicate

negatives on 35-mm film were kindly lent us by the Princeton University Observatory.

The photographs were taken in the green-yellow region with an exposure time of 0.0015

sec at a rate of a frame every 0.929 see; the highest spatial resolution attained is

0.112" or 271 fun on the Sun.

The time sequence lasts about ]6min (Frame Nos. 2145 - 3200) and refers to a

quiet region near the disk centre; some of the frames are illustrated in Bahng and

Schwarzschild (196;), where the observational data are also given, The search for ex-

ploding granules and the study of their evolution have been done on positive enlarge-

120

a

2 1 5 3

Ore06 s

2413

4m07 s

i b 2 1 8 8

0 m 3 8 s

C

2 2 5 9

l m 4 4 s

g 2 5 5 0

6m14 s

h

2 7 0 6

8 m 3 9 s

J

|

d 2311

2m:

2 8 0 3

10 m 0 9 s

e "~- > 1 0 m 5 7 s

2 3 5 3

3 m l l s 0 5 10" I . . . . I , , , , I

Fig. I. Evolution of a typical exploding granule (centre) - a time sequence from Stratoscope granulation photographs obtained on August I?, 1959. Indicated are frane numbers, relative times, the lifetime, and the distance scale (I" = 725 km). Note also remarkable stability of surrounding granules. (Small specks and dots in some pictures are blemishes.)

121

mants of about 50 selected frames (the image factor: l"= 4 mm for all and 1 ''= 6 mm

for portions including the granules treated here). The work was troublesome, because

the focussing of the image was widely variable and high-resolution pictures are not

uniformly distributed in the sequence. Furthermore, besides small shifts of the granu-

lation field in the frame, there is a drift of the solar image that divided the time

sequence into two parts, one lasting about 10 min and the other 6 min, with some spa-

tial overlapping (cf. Fig.2 below).

In this study seven exploding granules were examined. Fig. l shows a series of pic-

tures for a typical example. Although the picture quality is somewhat variable, the

series illustrates nicely its evolution (cf. Carlier et al.,196g and Musman, 1972).

On the positive prints we measured the area of granules as a whole and expressed them

in terms of the average diameter (D). For the pictures shown in Fig. l microphotometry

has been carried out with a Joyce Isodensitracer.

3. RESULTS

Although individual exploding granules behave rather differently, their evolution

pattern may be summarized schematically as follows. A granule appears as a bright spot,

which grows quickly. As the size increases, a vague shade forms over the central part

of the bright granule; it soon becomes a round dark hole - see Fig. l. As the central

darkening develops a few dark canals radiate from it toward the boundary. This may be

regarded as the primary splitting of the granule, usually into two or three smaller

granules. Around the time where the granule reaches its maximum extension the split

granules break further into several parts (the secondary splitting?), provided the gra-

nule is large. So, one can count from several to more than ten granules, split from a

single granule. The evolution after the fragmentation depends upon the granule. One of

the split granules may become a new exploding granule while others fade away. Carlier

et al. (1968) showed an example that developed three "generations" of exploding gra-

nule from a single entity. We should mention that the splitting is a very common pheno-

menon of granules: even for small granules (D < I") it is rather difficult to find such

granules that do not split.

From the study of seven exploding granules the following characteristic features

have been derived.

(a) Measured diameters are plotted in Fig.2 as a function of time. The maximum size

reached during the evolution was from 3.3" (2200 km) to 5.4" (4000 km) in diameter. The

expansion rate AD/At is more or less linear with a speed of ].3 to 3.3km/s. Hence, the

radial expansion rate AR/At ranges from 0.7 to 1.7 km/s, in ~greement with the earlier

estimates of 1.5 to 2 km/s (Carlier et ai.,1968) and of 1.8 km/s (Musman, ]972). These

values are only a fraction of the sound speed in the photosphere (~8 km/s), but they

exceed the horizontal flow velocity of 0.34 km/s (maximum) at the top of granules,

measured by Beckers and Morrison (1970).

Furthermore,there is a tendency that the expansion speed is proportional to the ma-

122

6'

5

t n,,

LLJ

F,3

I I I I I 2200 2400 2600 2800 3000 3200

TIME (Frame N o) ,,

Fig. 2. Growth of exploding granules: average diameter (I" = 725 km) vs, time (a fr~ne every 0.929 eec), The granulej illustrated in Fig. 1, is the third from the top.

x 1.0

E

",~ .s i .4

,2

I ! . . . . . I I I I I I I

Dmax I

1 i i i i i i i i i O. 0 -8 -6 -4 -2 0 +2 +4 .6 *8 +I0

TIME (minutes)

Fig. 3. Relative growth of the exploding granules, derived from Fig. 2. The ~verage rate of growth &In D/At ~ O.045/min.

I23

ximumdiameter reached. In Fig.3, where the mean diameter relative to the maximum dia-

meter is plotted versus time, all the growing branches cluster around a line (not drawn)

with a slope

A in D O.045/min. (I)

At

(b) The above relation suggests that in the sub-photospheric layer the upper part of

the granules may look like huge cones, as sketched in Fig.4.

(c) The central darkening occurs only when the granule is roundish, uniformly bright

and exceeds a certain critical diameter D c ~2.3" (1600 km). It does not show up in

large but elongated granules with a width smaller than this value.

The isophotometry of the pictures shown in Fig.] yielded interesting data about the

development of the central darkening for this particular granule. In the interval from

picture b to picture e, the central darkness increases linearly with time from 13 to

25% of the surrounding brightness and the diameter at half the central depression grows

from 700 to II00 km at a linear rate of 3.0 km/s while the granule expands at a rate of

3.2 km/s in the same period.

(d) The lifetime for the individual granules has been found to be much longer than

the"correlation"lifetime of 8.6 min measured by Bahng and Schwarzschild (1961). The

time sequence allowed us to determine only a lower limit of 10 - 11 min. But Fig.3

suggests'a lifetime of, say, half an hour. In Fig. I also the remarkable stability of

normal granules surrounding the exploding granule is apparent: many persist through

the |0-min time. This is the case also for small granules in the range of 0.8" ~ D

< ].7" (Namba and P. Provoost, ]973, unpublished).

(e) In the time sequence we counted about 20 certain and ]0 suspected exploding gra-

nules with the characteristic features (central darkening with or without radial canals)

over an area of roughly 70" x 90" at any given moment; they cover 2 to 3% of the total

area. In addition, thanks to the time sequence, we found about 10 granules, which be-

came exploding ones, and nearly twice as many already "exploded" granules.

The spatial distribution of exploding granules does not seem to be random, but

we lack further observational material to investigate whether there is any correlation

between the locations of exploding granules and the supergranulatlon pattern. Allen and

Musmen (1973) found no such correlation in their observations.

4. A POSSIBLE INTERPRETATION OF THE EXPLODING GRANULE PHENOMENON

Musman (]972) interpreted the phenomenon as follows. When a rising granule pene-

trates into the overlying stable region, the internal g=anular motions and the conser-

vation of angular momentum act to change the form of the granule into a vortex ring,

which is stretched out horizontally and breaks. This"smoke rlng model, however, meets

difficulties in explaining some observed facts, for example, why some granules are of

the exploding type while man~ others are not, and why a part of the broken ring may

124

a photospheric

level-h . . . . - - . . . ~

b = 8 0 0 km

C "iO Fig. 4. A possible interpretation of the exploding granule phenomenon -

development of the central darkening (vertical cross section). Cf. Fig. I. a: as the granule rises the top layer is cooled instantly; the cold-matter flows outj b: since the granule grows fast and large (D >~ 1800 kin) the cold matt'er is left behind the expanding granule boundaryj and the central darkening begins to form. It develops parallel to the growth of the granule~ c: when the mass of the cold matter in the centre becomes too heavy,-~t sinks, breaking through the granule. The development is accompanied with splitting of the granule. Borne of the split granules may grow further while others fade away.

125

become a new exploding granule.

The evolution pattern reported here suggests an alternative aualitative interpre-

tation which is illustrated in Pig. 4. As soon as a hot granule reaches the photospheric

level its top is cooled. At first the cooled matter flows horizontally (with a velocity

of ~ 0.5 km/s) and then downward to join the surrounding intergranular region (Fig.4a).

The upper layer of the steadily growing granule is removed continuously in this way.

This process goes smoothly as long as the granule size does not reach the critical dia-

meter mentioned above.

However, when the granule expands with a speed faster than the horizontal flow velo-

city, the cold matter can never reach the granule boundary and is left behind, and con-

sequently the central darkening begins to form (Fig.4b). By this time the granule dia-

meter may have reached the critical value D c. The central darkening develops parallel

to the growth of the granule (paragraph 3c above).

The mass of the cold matter accumulated in the centre of the granule is supported

by the buoyancy and the upward motion in the granule, until its total weight becomes

too large. Then the balance is lost and the cold mass sinks, breaking through the gra-

nule (Fig.4c).

The development of the central darkening may be accompanied with the fragmentation

of the granule. The onset of the downward streaming at the centre might induce the

secondary splitting.

Our interpretation predicts a downward motion in the central darkening after some

moment. It is of particular interest to determine the Doppler velocity in the central

darkening and its change during the evolution of exploding granules.

ACKNOWLEDGEMENTS

Our thanks are due to the late Dr R.E. Danielson of the Princeton University Ob-

servatory, who kindly supplied us with the valuable Stratoscope material. The Project

Stratoscope I was sponsored by NSF, ONR, and NASA, U.S.A. We thank Mr J.L.A. van

Hensbergen for his assistance in photographic work. We acknowledge Drs A.G. Hearn,

G.D. Nelson and C. Zwaan for stimulating discussions.

REFERENCES

Allen, M.S. and Musman, S., 1973, Solar Phys., 32, 3]]

Bahng, J. and Schwarzschild, M., 1961, Astrophys. J,, 134, 3]2

Beckers, J.M. and Morrison, R.A., 1970, Solar Phys., ]4, 280

Carlier, A., Chauveau, F., Hugon, M. and Rosch, J., 1968, C.R.Acad.Se.Paris,226,199

Musman, S., 1972~ Solar Physo, 26, 290

Namba, O. and Diemel, W.E., ]969, Solar Phys., !, 167

GRANULATION OBSERVATIONS

A. Nesis

Fraunhofer-lnstitut

F r e i b u r g i~Br., FaG

The a n a l y s i s o f h i g h r e s o l u t i o n s p e c t r o g r a m s t h a t were o b t a i n e d in

Capr i w i t h t h e Coud6 r e f r a c t o r gave t h e f o l l o w i n g r e s u l t s r a g a r d i n g

the g r a n u l a r f i e l d s ,

a) The rms v e l o c i t y of t he g r a n u l a t i o n was 300 m / s e c . Th i s v a l u e r e -

p r e s e n t s t h e raw d a t a and s h o u l d be s u b s t a n t i a l l y i n c r e a s e d to a c c o u n t

f o r i n s t r u m e n t a l and s e e i n g e f f e c t s (W. M a t t i g e t a l . , 1969) . Note

t h a t t h e o s c i l l a t o r y component of t he t o t a l v e l o c i t y f i e l d has been

e l i m i n a t e d by a n u m e r i c a l f i l t e r °

b) In the photosphere the intensity and velocity fluctuations show a

similar structure. The size distributions of the observed Doppler

shifts and intensity variations have been studied by means of power

spectrum analysis (Fig.l, ~. Mattig and A. Nesis, 197~)o These spectra

are not corrected for finite instrumental resolution or for atmospheric

seeing.

The question that we put to the theorists is which of the current

models of convection is able to interpret these observations?

Should one resort to the anelastic approximation for convection to

study our data, or must one work with the thermals picture, t o limit

oneself to just two examples?

Since we can arrive at an understanding of the granulation only by

careful and systematic observations, I believe that a practically

orientated dialogue between theory and observations is indispensable.

References

Mattig, W., Mehltre%ter, J.P., and Nesis, A.: 1969, Solar Phys. I__0, 25~

Mattig, W., and Nesis, A.: 1974, Solar Phys. 36, 3

127

Io 9 P{K}

3-

LM -\ \ \

k , , r - " - ~

13."8 3."8 1:9 1"2 0~85 0."75 i i i I I ' i

23 46 k 69 92 115 x lO'"km "~

Fig. J. The power s p e c t r u m of t h e i n t e n s i t y I ( x ) and o f t h e v e l o c i t y

v ( x )

LoK.: c o n t i n u u m l e v e l s ~ L, Mo: Line c e n t e r

SOME ASPECTS OF CONVECTION IN METEOROLOGY

R. S. Lindzen

Center for Earth and Planetary Physics

Harvard University

Cambridge, Massachusetts 02138

ABSTRACT

Various aspects of convection in meteorology which may have some relevance for astro-

physics are discussed. In particular the role of convection in determining the gross

thermal structure of the atmosphere, the treatment of convective turbulence in the boun-

dary layer, and the larger scale organization of convection are dealt with. ~

I. INTRODUCTION

As noted by Prof. Biermann during this conference, a number of seminal approaches

to convection in astrophysics such as convective adjustment and mixing-length theory

originated in meteorology. The purpose of this paper is to briefly review the current

status of such notions in meteorology as well as to report some relatively recent approa-

ches to meteorological convection which may prove useful in astrophysics.

Since most of what I discussed at the conference has appeared in the meteorological

literature 9 1 will tend to use this written version of my lecture as a selective anno-

tated directory to this literature rather than a complete version of the lecture. No-

thing remotely approximating a complete review of convection in the meteorological li-

terature will be attempted.

In section 2, I will describe the observed thermal structure of earth's atmosphere

and explain why simple radiative models with convective adjustment prove inadequate --

qualitatively and quantitatively. I will also outline our current understanding of

the observed structure -- though this understanding is by no means complete.

In section 3, I will introduce a current phenomenological approach to penetrative

convection in the atmosphere which may prove a more consistent alternative to convective

adjustment in astrophysics. In both sections 2 and 3~allusions will be made to the

fact tha~in the atmosphere, convection often occurs in relatively narrow plumes, and

that such convection is generally associated with local static stability. The point

which may be relevant to astrophysics is that convection is not necessarily related

to local temperature gradients in a simple way.

129

Section 4 deals with a particular observed feature of atmospheric convection :

namely, when broad regions (for example the maritime tropics equatorwards of 20-30 =)

are uniformly unstable (or conditionally unstable) convection does not occur in a uni-

formly distributed manner. Rather, the convecting system itself appears to be unstable

to so-called mesoscale systems (physically akin to internal gravity waves) which in turn

organize the convection on scales much larger than the scale of the convective elements.

The mechanism for this organization appears to be different when the convection is deep,

extending almost from the ground to the tropopause (cumulonimbus convection), and when

convection occurs within the middle levels of the troposphere; both are discussed in

section 4. The point, however, is that in the earth's atmosphere convection rarely if

ever manifests itself in the appearance of a pattern characterized by the scale of the

convective elements alone; larger scales of organization and motion almost invariably

appear. This may offer some insight into such phenomena as supergranules on the sun.

2. THE LARGE SCALE THERMAL STRUCTURE OF THE EARTH'S ATMOSPHERE

In Figure ! a somewhat smoothed picture of the height and latitude structure of

longitudinally averaged temperature is presented (only heights up to |6 km are shown).

In Figure 2 several height profiles of temperature appropriate to various latitudes are

shown. From Figure 2 we may note a certain similarity of profiles from various latitudes.

In most cases temperature decreases with height above the ground with a gradient of

about 6°/km until some level (known as the tropopause) where the temperature gradient ~T

goes to zero or becomes positive. The relatively uniform lapse rate (- ~-~ ) of the

lower atmosphere as well as the observation that radiative equilibrium profiles for the

lawer atmosphere are statically unstable led early on to the use of a convective adjust-

ment model to explain the thermal structure of the lower atmosphere (Gold, 1909; Emden,

1913; Goody, 1949). Several important difficulties have, however, arisen in connection

with such models (some were recognized almost immediately) :

i) Convective adjustment would lead to the adiabatic lapse rate in the lower atmos-

phere. This is 9.g°/km not 6=/km. It is sometimes suggested that the atmosphere adjusts

instead to the moist saturated adiabatic lapse rate. While this lapse rate is on the

order of 6°/km near the ground, by 4km the atmosphere, because of its low temperature,

saturates with very small amounts of water vapor and the saturated lapse rate differs

little from the dry value. Thus, the use of the saturated lapse rate is no solution to

this problem -- even ignoring the fact that the atmosphere is rarely saturated. It is

interesting to note that the application of convective adjustment to the earth's atmos-

phere leads to errors on the order of 30=K or 10% in absolute temperature.

ii) Convective adjustment in no way can account for the abrupt change in tropopause

height near 30 = latitude: equatorwards it is near |6 km, polewards it is near 12 km.

1 3 0

iii) Radiative-convective equilibrium does not account for the fact that in the

nelghbourhood of the tropopause, a temperature minimum exists at the equator.

iv) Finally and revealingly, close scrutiny of Figures ] and 2 indicates that the

lapse rate of the tropopause is not really uniform,especially at high latitudes.

MB - : |k in . . . . . . . . . . . ; 6

~ 0 0 ~ . . . . . . . L - - k . . . r - - - - o - ~ . . . . . . . . . . . - - - , ~ - ' ~ ~ *-~-45 . . . . . 4 - 1 2

-~5_b--~. . ,> '~ I . - L ~ ~ " - ' ' : ~ ' ~

. . . . ~.~ ,-~. ~ - - - . " - ~ 7 ~ . : - - - - - C " ---------_ ~ - 4 ~ ~ ~ 8

500 4

7 0 0 . - - 2 5 " - - ' " . . . . ;0 . . . .

~O~N ~ 0 o 3 0 ° 0 o S~ME.

( 0 ) H E'~I~jTEH'~R E JANUARY HEMISPHERE

100

20O

3 0 0

5 0 0

7 0 O M B

1 0 0 0 9 0 ° N

( b }

6 0 ° 3 0 ° 0 = 3 0 o 6 0 == SUMMER WINTER

HEMISPHERE HEMISPHERE J U L Y

12

8

4

0 9 0 °

Figure ]. Zonally (longitudinally) averaged temperature as a function of height

and latitude. Contours are lines of constant temperature (°C). After Palmen and

Newton (1969).

131

4C . . . . . . . .

~ 5 C - - - -

E

b.,,I C~ :::) 20 l - - i I-- .-I

1 0

180 5 ~

I i 45t /~1~..,,,,P I JANUARY AND

60£~.. ~/~./~/i/. 2" ~ SPRING/FALL,

,~/j -~---~ . . . . . . . . . ~- T

I / o |

\ h I

i -

"

200 220 2 4 0 260 2 8 0 TEMPERATURE (OK)

Figure 2. Zonally averaged temperature as a function of height for various lati-

tudes. After U.S. Standard Atmosphere Supplements (1966).

Our current understanding of the atmosphere's structure suggests no uniform expla-

nation for the whole globe. Recent work (Schneider and Lindzen, 1976; Schneider, 1976)

shows that within a certain neighbourhood of the equator (extending to about 30 ° lati-

tude) the atmosphere cannot sustain significant horizontal temperature gradients (in

many respects this region is similar to a spherically symmetric atmosphere where rota-

tion is not of great importance). Large scale dynamic effects in this region serve

primarily to homogenize (horizontally) the temperature in this region, and as a result

the vertical temperature structure of this region is indeed describable in terms of

radiative-convective equilibrium. However, because the convection occurs in relatively

narrow cumulonimbus towers, it leads to finite stability rather than neutral lapse rates.

How this occurs is outlined in Appendix ]. From about 30-70 ° latitude, horizontal tem-

perature gradients are significant and rotation is of basic importance. It is generally

believed that convection in this region is due to baroclinic eddies whose energy is

drawn from horizontal temperature gradients. These eddies tend to carry heat upwards,

132

and the rate at which these eddies stabilize the atmosphere is much greater than the

rate at which radiation acts to destabilize the atmosphere~so the question of convec-

tive adjustment does not arise. The stability achieved in this region is primarily

r~lated to the north-south temperature difference, and at the moment there does not

appear to be any basic reason why temperature lapse rates at middle latitudes should

be the same as they are in the tropics. A discussion of how baroelinio eddies act to

establish the lapse rate in middle latitudes may be found in Stone (1972, 1973). The

relevance of this process to astrophysics is not at all clear. Finally, the arctic-

antarctic ice and snow cover lead to high surface albedos and radiation tends to stabi-

lize rather than destabilize the atmosphere. This, in turn, tends to suppress baro-

clinic eddies. A comprehensive discussion of terrestrial atmospheric stability based

on numerical simulation may be found in Reid (1976).

3. PENETRATIVE CONVECTION AND MIXED LAYERS

One may reasonably ask, at this stage, whether convection in the earth's atmosphere

ever leads to a neutral lapse rate. The answer is almost certainly yes, but it is not

clear that even in these instances, convective adjustment is the correct approach.

We shall, in this section, look at one of the more extensively studied examples of

convective mixing: namely the convective mixing of the air near the ground where the

convection is forced by solar heating of the surface. A substantial number of pheno-

menological theories exist for this process and there is still a measure of controversy

surrounding them. I will sketch one typical example of such theories due to Tennekes

(1973). The geometry of the situation is shown in Figure 3 where profiles of both po-

tential temperature and convective heat flux are presented. At the bottom of the mixed

layer there is a thin superadiabatic layer dominated by mechanical turbulence. The

nature of this layer is ignored except insofar as it delivers a heat flux (O-W)o to the

interior; this heating forces the convective mixing which proceeds over a finite layer

of thickness, h, topped by an inversion layer with temperature jump, A. The region dO

above this jump is stably stratified with~-~z= ~. As heating continues, h increases

with time -- whence the name "penetrative convection". The picture thus far is reaso-

nably well observed over land in middle latitudes. As the mixed layer rises into a

warmer environment, the cooling of the entrained warmer air must give rise to a nega-

tive flux (~)i beneath the inversion. This is mathematically expressed as follows :

dh (3. I) (Ow)i = ~ d--f •

An equation may be written for the time evolution of A, on noting that the penetration of

the mixed layer into the stable interior tends to increase A, while the heating of the

mixed layer tends to decrease A :

dA _ dh ~O d-{- Y ~ " - ('~t') b . l . (3.2)

133

we ignore radiative processes, Ob.l. satisfies a simple budget :

__ Cp0( )b.]. = - 3--~ ( Cp 08w),

orjintegratlng over the mixed layer, we find

(3.3)

(~w) o - (e-~) i 38 (TC)b. i . : h

(3.4)

Substitution of (3,4) into (3.2) yields

dA dh (E~;)o (@w)i dt ? dt h + h

• (3.5)

and (3.]) together with (3.5) are generally taken as the basic equations for the sys-

tem. (~)o is given, and (3.1) and (3.5) then form 2 equations in 3 unknowns : A, h

and (ew) i . Clearly another relation is needed (and it is at this point that the bulk

of the controversy is engendered). Tennekes (1974) first considers the turbulent

energy budget near the inversion:

# - q2w) ÷ * , (36) o o

where q is the magnitude of turbulent velocity fluctuations and ~ is a dissipation

rate which is empirically found to be negligible near the inversion. (T o is a mean

temperature.) Thus (3.6) suggests that the kinetic energy generated by buoyancy is

consumed in bringing heat down through the inversion. Since buoyancy tends

to generate vertical velocity, and buoyancy acts th~Dughout the mixed layer,

(-~ q2w) ought to scale as follows:

3 ~z (2 q2w) ~ - 0 (-~), (3.7)

where o W

is the vertical velocity variance, and

T 3 -(~)i ~ o ~ (3.8)

g h

In addition since o~ is generated by (~)o

G 2

O,.v * gh 0

we have

and ow 3 ,x, gh ( ~ ) o (3 .9 ) T

0

134

Combining (3.8) and (3.9) we have

- (~)w)i = k (SW) o : (3.10)

k is a constant which is empirically found to be about 0.2. Equation (3,10) closes the

system described by (3.]) and (3.5). The resulting equations have been used (with

moderate success) to describe a variety of convective boundary layers, For the diur-

nal boundary layer, surface heating during the day causes h and Qb.|. to increase;

the heat thus deposited is carried away by radiation during the night when (~)o is

zero. This incidentally explains how there can be a turbulent heat flux into the atmos-

phere in the mean even though the mean stability may be positive.

d8

d--fiY Z Z

t+dt

~e 8W

Figure 3. The vertical distributions of potential temperature and turbulent heat

flux in and above a convective boundary layerl after Tennekes (]973),

To be sure, the concept of a diurnal boundary layer is hardly applicable in astro-

physics. However, the above approach has also successfully accounted for the semi-

permanent mixed layer of the tropical maritime atmosphere (Sarachik, 1974). In that dh particular case there exists a between-cloud subsidence which causes ~ in equ, (3.])

dh and (3.5) to be replaced by (~ - w) and an equilibrium solution exists wherein

135

dh d-~ = O. More germain to astrophysics would be the inclusion of radiation in the above

picture. Equ. (3.]), (3.3), (3.6) and (3.7) would all need modification since radia-

tion would not only alter the gross budgets but would also act to dissipate buoyancy.

1% is also conceivable that convection, if it were to occur in plumes~would not lead

to an adiabatic lapse (well mixed potential temperature) (see Appeudix ]). This might

affect the validity of (3.8) and (3.9) since the mean stability would inhibit buoyant

acceleration. The above, of course, all remains to be done, but it might conceivably

form a more satisfactory alternative to convective adjustment. The possibility of con-

vection leading to inversion "discontinuities" etc. might have significant implications

as well.

4. MESOSCALE ORGANIZATION OF COI'~ECTION

We turn now to a last and somewhat different aspect of atmospheric convection.

Even when rather broad regions of the atmosphere are relatively uniformly unstable

(or more typically conditionally unstable with respect to moist processes), convec-

tion (in the form of cumulus clouds) rarely if ever occurs in a uniformly distribu-

ted manner. Instead, the convection is almost always organized into systems whose

scale is typically I-2 orders of magnitude larger than the scale of the cumulus

clouds themselves. The larger scale (100-4001~) is referred to in meteorology as

the mesoscale. Cloud clusters and squall systems are examples of mesoscale systems.

Mesoscale organization appears to be an intrinsic feature of atmospheric convection.

For certain types of atmospheric convection the relation to mesoscale organization

seems reasonably clear. In these cases moisture is concentrated near the surface

(in the first 2 kilometers of the atmosphere typically) and virtually the entire

depth of the troposphere is conditionally unstable. Such situations tend to be

characterized by intense cumulonimbus convection. The rainfall in such situations

tends to satisfy a simple moisture budget where the rainfall (and hence latent heat

release) is proportional to the convergence of moisture (plus evaporation where this

is relevant). Moreover~ since the moisture tends to he confined to Z < Z T (where Z T

is typically 2 km), the convergence of moisture tends to be proportional to the ver-

tical velocity at Z T. Finally the latent heat release is significant for the larger

scale motions. In the presence of an internal wave perturbation (which produces

convergence) one can imagine an interaction of the sort indicated below:

Latent Heating

J \ Surface convergence ~-" -- Internal waves

136

If the internal waves produced by latent heating produce more surface convergence

(in the proper phase) than is needed to maintain the wave, the system will be un-

stable. This mechanism is referred to as wave CISK (conditional instability of the

second kind), and is described in greater detail in Lindzen (1974). CISK is used

to describe any collective instability of cumulonimbus convection and larger scale

motions. The concept was introduced by Charney and Eliassen (1964) in connection

with hurricane generation. The mathematical problem in the present instance con-

sists simply in the solution of the equation for thermally forced internal gravity

waves which takes approximately the following form:

d2w+ ~2w = Q(z) (4.1)

dz 2

where are proportional to elk(x-ct).'" w is the vertical velocity, Q is all fields

proportional to heating, x is a harizontal coordinate, k is a horizontal wavenumber,

and c is a horizontal phase speed which may be complex (for unstable solutions).

For our purposes N2

~2 ~ C 2 (4.2)

where N is the Brunt-Vaisala frequency. Now it is an easy matter to write the solu-

tion for w (satisfying suitable boundary conditions) as functional of Q(z):

w ffi F c [Q] ( 4 . 3 )

where w depends on c (and z) as well as Q, But Q is proportional to W(ZT), and (4.3)

becomes

w(Z) = F c [q(Z')w(ZT)] (4.4)

where q(Z) is a specified function. At Z = Z T (4.4) becomes

w(Z T) = F c [q(Z')W(ZT)] (4.5)

which proves to be possible only for certain values of c--one of which is typically

associated with the greatest degree of instability. Current calculations indicate

that the imaginary part of c is much smaller than the real part and that for common

terrestrial situations Re(c) ~ 15m/s. Since solutions are of the form e Ik(x-ct),

growth rates are equml to k x Im(c) and one might infer that maximum growth rates

are achieved as k ÷ ~(and as the frequency k Rec ÷ ~ also). This, however, is in-

consistent with the fundamental premise of C!SK: namely, that convection is organ-

ized by large scale convergence. Clearly such organizatiun cannot be achieved on

time scales shorter than characteristic development times for the clouds. For ex-

ample in the tropics cumulonimbus clouds have a characterlstic time scale of about

i hour, which suggests a maximum frequency, ~, of about (I hour) -I. Now

1 ~ k x 15 mls. ~ kc ~ 3600 see

137

1 Hence k

3600 x 15 m 2~

and horizontal wavelength ~-~ 2~ x 3600 x 15 m ~ 339~. (4.6;

In fact, both this wavelength, and the predicted phase speed are characteristic of

tropical mesoscale disturbances, implying that the maximum frequency suggested above

is, in fact, what is realized. A similar approach has been used by Raymond (1975)

to account for the structure and evolution of intense co=vective storms in the mid-

western United States.

The relevance of wave-CISK for astrophysics is questlo=able since there appears to

be no astrophysical counterpart to rainfall. However, it is also observed in the

earth's atmosphere that cumulus convection which is restmicted to relatively shallow

layers within the middle of the troposphere and which is associated with little

(and sometimes no) rainfall is also organized into mesoscale patterns. Latent heat

does not appear at first sight, to play a major role in forcing these mesoscale sys-

tems. In a recent paper, Lindzen and Tung (1976) have shown that the near neutral

Stability created by mid-level cumulus activity helps trap internal gravity waves in

the stable region below the clouds~ creating a duct wherein wave modes may exist

without significant forcing. The phase speeds of these ducted modes (determined pri-

marily by the thickness of the stable region below the clouds) are in good agreement

with observations. Furthermore, observed periods appear to satisfy the relation

2w (4.7) T~ave rcloud

JUSt as in the case of wave CISK disturbances. Given a duct phase speed, c, and a

characteristic cloud time scale Tclo~d, the mesoscale wavelength is again

wavelength ~ 2~ c Tclou d (4.8)

The means for interaction between the waves and the cloud field are not entirely

dear in this case. However, the period given by (4.7) is still the shortest period

on which any interaction could take place. Moreover, the well known degeneracy of

such features of convection as its plan form suggests that the organization of con-

vection might be responsive to relatively weak perturbations. Similary, the waves,

being dueted, call for only small forcing.

At this point it is worth noting that the earth's atmosphere can sustain a class

of free oscillations (Lamb waves) which do not require explicit ducting. These

waves are, essentlally, horizontally propagating acoustic waves %~th c ~ 319 m/s.

By the above arguments we ought to expect organization of convection with wavelengths

given by (4.8) based on the speed of sound and Tclou d. There is no clear cut ohser-

vatlonal evidence available for this suggestion. However, the wavelengths obtained

are on the order of several thousand kilometers, and on the earth, regions on this

scale with relatively uniform conditional instability are rare. The situation ap-

pears somewhat more congenial on the sun where a convective layer exists over the

entire star. Identifying the convective elements with granules for which T ~ 5 min-

138

utes and taking c ~ I0 km/sec one obtains from e~. (4.8) that the dominant wave-

length ought to be 40,000 km. Whether it is purely an accident that this is also

the scale of supergranules remains to be seen. Less arguably, the above discussion

demonstrates rather clearly that the appearance of structures of a given horizontal

scale need not imply vertical scales of the same order. Similarly, terrestrial ex-

perience suggests that convectio~ rarely involves merely a single horizontal scale.

ACKNOWLEDGEMENTS

The author wishes to thank E, Spiegel for encouraging the preparation of this

manuscript 9 and the National Science Foundation for its support under Grant ATM-75-

20156.

REFERENCES

Arakawa, A. and W.H. Schubert, 1974 : Interaction of a cumulus cloud ensemble with the large scale environment. J. Atmos, Sci., 3_[], 674.

Charney) J. and A. Eliassen, ]964: on the growth of the hurricane depression. J. Atmos. Sci., 2_[I, 68

Emden, R., 1913: Uber Strahlungsgleichgewicht und atmosph~rische Strahlung. Sitz. d. Bayerische Akad. d. Wiss., Math. Phys. KlasRe, p. 55.

Gold, E., 1909: The isothermal layer of the atmosphere and atmospheric radiation. Proc. Roy. Soc. A, 82, 43.

Goody, R.M., 1949 : The thermal equilibrium at the tropopause and the temperature of the lower stratosphere~ Proc. Rcy. Soc. A, ]97, 487.

Held, I.M., 1976 : The Tropospheric Lapse Rate and Climate Sensitivity, Ph.D. The- sis, Princeton University, 2]7 pp.

Herman, G., and R.M. Goody, ]976: formation and persistence of summertime arctic stratus clouds. J. Atmos. Sci., 33, ]537-1553.

Lindzen, RoS., ]974: Wave-CISK in the tropics. J. Atmos. Sci., 3~], ]56.

Lindzen, R.S., and K.-K. Tung, 1976: Banded convective activity and ducted gravity waves. Mon. Wea. Rev., 104, in press.

Palmen, E. and C. W. Newton, ]969: Atmospheric Circulation Systems, Academic Press, New-York, 603 pp.

Raymond, D. J., ]975: A model for predicting the movement of continuously propagating convective storms, J~ Atmos. Sci.~ 32, 1308.

Sarachik, E.S., ]974: the tropical mixed layer and cumulus parameterization. J. Atmos. Sci,, 31 , 2225.

139

Schneider, E.K., 1976: Axially sy~netrie steady state models of the basic state for instability and climate studies. Part II: Nonlinear calculations. J. Atmos. Sci., 3_33, in press.

Schneider E.K., and R.S. Lindzen, ]976: Axially symmetric steady state models of the basic state for instability and climate studies. Part. I; linear calculations. J. Atmos. Sci., 33, in press.

Stone, P.H., ]972: A simplified radiative-dynamical model for the state stability of rotating atmospheres. J. Atmos. Sci., 29, 405.

Stone, P.H., ]973: The effect of large scale eddies on climatic change. J. Atmos. Sci., 3__00, 521.

Tennekes, H., ]973: A model of the dynamics of the inversion above a convective boundary layer. J. Atmos. Sci., 30, 558.

U.S. Standard atmosphere supplements, ]966: available Superintendant of Documents U.S. Government Printing office, Washington, D.C. 20402.

APPENDIX ]. HEAT TRANSFER BY THIN PLUMES

The following discussion is based on work by Arakawa and Schubert (]974) concerning

cumulonimbus clouds. The present discussion, however, ignores moisture (both for sim-

plicity and because of its irrelevance to astrophysical problems). We shall consider

convection which occurs in plumes which occupy a small fraction of the total horizon-

tal area and which despite their small area contribute significantly to the mean ver-

tical mass flow. By "mean" we shall always refer to an average over an area large

compared to the cross-sectional area of plumes, but small compared to any large scale

flow. Our aim will be to parameterize the effect of plumes on this large scale flow.

Means will be indicated by overbars. The approach will be analogous to the use of

Reynold's averaging where the eddies will be convective plumes.

We will first partition the mean vertical mass flux into plume and environmental

(non plume) contributions: %

p-'w = Mp + M, (A.])

where p = density, Mp = plume mass flux and M = environmental mass flux.

poses the following quasi-Boussinesq continuity equation will suffice:

V o (pl) + ~ (~-6) : 0

For our pur-

(A.2)

(V • ~) will here refer to horizontal divergence of q.

consider an ensemble of plumes where

It will also prove useful to

Mp = EMi0 (A.3) i

140

Each plume may either be entraining mass from its environment in which case

,~Mi ~oi, ~Mi 8oi Ei = i--~z + P =~t )' -~z + ~ > 0 (A.4a)

or detraining into the environment in which case

,~Mi "-~)~°i'' ~Mi i D i = - t--~ + --~ +--~ < O; (A.4h)

o. is the fractional area occupied by the i th 1

p ied by a l l plumes i s

plume and the fractional area occu-

= Z o.. (A.5) p .l i

Mp satisfies the following budget :

~Mp Bop E - D = BT + P ' - - ~ , (A.6)

where

The static energy:

= E Ei entraining

plumes

= S Di

detraining plumes

s = c T + gZ (A.7) P

is conserved during adiabatic processes. The budget for s in the environment is given

by

op)psJ " ~ (M s) + QR (A.8) ~--- [(] - ~ = - V ~ (PVs) - Es + S DiSDi - --Bz , ~t d,p,

where l refers to a sum over detraining plumes and SDi is the static energy of

the d.p, i th detraining plume; QR represents radiative heating in the environment.

Using equ. (A.]), (A.2) and (A. 6) we may easily transform (A,8) to the following:

+d~p. Di (SDi - ~) + Mp ~- + Qr . (A.9)

We will now assume the following to be adequate approximations:

v • pv ~ V (~-v) , (A.!Oa)

v ' pvs ~V • (p v s) . (A.]0b)

141

Also, for P

and

<< 1, it is readily shown that (] - ~p) ~ 1,

s,

~r ~

Equ. (A.9) then becomes

27 - 37 27 p -~ + 0 v • 7 ~ + p~z = Mp ~ + dip. Di(SDi - ~) + Qr. (A. II)

Let us finally assume each plume detrains at precisely that level where its static

energy equals that of the environment (i.e., where it looses buoyancy: this is con-

sistent with the known instability of decelerating jets). Then

Z Di(Ssi - ~) = 0 d.p.

and (A. II) becomes

~-~ + p ~ ~ . V ~ + pw ~ z = Mp ~ + ~. (A.12)

We see from (A.]2) that the primary effect of convective plumes on the environment 37

is to introduce a heating term Mp ~z " This term is easily interpreted: a portion

of pw (i.e., Mp) which rises in plumes does not give rise to adiabatic cooling in

the environment - and appears, therefore, as a heating term. If we ignore p-w, Mp must

be compensated by equal subsidence which does, in fact, lead to eompressional heating

in the environment. A crucial point which may be made from (A.12) is that if convec-

tive plumes are to supply heat which is then carried off by radiation, ~z must be

positive!

NUMERICAL METHODS IN CONVECTION THEORY

N. O. Weiss

Department of Applied Mathematics and Theoretical Physics

University of Cambridge

SUMMARY

Two and three-dimenslonal computations have enlarged our understanding of non-

linear convection, particularly in Boussinesq fluids. However, we cannot

adequately predict the relationship between convective heat transport and the super-

adiabatic temperature gradient. Nor is there any indication of a preferred length

scale, other than the depth of the convecting layer, in a compressible fluid.

I. INTRODUCTION

The standard procedure for calculating the structure of stellar convection

zones is to use mixing length theory, calibrated to fit the sun and neighbouring

stars on the main sequence. Mixing length theory is based on plausible physical

assumptions and seems to provide qualitatively acceptable results but, as Dr. Gough

has emphasized, it lacks any firm theoretical basis. The principal need in astro-

physical convection is for a soundly based theory that can confirm, or replace , the

procedure now adopted. In particular, we would like to establish the functional

relationship between the convective heat transport and the superadiabatic temperature

gradient, and to determine the preferred scale of convective motion. Spiegel's

(1971b, 1972) excellent review of astrophysical convection contains a thorough dis-

cussion of the basic fluid dynamical problem and includes a full list of references,

which has been brought up to date by Gough (1976). go I shall limit myself to

describing recent progress towards understanding nonlinear convection by solving

model problems numerically on a computer.

Direct observation of solar convection reveals cellular patteraa. Hot gas

rises, cold gas sinks and the lifetime of an individual cell is of the same order as

the time taken for fluid to turn over in it. The photospheric granulation has a

horizontal scale similar to the local density or pressure scale height, and comparable

with the thickness of the strongly superadiabatic layer at the top of the convective

zone. Supergranulea have diameters about 15 times larger; their relationship to

features with strong magnetic fields implies that they correspond to more deep-

seated convection. There are also suggestions of motion on a scale comparable to

the depth of the convection zone, while speckle photometry indlcates that there may be

large scale convective cells in the outer layers of red giants.

Observations provide few constraints on the relationship between heat flux and

temperature gradient. Nor can the parameters appropriate for astrophysical convection

be modelled in laboratory experiments. Hence we must attempt to solve the governing

equations which, since they are nonlinear, have to be tackled on a computer. The

143

full problem is still too difficult. So it is necessary to make various simplifying

geometrical and fluid dynamical approximations. We hope that a better description

of astrophysical convection will eventually emerge from the results of a sequence

of idealized numerical experiments.

2. THE IDEALIZED PROBLEM

Let us consider convection in a horizontal layer, heated uniformly from below

and confined between the planes z = O, d, where the z-axis points vertically

upwards. In the absence of motion the superadiabatic temperature gradient

~ ]

where T is the temperature and the adiabatic gradient

Cp Here g is the gravitational acceleration, Cp the specific heat at constant pressure

and = the coefficient of thermal'expansion (for a perfect gas = = l/T). In the

Boussinesq approximation we assume that the layer depth d is much smaller than the

temperature scale height Cp/g~ and that the Mach number U/e s <<I (where U is a

typical velocity and c s is the velocity of sound): then the velocity ~ and density

satisfy the equations

where %, T O are constant, and the configuration is described by two dimensionless

parameters, the Rayleigh number

R =

and the Prandtl number ~ = v/X , where ~, ~ are the thermal and viscous diffusivl-

ties. The heat flux can be expressed in terms of the dimensionless number

where the total heat flux is Cp~ F. For an infinite layer N is a function of R

and ~ only.

In most laboratory experiments the convecting fluid is confined between rigid

boundaries at which ~ vanishes, and the resulting flow is dominated by viscous

boundary layers. These boundary conditions are inappropriate for stars and it

usual to assume that the tangential stress and normal velocity vanish at the

surfaces z = O, d, which are held at fixed temperatures T O + ~d, T O . These "free"

boundary conditions are dynamically fairly passive and mathematically convenient.

Nevertheless, any technique or theory should be capable of describing experimental

results correctly before it is applied to astrophysical convection.

A bewildering array of power laws has been put forward for the function

N(R, ~). For R >> 1 an asymptotic upper bound with N ~ R ½ has been established

(Howard 1963, Busse 1969). At high Prandtl numbers NNR I/3 for free boundaries

but the radiative conductivity is high in stars and the Prandtl number is therefore

small~ For ~ << I we might expect that the energy flux should not depend explicitly

144

on the viscosity W , so that N = N(S), whore S = ~R. In the sun, ~m 10 -9 but

S is typically of order 1012 . Arguments can be found for suggesting asymptotic

power laws of the form N ~ S r with r = 2, ½, 1/3, 1/4, 1/5 (Spiegel 1971 a,b;

Gough and Weiss 1976; Jones et al. 1976; Gough et al. 1975) but it is not obvious

which, if any, of these exponents is correct.

3. BOUSSINESQ CONVECTION

In the Boussinesq approximation the pressure can be eliminated by taking the

curl of the equation of motion. The time-dependent equations then become

and ~' : V^(~^~ - ~T^ 9 * ~V~

together with V.H = O, where the vorticlty ~ : V^~ and e, is a unit vector in the

z-direction. In two dimensions, with motion confined to the xz plane and indepen-

dent of the y co-ordlnate, the vnrticity has only a y-component and the velocity

can be expressed in terms of a stream function ~ such that

where @~ is a unit vector in the y-direction.

The vorticity equation then reduces to

~ _ ~ ~ ~ V~ .

3.1 Rigid boundaries

Convection sets in at the critical Rayleigh number R c and two-dimensional

solutions for R ~ I000 R have been available for some time (e.g. Fromm 1965, c

Schneck and Veronis 1967, Plows 1968). Busso (1967) first showed that two-dimenslonal

solutions at infinite Prandtl number may be unstable to three-dimensional pertur-

bations and the development of rolls into three-dimensional and time-dependent regimes

has been studied experimentally (e.g. Busse and Whitehead 1974) for fluids with high

Prandtl numbers. The stability of two-dimensional rolls was systematically inves-

tigated by Clever and Busse (1974): for Prandtl numbers of order unity, the rolls

develop a wavelike oscillatory instability when R ~ 3.5 R c. The most thoroughly

investigated case is convection in air ( ~ = 0.7) for which Veltishchev and Zelnin

(1975) and Lipps (1976) have computed three-dlmensional solutions with R 4 15 R c.

At low Rayleigh numbers Lipps' numerical experiments show the development of rolls

whose preferred width differs from that which maximizes the heat transport. As R

is increased, the oscillatory instability appears and solutions become time-

dependent. For R ~ 15 Rc, motion is three-dimensional and aperiodic. However the

change from two to three dimensional convection does not greatly affect the time-

averaged Nusselt number. These numerical results are all supported by experiments

(Willis and Deardorff 1967, 1970; Krishnamurti 1970a,b, 1973; Brown 1973).

Unfortunately, apart from the experiments by Rossby (1969), few results are available

for low Prandtl number convection.

145

3.2 Free boundaries

Convection in two-dimensional rolls has been studied in numerical experiments

with R 6 IO00 Rc (Fromm 1965, Veronis 1966, Moore and Weiss 1973). For high Prandtl

number ( V >> R ~) there are steady solutions with N~R I/s, which ere apparently

stable (Straus 1972). For V << i, the Nusselt number depends only on R and for R>>]

N ~ R 0"36 (Moore and Weiss 1973). In these laminar solutions the vorticity ~ is

nearly constant on the streamlines. The nonlinear term in the vorticity equation

remains small even when the Reynolds number is large: the rolls behave like flywheels

and are slowly accelerated until, after they have turned over many times, the buoyancy

torque is eventually balanced by friction. However, the oscillatory instability sets

in near the critical Raylelgh number for v << I (Busse 1972) and the rolls should

break down into three-dimensional cells.

In the two-dlmensional solutions, rising and falling plumes are exactly symmetrical

but this syrmmetry is no longer present in, say, a hexagonal cell where fluid can rise

in a central column and sink around the perimeter of the cell. It was conjectured that

this geometrical change might affect the physics so that N could depend on S for

<< i. So we investigated axisymmetric convection in a cylindrical cell (Jones et al.

1976). This idealized model is mathematically two-dlmensional but geometrically three-

dimensionalj though the cells cannot be packed together to fill a plane. Referred to

%ylindrical polar co-ordinates (r,~,z) the velocity is given by a Stokes stream

function ~ (r,z) such that

where e~ is a unit vector in the ~-dlrectlon, and the vortlcity

= r~l _e e ,

where

~'-~ - r ~ '7

We found, however, that the convective flux was similar to that for two-dimensional

rolls. For high Prandtl numbers N~R I/3 again, while for f<< i N ~R 0"4 approximately.

So the Nusselt number is still independent of • and approaches closer to the upper

bound. The form of the solutions is displayed in Fig. i, which shows streamlines,

isotherms and profiles of the modified vorticity~ for a steady solution with

= O.O1, R = IOO R . Vortex tubes are stretched as they move away from the axis with c

the fluid and~l is nearly constant along streamlines. So flywheel solutions exist in

a cylinder and would also, presumably, appear in hexagons.

Are these solutions stable? Jones and Moore (1977) have recently shown that the

axis~mmetric flow is Unstable to non-axisymmetric perturbations. Thus a cylindrical

cell can fragment into sectors, like unstable vortex rings in laboratory experiments

(Widnall and Sullivan 1973, Widnall, 1975). As the Reynolds number increases, three-

dimensional convection cells should therefore become unstable and split up. Such e

phenomenon is observed in the sun: large granules explode and break up into smaller

146

TZ

F (a)

F

(b)

Figure i. Axisymmetric convection in a cylindrical cell. Results for R = i00 Rc, p = 0.O1. (a) Isotherms and streamlines: equally spaced contours of T(left) and ~ . (b) Profiles of the modified vorticlty~, which is nearly constant along the streamlines.

147

cells (Musman 1972). In a tesselated convection pattern there may also be collective

instabilities which allow vortex rings to reconnect, so that cells are swallowed up

and disappear.

3.3 The modal approximation

An alternative approach to three-dimensional convection has been to adopt a

truncated modal expansion. For example, the vertical velocity w can be expanded in

eigenfunctions of the two-dimensional Laplacian operator:

,

The single mode expansion, normalized so that ~= O, f2 = i, 7 ~ 2C (where the bars

denote horizontal averages) has been studied in great detail (Gough et el. 1975;

Toomre et ai. 1977) and numerical solutions have been obtained for Rayleigh numbers

up to 1025. In this approximation the plan form of a convection cell is prescribed

by the linear eigenfunction, and enters the equations through the parameter C. For

two-dimensional rolls C = 0 and the equations reduce to the mean field approximation;

for cylinders C = 0.18 and for hexagons C = 0.41.

With rigid boundaries the results for a single mode agree quite well with experi-

ments; with free boundaries N~R I/3 when ~ >> I but N N (S in S) }/5 for R -| << ~ << |.

The imposed plan form generates a large nonlinear term in the vorticity equation. If

the flow is constrained only to be laminar and steady then it can adjust its plan

form to make ~^(~^~) very small and the effective dissipation can therefore be

reduced.

The modal expansion and the flywheel solutions are two extremes. We might expect

that instabilities would limit the lifetimes of three dimensional convection cells,

so that they are comparable with the turnover time and laminar flywheel solutions

cannot be attained. Then N should depend on S, though it is not clear what power law

would hold. This problem will not be resolved until the results of fully three-

dimensional computations have become available. Meanwhile, the power law derived

from mixing length theory (NNS ½ for S >> i) remains as good as any other.

4. COMPRESSIBLE CONVECTION

The Boussinesq approximation is manifestly inadequate for stellar atmospheres

that extend over many scale heights (the density increases by a factor of 106 in the

solar convection zone). It i8 commonly supposed that the dimensions of convection

cells should be of the same order as the local density or pressure scale height.

This assumption fits the photospheric granulation and some physical arguments can be

adduced to support it (Schwarzschild 1961; Weiss 1976). In mixing length theory

(which is essentially Boussinesq) the mixing length is generally set equal to some

multiple of the local pressure scale height. It would be comforting to have some

theoretical justification for this choice of length scale.

Linear theory gives no help: in a polytropic atmosphere convection sets in with

a horizontal scale that is comparable with the layer depth, even for the complete

148

atmosphere where the scale height shrinks to zero at the upper boundary (Spiegel

1965; Gough et al. 1976; Graham and Moore 1977). At supercritical Rayleigh numbers

modes with smaller horizontal scales have higher growth rates, at least when dis-

sipation is ignored (see Spiegel 1972). B~hm (1967) discussed the growth rates of

linear modes in a model of the solar convection zone, neglecting turbulent viscosity,

and found that the growth rate increased monotonically with the horizontal wavenumber.

Vandakurov (1975a,b) has included the effects of an eddy viscosity and found a

maximum growth rate for cells with a horizontal scale intermediate between those of

granules and supergranules. In these gravest modes there is no reversal

of the velocity, though the amplitude is strongly peaked near the surface. A

preliminary study of the marginal stability problem (Bbq~m 1975) indicates that there

may be internal nodes but their interpretation is obscure. (The reversal in the

temperature perturbation reported by Vickers (1971) is apparently due to

numerical error.) Smaller length scales seem to be produced not by the density

variation but by the strongly superadiabatic gradient, coupled with ionization, near

the top of the convective zone.

In nonlinear studies sound waves can be filtered out by using the anelastie

approximation (Gough 1969) which is valid provided the Mach number remains small.

This has been applied, using the modal approximations to study (inefficient) con-

vection in A-type stars (Latour et al. 1976; Toomre et al. 1976). However, no

careful study of the transition from Bousslnesq to compressible convection has yet

been carried out.

Dr. Graham will describe his numerical experiments on fully compressible non-

linear convection in two and three dimensions. For steady convection in two-

dimensional rolls the eye of an eddy is no longer at the centre of the cell but is

displaced downwards and towards the sinking plume (Graham 1975). This asymmetry is

observed in solar granules, which show a broad column of hot gas, rising at their

centres, surrounded by narrower, more rapidly sinking ring of cold material (Kirk

and Livingston 1968; Deubner 1976). Graham finds no evidence for small scale motion;

convective cells extend across the entire layer, even when the density varies by a

factor of 50. In studying compressible convection it is most straightforward to

assume that the dynamic viscosity ?g is uniform. Then the viscous term dominates

the equation of motion near the upper boundary, where the density is small (Gough

et al. 1976). If the aim is to represent turbulent dissipation by an eddy viscosity,

then the kinematic viscosity ~ can be obtained from a model of the convection zone.

For the sun~ ~ is roughly constant (Cocke 1967, B~hm 1975). However, Graham finds

that the cell size is not altered by setting ~ constant across the convecting layer.

So fare the only suggestion of small scale motion has come from some nonlinear

calculations by Deupree (1976), whose resolution is too coarse for the results to

be credible. Unless further computations on compressible convection reveal some

new pattern of behaviour, we shall have to suppose that the observed scales of con-

vection in granules and supergranules are caused by boundary layers near the surface

149

of the sun, rather than by the changing density scale height. If so, the mixing

length cannot be locally determined and mixing length theory is, at best, reliable

only near the surfaces of main sequence stars.

REFERENCES

B~hm, K.-H., 1967. Aerodynamic phenpmena in stellar atmospheres (IAU Symp. No. 28) ed. R. N. Thomas, p. 366j Academic Press, London.

g~hm, K.-H., 1975. Physique des mouyements dans les atmosph_~r@s st ellaires , ed. R. Cayrel and M' Steinberg, p. 57, CNRS, Paris.

Brown, W., 1973. J. Fluid Mech. 60, 539.

Busse, F. H., 1967. J. Math. Phzs. 46, 140.

Busse, F, H., 1969. J. Fluid Mech. 37, 457.

Busse, F. H., 1972. J. Fluid Mech. 52, 97.

Busse, F. H. and Whitehead, J. A., 1974. J. Fluid Mech. 66, 67.

Clever, R. M. and Busse, F. H., 1974. J. Fluid Mech. 65, 625.

Cocke, W. J., 1967. A strophys. J. 150, 1041.

Deubner, F. L., 1976. Astr. Ast[ophys. 47, 475.

Frorm, J. E., 1965. Phys. Fluid s 8, 1757.

Gough, D. 0., 1969. J. Atmos. Sci. 26, 448.

Gough, D. O., 1976. Trans. IAU 16A Part 2, 169.

Gough, D. O., Moore, D. R., Spiegel, E. A. and Weiss, N. 0., 1976. Astrophys. J. 206, 536.

Gongh, D. O., Spiegel, E. A. and Toomre, J., 1975. J. Fluid Mech. 68, 695.

Gough, D. O. and Weiss, N. 0., 1976. Men. Not. R. Astr. Soc. ll~76l, 589.

Graham, E., 1975. J. Fluid Mech. 7qO, 689.

Graham, E. and Moore, D. R. 1977. In preparation.

Howard, L. N., 1963. J. Fluid Mech. 17, 405.

Jones, C. A., Moore, D. R. and Weiss, N. O., 1976. J. Fluid Mech. 73, 353.

Jones, C. A. and Moore, D. R., 1977. In preparation.

Kirk, J. G. and Livingston, W., 1968. Solar Phys. 3, 510.

Krishnamurtl, R., 1970a. J. Fluid Mech. 42, 295.

Krishnamurti, R., 19705. J. Fluid Mech. 42, 309.

Krishnamurti, R., 1973. J. Fluid Mech. 60, 285.

Latour, J., Spiegel, E. A., Toomre, J. and Zahn, J.-P., 1976. As troph~s. J. 207, 233.

150

Lipps, F. B., 1976. J. Fluid Mech. 75, 113.

Moore, D. R. and Weiss, N. 0., 1973. J. Fluid Mech. 58, 289.

Musman, S., 1972. Solar Phys. 26, 290.

Plows, W., 1968. Phys. Fluids II, 1593.

Rosshy, H. T., 1969. J. Fluid Mech. 36, 309.

Schneck, P. and Veronis, G., 1967. Phys. Fluids iO, 927.

Schwarzschild, M., 1961. Astrophys. ~.t 134, I.

Spiegel, E. A., 1965. Astroph~s. J. 141, 1068.

Spiegel, E. A., 1971a. Comm. Astrophys. Space Phys. 3, 53.

Spiegel, E. A., 1971b. Ann. Rev. Astr. Astrophxs. ~, 323.

Spiegel, E. A., 1972. Ann. Rev. Astr. Astrophys. iO, 261.

Straus, J., 1972. J. Fluid Mech. 56, 353.

Toomre, J., Gough, D. O. and Spiegel, E. A., 1977. J. Fluid Mech. 79, i.

Toomre, J., Zahn, J.-P., Latour, J. and Spiegel, E. A., 1976, Astrophys. J. 207, 545.

Vandakurov, Yu. V., 1975a. Solar Phys. 40, 3.

Vandakurov, Yu. V., 1975b. Solar Phy.~. 45, 501.

Veltishohev, N. F. and ~elnin, A. A., 1975. J. Fluid Mech. 68, 353.

Veronis, G., 1966. J. Fluid Mech. 26, 49.

Vickers, G. T., 1971. Astrophy s. J. 163, 363.

Weiss, N. 0., 1976. Basic mechanisms of solar activity (IAU Symp. No. 71), ed. V. Bumba and J. Kleczek, p.ZZ9, Reidel, Dordrecht.

Widnall, S., 1975. Ann. Rev. Fluid Mech. ~, 141.

Widnall, S. and Sullivan, J., 1973. Proc. ROy. So~. A 332, 335.

Willis~ G. E. and Deardorff, J. W., 1967. phys. Fluids IO, 931.

Willis, G. E. and Deardorff~ J. W., 1970. J. Fluid Mech. 44, 661.

COMPRESSIBLE CONVECTION

Eric Graham


University of Cambridge, England*

I . INTRODUCTION

Stellar convection zones often extend over several pressure scale heights and

convective velocities can be comparable to the local sound speed. Neither laboratory

convection experiments~or analytic solution of the non-linear equations are feasible

in such regimes. In order to gain insight into the details of s te l lar convection we

are obliged to use numerical simulations. At the present time, even this approach

cannot be applied to the parameter range typical of s te l lar interiors~ however sol-

utions can be obtained which extend over many scale heights and have non-negligible

Mach numbers. Under these conditions i t is necessary to employ the fu l l compressible

equations rather than the anelastic approximation (Gough [1] ) or the Boussinesq app-

roximation (Spiegel & Veronis [~ ).

2. THE PROBLEM

Rather than attempting to model a complete star, we w i l l employ a simplified

geometry. In this way we can fac i l i ta te the numerical calculation, while avoiding

the complexities of treating the transition between the convective zone and the opt-

ica l ly thin region. As a standard problem, we consider a gas confined in a rectangular box with slipp-

ery walls. The upper and lower faces are maintained at fixed but different temperat-

ures, T u and T I . The side walls are thermally insulating, A constant gravitational

f ie ld is imposed which has suff ic ient magnitude to produce a signif icant density var-

iation with height. The equation governing the problem are

and

Present address: National Center for Atmospheric Research, High Alt i tude Observatory,

P.O. Box 3000, Boulder, C'olorado 80303, USA.

t52

where ~ j = ~ ( ~ H- ~)X; -- ~- ~Xs" ,

S is the specific entropy,~} is the coefficient of viscosity, K is the thermal conduct-

i v i t y and al l the other symbols have their usual meaning.

Even i f we prescribe the equation of state, the functional forms of the conduct-

i v i t y and the viscosity and the aspect ratios of the box, we s t i l l have f ive degrees

of freedom in setting up the problem (see Graham E~). In addition we can choose our

i n i t i a l velocity, density and temperature distribution.

Numerical solutions for the two-dimensional problem have been presented by Graham

[3]. The most important parameters are found to be the Rayleigh number, R, the Prandtl

number,Or , and the layer depth parameter, Z, given by

= o p t / K , .

I t is sometimes useful to have solutions of the linear equations for the onset of

convection. These equations have been treated by Spiegel[4], Gough et al[5] and by

Graham and Moore[61. A relative Rayleigh number,~ , can be defined by scaling R by

the cr i t ical Rayleigh number of the linear problem.

3. NUMERICAL METHODS

A variety of numerical methods have been used for compressible convection. The

f i r s t solutions were obtained for two-dimensional motions using a modified Lax-Wendroff

f in i te difference scheme. This method is described by Graham [31. The scheme has

been generalised to three-dimensional flows. An alternative to the f in i te difference

method is the pseudo-spectral or collocation method. This has been successfully used for compressible convection equatiomby Graham (unpublished). Each dependent variable

is approximated by a truncated Chebyshev series in two or three space dimensions. The

series are substituted into the dif ferential equations. The time derivatives of the

coefficients of the series are determined by requiringthat the differential equations

be satisfied at selected collocation points. Because Chebyshev transforms can be cal-

culated using fast Fourier techniques, the method is economical. Both the Chebyshev

scheme and the Lax-Wendroff scheme suffer from numerical s tab i l i ty problems for low

Prandti numbers and large values of the Rayleigh number. Solutions have been obtained

withX =100,c=0.I and Z=IO. Current work is directed at developing an alternating

direction implici t f in i te difference scheme.

153

4. THE RESULTS

Because of the computational labour involved in obtaining three-dimensional sol-

utions, most of the calculations are restricted to two dimensional flows. The calculations reported by Graham [3] relate to a perfect gas law and constant K and n~

A number of general results were found. 1. Two-dimensional solutions evolve to steady state flows. The time taken to reach a

steady state increases with increasing horizontal box dimension and decreasing or. This suggests that for more extreme configurations, there may be no steady solution.

2 '

Uu PP er/u lower

_ ~ t a n t V

constant

I I0 I00

Udownwa rd/uu pwa rd

, , J

1 I0 I00

Figure la Figure lb

0

0. :

Uupper/Ulower

const

c o n s t a n t "r I

1 10

Udownward/Uupward

constant

constant -~ " -

0 ,,,, ,, , I

0.I I

O-

10

Figure Ic Figure ld

154

2. There is an asymmetry between upward and downward velocit ies, downward velocities

usually being larger. Horizontal velocities are similar at the upper and lower surfaces,

with the lower velocity often being s l ight ly larger. This is a surprising result, par-

t icu lar ly for large values of Z, because continuity arguments have been proposed to su-

ggest that convective velocities are larger in low density regions.

3. When the horizontal box dimension is large enough to permit several convective ro l ls ,

the horizontal wavelength differs signi f icant ly from that which would maximise heat

f lux.

4. Convective cells extend over several pressure scale heights in the vertical direction.

No cases were found where the flow breaks up into several rol ls in the vert ical.

Further calculations have been performed with a constant kinematic v iscosi ty , ] / ,

rather than a constant dynamic v iscosi ty ,~. I t had been conjectured that the increase

of l / near the surface reduced the upper horizontal velocity. Figure la shows the ratio

of upper to lower velocity as a function of ~ with Z=10. We see that i t is only for

small values of ~ that the upper velocity is enhanced. Figure lb shows the correspond-

ing behaviour of the ratio of downward to upward velocit ies. The variations with Pran-

dtl number is shown in figures lc and Id. The general conclusion is that the solutions

are insensitive to the form of the viscosity law in the cases of large R and small G ,

which is the regime found in ste l lar convection zones.

1 \(

I I

j7

_ ::):

i

; J ,/

J /

Figure 2

155

Relatively few three-dimensional calculations have been performed. I f the horizontal

box size is comparable to the vertical size, two-dimensional flow patterns are found.

As the horizontal size is increased, the flow pattern becomes time dependent even for

modest values of ~ . Figure 2 shows a velocity f ie ld for Z=I and~ lO . In this per-

spective picture, a rectangular bite has been removed from one corner of the box to

reveal the interior. The arrows represent velocity components parallel to the faces.

The arrows are distributed at random with a probability proportional to the density.

The cut away portion shows that the f lu id has significant vertical vor t ic i ty . Such

regions are observed to be short lived, being dissipated and then reforming in a new

position.

5. CONCLUSIONS

Numerical simulat ion of compressible convection provides a way of obtaining a

detai led picture of s t e l l a r convection. At the present time, solut ions are s t i l l far

from the parameter range found in s t e l l a r i n te r io rs . However the solut ions are well

removed from the Boussinesq l i m i t of laboratory convection experiments. I t is to be

hoped that future developments in the form of more e f f i c i e n t algorithms for computat-

ional f l u i d dynamics, turbulence theories for handling the f ine scale features of the

flow and increases in the available computing resources wi l l al l help in attempts to

construct more real ist ic models of ste l lar convection zones.

REFERENCES

1 Gough, D.O., The anelastic approximation for thermal convection, J. Atmospheric

Sciences 26 (1969) pp. 448-456 2 Spiegel, E.A. & Veronis, G., On the Boussinesq approximation for a compressible

f luid, Astroph~s. J. 131 (1960) pp.442-447 3 Graham, E., Numerical simulation of two-dimensional compressible convection, J .

Fluid Mech. 70 (1975) pp. 689-703 4 Spiegel, E.A., Convective instabi l i ty in a compressible atmosphere I, As trophys.

J. 141 (1965) pp. 1068-1090 5 Gough, D.O., Moore, D.R., Spiegel, E.A. and Weiss, N.O. Convective instabi l i ty in

a compressible atmosphere I I , As__trophys. J. 206 (1976) pp. 536-542 6 Graham, E. & Moore, D.R., The onset of compressible convection, To appear

CONVECTION IN ROTATING STARS

F.H. BUSSE University of California, Los Angeles

SUMMARY

It is shown that many features of convection in rotating spheres and spherical

shells can be understood on the basis of plane layer models. The phenomenon of

differential rotation generated by convection is emphasized. The potential applications

and limitations of analytical and numerical models for problems of astrophysical

interest are briefly discussed.

I INTRODUCTION

In thinking about the effects of rotation in stars a variety of thoughts comes

to mind. Some are more negative: Rotation is a breaker of symmetry. It spoils our

notion of a star as an ideal spherically symmetric body which is in static equilibrium

except for the convection zones. Even the latter can be regarded as spherically symme-

tric with respect to their gross properties in the absence of rotation. Although the

deviations from spherical symmetry are small in most rotating stars, the effects of

rotation are only partially known and continue to irritate the theoretician involved

in computations of stellar evolution.

On the other hand stellar rotation is an exciting subject because of the variety

of interesting phenomena associated with it. The generation of magnetic fields is in

general connected with rotation. The shape of surfaces of equal potential in a

rotating star may become unstable in phases of contraction. Rotation can cause

meridional circulations and mixing processes and, in addition, there is a variety of

phenomena connected with differential rotation.

For the theoretical fluid dynamicist rotation brings to mind still other thoughts.

Whenever the Coriolis force becomes dominant the dynamics of fluids are profoundly

altered. The intuition developed from experience with hydrodynamics in non-rotating

systems is no longer valid. Intuitive concepts like mixing length theory appear to be

even less applicable in the case of low Rossby number convection, i.e. when the vor-

ticity of motion relative to the rotating system is small compared to the rotation

rate. On the other hand, theories developed for small amplitude convection appear to have

a much larger range of validity than in a non-rotating system. The two-dimensionality

enforced by a dominating Coriolis force tends to suppress instabilities and restricts

the degree of freedom for turbulent motion.

In the following theoretical and experimental results for the small Rossby number

N

o)

O)

a) Convection columns aligned with the axis of rotation.

b) Convection rolls in a

layer rotating about a

vertical axis.

FIGURE

]

158

case will be presented. Among the nonlinear phenomena caused by convection in rotating

systems we shall emphasize the generation of differential rotation. In discussing the

application to rotating stars we shall restrict our attention to the Sun and Jupiter.

The detailed surface observations available in both cases offer the best hope for

eventual quantitative tests of theoretical concepts.

2 BASIC EFFECTS OF ROTATION ON CONVECTION

The dynamics of nearly stationary motions in a rotating system are

governed by the Proudman-Taylor theorem which states that a small amplitude stationary

velocity field of an inviscld incompressible fluid must be independent of the

coordinate in the direction of the axis of rotation. It is of interest for astrophy-

sical applications that the theorem holds for barotropic fluids as well if the velocity

vector ~ is replaced by the momentum vector p~: By taking the curl of the equation

of motion

2 ~ x pv = - Vp- pV~

and using the equation of continuity

V- p~ = 0

the relationship

2~" V~ = 0 (I)

is obtained. In the following we shall restrict our attention, however, to the

case of incompressible fluids, or, more exactly, Boussinesq fluids for which the

temperature dependence of the density is taken into account in the gravity term

only.

We start the discussion of convection in rotating systems by considering two

simple cases as shown in Figure 1. In case (a) the vectors of gravity and rotation

are at a right angle and convection solutions satisfying the Proudman-Taylor theorem

are possible. The Coriolis force is entirely balanced by the pressure gradient in

that case and the critical value of the Rayleigh number for the onset of convection

becomes the same as in a nonrotating system. Since the Coriolis force always increases

the critical Rayleigh number unless it is balanced by the pressure, the solution

corresponding to convection rolls aligned with the axis of rotation is physically

preferred. It can be easily realized in the laboratory by heating a cylindrical

rotating annulus from the outside and cooling it from the inside and using the

centrifugal force as gravity (Busse and Carrigan, 1974).

While the stabilizing effect of the Coriolis force vanishes in case (a) it

reaches its maximum in case (b) when the vectors of gravity and rotation are parallel.

This is realized when a fluid layer heated from below is rotating about a vertical

axis. Release of potential energy by convection requiresa vertical component of

I

I I .~

..4 .

...

i -

s

(o)

g---

~

Convection columns in an annulus with

inclined top and bottom boundaries.

Fig

ure

2

: Convection layer inclined with

respect to axis of rotation.

f

/

(C)

Convection layer with

changing light.

160

motion which cannot occur without violating the Proudman-Taylor theorem. In order to

overcome the constraint of the Proudman-Taylor theorem viscous friction must become

sufficiently strong, thus playing a destabilizing role in this case. The non-dimen-

sional number describing the ratio between viscous friction and Coriolis force is

the Ekman number

E = ~ / ~ d 2 (2)

where d is the thickness of the layer and ~ is the kinematic viscosity. Since E is

very small in most applications, the horizontal scale of convection must become much

smaller than the vertical in order to increase friction. More detailed analysis (we

refer to Chandrasekhar's (|961) book) shows that the horizontal scale decreases like

E I/3 and the Rayleigh number for onset of convection increases like E -4/3 for small

E. Besides the Ekman number and the Rayleigh number, which is a measure of the buoyan-

cy, the Prandtl number is the third dimensionless parameter of the problem. It des-

cribes the ratio between thermal and viscous time scales of convection. For Prandtl

numbers less than a value of about I, oscillatory convection offers an alternate way

to overcome the constraint of the Proudman-Taylor theorem wihtout changing, however,

the power laws in the dependences on E.

3 EFFECTS OF INCLINED BOUNDARIES

The two extreme cases (a) and (b) of Figure I obviously correspond to equatorial

and polar regions, respectively, of rotating spherical fluid shells heated from within

and subjected to spherically symmetric gravity. There are, however, some important

deviations because of the finite dimensions of the spherical shells. To discuss these

effects let us consider the influence of inclined boundaries in (a) and (b). If top

and bottom boundaries are added in case (a) convective motions satisfying the Proud-

man-Taylor theorem are still possible as long as the boundaries are parallel and vis-

cous friction is negligible. Boundaries inclined with respect to each other, however,

require a dependence of the velocity field on the coordinate in the direction of the

axis of rotation, which we shall cal z-coordinate. A typical example is shown in

Figure 2 (a). The deviation from the Proudman-Taylor condition is accomplished by a

combination of time dependence and viscous friction in this case: Convection still

has the form of columns aligned with the z-axls, but the columns are travelling like

Rossby waves in the prograde or retrograde azimuthal direction depending on whether

the distance between top and bottom boundaries decreases or increases with distance

from the axis. In addition the azimuthal wave number ~ becomes large in order to

increase frictional effects. In the limit of small values of E we find

R = ( ~ ~ )4/3 , ~ = ( ; )I/3 , -I

(3)

/-

0 Figure 3:

Buoyancy force A and inhibition C

caused by inclined boundaries in

a sphere as a function of distance

S from the axis.

Figure 4:

Sketch of motion at the onset of

convection in a rotating sphere.

162

rotation rate, and n is the tangent of half of the angle between the inclined

boundaries. A detailed theory and an experimental study of the stabilizing effect

of inclined boundaries can be found in the papers by Busse (1970b) and Busse and

Carrigan (1974). In section 4 we show how the theory can be applied more or less

directly to the case of convection in a sphere.

Since convection at the mid-latitudes of a rotating spherical shell corresponds

to intermediate angles between gravity and the rotation vector, it may be anticipated

that it shows properties intermediate to those of the extreme cases (a) and (b) of

F~gure I. Indeed, it is easily shown (Chandrasekhar, 1961) that both the Rayleigh

number and the wave number at the onset of convection are governed by the expressions

derived for case (a) if the rotation rate ~ is replaced by its vertical component

cos ~ . Convection occurs in the form of rolls aligned with the horizontal component

of ~ , as indicated in Figure 2(b). Accordingly, the component of the Coriolis force

proportional to ~ sin ~ is balanced by the pressure and drops out of the dynamical

considerations.

When applying the theory of plane parallel convection layers to spherical shells

the strong dynamical coherence of the fluid along any line parallel to the axis of

rotation must be kept in mind. For this reason convection in a spherical shell

exhibits the effects of non-parallel boundaries even though the distance between the

boundaries is constant. Since the tangential surfaces to the spherical boundaries

of the shell are not parallel at the points intersected by the same line parallel to

the z-axis, the dynamics of convection exhibit the same effects as in the case of the

convective layer shown in Figure 2(c). The variation of "height" with distance from

the axis of rotation induces a wave propagation property of the convective motions

similar to that of the convection columns in Figure 2(a). Because of the particular

phase relationship between buoyancy force and motion the phase propagation velocity is

opposite that of Rossby waves, at least for Prandtl numbers of the order I/3 and

larger (Busse and Cuong, 1976).

4 CONVECTION IN ROTATING SPHERES AND SPHERICAL SHELLS

The problem of convection in a self-gravitating rotating fluid sphere has been

traditionally considered for the case of homogeneous internal heating. Both gravity

vector and temperature gradient vary linearly with distance r from the center in this

case. Roberts (1968) gave a detailed mathematical analysis of the problem. The physically

realized mode was determined by Busse (1970b).

An approximate solution of the problem can be obtained without any numerical

analysis by applying the concept of convection in a rotatin~ annulus, as shown in

Figure 2(a). Because of the coherence in the z-direction enforced by rotation and the

small length scale of the convection columns in the perpendicular direction,

convection in any cylindrical section of the sphere behaves as in the corresponding

163

Figure 5: Laboratory simulation of convection in a rapidly rotating sphere. The motions are made visible by small flaky particles which align with the shear.

164

annulus problem. In Figure 3 the stabilizing effect C of the Coriolis force owing

to the inclination of the boundary has been plotted as a function of the distance

s from the axis together with the buoyancy force A~ which is given by the product of the s-

components of gravity and temperature gradient, since the z-component of the buoyancy

force has little effect on the convection motion. The minimum of C/A at a distance s

of about half the radius indicates the cylindrical surface where the onset of

convection will occur as the critical value of the temperature gradient is reached.

Figure 4 gives a qualitative sketch of the solution of the problem.

The fact that only the component of gravity perpendicular to the axis of

rotation enters the dynamics in a first approximation is the basis for the laboratory

simulation of the convection process (Busse and Carrigan, 1976). By using centrifugal

force in place Of gravity and by cooling the sphere from the inside and heating it

from the outside the convection flow described above can be realized in a laboratory

experiment. The onset of convection occurs in the form of regularly spaced columns,

as shown in Figure 4. When the buoyancy force increases beyond the critical value,

the region of convection is extended until the entire sphere is filled by convection

columns. While amplitude fluctuations and the difference in the speed of propagation

cause deviations from the regular picture at low amplitudes the perfect alignment of

the columns persists, as shown in Figure 5.

The analysis of the spherical case applies directly to the equatorial region of

spherical shells outside the cylindrical surface touching the inner boundary at the

equator. In all cases the Rayleigh number for the onset of convection is lower in that

region than in the other parts of the fluid shell. Inside the cylindrical surface the

onset of convection can be described approximately by applying locally the theory of an

inclined convection layer if the effects discussed in connection with Figure32(b) and

2(c) are taken into account. Of particular interest is the prograde propagation of

convection modes everywhere except at the poles. An asymptotic analysis for different

radius ratios and for varying Prandtl number P is given by Busse and Cuong (]976).

Figure 6 shows the local Rayleigh number for onset of convection as a function of the

distance from the axis in a typical case. The corresponding wave number and frequency

of convection are also shown. The asymptotic results agree reasonably well with the

earlier numerical results obtained by Gilman (1975) at finite values of E in the case

P= I and for a radius ratio ~/~ = 0.8, which is appropriate for the solar convection

zone.

Figure 7 illustrates the most important feature of convection in a rapidly

rotating spherical shell: The change in the character of convection across the

cylindrical interface s = r i. While the vorticity of the motions is nearly z- independent

for s > r i the z- component of vorticity changes sign between lower and upper parts

of the convection cell for s < r.. This change in the symmetry of convection has z

important effects on the nature of the differential rotation generated by convection

and on the heat transport.

165

t00

RE-Z/3

50

Figure 6:

R a

\

\ ,

/

a i # / ,'~

,, . . - ; : q ~t ~

1 ; J

/

" ' ' " I I , i t I

.5 S

R

1 I I I

2

S ,,,"?'./wE-

Rayleigh number R for the onset of convection in a spherical shell with radius ratio r4/r~ = 0.6 as a function of distance S from t~e ~xis. Wave number a and frequency e of convection columns are shown by dashed lines.

I66

j/

I

/

Figure 7: Sketch of convection modes in a spherical shell.

iO s

RO

TATI

ON

AL

REG

IMES

D

UE

TO

CO

NVE

CTI

ON

IN

R

OTA

TIN

G

SP

HE

RIC

AL

SH

ELL

SO

LID

R

OT

AT

ION

R

~

T.8

T z

/3

I r,."

10 4

bJ

El

Z

I (3

Ld

.d

10 3

>

- S

OL

ID

RO

TA

TIO

N

R ~

0

,84

T z

/3

Pra

nd

tl

Num

ber

P=

1 S

tre

ss

fre

e

bo

un

da

rie

s

I0 z I0

z

IO 3

I0 4

I0 s

TA

YL

OR

N

UM

BE

R

T -"

" Figure 8:

Regimes of differential rotation from Gilman

(1976a).

IO s

168

5 NONLINEAR ASPECTS

The phenomenon of solar differential rotation has stimulated much of the

recent effort to understand convection in rotating spherical shells. It was first

shown by Busse (1970a) that convection in a spherical shell can generate a differen-

tial rotation of the same form as that observed on the Sun. While Busse used an

analytical perturbation method in the thin shell limit, Durney (1970) independently

developed a mean field approach for the solution of the problem from which he

obtained--after using the wave propagation property demonstrated by the analytical

theory--essentially the same results. The exciting aspect of the observed solar

phenomenon as well as of the theoretical results is that a prograde differential

rotation occurs at the equator. This eontradicts the earlier notion of angular mixing

by convection which would have led to a deceleration of the equatorial region.

That the hypothesis of angular momentum mixing by convection is incorrect

can easily be demonstrated in the case of convection in a cylindrical annulus

discussed earlier. Since the Coriolis force can be entirely balanced by the pressure

in this case, the influence of rotation disappears from the full nonlinear equation

for two-dimentional convection rolls. Differential rotation cannot be a part of the

solution since the basic equations are identical to those in a nonrotating region in

this case and since a preferred azimuthal direction cannot be distinguished. Generation

of differential rotation obviously depends on secondary features such as the curvature

of the boundaries, and cannot be predicted by simple physical arguments.

How complicated the phenomenon of differential rotation in a convecting

spherical shell can become at higher Rayleigh and Taylor numbers is evident from

the numerical computations of Gilman (]972, 1976a,b). Because both the Reynolds stresses

of the fluctuating convection velocity field and the meridional circulations caused

by the inhomogeneity of convection contribute to the generation of differential rotation,

small changes in the parameters of the problem may change the form of differential

rotation dramatically. Figure 8 from Gilman (1976a) shows how the equatorial maximum of

angular velocity changes into a relative minimum as the Rayleigh number is increased.

The influence of boundary conditions also appears to be important. The almost exclusively

used stress-free boundaries actually represent a singular case in the thin shell limit

(Busse, ]973) since an equilibration between Reynolds stresses and viscous stresses can

take place only in the latitudinal direction.

In order to investigate the generation of differential rotation in a conceptually

simple ease, the problem of convection in a rotating cylindrical annulus has recently

been studied both experimentally and theoretically. Since the measurements are still in

progress we restrict our attention to the qualitative picture, as shown in Figure 9. No

differential rotation is generated in the case of straight top and bottom boundaries

S

/

/

---~--~--Figure 9:

Differential~

rotation generated by convection

~-in

a rotating cylindrical annulus.

170

of the annulus. The experimental observations show an increase of the gradient of

angular momentum for convex boundaries and a decrease for concave boundaries, in

agreement with theoretical predictions.

Meridional circulation and latitudinal variation of the convective heat

transport are other important nonlinear properties of convection in spherical shells.

Both phenomena are closely linked since the variation of the mean temperature caused

by an inhomogeneous heat transport is the most important cause of meridional circulation.

The lack of observational evidence for either phenomenon on the solar surface has been

a source of controversy in the interpretation of theoretical models. We shall return

to this point in the next section.

6 APPLICATIONS TO THE SUN AND JUPITER

It is fortunate for the theory of convection in rotating stars that there

exist two quite different celestial bodies for which detailed surface observations are

available. In the case of the Sun the influence of rotation is relatively small: The

Rossby number is large compared to unity at least for the velocity field in the

upper part of the convection zone. Jupiter represents the opposite case of a rapidly

rotating system characterized by a small Rossby number. Although about half of the

energy emitted from the surface of Jupiter is received from the Sun, the convective

heat transport required for the other half is the dominating source of motions in the

Jovian interior. In this respect Jupiter does indeed represent a low Rossby number

example of a rotating convecting star.

The application of theoretical models which are valid at best for systems of

laboratory scales to systems of stellar dimensions faces obvious difficulties. It is

con~mon practice to take into account the effects of turbulence owing to motions of

smaller scale than those considered in the form of an eddy viscosity ~e which

replaces molecular viscosity in the equations of motion. The main justification for

this procedure is that it appears to work well in many cases.

If ~ is chosen sufficiently large that the Rayleigh number and Taylor number e

4E -2 are not too large the differential rotation observed on the Sun resembles that

predicted by the theoretical models fairly well. There is also evidence for the large-

scale convection cells, often called giant cells, girdling the equator like a cartridge

belt (Howard and Yoshimura, 1976). Figure I0 shows a laboratory simulation. The radius

ratio in the laboratory experiment is closer to unity than in the solar case and the

number of cells is correspondingly larger. Otherwise the cells show a surprising

resemblance to those observed on the Sun by Walter and Gilliam (1976). Because the

latter authors show magnetic regions a direct physical interpretation of the phenomeno-

logical resemblance is difficult, especially since the simultaneous occur~neeOf magne-

tic features which are syrmetric or antisymmetric with respect to the solar equator is

not well understood.

171

Figure 10: Laboratory simulation of convection in a rotating spherical fluid shell with inner radius r i = 4.45 cm and r = 4.77 Cm.

o

172

The measurement of the Coriolis deflection of the horizontal motion in

supergranules by Kubicela (2973) appears to be the only direct determination of the

effect of rotation on solar convection. Kubicela interprets the observed deflection

of the velocity as the Coriolis acceleration multiplied by the lifetime of a super-

granule. Using a lifetime of 20 h he finds reasonable agreement with the measurements.

Since the supergranular velocity field is defined as the mean over a field of highly

fluctuating granular motions, the eddy viscosity concept can be used as an alternative

possibility of interpretation. Using the linear solution for a convection cell in a

rotating layer with stress-free boundaries (Chandrasekhar, 196]) we find the expression

2e tgy

~ 2d2 e

for the angle 7 of deflection, where d is the depth of the supergranular layer.

For simplicity we have assumed that the horizontal wavelength of the cells is large

in comparison with d. Using ~ ]09 cm and ~ = 2.6 " 20 -6 sec -2 we derive from the

observed angle y ~ 20 ° an eddy viscosity of the order 2 • j0]2cm2sec -2 , which is in

reasonable agreement with values derived from other more heuristic considerations.

For the larger scale of giant cells a slightly higher value of ~e appears to be

appropriate yielding an Ekman number of approximately ]0 -2, which is of the same order

as the value used by Gilman (]976b) in his numerical simulation of the solar convec-

tion zone.

It should be mentioned that earlier theories of the solar differential rota-

tion by Kippenhahn (2963) and others used the concept of an anisotropic eddy viscosi-

ty proposed by Biermann (1958). This concept often mimics the anisotroplc dynamical

influence of large-scale eddies. If the deviations from rigid rotation are described

in terms of an anisotropic viscosity it would seem reasonable in view of the more

detailed theory described above to use a latitude-longitude anlsotropy rather than

a horizontal-vertical anisotropy as proposed by Biermann.

Raylelgh numbers for stellar convection zones are based on the superadiaba-

tic part of the temperature gradient, which amounts in general to only a small frac-

tion of the total temperature gradient. A small change in surface temperature causes

a disproportionately large change in the Rayleigh number and an even larger in the

convective heat transport. The convection zone reacts like a highgain amplifier to

any change of the temperature at the surface and it is not surprising that no subcri-

tical large-scale variations of the solar surface temperature are observed. Since the

temperature determines the energy emission, the convective heat flux must adjust itself

to a uniform value. Ingersoll (]976) has emphasized this point in the case of Jupiter,

where the convection heat transport adjusts itself in such a way that large-scale va-

riations of the surface temperature vanish.

For this reason the heat flux variations and associated meridional circulations

N

Ob

serv

ed

C

om

pu

ted

,, -

' ,

~ _

l\

~~

C~

:~

'

~ \ \

\ \ \

~ \\\

~\

\ ~\ \

\ ~ \

\ 1_

1 ~r~

I

1\

\ k i,

\\ \

\\\

\ \~

\\\:

\\\

\\\x

t\\'

\l

\'t\~

l\~lk

"

l \

II

I1

\

..... I,

i ..

..

.

i .

..

..

..

..

..

..

..

Figure ii:

Comparison between theoretical predictions and observations

of bands on Jupiter (from BUSSe, 1976).

174

of low Rayleigh number models do not have much meaning for high Rayleigh number

stellar convection zones. Even in laboratory experiments it is apparent that the

inhibiting influence of rotation on the convective heat transport reverses itself

with increasing Rayleigh number. Rossby's (1969) measurements even show a slight

increase in heat transport owing to rotation at high Rayleigh numbers. The

generation of differential rotation, on the other hand, depends on the alignment

effect rather than the inhibition effect of rotation. It seems intuitively reasonable

that the former effect, which does not have direct energetic consequences, persists

at high Rayleigh numbers, while the latter effect is diminished by nonlinear processes.

Because of its low Rossby number, convection in the planet Jupiter may be

more accessible than solar convection to theoretical analysis. A simple model has

recently been proposed (Busse, |976). It is generally believed that a transition

from molecular to metallic hydrogen occurs at a radius of about 5/7 of Jupiter's

outer radius and that the interface inhibits penetration by convection. Accordingly

we are faced with the problem of convection in a rotating shell as sketched in Figure 7,

which was actually drawn to apply to Jupiter. The fact that a relatively sharp transition

from the low latitude band structure to the polar region of random eddy motion is

observed on Jupiter at about 45 ° latitude appears to be the strongest argument for a

dynamical influence of rotation along the lines outlined in this paper. To obtain a

more detailed comparison as shown by Figure 11 the concept of an eddy viscosity must

be invoked again. The value of ~ required for a fivefold layer of convection columns e

is in good agreement, however, with the eddy viscosity deduced from convection models

for the heat transport. More elaborate models are clearly possible and Jupiter may

well become the testing ground for future theories of convection in rotating stars.

175

REFERENCES

BIERMANN, L. 1958 IAU Syrup. N ° 6, 248

BUSSE, F.H. 1970a Astrophys. J. 159, 629-639

BUSSE, F.R. 1970b J. Fluid Mech. 444, 442-460

BUSSE, F.H. ]973 Astron Astrophys. 27, 27-37

BUSSE, F.H. 1976 Icarus, in press

BUSSE,F.H. and CARRIGAN, C.R. 1974 J. Fluid Mech. 62, 579-592

BUSSE F.H. and CARRIGAN, C.R. 2976 Science 192, 81-83

BUSSE, F.H. and CUONG, P.G. 1976 Geephys. Fluid Dy., in press

CHANDRASEKHAR, S. 1961 Hydrodynamic and Hydromagnetic Stability Oxford Clarendon Press

DURNEY, B.R. ]970 Astrophys.J. 26]_, ]II5-]127

GILMAN, P.A. ]972 Solar Phys. 27, 3-26

GILMAN, P.A. 1975 J. Atmos. Sci. 32, 1332-2352

GILMAN, P.A. 1976a Proc. IAU Symp. 71, in press

GILMAN, P.A. 1976b J. Fluid Mech., submitted

HOWARD, R. and YOSHIMURA, H. 1976 Prec. IAU Syrup. N ° 71, in press

INGERSOLL, A.O. 1976 Icarus, in press

KIPPENHAHN, R. 2963 Astrophys.J. 137, 664

KUBICELA, A. ]973 P roe. 1st European Astr~ Mt~. Solar Activity and Related Interpla-

netary an d Terrestrial Phenomena, J. Xanthakis, ed. Springer

ROBERTS, P.B. |968 Ph~l. Trans. Roy. Soc0 London A 263, 93-I]7

ROSSBY, H.T. ]969 J. Fluid Mech. 36~ 309-335

WALTER, W.T. and GILLIAM, L.B. 1976 Solar Phys., in press

MAGNETIC FIELDS AND CONVECTION

N. O. Weiss



SUMMARY

In a highly conducting plasma convection is hindered by the imposition of a

magnetic field. Convection may set in as direct or overstable modes and behaviour

near the onset of instability depends on the ratio of the magnetic to the thermal

diffusivity. Vigorous convection produces local flux concentrations with magnetic

fields that may be much greater than the equipartition value. The interaction

between magnetic fields and convection can be observed in detail on the sun and is

essential to any stellar dynamo.

I. INTRODU6TION

Magnetic fields - whether primeval or maintained by dynamo action - are

ubiquitous. Any rotating, convecting star seems able to generate a magnetic field,

though the interaction between convection, rotation and magnetic fields bristles with

problems for the theorist. We can usefully distinguish between the problem of

maintaining large scale fields by dynamo action, which will be discussed by Dr

Childress, and that of the interaction between small scale convection and an imposed

magnetic field. I shall assume that any convective timescale is short compared with

the lifetime of large scale magnetic fields and I shall not concern myself with their

origin.

The scale of ordinary laboratory experiments is too small for them to model

hydromagnetic behaviour in astrophysical plasmas. However, the sun provides a

marvellous laboratory where such phenomena can be observed. Sunspots are dark

because normal convection is suppressed by the strong magnetic fields; on a smaller

scale, it is now possible to resolve features a few hundred kilometres across and

to follow the interaction between weak fields and granular convection.

This increase in resolution has revealed more magnetic structures and strongsr

magnetic fields than had been expected.

The theoretical description of a convecting system is particularly rich when

stabilizing and destabilizing effects compete in it (Spiegel 1972). Dr Huppert has

reviewed thermohaline convection; the nonlinear Lorentz force makes magnetic con-

vection yet more complicated. I shall first summarize the results of linear theory

and then discuss various nonlinear problems: is motion steady or oscillatory? are

there subcritical instabilities? how is energy transport affected by the field?

what limits flux concentration between convection cells and how strong are the fields

177

produced? Not all these questions are yet answered but nonlinear magnetic con-

vection is gradually being understood. Finally, I shall try to relate this theory

tosolar magnetic fields and to some aspects of the dynamo problem.

2. LINEAR THEORY

In the absence of a magnetic field a stratified gas is stable to adiabatic

perturbations if Schwarzschild's criterion is locally satisfied. The imposition

of a unlform magnetic field inhibits the onset of oDnvection: a plane, perfectly

conducting layer is eonvectively stable if

(Gough and Tayler 1966), where B O is the vertical component of the magnetic field,

T is the temperature, p the pressure, ~ the ratio of specific heats, /x the

permeability and the adiabatic gradient (dlnT/dlnP)ad = (~-i)/~ for a perfect gas.

Strong magnetic fields can therefore hinder the onset of convection in a star,

though the difference between the adiabatic and the radiative gradient is usually

large enough for instability to occur before the latter is attained (Moss and

Tayler 1969, 1970; Tayler 1971).

When the conductivity ~ is finite, plasma can move across the lines of force

and the stabilizing effect of the magnetic field is relaxed. What happens depends

on the relative values of the magne~c diffusivlty ? = ~#~)" and of the thermal and viscous diffusivities ~ and ~ . In typical stellar conditions, ~ q ~ ~.

The onset of instability in a Boussinesq fluid has been studied in detail (Thompson

1951; Chandrasekhar 1952, 1961; Danielson 1961; Weiss 1964a; Gibson 1966). For a

plane layer of depth d the stabilizing effect of a uniform magnetic field is

measured by the dimensionless Chandrasekhar number

_ ~ a ~

which is the square of a Hartmann number and can be regarded as a "magnetic Rayleigh

number" (Spiegel 1972). A configuration is defined by Q, by the RayleiKh number

R = g~d4/~9 , where ~ is the coefficient of thermal expansion and ~ the super-

adiabatic temperature gradient, and by the Prandtl number ~= ~/~ and the

magnetic Schmidt (or Frandtl) number

=

If, for simplicity, we adopt "free" boundary conditions (Chandrasekhar 1961,

Gibson 1966) then the linear modes have the form

, , , , = Wc ) e ,

where W(z) = WosinrCz (0 4 z ~ d) and

178

with W and a constant, referred to cartesian co-ordinates with the z-axis vertical. o

If ~ q (~)T) linear instability sets in as in ordinary Rayleigh-B6nard

convection. The growth rate s is real and instability sets in as a direct mode~

corresponding to an exchange of stabilities, when R = R Ce) . (Semantics are

succinctly sunnnarized by Spiegel, 1972.) For large Q, R (e) is a minimum when the

dimensionless horizontal wavenumber

so convection first appears in vertically elongated cells at R = R (e~ ~ ~2Q. c

Standing hydromagnetic waves in an unstratified fluid produce oscillations

which are damped by ohmic and viscous dissipation. When ~ >~ these oscillations

may be destabilized by the thermal stratification (Cowling 1976a), so that con-

vection sets in as overstable oscillations when R = R t~ For sufficiently large

Q, overstability first occurs in elongated cells, when

When ~< ~ , therefore, R (°) ~ R (~ and instability first appears as overstable e c

oscillations. At R = R (~ there are two complex conjugate growth rates but as the c

Rayleigh number is raised IIm (~[ decreases until for some R = R ~O the growth

rates are purely real. Thus convective instability sets in with direct modes at

R=RC° AsQ+ ~ , for ~<<~<' (~?,>~) ~ ~CZ~Q, =(~)~ and the minimum value of R ~+) is ~++) ~ (~/=) =~ ~ (Danielson 1961; Weiss

1974a); thus R +~ <~ RIO << R+~ For R +O ~ R < R +~ there are two distinct c c c

positive real growth rates. One of these changes sign when R = R(e) but this

exchange of stabilities has no physical significance.

So far we have considered only free boundary conditions. Analogous results

hold also for other boundary conditions (Chandrasekhar 1961, Gibson 1966). In

particular, the effect of superposing a stable layer on top of the unstable region

has been investigated by Musman (1967) and Savage (1969). The treatment has also

been extended to include some effects of compressibility (Kato 1966; Syrovatsky and

Zhugzhda 1967, 1968; Saito and Kato 1968). If the Alfv6n speed is small compared

with the sound speed, slow magnetosonic oscillations become overstable; if the

Alfv~n speed is large, the fast magnetosonic mode can be destabilized (Cowling

1976b).

3. NONLINEAR cONVECTION

In a Boussinesq fluid the magnetic field satisfies the induction equation

~B

~t

179

while the vorticity _&) and the temperature T are governed by the equations

~ . g - - - - ~ - - - -

"~T

and

~.~ ~ 0 , V_B = O.

Here ~ = ~-J ~zA~ - is the electric current, ~ is the velocity and ~ the density.

For two-dimensional convection, with ~ and B confined to the xz-plane and independent

of y, B can be described by a flux function (the y-component of the vector

potential) A such that

and

while the vorticity equation reduces to

The most convenient boundary condition on the field is obtained by setting B x = 0

at z = O, d. This is somewhat artificial but corresponds to the free boundary

conditions adopted for linear theory.

Near the exchange of stabilities (R = R (e)) finite amplitude solutions can be

constructed using modified perturbation theory. Veronis (1959) observed in a

footnote that subcritical instabilities were possible when ~ . Busse (1975) has

considered a two-dimensional model in which the magnetic field affects the

amplitude, but not the form, of the motion. For R near Rc, the critical Rayleigh

number in the absence of a magnetic field, he combined a perturbation expansion for

the velocity with a computed solution for the distorted magnetic field. He showed

that when ~ >> ~ stationary convection is possible with R c ~ R ~ R :~ In

these solutions the magnetic Reynolds number

U~

(where Q is a typical velocity) is large and the magnetic field is confined to

narrow regions so that its overall stabilizing effect is correspondingly reduced; an

analogous argument applies to the thermohaline problem that Dr Huppert has described.

Since subcritical convection appears when ~4 T and R~)~< R ~) this technique

cannot rigorously establish whether steady finite amplitude solutions are possible

before the onset of overstability. However, Busse's results do suggest that such

subcritical instabilities may occur and Proctor (private communication) has developed

a simplified model of magnetic convection which shows steady motion when R is close

t o R . e

180

Perturbation methods are reliable only while the P~clet number (U d/w) is low

and the convective energy transport relatively small. The efficiency of convection

is measured by the Nusselt number N = (F -~ad)/~, where F is the total thermo-

metric flux and ~ad the adiabatic temperature gradient. The effect of a magnetic

field on the Nusselt number was investigated by van der Borght, Murphy and Spiegel

(1972)~ using the mean field approximation (Spiegel 1971). They considered only

steady convection which, in this approximation, is independent of both ~ and ~ .

For a fixed Rayleigh number, N decreased monotonically and smoothly with increasing

Q until the exchange of stabilities was reached~ thereafter convection was completely

suppressed. Van der Borght (1974) has also attempted to describe the time dependent

problem.

Fully nonlinear two-dimensional computations show a different range of

5ehaviour (Weiss 1975). For free boundary conditions with ~= I, ~ ~ i there is

a general tendency to generate nonlinear oscillations. When ~ = 5, convection first

appears as overstable oscillations in accordance with linear theory. If Q is then

decreased, while R is kept fixed, these oscillations are stabilized at some small

but finite amplitude and convection remains comparatively inefficient. As Q is

further reduced~ R ~) becomes less than R and linear theory predicts direct,

exponentially growing modes. These appear in the numerical solutions but eventually

develop into periodic nonlinear oscillations. The Lorentz force is quadratic in

and linear theory underestimates the restoring force. Hence nonlinear magnetic

convection differs from thermohaline convection, where the stabilizing force remains

linear. If Q is decreased yet further the oscillations develop into irregular

aperiodic motion and, eventually~ into steady convection. The tlme-averaged Nusselt

number rises monotonically as Q decreases but there are no noticeable discontinuities,

nor could any hysteresis be detected,

Dr. Galloway will describe his numerical study of axisymmetrlc convection in a

magnetic field. The results are qualitatively similar, though he found some

hysteresis~ indicating different solutions when Q was decreased from the critical

value and subsequently increased. So far, however, numerical experiments have

provided no evidence of any Jump in the Nusselt number or of any metastable conducting

state associated with suberitlcal convection. Computations on thermohaline convection

(~uppert and Moore 1977) demonstrate that such phenomena can occur. Busse's finite

amplitude results and Proctor's simple model both suggest that, with suitably

chosen parameters~ metastable magnetic configurations should exist. Further com-

putations, with a wider range of diffusivities, are needed to establish w~ether

subcritical convection can be found. It is obviously important to determine what

parameter ranges allow metastable states and whether linear stability theory has any

relevance to convection in a strong magnetic field in a star.

4. FLUX CONCENTRATION

In the limit when Q is su~dclently small the magnetic field is weak and the

Lorentz force has no dynamical effect. The velocity R can then, in principle,

181

be derived from some theory of ordinary Rayleigh-B~nard convection. If ~ is then

fed into the induction equation the kinematically distorted magnetic field can be

calculated. This has been done for various plausible velocity fields (Parker

1963; Clark 1965, 1966; Weiss 1966; Clark and Johnson 1967; Busse 1975). When the

magnetic Reynolds number is high, magnetic flux is rapidly swept to the edges of

convection cells to form ropes. Within a ceil, the field is wound up until the

lines of force eventually reconnect and magnetic flux is expelled. If R >> i the m

pattern of motion must persist for many turnover times before the expulsion process

is completed. However, ropes are formed between the cells by the time they have

turned over once. Within these ropes the field strength has an approximately

Gaussian profile.and the peak field

B* ~ R ½ B in two dimensions mo

R B in three dimensions~ mo

where B is the average initial field (or the field in the absence of convection). o

Astrophysical length scales are large and ~l is big enough for enormous

magnetic fields to be produced locally if concentrations were purely kinematic.

Eventually, the J^B force in the flux rope must become powerful enough to halt the

concentration: amplification of the field is then dynamically limited. But it is

not immediately obvious what limiting field strength can be produced. Partly on

dimensional grounds, it has been popularly supposed that the local field strength

cannot exceed the equipartition field Be, where

~ : , U ~ = ~

The principal argument for this limit depends on considering pressure fluctuations

associated with convection but in a Boussinesq fluid the pressure can be eliminated

from the equations and the equipartition limit should therefore be irrelevant.

Busse (1975) showed that for small amplitude two-dimensional convection

B*~ (~)i/4. Hence B*/B e could be made arbitrarily large by a suitable choice of

. The full two-dlmensional problem has been investigated numerically for con-

vection driven by imposed horizontal temperature gradients (Peckover and Weiss

1977) and by heating from below (Weiss 1975), and Dr, Galloway has computed solu-

tions for axisymmetric convection. The maximum value of the peak field B* can be

estimated by a simple argument. The two-dlmensional results show that kinematic

amplification is halted when ohmic dissipation in the flux rope becomes comparable

with viscous dissipation throughout the convection cell. It follows that the

maximum field B* ~.~/4,-- a result confirmed by the computatio~ In three dim- max

ensions a similar argument yields a maximum field B*max~'~'(galloway etal. 1977).

By choosing ~" sufficiently large, B* can be made much greater than B and solutions e

have been obtained with B*/Bem 5 (though the particular value has no significance).

Once the magnetic field becomes dynamically significant, vorticity is

generated in the flux ropes, where there is a local balance between the magnetic

and viscous terms in the vorticlty equation. The buoyancy force generates vorticity

182

with one sense, while the Lorentz force generates vorticity with the opposite sense

and viscosity maintains a balance. The resultant vorticlty distribution corresponds

to a velocity field with a reduction in the transverse flow that concentrates the

flux. A simple physical description confirms that two-dimensional amplification

is halted as ohmic dissipation reduces the overall flow. In three dimensions motion

can be excluded from the flux rope slightly earlier (Galloway, private communication).

As B is further increased, the flux ropes grow broader and develop a different o

structure. The field within a rope is more nearly uniform, dropping abruptly near

its boundary. The ensuing current sheath produces a Lorentz force which prevents

the motion from entering the flux tube. Ultimately the layer separates into con-

veering cells, from which magnetic flux has been expelled, and stagnant flux ropes

in the interstices between them.

5. SOLAR MAGNETIC FIELDS

Cowling (1953, 1976a), Sweet (1971) and Mullan (1974b) have reviewed magnetic

fields in the sun. The discussion of flux concentration relates most directly to

intense, small scale magnetic fields in the photosphere (Weiss 1977). Over the

last eight years ground-based observers have succeeded in resolving magnetic

structures with a scale smaller than that of the granulation and fields of up to

about 1500 G (Schr~ter 1971; Harvey 1971, 1977; Dunn and Zirker 1973; Mehltretter

1974; Stenflo 1976). These features are formed between granules and have lifetin~s

similar to those of individual granules. The fields are much larger than the local

equipartition field (Be~5OO G) and the magnetic pressure alone is almost sufficient

to balance the external gas pressure. Such high fields can only be contained by that

gas pressure (Parker 1976a). A full theory of convective transport in strong

magnetic fields is needed to explain the formation of these flux ropes but a crude

extrapolation from ~he Boussinesq results indicat~that the field can be amplified

to reach the strengths observed (Galloway et al. 1977).

On a larger scale, magnetic flux is swept aside by supergranules and concen-

trated at their boundaries to form a network in which most of the small scale

features are located. Irregular small scale fields have recently been detected

within the network (Harvey 1977) but the flux involved is relatively slight. As

more flux is brought together the magnetic field inter~res with convection so that

the gas is cooled. Dark pores or sunspots then appear between the supergranules.

The magnetic flux that emerges through a sunspot is presumably assembled into a

rope deep in the convective zone, though supergranules certainly play a part in the

formation of a spot. Conversely, though small flux ropes can be shifted to fit the

pattern of supergranular convection, large sunspots are anchored deeper down and

long-lived, stable convection cells may form around or near them (Harvey and Harvey

1973; Livingston and Orrall 1974; Meyer et al. 1974).

In the umbra of a sunspot the magnetic field is nearly vertical and strong

enough to suppress convective instability, Following a suggestion by Biermann

183

(1941), Cc~ling (1953, 1976 a,b) argued that motion across the magnetic field is

inhlbitedj and that convection is limited to predominantly vertical, oscillatory

motion, in slender elongated cells. Theoretical models of sunspots (Chitre 1963;

D~inzer 1965; Chltre and Shaviv 1967; Yun 1970) show that radiation alone cannot

supply the energy emitted from the umbra and microturbulsnt velocities (Backers

1976) provide some observational evidence for convection. (Umbral dots are too

sporadic to be an essential feature of the transport process.) Various attempts

have been made both to relate linear stability theory to umbral and penumbral

structure in sunspots (Danielson 1961, 1965, 1966; Weiss 1964b, 1969; Musman 1967;

Saito and Kato 1968; Danielson and Savage 1968; Savage 1969; Mullah and Yun 1973;

Moore 1973) and also to study overstability in an isolated magnetic flux tube

(Parker 1974b, B. Roberts 1976, Defouw 1977).

Parker (1974a,h, 1975a, 1976a,b) has recently emphasized the importance of

mechanical energy transported by transverse hydromagnetic waves, which may escape

either upwards or downwards from the umbra. He suggests that thermal energy which

would otherwise have reached the photosphere is carried away by these waves, which

are so efficiently coupled to sub photospherlc convection (cf. Mullah 1974a) that

they refrigerate the sunspot, cooling by Alfv~n waves requires extreme efficiency

and this mechanism has been criticized by Cowling (1976b). Moreover, the corona

absorbs only a comparatively small amount of energy and excess X-ray emission is

associated with active regions, not specifically with sunspots. So magnetic

inhibition of convection still provides the most obvious explanation for the cooling

of pores and spots.

Unlike the umbra, the penumbra of a sunspot is essentially inhomogeneous, and

the radial filaments are correlated with convective motion (Backers and Schr~ter

1969, Schr~ter 1971). According to linear theory convection in rolls lying in the

plane of B is affected only by the vertical component of the field. The inclination

of the magnetic field increases across the penumbra until it becomes almost hori-

zontal at the edge of the spot. Danielson (1961) and Pikel'ner (1961) therefore

suggested that the filamentary structure is caused by convection in horizontal rolls

and this axplanation is qualitatively convincing. Linear theory indicates that the

penumbra may be convectively unstable, to direct rather than to overstable modes

(Danielson 1961; Musman 1967; Saito and Kato 1968; Savage 1969). Nonlinear results

imply that convective transport would then be significant, though motion might

still be periodic (Weiss 1975). These theoretical models are obviously oversimplified.

In particular, the boundary conditions are too stringent: one might, for instance,

expect that vigorously convecting plasma from below the penumbra would be able to

penetrate through the shallow magnetically dominated region (Meyer etal. 1977). A

more complete theory should also explain the Evershed outflow as a consequence of

convection (Galloway 1975).

184

6. CONVECTION AND DYNAMOTHEORY

There is some observational evidence that other stars with outer convection

zones have magnetic cycles llke the sun (Wilson 1976) and flux concentration is

inevitable in any stellar dynamo that is driven by convection, Turbulent motion

tends to remove magnetic fields from a convective zone: flux tubes may emerge from

the surface and be carried off by a stellar wind or they may be expelled downwards

into the radiative zone. Systematic differences in the velocity~ caused for example

by the radial density gradient, may pump flux preferentially in one direction (Moore

and Proctor 1977). A more important topological effect was pointed out by

Drobyshevsky and Yuferev (1974). In three dimensional convection with upward motion

at cell centres, the sinking fluid forms a continuous network, while regions ef

rising fluid are separated from each other. Since a flux tube can wind continuously

through downward moving gas there is a tendency to pump flux downwards and to concen-

trate the field at the base of a convecting layer. Topological pumping competes with

magnetic buoyancy (Parker 1955j 1975b; Gilman 1970; Unno and Rihes 1976). If the

field in the flux rope has the equipartitien value then the rope floats upward

relative to the ambient gas at about the Alfv~n speed, which is equal to the downward

convective velocity. Se the net motion of the flux tube cannot readily be estimated,

though it seems unlikely that flux can remain within the star unless it is surrounded

by sinking gas. A proper description of the inhomogeneous magnetic field must he included in any

realistic dynamo model. The theory of turbulent dynamomhas often been reviewed (eg.

Parker 1970; P. H. Roberts 1971; Vainshtein and Zel'dovich 1972; Gubbins 1974; Mestel

and Weiss 1974; Moffatt 1977). Without systematic helicity, homogeneous turbulence

is unlikely to maintain a field (Moffatt 1977), and helicity is caused by rotation.

The Coriolis force, like the Lorentz force, tends to inhibit convection but these

costraints may be relaxed if both are simultaneously ~resent (Malkus 1959;

Ohandrasekhar 1961; Eltayeb and Roberts 1970; Eltayeb 1972~ 1975; van der Borght and

Murphy 1973; Roberts and Stewartson 1974, 1975). Attempts to solve the full hydro-

magnetic dynamo problem will be discussed by Dr. Childress.

REFERENCES

Backers, J. M., 1976. Astrophys. J. 203, 739.

Backers, J. M. and Schr~ter, E. H., 1969. Solar Phys. i_O0, 384.

Biermann, L., 1941. yiertel~ahrscPr. Astr. Ges. 76, 194.

van der Borght, R., 1974. Men. Not. R. Astr. Soc. 166, 191.

van der Borght, R., and Murphy, J. 0., 1973. Austr. J. Phys. 2~6, 617.

van der Borght, R., Murphy, J. O. and Spiegel, E. A., 1972. Austr. J. Phys. 2~5, 703.

Busse~ F. H., 1975. J. Fluid Mech. 7~i, 193.

Chandrasekhar, S., 1952. Phil. Mag. (7) 4~3, 501.

185

Chandrasekhar, S., 1961. Hydrodynamic and hydromagnetic stability. Oxford.

Chitre, S. M., 1963. Mon. Not. R. Astr. Soc. 126, 431.

Chitre, S. M. and Shavlv, G., 1967. Solar Phys. ~, 150.

Clark, A., 1965. Phys. FIo ~, 644.

Clark, A., 1966. Phys. FI. ~, 485.

Clark, A. and Johnson, A. C.+ 1967. Solar Phys. ~, 433.

Cowling, T. G., 1953. The sun, ed. G. R. Kulper, p. 532. University of Chicago Press.

Cowling, T. G., 1976a. Magnetohydrodynamics. Hilger, Bristol.

Cowling, T. G., 1976b. Mon, Not. R. Astr. Soc. 177, 409.

Danielson, R. E., 1961. Astrophys. J. 134, 289.

Danielson, R. E., 1965. Solar and stellgr magnetic fields (IAU Symp. No. 22), ed. R. LUst, p. 314. North-Holland, Amsterdam.

Danielson, R. E., 1966. Atti de l c0+nve~no sulle macchie solari, ed. G. Righini, p. 120, Barbara, Florence.

Danielson, R. E. and Savage, B. D., 1968. Structure and development of solar active re~ions (IAU Symp. No. 35), ed. K. 0. Kiepenheuer, p. 112. Reidel, Dordrecht.

Defouw, R. J., 1977. Astrophys. J. (in press).

Deinzer, W., 1965 Astrophys. J. 141, 5~8,

Drobyshevsky, E. M. and Yuferev, V. S., 1974. J. Fluid Mech. 65, 33.

Dunn, R. B. and Zirker, J. B., 1973. Solar Phys. 33, 281.

Eltayeb, I. A., 1972. Proe. Roy. Soe. A 32__~6, 229.

Eltayeb, I. A.~ 1975. J. Fluid Mech. 71, 161.

Eltayeb, I. A. and Roberts, P. H., 1970. Astrophys. J. 162, 699.

Galloway, D. J., 1975. Solar Phys. 44, 409.

Galloway, D. J., Proctor, M. R. E. and Weiss, N. 0., 1977. Submitted to Nature.

Gibson, R. D., 1966. Prec. Camb. Phil. Soe. 6_22, 287.

Gilman, P. A., 1970. Astrophys. J. !62, 1019.

Gough, D. O. and Tayler, R. J., 1966. Mon. Not. R. Astr. Soc. 133, 85.

Gubbins, D., 1974. Key. Geophys. Space Phys. 12, 137.

Harvey, J., 1971. Publ. Astr. Soc. Pacific 83, 539.

Harvey, J., 1977. Highllghts 0f astronomy, ed. E. MUller, Reidel, Dsrdrecht.

Harvey, K. and Harvey, J°, 1973. So!at Phys. 28, 61.

Clarendon Press,

186

Huppert, H. and Moore, D. R., 1977. J. Fluid Mech. 78, 821.

Kato, S., 1966. P ubl. As tr. Soc. Japan, 18, 201.

Livingston, W. and Orrall, F. Q., 1974. Solar Phys. 39, 301.

Malkus, W. V. R., 1959. ~ . 130, 259.

Mehltretler, J. P°, 1974. Solar Phys. 38, 43.

Mestel~ L. and Weiss, N. 0., 1974. Masnetohydrodynamics. Swiss Soe. Astr. Astrophys., Geneva.

Meyer, F., Schmidt, H. U. and Weiss, N. 0., 1977. Mon. Not.R. Astr. Soc. (in press).

Meyer, F., Schmidt, H. U., Weiss, N. O. and Wilson, P. R., 1974. Mon. Not. R. Astr. Soc. 169, 35.

Moffatt, Ho K., 1977. Masnetic field seneration in electrically, conduetin$ fluids. Cambridge University Press.

Moore~ D. R. and Proctor, M. R. E., 1977. In preparation.

Moore, R. L., 1973. Solar Phy s . 30, 403.

Moss, D. L. and Tayler, R. J., 1969. Mon. Not. R. Astr. Soc. 145, 217.

Moss, D. L. and Tayler, R. J., 1970. Mon. Not. R. Astr. Soc. 147, 133.

Mullah, D. J., 1974a. Astrophys. J. 187, 621.

Mullan, D. J., 1974b. J~ Franklin Inst. 298, 341.

Mullah, D. J. and Yun, H. S., 1973. Solar Phys. 30, 83.

Musman, So, 1967. Astro~hys. J. 149, 201.

Parker, E. N., 1955. Astrop y .h s J. --121, 491.

Parker, E. N., 1963. Astrophys. J. 138, 552.

Parker, E. N., 1970. Ann. Rev. Astr. Astro~h~s. 8, i.

Parker, E. N., 1974a. Solar PhTs. 36, 249.

Parker, E. N., 1974b. Solar Phys. 37, 127.

Parker, E. N., 1975a. Solar Phys. 40, 275.

Parker, Eo N., 1975b. Astrophys. J. 198, 205,

Parker, E. N., 1976a. ~ . 204, 259.

Basic mechanisms of solar activity (IAU Symp. No. 71), ed. Parker, E. N., 1976b. V. Bumba and J. Kleczek, Reidel, Dordrecht.

Peckover, R. S. and Weiss, N. 0., 1977. In preparation.

Pikel'ner, S. B., 1961. Osnovy kosmieheskoy elektrodinamiki.

Roberts, Bo, 1976. ~ . 204, 268.

Nauka, Moscow.

Roberts, P. H., 1971. Mathematical prob!ems in the geophysical sciences, ed. W, H. Reid, p. 129, Am. Math. Soc., Providence.

187

Roberts, P. H. and Stewartson, K., 1974. Phil. Trans. A 277, 287.

Roberts, P. H. and Stewartson, Ko, 1975. J. Fluid Mech. 68, 447.

Saito, M. and Kato, S., 1968. Solar Phys. ~, 531.

Savage, B. D., 1969. Astroph~s. J. 156, 707.

Schr~ter, E. H., 1971. Solar magnetic fields (IAU Symp. No. 43) ed. R. Howard, p. 167, Reidel, Dordrecht.

Spiegel, E. A., 1971. Ann. Rev. Astr. Astrophys. 9, 323.

Spiegel, E. A., 1972. Ann. Rev. A str. Astroph~s. I__OO, 261.

Stenflo, J. O., 1976. Basic mechanisms of solar activity (IAU Symp No. 71), ed. V. Bumba and J. Kleczek, p. 69. Reidel, Dordrecht.

Sweet, P. A., 1971. Solar magnetic fields (IAU Symp. No. 43), ed. R. Howard, p. 457, Reidel, Dordrecht.

Syrovatsky, S. I. and Zhugzhda, Y. D., 1967. Astr. Zh. 44, 1180 (Soy. Astr. II, 945, 1968).

Syrovatsky, S. I0 and Zhugzhda, Y. D., 1968. Structure and development of solar active re$ions (IAU Symp. No. 35), ed. K. 0. Kiepenheuer, p. 127. Reidel, Dordrecht.

Tayler, R. J., 1971. 9" J" R. Astr. Soc. 12, 352.

Thompson, W. B., 1951. Phil. Ma~. (7) 4-2, 1417.

Unno, W. and Ribes, E., 1976. Astrophys. J. 208, 222.

Vainshtein, S. I. and Zel'dovich, Y. B., 1972. Usp. Fiz. Nauk 106, 431 (Sov. Phys. Usp. 15, 159).

Veronis, G., 1959. J. Fluid Mech. 5, 401.

Weiss, N. 0., 1964a. Phil. Trans. A 256, 99.

Weiss, N. 0., 1964b. Mono Not. R. Astr. Soc. 128, 225.

Weiss, N. 0o, 1966. Proc. Roy. Soc. A 293, 310.

Weiss, N. O., 1969. Plasma instabilities in astrophysics, ed. D. G. D. E° Tidman, p. 153. Gordon and Breach, New York.

Wentzel and

Weiss, N. O., 1975. Adv. Chem. Phys. 32, IO1.

Weiss, N. O., 1977. Highlights of astr0nom[, ed. E. MUller, Reidel, Dordrecht.

Wilson, O. C., 1976. Basic mechanisms of solar activity (IAU Symp. No. 71), ed. V. Bumba and Jo Kleczek?Reidel, Dordrecht.

Yun, H. S., 1970. Astrophys. J. 162, 975.

AXISYMMETRIC CONVECTION WITH A MA~ETIC FIELD

D. J. Galloway Astronomy Centre

University of Sussex Brighton, BNI 9QH

England

Summar~

The n o n - l i n e a r B o u s s i n e s q e q u a t i o n s d e s c r i b i n g a x t s y m m e t r i c c o n v e c t i o n i n a c y l i n d e r

w i t h an i n i t i a l l y u n i f o r m m a g n e t i c f i e l d have been i n t e g r a t e d f o r w a r d i n t i m e

n u m e r i c a l l y . When t h e f i e l d i s weak a s t r o n g c e n t r a l f l u x r o p e i s formed a t t h e

axis. In this case the maximum field strength can be limited either klnematlcally

or by dynamical effects, and the equipartitlon prediction B 2 ~ 4~pu ~ is eg~ily max

exceeded. If the field is strong oscillations can occur and hysteresis is posslble

as the field is increased and decreased.

i. Introduction

The interaction between convection and a magnetic, fleld determines many features

observed in the solar photosphere. Sunspots and smaller scale magnetic fleld

elements are symptoms of the ability of convection to concentrate a weak average

field into strong fluxropes. Oscillatory phenomena such as running penumbral waves

can occur in the presence of a strong field. To study such effects it is necessary

to solve non-linear problems, and clearly to do So in three dimensions if at all

possible. The recent work of Jones, Moore and Weiss (1976) on axlsymmetric con-

vection is easily extended to include the presence of a ma~etlc field with average

strength B . This problem is geometrically three-dlmenslonal but depends mathe- o

matlcally on only two variables, thereby rendering Itself tractable to nnmsrlcal

computation. The normal equations of Bousslnssq convection are modified to include

the effect of the Lorentz force in the vortlclty equation, and in addition the

electromagnetlc induction equation is solved to update the magnetic fleld as the

system is integrated forward in time.

S. The Problem

To solve the equations it is convenient to set up stream functions for the

velocity and magnetic fields thus:-

u= Va(o, , o) = (- r ~z o, r ar ,

B= V^(O, ~r' O) = (- r ~z O, r az "

We use cylindrical polar coordinates (r, 8, z). The geometry of the problem is

shown in figure 1.

189

Bo- In~t~a ll~.,, =,q~co,',~.

F=O

_

ff"e:rn p. T : T,,+AT

Temp.T--To

ease r'= a.

Fig. 1

Geometry for the axisymmetrie problem, showing basic cylinder on left and

axis-edge cross-section on right.

The equations V.~ = 0 and V.~ = 0 are automatically satisfied by Zhe above

and B flelds. Those remaining can be non-dimenslonalized and put in the following

form:

~-~ = -- p3 R -- - r - ~ r + ( V . ( r V ( r 2 ~ ) ) , ( 2 . 1 )

~_TT = -V. (Tu) + 1 V. (VT) Bt -- (pR)½

( 2 . 2 7

~t p3(pR)~ *

(2.3)

w h e r e ~ = (VA--u)o = - r ~ r2 - r ~ r ~z 2

J = (V^S--)e = - - r ~r ~z

There are five dimensionless parameters specifying each solution to the problem;

these are

2 2 __ 3 B d

R = g~wdz o K K~) ' Q = 4W~O~ ' p = ~ ' P 3 = ~ , a n d

a , t h e r a t i o o f c e l l w i d t h tO c e l l h e i g h t . The n o n - d i m e n s i o n a l i z a t i o n h a s b e e n t

c o n d u c t e d w i t h l e n g t h s c a l e d ( l a y e r d e p t h ) , t i m e s c a l e ( d / g o , AT) ~, f i e l d s t r e n g t h

scale B and temperature scale AT. The density and coefficient of volume expansion o

of the fluid are Q and ~ ;~ ,<, and i'] are its viscous, thermal and magnetic

190

diffusivities respectively, and g is the acceleration due to gravity.

The following boundary conditions are used:

T = T O + AT, B r = O, @ = O, ~= 0 (z = O)

T = T , B = 0, @ = 0, ~= 0 (z = 1) o r

~T - - = = ~r O, X const,, ~ = O, ~ = 0 (r = a)

~T ~--~= O, × = O, ~ = O, ~ = 0 ( r = O)

The conditions on the fluid are those commonly known as stress-free. The con-

straint X = const, at r = a fixes the total flux in tho cylinder and circumvents

Cowllng's theorem, so that steady solutions are possible.

The above equations and boundary conditions have been solved by finite-difference

methods similar to those described in Moore~ Peckover and Weiss (1973). The

equations were integrated forward in time until the solutions converged to a steady

state or a repeating oscillation. In many cases one solution was started from

another, and in this way the effects of continuously varying one parameter could be

investigated.

3. Discussion of Results

To correspond most closely with highly conducting astrophysical plasmas the

program WaS run with values of </n ranging from 10 to 50. For fixed and moderately

non-linear values of the Rayleigh number the followlng types of solution are found

as Q is increased.

i) For very weak fields the convection is unaffected and concentrates all the flux

kinematically into a central rope. The structure of this rope is fixed by the

balance between diffusion and ndvection in the induction equation. The maxi-

mum field strength B m is higher than the input field B ° by a factor of the order

of the magnetic Reynolds number, and the profile of the rope is Gaussian. Such

solutions have been described by Weiss (1966) and Clark and Johnson (1967).

ii) As Q is increased a regime ensues where the imposed field remains compressed in

a fluxrope hut can exert a dynamical influence on the convective flow. Within

the rope motion is minimal: at its edge, typically a few mesh points from the

axis, there is a shear layer and the velocity reaches a value comparable with

that in the absence of the field. The dynamics are dominated by a balance

between the total thermal and magnetic torques; dissipation can be ohmic or

viscous, and the maximum field can be Successfully predicted by s power law of

the form

191

Bo ~ The numerical experiments yield average ~ = ~= 0.63 and ~ = 0.77; the power-law

behaviour extends over typically two orders of magnitude. The greater ability of a

three-dimensional geometry to concentrate flux means it is far easier to chart this

regime in the axlsymmetric case than for two-dimenslonal rolls. It is also possible

to advance physical arguments based on a boundary-layer structure (Galloway, 1976),

and predict a law similar to (3.1). The exact values of ~,8 and 7depend on whether

viscous or ohmic dissipation is dominant, and the formulae involve weakly varying

logarithms, but agreement with the numerical experiments is generally very good.

An example of one of these dynamically limited solutions is shown in figure 3,

which is the case Q = i00, R = 20Rc, p = 1, P3 = I0, and a = 4/3. (Here R c =

(27/4)~4). The fluxrope is almost stagnant; st its edge there is a current

sheet which generates a large localized amount of negative vortlcity. Within the

fluxrope horizontal temperature gradients cause a very weak countercell to develop;

this has an advective effect on the field and causes the fluxrope to develop a

maximum some distance away from the axis. The run of the fluxrope profile as Q

increases is shown in figure 2.

Q = 1 q = 5 Q = 20 Q = I00 Q = 1000

Fig. 2. Fluxrope profiles for R = 20R , p = I, </n = I0, a = 4/3 e

i i i ) The c e n t r a l f l u x r o p e b r o a d e n s a s Q i s f u r t h e r i n c r e a s e d a n d e v e n t u a l l y i t

o c c u p i e s a b o u t a h a l f o f t h e r a d i u s o f t h e c e l l . At t h i s s t a g e t h e r o p e b e g i n s t o

o s c i l l a t e w h i l s t t h e o u t s i d e c i r c u l a t i o n r e m a i n s s t e a d y , a n d t h e r e i s a c o r r e s -

p o n d i n g v a r i a t i o n i n t h e h e a t t r a n s p o r t , T h i s f o r m o f s o l u t i o n h a s a n a t u r a l

e x p l a n a t i o n . T h e i n i t i a l l y i m p o s e d f i e l d B ° i s w e a k e n o u g h t o a l l o w s t e a d y c o n -

v e c t i o n ~ w h i c h c o n c e n t r a t e s t h e f l u x i n t o a r o p e o f s t r e n g t h B a n d r a d i u s ~ a . F l u x m

conservation suggests Bm % 4]30, and this means that, considered in isolation, the

central rope is overstable to linear theory. The frequency of the computed solution

agrees moderately well with such a linear prediction. When the senses of the two

192

a) magnetic lines of force

f

b) streamlines

c) isotherms

I///~ "///.////~////~/ d) vortlcity ~ (axis on left)

193

circulations are opposite, upward moving plumes are adjacent and the heat transport

is a maximum, When the senses are the same, the cold downdraught of the countereell

is next to the hot updraught of the main flow and lateral diffusion reduces the heat

transport to a minimum.

iv) Finally Q is so strong that only finite-amplltude oscillations are posslble.

These are confined mainly to the outer half of the radius, so that the solutions

are quite different to the elgenfunctions of linear theory. Periods are typically

10% - 20% faster than the linear values, presumably because the essentially quad-

ratic Lorentz t e rm in (R.l) is badly underestimated in the linear approximation.

The nature of the solutions depends on the five dimensionless parameters defined

earlier. However there are also occasians when the system adopts a configuration

dependent on the initial conditions, so that hysteresis occurs. This effect is

encountered when the field is fairly strong. A solution with given (Q,R,p,~/~,a)

can then be steady if it is part of a branch with Q inereasin~and oscillatory if

part of a branch with Q decreasing - the system remembers what it was doing for

earlier values of Q. This effect can be quantified by using the Nusselt number N,

averaged in time if necessary, as a measure of the amplitude. A graph showing the

variation of N with Q as the latter is increased and decreased is shown in figure

4. For this example, Q = 10,300 marks the onset of overstability and Q = 2535 the

transition from steady to oscillatory modes according to linear theory. The slight

increase in N as Q increases in the lower branch at Q = 4000 appears real.

5

3

!

N

"\

i 0 | 0 0 i O 0 0 LO)O00

Fig. 4. Variation of N against log Q for R = 20Rc~p = I, </q = i0, a = 4/3.

...... oscillatory solution -.-.-.- mixed steady-oscillatory solution --steady solution.

194

It is interesting to compare these hysteresis effects with the results of

Huppert (1978) on double-diffusive convection, also described elsewhere in these

proceedings, Broadly slmilar results are obtained but the magnetic results are

more regular and do not show the sudden Jumps in N found in the salt case. Further-

more no subcritieal instabilities have yet been found in the present study.

Applications of this work to the production of intense solar magnetic fields

are discussed in Galloway, Proctor and Weiss (1976). The fluxrope solutions give

fields limited either by the magnetic Reynolds number or by formula (3.1); in a

Boussinesq fluid the equipartition argument B 2 % 4~Wpu 2 is quite irrelevant since max

the pressure can adopt arbitrarily high values. The numerical results give fields

up to six times greater than this prediction. We conclude that in the solar photo-

sphere the maximum field strength is limited by the gas pressure.

Acknowledgements

I am very grateful to Dr D. R. Moore, who wrote the program on which this work

was based, and to Dr N. O. Weiss for many helpful discussions. I thank the Science

Research Council and Trinity College Cambridge for financial assistance.

References

Clark, A. and Johnson, A.C., 1967. Sol.Phys.2, 433

Galloway, D.J., Proctor, M.R.E., and Weiss, N.O. 1976. Nature, to be published

Galloway, D.J., 1976. Ph.D. Thesis, University of Cambridge

Huppert, H.E., 1976. Nature, 263, 20

Jones, C.A., Moore, D.R. , and Welss, N.O., 1976. J.Fluid Mech. 73, S53

Moore, D.R., Peckover, R.S., and Weiss, N.O., 1973. Computer Phys,Comm. 6, 198

Weiss, N.O., 1966. P roc0. Roy. Soc. A , 293, 310

CONVECTIVE DYNAMOS

S t e p h e n C h i l d r e s s Courant Institute of Mathematical Sciences

New York University New York, N. Y. 10012

i. INTRODUCTION

Convective dynamo theory can be regarded as combining two kinds of

physical problems, each involving an electrlcally conducting fluid

medium, but differing in the role of the magnetic field and in the

physical processes described. On the one hand, if the fluid is taken

to be permeated by a prescribed magnetic field B, under suitable

conditions, involving a sufficiently strong flux of heat for examplej

convective motion of the fluid will ensue. On the other hand, kine-

mGtic dynamo theory insures that a sufficiently compllcated fluid

motion u can sustain or excite a magnetic field. In a convective

dynamo the origin of the magnetic field is internal and we must regard

the applied and excited fields as one and the same (Figure i). In the

present paper we shall outline some of the current work on such sys-

tems. The research has been motivated primarily by the search for

tractable models of planetary and solar magnetism, and the focus in

this paper will be on models of the geodynamo. For simplicity we

restrict attention to Boussinesq fluids and emphasize asymptotic solv-

able problems rather than a realistic description of the Earth's core.

We shall, however, require that the dynamo be essentially convective,

in that no auxiliary driving forces are needed. (The convective

process could of course involve any advected, diffusing substance

which changes the weight of a fluid element.)

u

Figure i.

~ rotatlon?

kinematic induction

magnetic convection

heat~flux

J

The Convective D~namo Cycle.

The simplest physical system admitting a convective dynamo cycle

is not obvious (to this author), although in the case of the geodynamo

it would appear that large-scale rotation of the fluid is sufficient

196

if not essential. In the models discussed below it is precisely the

combination of bouyancy and Corlolls forces which create the neces-

sary flow structure, so in this respect at least they may be relevant

to the processes at work in the Earth's core.

2. BOUNDS AND ESTIMATES

One way to define dynamo action is simply to require that the mean

magnetic field of the system, obtained by appropriate integrals over

space and time, b e positive. With this definition we can, in a sense,

"prove" dynamo action by showing that the convective system with B - 0

is unstable to magnetic fields; that isj small "seed" fields are

always ampllfled. This property can be tested rather easily since the

two parts of the system depicted in Figure 1 decouple when the magne-

tic field is weak. Now it is an essential feature of our problem that

the mean magnetic energy of the ultlmate state(s) of the dynamo is

an internal property of the system~ although we may assume that

such a mean energy may be defined and that it will depend upon the

various parameters~ the geometry, etc. This being the case, it is of

interest to determine, without studying the evolution of the system in

detail, an u pr~oPi upper bound on mean magnetic energy.

This intriguing question was apparently first studied only recent-

ly by Kennett (1974) in the case of B~nard convection between free,

perfectly conducting plane isothermal boundaries rotating about a

vertical axis (cf. Section 5 below). We will use the following nota-

tion: ~ - magnetic permeabillty~ p = density, ~ = kinematic viscos-

ity, K - thermal dlffuslvity, n = magnetic dlffusivity, a - coeffi-

cient of thermal expansion, all of the above being taken to be

constant~ P = ~/~ = Prandtl number, P = ~/K, R = Rayleigh number 4

~ased on a temperature gradient 7) = uyEL /K~ , M = Hartmann number

- BL/(~p~) 1/2, Ta = Taylor number = 4~2L4/~2~ where G ~s the

angular speed of the system. If E B denotes the time and volume mean

of B 2 Kennettls result may be written i

_3/2.2,^~2p2 E B ~ 4~ .01~ ~, B 0 - (~pV)I/2/L . (i)

This estimate is obtained by equating the mean dissipation to the

mean work done by the gravitational forces, and involves extensions of

the familiar power integrals of the B~nard problem.

Although as a general rule analysis of this kind rather severely

overestimates energies, (i) is interesting as an indication of the

influence of the various material properties. The parameter

197

P is evidently significant in determining the magnetic energy

realized by convection at a given Raylelgh number, Incidentally, P

has a value of about 106 in the earth's core, but may be as small as

10 -5 in stars because of radiative cooling, so that ideally we would

like a dynamo model to retain Pn as an arbitrary parameter. The fact

that the bound (i) diverges as n ÷ 0 probably ref12cts the infinite

amplification that can be achieved in a perfect conductor by the

twisting and stretching of field lines. But note that the bound also

diverges llke V -I/2 in the limit of small viscosity. While such a

divergence might be expected for convection between isothermal bound-

aries in the limit of zero Prandtl number, it seems unlikely for

Systems driven by a fixed rate of heating; in this case E B should be

bounded independently of the viscosity.

Such a result is in fact implied by the interesting thermodynamic

arguments of Malkus (1973), and Hew~tt, McKenzle, and Weiss (1975).

Following these authors we consider a spherlcal region of (current)

conducting fluid surrounded by a rigid non-conductor. Let the bound-

ary r = L be held at a fixed temperature T O and the interior be

heated uniformly at the rate qo" We seek a bound for the magnetic

energy in terms of the material constants, L, and q0" Let . E B , qj ,

qv • and W now denote the time averages of global B 2, Joule dlsslpa-

tlon, viscous dissipation, and bouyancy work, respectively, all

normalized by the volume V of the sphere. Assuming internal energy

is bounded in time the first law requires

w = qj + qv " qj Z o, qv S 0 . (2)

On the other hand, a well-known property of currents in a homogeneous

spherical conductor is (see e.g. Backus 1958)

qj ~ n~2EB/~L2 (3)

Then f r o m (2) and (3)

E B ~ ~L2Wln w2 (4)

To estimate W we use the temperature equation in the Bousslnesq limit,

8T 8-q.+ ~-vr - K72~ - qo/Oep . (5)

Le t , wi th ~-~ = u z r g o / L ,

198

L

W = (4~spg0/LV) I r3 w( r ) dr (6)

0

so that w is the time average of the spherical mean of u T. If 8 r

denotes t he same operation on T, (5) yields upon integration the flux

balance

3 3 dO 4q0 /30c p r w - Kr ~ = r (7)

o

We may measure 8 in K. Integrating (7) from r = 0 to r - L and

using 8 ~ 0 we have

L

r3w dr ~ Lbq0/15pe p + ~L3T0 (8)

0

Combining (4), (6) and (8) there results

nEB/~L2q 0 ~ (8/5~2)(I + 15K0CpT0/L2q 0) (9)

Here 8 ffi ~g0L/Cp is the ratio of L to the temperature scale height

and is necessarily a small number in the Bousslnesq approximation.

(The inclusion of the dissipation terms on the right of (5) changes

(9) by terms O(8~.) But the left-hand side of (9) should be independent

of the origin of the temperature scale and, indeed, it can be shown

that for uniform heat addition (9) holds with T o = O. Qe therefore have

nEB/~L2q0 ~ 8/5~ 2 (i0)

This provides us with a useful (and small) measure of the efficiency

of a convective dynamo. Other estimates of this kind are contained in

the references cited above.

If we introduce the Raylelgh number

Rq = ~g0qoLb/ocp~2~ ,

then (I0) may be rewritten

E B < R B~/5~2p 2 (ll) -- q o

and thus has the form of (I) reduced by a factor 9/(20 RI/2), the

bound now being independent of the viscosity.

In the above we have dropped viscous dissipation ~ecause of the

199

inequality (3), but if this term is now retained one obtains

L4qv/P n 2 + ! q/Sp 411.)

in place of (ii). If, relative to a rotating frame, the no-sllp

condition is satisfied on the boundary, qv can be bounded from

by a multiple of p~L-2Eu , in which case the terms on below the left

of the inequality would be comparable provided the ratio of kinetic

to magnetic energy is roughly P /P. It is known (Childress 1969a)

that L2E / 2 must exceed a fixed positive bound for a dynamo effect u

to be possible in a given domain of fluid. Consequently in the

El/2 E I/2 plane a convective dynamo must lie within the first B - u

quadrant of an ellipse, and to the right of a vertical line determined

by the dynamo condition. In reality, of course, the radius of the

ellipse should be altered to express the existence of a critlcal

Rayleigh number, and it would be of interest to extend (Ii*) to

account for this shift, perhaps by applying the method used by

Kennett (1974).

Since the above arguments completely ignore the dynamical process

by which the dynamo effect is realized, 411) tells us little about the

behavior of any given system. Suppose, however, that some dynamics

allows the bound (11) to be obtained, and take 1 ~ P~ > 1. (Through-

out we use the symbol ~ as follows: a ~ b if a = O(b) and b = O(a).)

Then 411) implies M 2 ~ R , which is reminiscent of the relation q

R e ~ M 2 (M >> i) obtained for the or~ti~al Rayleigh number for con-

vection between isothermal planes dominated by the magnetic field

(Chandrasekhar 1961). That is, the "optimal state" of the convective

dynamo is close to marginal in the context of linear stability theory.

If the process by which the optimal state were reached involved rapid

rotation (Ta >> i), the analogous stability results of Eltayeb and

Roberts 41970) and Eltayeb (1972) show that if P > .67659, then

Rc(M,Ta) is minimized when M 2 ~ Ta I/2, R c ~ Ta I/2, which is again

compatible with (II) when P > 1. (And note that M4/Ta is also

independent of viscosity.) The Eltayeh-Roberts ordering may be loose-

ly interpreted to imply that magnetic energy with Hartmann number T I/2 a

would be acquired by a rapidly rotating body once the Raylelgh number

was raised to a value ~ T 1/2 But again the argument assumes the a necessary dynamo action by the convection. In particular t~ere is no

implied critlcal rotation rate.

If both fluld inertia and viscous stresses can be neglected (as

seems to be the case in the Earth's core outside Ekman layers~ cf.

Roberts and Soward 1972), the dimensionless parameters of the heated

200

convective system may be reduced to a "Raylelgh number"

2 1/2 = Rq/P Ta (12)

together with P~, by the choice of ~/L, (2~pn) I/2, and q0L2/Pep~ as

units of speed, magnetic field strength, and temperature respectively.

The dimensionless equations are then

÷ ÷ _~g;l Vp + ~ - 1 ~ + Bx(V×B) = ~z, V-~ = Or (13)

, ,

B_~T + ~-VT - p-Iv2T = i . (15)

For given P,

ible with boundary conditions, and from these determine the one

with maximum E B (now dimensionless). By (11) this value cannot

exceed R/5~ 2. Such an operating state, where the mean magnetic energy

is as large as possible for a given heating rate, may be taken as

"optimal", since it is presumably stable locally and can only lead

to smaller energy under a finite perturbation. Of course it is not

clear that the system admits uny nontrlvial solutions (5 # 0).

Note that if such a solution existed for some R, and if it were

known that rotation was essential for a dynamo effect, then it would

be necessary that solutions terminate for sufficiently small Gnd

sufflciently large values of R.

The existing theory of convective dynamos has concentrated on

cases whlchare probably far from optimal in the above sense. Viscous

effects are freely admitted, parameters and geometry chosen to a11ow

the convective modes to be determined by considerations at marginal

stability, and various devices are used to simplify the analysis of

electromagnetic induction. In the following sections we study various

aspects of these highly idealized models, but return to some of the

questions raised above at the end of the paper.

we can seek solutions of (13)-(15) which are compat-

201

3. KINEMATIC INDUCTION

This aspect of the problem has a large recent literature (see e.g.

the reviews of Roberts 1971, Weiss 1971, and Gubbins 1974). One of

the more direct evaluations of the regenerative effect is possible if

the fields are taken to be periodic in space and time. (The spatially-

periodic case was treated by Childress 1967, 1969b, 19701 the theory

in its most general setting was developed by G. O. Roberts 1969, 1970,

1972.) This situation arises naturally in planar or almost-planar

models involving simple boundary conditions. The analysls is facili-

tated (and can be made explicit) if the two dimensionless numbers

r = ~ /tlk~ r k = U/~k, (16)

where U, k , and ~ are the speed, wavenumber, and frequency charac-

teristic of the velocity field, satisfy

r k = o(1), r - o(1). (17)

That is, the magnetic Reynolds number of a fluid eddy must be small,

and the time scale of the motion should be of the order of the decay

time of a magnetic field structure of the same size. With (17) it

becomes rather easy to demonstrate self-excltatlon of a magnetic field

which is slowly-varying relative to the scales k, ~. (I¢ is unlikely

that the first of (17) is satisfied in the Earth's core, but the basic

inductive mechanism, which goes back to the pioneering paper of

Parker (1955), can in fact he deduced without such a restriction

(G. O. Roberts 1970).)

We return to the dimensional induction equations, which are

V-~= 0

(is)

(19)

Consider the solenoidal velocity field

~(a) = U(O, sin o, siu(~+~)), a = kx + ~t, (20)

and suppose t h a t ~ has t he d e c o m p o s i t i o n

= ~ + g, g = o(i), ~ slowly-varylng. (21)

202

Uslng (20) and (21) in (18), (19), one sees that the part ~ will

approximately satisfy

- f-Vu ,

so that fl k d~

g = (nk 2 ~+ ~u) n2k4+~ 2

The slowly-varylng component will then satisfy

-- - = V x ( u x g ) = ~ t

422)

( 2 ~ )

V"~ = 0, (24)

where the overbar denotes the o-average and A is a constant pseudo-

tensor. For (20) the only non-zero component of A is

All = . (qk3U2(sln ¢)/(~2k4+~2)) ~ (25)

Thls additional contribution to mean electromotive force is usually

referred to as the "a-effect". The most general Q-effect, involving

arbitrary symmetric A, can be created by suitably combining linearly

independent modes of the form (20). It is easily seen that, by exam-

ining the case of diagonal A, that (24) can be made to admit exponen-

tlally-growing spatially periodic solutions (and note that (17)

insures that they will be slowly-varying).

Let us look more closely at the underlying inductive mechanism

when ¢ - ~/2. From (22) it is clear that the source of small-scale ÷

magnetic structure is proportional to the x-derivative of u, l.e. to

the sheur of the flow. Now trigonometric spatial modes of the diffus-

ion equation decay without change of shape, but there is a phase shift

between the solution and moving sources. Combining this shift with

that introduced by differentiation, we see that ~ is proportional to

~(O + $)where ~k2/~ - tan ~. As O varies, ~ and ~ rotate in the yz

plane while maintaining thls phase difference, so the induced current,

obtained as a cross product, is independent of o and proportional to

sln ~.

For a given mode 420) the corresponding entry in A is maximized

when ¢ = ~ TM W/2 s l.e. the motion is both quasl-steady and Beltraml

(vortlcity and velocity everywhere parallel). Thls maximizes

the mean heZ~u~ty (Moffatt 1968), defined as the volume average of

u. Vxu, for a given mean kinetic energy. Note that the mean hellclty

is opposite in slgn to ~ for these elementary Beltrami modes.

To summarizer time-lndependent velocity modes having the property

203

that the velocity is orthogonal to the wavenumber vector and the

two orthogonal components are 90 ° out of phase, provides a basic

element of a particularly efficient kinematic dynamo process, charac-

terized by a constant mean heliclty. A variety of other, less effici-

ent dynamo mechanisms (involving an A which either vanishes or has

rank 1~ see e.g. case IV in G. O. Roberts 1972) can he studied by a

refinement of these procedures~ but in the present context it is

rather a slightly different point of view which is needed, since the

relevant convective modes cannot be regarded as exclusively small-

scale. We accordingly consider next the dynamo mechanisms which are

compatible with the dynamics of convection.

4. DYNAMICS

The efficient kinematic dynamos considered above are very special

in that the hellclty of the flow may be averaged over space and time

to obtain a non-zero pseudo-scalar H. It appears to be difficult,

however, to find physical systems which will exhibit this property.

For example, owing to dissipative processes we expect a rotating

sphere of heated fluid to settle down so that H can be defined Indepen-

dently of the initial conditions. Now restart the system but with

initial conditions T(-~,0), -~(-~,0). If the magnetic field is zero

the Bousslnesq equations are invarlant u~der this reflection (recall

g - g0r), so the system will evolve toward a mean hellclty -H = H,

implying H = 0. To argue this in a different way, the mean state of

the system should depend only upon the mean heating rate and various

material parameters (all scalars) and the pseudo-vector ~, from which

it is impossible to construct a pseudo-scalar. If the system is +

endowed with a magnetic field having mean dipole moment m (a vector) ~

H could be expressed as an odd function of m.~, but the record of

magnetic reversals suggests that for the Earth there is no preferred

polarity and therefore m = O!

Ifj nevertheless, rotation is to be regarded as essential to the

convective dynamo, its action must be not to create mean hellclty,

But rather to "polarize" heliclty in space (or time) in such a way

that the resulting pattern of induced currents can be self-exclted.

This self-excltatlon is difficult to visualize and compute when the

length and time scales are unique (as in the system (13)-(15)) since

the induced currents resulting from the polarlzatlon are bound up

closely with the "eddy" currents which dissipate the field. For the

purpose of analysis of the effect it is therefore fortunate that

sufficiently rapid rotation of the fluid introduces two spatial scales

204

into the marginal stability problem, at least for sufficiently weak

magnetic fields (Chandrasekhar 1961), so that large-scale induction

and small-scale dissipation can be clearly distinguished.

To take a concrete example of a zero mean hellcity dynamo, con-

sider the solenoidal fleld

u = (- sin kx cos az, sin kx cos az, cos kx sin az). (26)

If k >> a the induction problem can be solved approximately as in

Section 3, with similar results except that now there is an additional 1 factor ~ sin 2z in A. But note that (26) can also be written

g . sln(k +az) ÷ sln(k -a )+ , ( 2 7 )

t h a t i s , a s a sum o f two O ( l ) r a p i d l y - v a r y i n g f i e l d s , e a c h h a v i n g c o m -

p o n e n t s in p~ase, so that each fails as an "efficient" dynamo of the

kind considered above. An analogous calculation can be carried out

for the standing wave

= (0, sin kx cos at, cos kx sin at), a << nk 2 (28)

to obtain helicity varying as sin 2at, and (28) can be expressed as

a sum of two progressive waves, moving in opposite directions with

phase speed a/k.

More generally, let

= ~ s i n ~ + ~ ' s i n ~ ' ~ - + ~ t , ( 2 9 )

and aseume Lk - k'I << k, I= - ='l <<~k 2" One them finds, using the

n o t a t i o n of Section 3,

A - ~k2 ,In~-a') (~,x~)o~ (30) n2k4 + 2

The operation k + k' can be thought of as a reflectlon across a

plane normal to ~ - ~', and if = = =' the dispersion relation for

the modes must be invarlant under this reflection; in addition, from

(30) it is clear that the two corresponding amplitudes must not be

parallel. The special case ~ = ~', ~' = -~, which could arise in

a conservative system, gives the tlme-periodlc induction.

These properties pertain to infinite fields having the proper

structure. In oan#u~ned rotating fluids hellclty can also be polarl-

205

ized as a result of "Ekman pumping" into a quasi-geostrophic flow. If

the latter has wavenumber k, the secondary flow set up by the Ekman

layer is of magnitude ~ Ta-i/dkL, L being the length scale for the

container, and the resulting helicity may or may not he comparable to

that introduced by other processes, depending upon the magnitudes of

k and Ta. Under certain conditions it can be demonstrated that the

Ekman layers are essential to a convective dynamo effect (see Roberts

and Soward 1972), but in certain idealized models (Section 5) they

definitely are not.

The polarity of the resulting hellcity is fixed by the direction

of rotation and is easily c~puted. At a point on the boundary with

outer normal ~, the nearby heliclty has the sign of -~-~. For the

rapidly rotating Benard layer (case I below), the polarity is the same

as that introduced globally by the convection mode for free boundar-

ies, but the distribution is different and (as Just noted) it is

smaller, by a factor Ta -I/12.

We turn now to the situation in rotating convection. We have in

mind, of course, convection in a heated sphere or spherical annulus,

but to get a qualitative picture it is helpful to consider several

planar "approximations" to parts of a spherical annulus, which we

indicate in Figure 2.

Figure 2. Planar "approximations". The heavy lines indicate isothermal boundaries to be represented by tangent planes. In IV the included angle is made small. The polarization of helicity is indicated for rotation in the direction shown at the top. In a homogeneous sphere~region ~ can be regarded as extending from top to bottom.

206

We consider these regions in turn. Letting g and ~ be parallel and

oonstant~ and choosing units of length, time, and speed to be L, L2/~

and K/L, we find the dimensionless linearized equations to be

8~ + Vp + Ta I/2 ÷ ÷ ffi T' ~--[ i~u - v2~ - R l g , v.~ = o (31)

!

@T' - V2T = 0 (32) ~ T i - ÷ ~ - I g

where ~g and ~ are unlt vectors, T' is the temperature perturba-

tion, and we take Ta >> i.

Case I. This classical problem is treated by Chandrasekhar

(1961). In the limit Ta ÷ ~, provided P exceeds about 0.68, convec-

tion ensues as a small-scale pattern. The critical wavenumber vectors

(L is now the layer thickness) are almost perpendicular to ~G, the

motion is quasl-geostrophic, and under reflection across the plane of

the layer the vertical velocity component changes sign. There is

accordingly polarization of the helicity along the axis of rotation,

in a manner similar to that obtained for the motion (26). The criti-

cal parameters are

R = 3(~2/2)2/3Ta 2/3, k ~ (2Ta/2)l / 6, (33) c e

Using the term "roll" to denote the convection field corresponding to

a given wavenumber vector ~ in the plane of the layer, and setting

~G = (0,0,1), a single roll has, in the case of free boundaries, the

form

~ sin ~z cos • ~3 + (~2Tal/2/k4) ÷ ÷ cos

E a c h s u c h r o l l c o n t r i b u t e s a n e n t r y i n t h e u p p e r l e f t 2 x 2 s u b m a t r i x

o f A, w h i c h i s a n e g a t i v e m u l t i p l e o f s i n 2 ~ z . The c o r r e s p o n d i n g

h e l i c i t y h a s t h e p o l a r i t y s h o w n i n F i g u r e 2 . F o r r i g i d b o u n d a r i e s t h e

results are similar over the interior, the secondary Ekman flow being

smaller by a factor k Ta -1/4 ~ Ta -1/12 according to (33). c

From the point of view of dynamo action, the essential feature of

this ease is the high degeneracy of the geostrophic flow, allowing

rolls of arbitrary direction. The fact that A can be made to have

rank 2 implies a relatively efficient dynamo process based exclusively

on small scale motions (see Section 5). This realizes in a simple

planar geometry the so-called ,, 2,, kinematic dynamo (Roberts 1971).

On the other hand region I can hardly be regarded as typical of the

sphere as a whole, especially sinoep strictly speaking, the geostro-

207

phic contours have this degeneracy only at the poles!

Case II. Here gravity and i~ = (0,0,1) are orthogonal, and the

choice of geostrophic velocity ~ = Y×$(x~y)~ 3 reduces the problem

to classical B~nard convection without rotation. There is no dynamo

effect from these rolls since particle paths lie in planes. (Indeed

from (18) it follows that B 3 decays, and the resulting two-dimenslon-

al non-dynamo can be regarded as a special instance of Cowling's

theorem, of. Roberts 1967.)

However~ these special solutions completely neglect the presence

of "sidewalls" which might represent the effect of the sloping

spherical boundary. As Busse (1970) has emphasized, the sidewall

constraint drastically upsets the geostrophic balance and the near-

equatorial region is best approached through case IV below.

Case Ill. We let ~g = (0,0,-i), ~ = (sin ~,O,cos ~). For

steady convection the Rayle~gh number is g~ven by

R = [k 6 + Ta(klSln ~ + k3cos ~)2]/-(k 12+. 2K2 )

where ~ is now an arbitrary vector. To make this expression an even

function of k 3 and therefore obtain modes of the kind needed to

satisfy conditions at the plane boundaries, it is seen that k I must

vanish, in which case the problem reduces to case I above but with Ta

replaced by Ta cos2~. The effect of the obliqueness is therefore to

reduce the critical Rayleigh number somewhat, and to restrict the

locally horizontal wavenumber vector to be nezrly perpendicular to

the plane of ~ and ;. Thus the 2 dynamo of Case I is reduced to an

incomplete or near incomplete s-effect, strongly biased toward induc-

ing i 2 current from i 2 fleld. In kinematic dynamo theory this induc-

tion is nevertheless essential to the success of the "~" dynamo

(Roberts 1971)~ as orlglnally envisaged by Parker (1955). In the

a~ mechanism the a-effect is supplemented by large-scale shear~ which

here replenishes ~2 field. Thus case III, which might be taken to the

be typical of a large fraction of spherlcal annulus~ suggests a

natural mechanism for obtaining "one-half" of the dynamo effect from

convection.

Case IV. Here one seeks to represent the effect of the sloping

sidewalls. A class of such models was studied by Busse (1970) and

subsequently used in a convective cycle (Busse 1975p see Section 6

below). The model has the advantage of describing rather closely, in

a simple geometry~ the essential physics of the convective instability

in a rapidly rotating, heated homogeneous sphere (Busse 1970).

The sidewalls upset geostrophy through Ekman pumping as well

208

as by their inclination to the axis of rotation~ but the latter

effect can be made to dominate. In this case Busse's results can be

obtained rather easily by inverting a device of the oceanographer and

replacing a slow decrease of the depth by a slow increase of ~. Let

= (-l,0,0) ~ - (0,0,i) and replace Ta I/2 by Tal/2(l" + Xx) in g , 3

(31). L e t t i n g

ffi V x ~(x,y,t)~ 3 , p - -(I +Xx)Ta I12 ~ + p'

and eliminating pt by cross-dlfferentlation we obtain

2 2 (-~ - V2)V2, = -~y (XTal/2* + RT') •

~T' V~T' ~-~ 2 ~2 ~2 _ = = +

P ~t ~y • V2 ~x 2 ~y2 "

If we look for modes proportional to exp(i~t + lax + iby)

( P i e + c 2 ) ( i ~ c 2 + c 4 + i a A T a 1 / 2 ) = Ra 2 ,

(35)

( 3 6 )

we have

( 3 7 )

where c 2 ffi a 2 + b 2. Thus

ffi - ( a A T a l / 2 ) / ( p + 1 ) c 2 • a2R ffi c 6 + (P/(P+I))2a2ATa/c 2

a n d t h e c o n d i t i o n s f o r m a r g i n a l s t a b i l i t y b e c o m e

= 2 - 1 1 6 ( ~ P l ( t + p ) ) l l 3 T a 116 , b = O, R = 3 ( X P I ( l + p ) ) 4 1 3 ( T a 1 2 ) 2 1 ~ ac c c

(38)

If X is regarded as small• the perturbation is on the strict geostro-

phic equilibrium of case If, but it is a singular perturbation with

strong roll selection. If A is regarded as % i, the ordering (38) is

in accord with that of cases I and IIl~ although the mode in the

present case has some features of a Rossby wave. In Busse's formula-

tion A is a typical sidewall slope and should be taken as positive,

so the phase speed given by (38) is "eastward". (If one regards the

observed "westward drift" of the non-dlpole geomagnetic field as a

phase speed of a flew field associated with the ~-effect, this result

is disappointing. However• as Busse notes the g~oup velocity ~s west-

ward, and this raises the question of whether sufficiently super-

critical convection in this system might not take the form of inter-

mittent wave packets.)

As we have noted the model can be interpreted as an appropriate

local section of a sphere, parametrized by the latitude of the inter-

209

section with the boundary andjif this is done~reasonable agreement is

obtained with the stability analysis of a rapidly rotating heated

sphere (Roberts 1968, Busse 1970). Roberts found that convection

begins on a cylindrical annulus oriented parallel to the axis of rota-

tlon~ with radius about half that of the sphere, and consists of

vertical rolls of dimension % Ta -I/6 along the aximuth, the radial

thickness of ~he convection zone being % Ta -I/9 both relative to the

radius of the sphere. It would appear that in the planar model the

radial structure is lost, or rather reflected only in the vanishing

of b c in (38). Certainly the spherical case should have a local

approximatlon with radial structure, which could be incorporated into

a dynamo model. Soward's remarks at this meeting concerning his

current calculations of the localized D-effect point toward such a

possibility, which would in effect open the way to an asymptotic

analysis of the spherical dynamo.

5. B~NARD-TYPE MODELS

We c o n s i d e r i n t h i s s e c t i o n t w o e x a m p l e s o f a e o n v e c t l v e d y n a m o

cycle based upon a classical B~nard layer. Busse (1973) considers

B~nard convection without rotation. It is assumed that only one set

of rolls is present, so hellclty is created by adding a shear flow

alone the axis of the rolls. Such a flow is somewhat artificial for

a B~nard layer, but it can in principle be driven by a modification

of the mean temperature profile. Moreover, such distortions might

well arise in a different geometry through convective heat transport

obllque to the direction of gravity. To effect a scale separation it

is assumed that the (spatial) mean magnetic field is dominated by a

component orthogonal to the roll axis and slowly-varylng along it.

In this model the mean hellclty vanishes, since the unidirectional

shear flow passes down rolls of alternating sign. The kinematic

dynamo effect is therefore not the first order D-effect of Section 3,

but rather a hlgher-order mechanism involving the spatial derivatives

of the mean magnetic field. (In the terminology of Roberts 1971 the

dynamo is of "B~" type.) Busse uses a numerical method to study the

equilibration of the system and the partitioning of internal energy.

While his approach, being essentially quasi-steady, does not deal with

the dynamics of equilibration, it does reveal an interesting balance,

similar to that suggested by (ii*), which is perhaps typical of near-

critical convective dynamos. Namely, if R - R > 0 is sufficiently c

+ bE B = constant, where a and be are posi- small, one finds that aE u *

tive constants, but that E must exceed a critical value E in order U U

210

to have a dynamo effect. The conclusion is that for E > E* the u u field energy will increase as the kinetic energy falls, while for

E < E the kinetic energy will rise as the field decays. Ultimately u u the magnetic field will be sustained at the maximum energy compatible

with the above dynamic constraint as well as stationary dynamo action.

The second model, put forward by Childress and Soward (1973) and

worked out by Soward (1974) for equilibrium at minimum field energy,

is based upon the rotating B~nard layer considered as case I in Sec-

tion 4. This model utilizes the rotation of the fluld to effect the

scale separation, in a manner which permits the dynamics of equilibra-

tion t o be followed in detail. We take P~ ~ i.

The "weak-field" case studied by Soward (1974) assumes that the

Hartmann number of the induced field is ~ I. This a pr~or~ hypothesis

on field intensity is then Justified by exhibiting consistent, appar-

ently stablejoperating states. The rolls have the form (34) with ~ an

arbitrary vector in the plane of the layer. The length L is here the

thickness of the layer, so that for regeneration of the field we must

have (of. (25) with ~ = 0)

R 2 ~ k ~ Ta I/6 , m c

where Rm = UL/~ is a magnetic Reynolds number based on roll ampli-

tude U. Thus Rm % Ta 1/12 and the small-scale field satisfies

g ~ Ta-I/12f. This ordering generates a series solution in powers

of Ta -1/12 .

The mean field equations are easily obtained for a discrete or

continuous distribution of rolls, and in the former case, with

= [Bl(Z,t),B2(z,t),O), take t h e form

~B i ~2B i + 2~A ~ [sin 2Wz Mij(t)B j] ...... 0 (39)

~z 2

where, in terms of the A in (24),

M = [ -A21A11 -A22 IA12 (40)

Here the unit of time is L2/n. Note that, since we have in mind that

roll structure is to be determined by auxiliary equations of evolution,

the discreteness or continuity of the roll pattern is determined by

the initial conditions. In (39) and (40) the normalization is such

that All + A22 = 2 so that A(t) i s a parameter proportional to

the kinetic energy of the flow.

For the weak-field solutions, it turns out that near marginal

211

instability A is fixed by the quantity (R - Rc)/R c. Thus the dynamics

of the model reduces to the study of how the magnetic field detez-

mines the partitioning of a fixed constant kinetic energy among the

various rolls. Soward was able to show that the roll structure indeed

evolves on the same time scale as the field. If q(t,~) denotes the

kinetic energy in a given r011, where ~ = ~/Ta I/6, the equation for

q takes the form

d ft. 2 ~ Q(K,K ) q ( t , K ) q ( t , K ) d t

1 + - [~(t,~) - ~ ~ ~(t,K') q(t,K')] q(t,K) = 0 , (41)

where Q(~,~') and B(t,K) are given explicitly, the former being skew

in its arguments. The magnetic field is contained in the quantity B,

which takes the form

= f(~) + g(~) . ZjKIKj ~(Bisj7 , (42) i i

~(f) = [ ( W2c°s2Wz - K2eln2~z)f dz i

0

Finally, the matrix A in (40) is obtained from the q's by

(t,~7 (43) Aij = ~ K 2 q K

where we may set K = K . c

The weak-field model is then given by (39)-(437 with R as para-

meter. Soward examinee 2 and 3-roll solutions of this system, as well

as an interesting continuous-roll solution, and finds a tendency for

the kinetic energy to localize itself at any one time in rolls near

a single direction, but the direction itself changes with time. In

fact, as the number of admitted discrete rolls is increased, the solu-

tions tend increasingly to resemble a single rotating roll, a property

then explicitly exhibited in the continuous roll example, where energy

is dispersed about an orientation which rotates with uniform speed.

Physically, the magnetic field, at each instant, favors rolls with a

certain orientation (determined by the term ~ in (41)). From (39) one

sees that the ~-effect then feeds energy into the component of the

field pGral~el to the preferred roll axis. If the field were indepen-

dent of z, the analysis of Eltayeb (19727 would apply and it could be

concluded that rolls with axis o~thogonu~ to the field are most

unstable. Thus both field and roll axis rotate, as the ~-effect keeps

212

up with the destabilizing effect of the field. This argument would

imply that rotation is opposite to the large-scale rotation of the

layer (of. M in (39) when A is diagonal), a property that was always

obtained in Soward's calculations.

The continuous-roll solution consists of at least two branches

when E B is plotted as a function of A, with subcritical bifurcation

occurring from E B = 0 , A = 1.5974, and it seems likely from the 2

and 3-roll calculations that some of these are stable on the time

scale of the model. (Soward establishes dynamic stability on the

relevant short time scale.) On the other hand, it is by no means

obvious that these solutions are stable to finite increases of

initial field energy, and indeed the Eltayab-Roberts ordering men-

tioned in Section 2 would suggest that they are not.

Preliminary attempts by the author to test the stability of the

dynamo by startin E it with magnetlc energy correspondin 8 to a Hartmann

number ~ Ta 1/12 (the Intermediate-field regime) have in fact uncovered

several kinds of instability, and recent unpublished calculatlons of

Yves Fautrell in the strong-field regime Ta I/6 < O(M) < Ta I/4 also

indicate instability under certain conditions. It is not definitely

known at this time whether or not there are regimes other than that of

weak field where local stability is obtained. It is possible that

stability is retained only at the "very strong field" level M ~ Ta I14,"

but there the multiple-scale procedure is ineffective since k c % i.

At the intermediate level One finds, first, that dynamic stability

on the fast time scale, shorter than L2/n by a factor Ta -I/12, is

upset. Examination of some two-roll solutions show that this

Instability represents collapse onto a single roll (without the dis-

persion of energy about a preferred direction which characterized the

weak-field ¢ontlnuous-roll solution), Single-roll solutions are

dynamically stable on the fast time scale at the intermediate level.

Single-roll solutions are found to be unstable, however, on the time

scala L2/~ ! To sea how this happens we write out the equation for q

at the Intermediate-fleld level:

Ta-i/6 d__qq = dt q[Cl + c2e(t) + c3 ~-(~'~)z)] ' (44)

where the cts are positive constants. Here GTa -I/6 is an amplitude of

the perturbation of the mean temperature, satisfylng

d._00 + c4 e = - c5 A (45) dt

where again the c's are positive constants. In particular, at the

213

fntermediate level A must he retained as a function of time even at

fixed Rayleigh number, Let us assume tha~ the system (39) p (44), (45)

is started "most stably" by adjusting the roll angle and 8 to make the

right-hand side of (44) take on its maximum value, and that this value

is zero. Assuming that this configuration (stable on short time scales)

is maintained, we may neglect the left-hand side of (44), solve for Q,

and use this expression in (45) to obtain an expression for A in terms

of B, which expression can then be used in (39) to obtain an equation

for ~.(This procedure also fixes the roll angle as a function of B.)

Numerical studies clearly show that in general this system allows solu-

tions which diverge in a finite time, owing to the quadratic dependence

of A upon B, and quite apart from the dynamo or non-dynamo property of

a single roll. In effect it appears that the field has destabilized

the convection to the extent that divergent behavior of the field is

caused by the rapid increase of the kinetic energy and heat flux, rather

than by dynamo action,

One reason for these difficulties may lie in the degeneracy allowing

multiple-roll solutions. Indeed, if a single roll direction could be

fixed by other considerations (as in Cases III and IV in Section 4),

the component of the field which is amplified by dynamo action is ortho- +

gonal to K, and so does not enter into (44). So long as more than one +

roll direction is permitted, however, the quadratic growth of A with B

would probably have the instability of the rotating roll.

It can also be asked whether the instability is not an indirect result

of the isothermal boundary condition, which more realistically should be

replaced by a condition of constant mean heat ~£ux. Some tentative work

at the intermediate level did not indicate a stabilizing effect, but the

question remains undecided at the higher field levels.

6. THE ANNULUS MODEL

As an outgrowth of his quasi-planar analysis of convection in case IV

above, Busse (|975) has considered a corresponding convective dynamo model.

His approach, as in the stability analysis, is to consider a simultaneous

expansion for large Ta and small %D, where I is a typical boundary slope

and D >> | is the width to height ratio to the annulus. The geometry is

summarized in Figure 3.

214

T = T 2

a ~ 0,112)

I T ~ T I < T 2

1 (DI2,0,0) ',4- g

Figure 3. Busse's annulus model; lengths are in units of the mean height. The almost horizontal boundaries are given by 2z = ~ e x p ( - ~ x ) .

The top boundaries are treated as rigid interfaces between conductor

and non-conductor, and on the vertical ones the temperature perturba-

tion as well as the x-component of velocity vanish.

The latter conditions are particularly important, since they

enforce a fixed roll structure on the convection, effectively remov-

ing the degeneracy of case I. Indeed, from the analysis of Section 4,

we see that a 0 and R e again have the asymptotic expressions (38),

but now b = w/D~ so the most unstable velocity mode has the form c

= ATal/2(sln(Wx/D + w/2) sin acY, a c

(46)

(W/acD) eos(wx/D + ~/2) cos acY, 0).

Of course, once R exceeds R a band of wavenumbers with continuously c

varying b will grow~ but rather than introduce a Fourier transform we

follow Busse and take (46), which can be expressed as a sum of two

rolls with wavenumber vectors a ± T/D, as typical of the convec- c

tire mode.

The hellcity corresponding to (46) vanishes Identlcally, so the

kinematic dynamo action rests on the combined effects of Ekman pumping

and sidewall slope. For suitable parameter values the former effect

can be made to predominate, and an a-effect is achieved, but one which

is strongly biased, the induced current associated with a y-component

of the mean field being (ae/bc)2 H 1/e 2 times that associated with a

215

comparable x-component. The steady-state mean field is thus found to

satisfy an equation obtainable from (39) by replacing sin 2wz by z,

and M by

I °I 0 l/e

where

(47)

m = [(A2a5Ta3/4)/8~ (w~ + a4p2/p23(w/D). c cq

(48)

These expressions can be derived using the familiar results for the

by the Ekman layer, together with (25). Since ~c/a~%p-l- flow induced

we see from (47) and (48) that the quantity

r = p2A2acTa3141D(I+P~) (49)

must exceed a positive number of order unity if we are to have a

dynamo effect.

A second condition is imposed by the two-scale expansion. The

small-scale magnetic field can be estimated from (22) as follows:

The dimensionless velocity amplitude is a A Ta 1/2, and since the c

x-component of the field predominates (see below), the relevant shear

is I/D times this amplitude. Thus

g/f % ac A Tal/2D-I 2 p2 2 4 -i/2 (mc + n /pac) << 1

is a condition on the expansion. Combining (50) with the condition on

r and using (38) we have

Tal/4zD >> 1 , (50)

and this inequality is easily met by the assumed ordering. On the

basis of his solution of the kinematic dynamo problem Busse concludes

t h a t if

acbc/P~Tall4 >> 1 (51)

(the inequality followlng equation (5.3) of Busse 1975), the expan-

sion is consistent. This adds a much stronger condition, which can

only be met, with a c given by (38) and b c ~ l/D, by making P smGlZ.

Since it is important to retain P~ as a large parameter in a geodynamo

model, it will be of interest to know if (51) can be relaxed while

216

maintaining a consistent expansion, or if (50) by itself might be

sufficient.

The equilibration of the system is studied using an equilibrium

calculation as for the non-rotatlng layer model of Section 5, with

similar results: For slightly supercrltlcal convection the magnetic

and kinetic energies are linearly related and the system equilibrates

as the dynamo effect becomes stationary.

The model has a number of advantages over those of B~nard type,

the foremost being that it is constructed to represent the convection

within a region of a homogeneous rotating sphere. Roll structure is

independently fixed by the boundary conditions, rather than evolving

in response to the mean magnetic field. The a-effect is of a new

type, induced by the response of the domain to a Rossby-llke wave.

As Busse notes, the rather stringent conditions on the parameters

can probably be considerably relaxed without affecting the qualitative

features of the model. Moreover, the behavior of the system as a weak

~2-type dynamo is probably of secondary importance compared to the

insight it gives into the possible origin of the ~-effect in a sphere.

In this connection it should be mentioned that the effect of the

boundary (absent in the realization of the case IV considered in Sec-

tion 4) enters into u as O(A2), and thUS is a reflection of boundary

our~Gture. This boundary contribution is independent of viscosity and

can be made to predominate over that due to Ekman pumping, although

Busse does not investigate the full dynamo cycle in that case.

On the other hand, certain features of the solution, imposed by

its asymptotic form, should be noted. First, the s-effect is such

that B 1 ~ E-2B2 , and since B 1 here represents the "meridlonal"

component of the field, the dynamo is characterized by a small

"toroidal" component. As Busse notes, the implication is, if one

accepts the model when ~ ~ I, that the two components are comparable,

but it is disturbing that this state is approached through an

asymptotic ordering that is usually regarded as improbable in the

Earth's core. A second point concerns the possibility of subcritical

instabilities. We have seen in the case of the rotating B~nard model

that the locally stable weak-fleld case may not be stable to finite-

amplitude perturbations in the magnetic field, and the question arises

as to whether or not a similar state of affairs prevails here. If one

examines the stability in case IV with an applied u~form magnetic

field of the form B(~ + E2J), it can be seen that (37) is replaced by

4 M2b 2 (IP~ + c 2) (iec 2 + c + + lalTa I/2) = Ra 2 (52)

217

where M is the Hartmann number based on B. From (52) it is easily

seen that convection at Raylelgh numbers ~ ATe I/2 can be realized

provided that a ~ b % 1 and that M 2 ~ lTa I/2. This is fully

analogous to the Eltayeb-Roberts ordering mentioned earlier, and we

suggest that there may be similar implications for the present model

at higher field energies.

7. MODAL EXPANSIONS

Numerical calculations utilizing truncated expansions in funda-

mental modes have played a prominent role in the kinematic dynamo

theory (we mention in particular the pioneering paper of Bullard and

Gellman (1954) and the recent study of Gubhins (1973)) as well as in

the simulation of thermal convection (GouEh , Spiegel, and Toomre 1975).

St is natural to consider the application of these methods to the

convective dynamo.

One immediate difficulty is the choice of appropriate "fundamental

modes", capable of representing the system at a rather low level of

truncation. The asymptotic models of the kind discussed above, which

have something of a "modal" character near the critical Eaylelgh

number, can be helpful here. The practical problem is, of course,

that if the asymptotic solution were to represent a globally stable

state, its finlte-amplitude modal counterpart offers a modest and

perhaps unnecessary extension. On the other hand, the value of the

modal approach lles in 8~muZut~o~ of the dynamo, and there the struc-

ture of the asymptotic solutions may be misleading. To take a speci-

fic example, in rapidly rotating non-magnetlc B~nard convection the

roll structure is given by (34). As we have seen, however, the

appropriate horizontal scale of the convection may increase dramati-

cally once a magnetic field is developed and M 2 ~ Ta I/2. Generally

the horlzontal scales of the modes are prescribed at the outset and

it is not obvious, u pP~or~,what value should be used.

In a rotating B~nard layer, the fundamental modes for velocity or

magnetic field will generally consist of a "pololdal" part

= Pz(Z,t)Vf(x,y) - F(z,t)V2f(x,y)~ 3

a n d a "toroldal" part

= C(z,t)~ 3 × Vg(x,y)

218

where f and g are functions chosen to represent the horizontal struc-

ture. The fields are built up from a finite number of such terms,

each corresponding to a choice of a,b in the equations q2f + a2f ffi 0,

V2g + b2g = O.

This approach has been applied by Baker (1973) to a B~nard-type

convective dynamo. Baker focuses on a "2-mode" closure (the number of

distinct F and G) and specifically on square convection cells generat-

ed by the choice: f = cos(ax), cos(ay). If the system does not

rotate, the model can be further reduced to "1-1/2 modes" by express-

ing the magnetic field in terms of one poloidal and one toroldal

component, and the velocity field in terms of two poloidal modes. One

then obtains six equations, second order in z and first order in t,

for the undetermined functions. The full 2-mode system includes

additional toroidal parts of the velocity field and takes account of

the influence of the Coriolls force, so it would appear to be the

simplest modal realization of a rotating dynamo.

In the l-I/2-mode closure dynamo action was found to occur over

a range of parameter values and for various boundary conditions.

In Figure 4 we show the energies developed in one of the oscillatory

dynamos. In this example the mean magnetic energy is about 2.5 times

the mean kinetic energy, and the tendency for the peaks to be out of

phase is consistent with Busse's quasl-equilibrlum analysis at margi-

10 4

W Z W

0

t o t a l

0 TIME

Figure 4. ~n oscillatory dynamo with 1 1/2 modes, for perfectly conductin~ rigid walls, R = 105 , P = 0.i, P = I, a - 3.I (from Baker 1973)).

219

nal stability (Section 5). The effect of rotation was not studied in

the same detail, but in some preliminary computations with 2-mode

closure the rotation was found to enhance the dynamo effect. Perhaps

most interesting is the fact that the calculations suggest the exis-

tence of a convective dynamo effect in a B~nard layer without rota-

tion, for square cells whose horizontal dimension i s comparable to

the thickness of the layer. Unfortunately, Baker notes a rather poor

convergence in going from 1-1/2 to 2 modes, so it must be regarded as

possible that the dynamo effect is illusory at this low level of

truncation.

Probably the simplest model of B~nard convection involves single

modes for velocity, perturbed temperature, and mean temperature, and

a further projection of the vertical structure onto a mode of the form

exp(Imwz). The resultlng system of three first-order ordinary differ-

ential equations in time is known as the Lorenz-Howard-Malkus or "ABC"

convection model (Lorenz 1963, Malkus 1972). Recently Kennett (1976)

has extended thle system to encompass magnetohydrodynamic convection,

by the addlt~on of terms representing the pololdal and toroldal field

components. The resulting "ABCDE" model can be thought of as a pro-

Jection of the vertlcal structure of Baker's 1-1/2 mode system, and

is simpler by one equation because of the absence of one pololdal mode

in the veloclty. Indeed it is probably the mlnlmal modal system for

a non-rotatlng convective dynamo. An interesting aspect of the form-

ulatlon is that it should allow systematic study of periodic and

aperiodic behavior; the latter is known to occur in the ABC model

(Lorenz 1963) as well as in other third-order systems (Baker, Moore,

and Spiegel 1971). The "CBE" part of the system, moreover, bears a

certain resemblance to the shunted dlsk-dynamo model studied by

Robblns (1975).

Kennett shows that the system admits equilibrium solutions with

non-zero magnetic field provided that

R > Cl +

where the o's are constants determined by the form of the horizontal

modes. For a range of parameter values these equilibria are unstable,

however, and by applying the method of averaging it is shown that

there exist in that case, in the limit of large time, nearby linearly

stable periodic solutions with non-zero magnetic field.

Because of their relative simplicity these systems are very useful

and deserve further study. It would be interesting to know, for

examplej what insight could be gained concerning the role of rotation

220

in the dynamo process, through the addition of a toroldal velocity

mode. Also, it is to be hoped that as our understanding of the pro-

cess deepens, a mode structure can be devised which converges rapidly

with the truncation level.

8. TOWARDS SIMULATION OF THE GEODYNAMO

We have not dealt in this paper with the well known solution of

the kinematic dynamo problem discovered by Braginski~ (see the review

of this work in Roberts 1971), since this approach was not exploited v

in the convective dynamos discussed above. The Braginskii dynamo has

the advantage of making no special assumptions regardin E the distri-

bution of spatial scales. Rather, a simplification is achieved by

requiring the magnetic Reynolds number of the velocity eddies to be

large. This enforces a certain symmetry on the fields (near axial

symmetry in a spherical core), which are then dominated by their

symmetric toroidal parts.

Recently Braginskii has initiated a study of the corresponding

spherical convective dynamo (Braginskii 1975). In this first paper

the fluctuating component is assumed given, so the problem reduces to

equations for the symmetric components of the fields. The questions

raised by the multi-scale convective dynamos, concerning the origin

of an s-effect and mean Lorentz force from the small-scale convection,

are thereby avoided, and the dynamic balance for the symmetric fields

can be studied at energies believed realistic for the geodynamo.

Braginskii proposes a solution in which the meridlonal magnetic field

within the core is predominantly parallel to the rotation axis. The

field is matched with its mantle counterpart through a magnetic boun-

dary layer at the core-mantle interface. Since the dynamo is of

"~" type, the azimuthal velocity which provides the "~-effect" must

be determined from the dynamics of the symmetric fields. As Roberts

and Stewartson (1974) have emphasized, this is a crucial step in the

construction once M 2 ~ Ta I~2. Braginski~'s model, which determines

the azimuthal flow by a process involving electromagnetic coupling of

core and mantle, thus confronts a problem not faced in the idealized

layer systems. (For a different approach to this question, devised

for u2-dynamics, see Malkus and Proctor 1975.) It is probably fair

to say that the convective origin of the s-effect is only one-half,

and perhaps the easier one-half, of the dynamical problem, and we

await with considerable interest the further development of this

approach to the spherical convective dynamo.

221

We conclude with a few general observations. For the sake of

argument we adopt a conservative attitude, as will be clear from the

following list of postulates for the geodynamo:

(I) The field is maintained by heating at a uniform rate in a

region of size L, fixed within the core relative to the rota-

tion axis.

(2) Within thls region, core motions are irregular (in particular

poloidal and toroldal components are comparable) and can be

characterized by a speed U and length scale L.

(3) Within this region the magnetic field is also irregular, with

field strength B > 0 and length scale L.

(4) Within this region the Coriolis, Lorentz, and buoyancy forces

acting on a fluid element are comparable.

(5) The system varies on a time scale of magnetic diffusion.

Given that heating is uniform and the system is Boussinesq, these

hypotheses are close to the "worst possible" if the aim is a pertur-

bational analysis. Indeed from (2) and the existence of a dynamo

effect it follows that R m - UL/~ ~ I, so that kinematic dynamo

problem is without a small parameter. Balancing the Corlolis and

Lorentz force we then have M 2 ~ Ta I/2, an ordering already encounter-

ed in Section 2. With (5) the units are fixed and, if it is addi-

tionally postulated that viscous and inertial forces are negligible,

the system reverts to the dimensionless form (13)-(15) (for example).

It is plausible that (with the possible exception of Ekman layer

effects and core-mantle coupling) the resulting equations contain the

relevant physics and the important matter is the ordering of terms.

In each perturbational model one or more of the above postulates is

relaxed.

A crucial question is the appropriate magnetic Reynolds number of

velocity eddies. Estimates range from 1 to 104 (Gubblns 1974). How-

ever, in view of the uncertainty over the possible size and location

of a convecting region in the Earth's core (cf. Busse 1975) a value

in the range i0-i00 is not unreasonable. This would tend to favor

Braginskii's ordering of the kinematic dynamo, but there is a

possible alternative, namely that the symmetry of the field with

respect to the rotation axis is a result of the location of the con-

vective region and the nature of the dynamo effect within it. In that

case (2) and (3) might reflect irregular motion with moderate concen-

trations of magnetic flux (Weiss 1966).

Regarding the induction problem, it is tempting to add a postulate

to our llst, namely that the dynamo is of "u~" type, even though if

Rm % 1 the ~ and e-effects are difficult to separate. We suggest that

222

the u-effect could be realized as in Busse's annulus model, or as in

case I~I of Sectlon 4. Busse's model is especially attractive, since

it also suggests how the corresponding w-effect could be developed.

Suppose we alter the direction of gravity to reflect the inclination

which occurs over most of the cylindrical annulus of rolls in the

marginally convective heated sphere. The convective heat transport

is then oblique to gravity, so the mean temperature field is distorted,

in such a way that the ~-effect arises from the "thermal wind". One

can Check that if the distortion is of the order of the equilibrium

mean temperature profile, the magnetic Reynolds number of the thermal

wind is indeed ~ i provided M and R are ordered as above. These q

estimates are likely to be modified somewhat if the convective zone is

only a small fraction of the electrically conducting region.

The geometry of the convecting zone relevant to the e-effect may be

significantly altered if, as Kennedy and Higgins (1973) suggest,

convection in the Earth's core occurs only near the inner core. In

that case the appropriate annulus model may involve a depth which

~norease8 with distance from the rotation axis, implying an a-effect

from ~estwurd-movlng waves.

Equations (13)-(15) indicate that Pn is a significant parameter in

our problem, a point that has been emphasized by Roberts and

Stewartson (1974) in their study of dissipative M.A.C. waves arising

in rotating magnetoconvectlon (cf. Roberts and Soward (1972)). It is

not clear whether ultimately the most profltable course will be to

take P~ ~ i, or rather to use the singular limit process P~ ~ (pre-

sumably leading to localized convective heat transport and a reorder-

ing of the variables) as intrinsic to the geodynamo.

One aspect of the problem which would appear to deserve further

study is the possibility of obtalnlnE more refined estimates of solu-

tions along the lines of the calculation of Kennett (1974), perhaps

wlth a view to maximizing magnetic energy in a system driven by

internal heating. It is likely that the Eeodynamo operates in a

state which is "optimal" in the realized mean magnetic energy (cf.

Section 2), and once the nature of this state is determined we can

expect, on the basis of the considerable advances made over the last

decade~ that it will then not be too difficult to secure a dynamical

model of the process.

ACKNOWLEDGEMENTS

The author is indebted to E. A. Spiegel for conversations, and to

L. Baker for Figure 4. This work was completed with the help of a

generous grant from the Guggenheim Foundation.

223

REFERENCES

Backus, G. (1958) Ann. Phys. 4, 372

Baker, L. (1972) Thesis, Astronomy Dept., Columbla University

Baker, N. H., Moore, D° W. and Spiegel, E. A. (1971). Quart. Jour. Mech. Appl. Hath. 2_~4, 391

Braginski~, S. I. (1975) Geomag. and Aaron. 15, 149

Bullard, E. C. and Gellman, H. (1954) Phil. Trans. Roy. Soc. Lend. Set. A 247, 213

Busse, F. H. (1970) Jour. Fluld Mech. 44, 441

Chandrasekhar, S. (1961) Hydrodynamic and Hydromagnetic Stabillty, Oxford University Press

Childrass, S. (1967) Courant Inst. Report AFSOR-67-0124

Childress, S. (1969a) Theorle magn~tohydrodyuamique de l'effet dynamo, Lecture Notes, M~canique theorique, Facult~s des Sciences, Paris

Childress, S. (1969b) in The Appllcatlon of Moder n Physics to the Earth and Planetar 7 Sciences , edited by S. K. Runcorn, Wiley, London

Childress, S. (1970) J. Math. Phys. 11, 3063

Eltayeh, I. A. and Roberts, P. H. (1970) Astrophys. Jour. 162, 699

Eltayeb, I. A. (1972) Prec. Roy. Soc. Lend. Ser. A. 326, 229

Geugh, D., Spiegel, E. A., and Toomre, J. (1975) Jour. Fluid Mech~ 68, 695

Gubbins, D. (1973) Phil. Trans. Roy. Soc. Lend. Set. A 274, 493

Gubbins, D. (1974) Rev° Ceophys° Space Phys. 12, 137

Hewitt, J. M., McKenzie, D. P. and Weiss, N. O. (1975) Jour. Fluid Mech. 68, 721

Kennedy, G. C. and Higglns, G. H. (1973) J. Geophys. Res. 78, 900

Kennett, R. G. (1974) in Notes on the Summer Study Program in Geo- physical Fluld Dynamics, Woods Hole, 94

Kennett, R. G. (1976) Stud. in AppI. Math. 55, 65

Lorenz, E. N. (1963) J. Atmos. Sci. 20, 130

Malkus, W. V. R. (1972) M~m. Soc. Roy. Sci. Liege, 6th series, 4, 125

Malkus, W. V. R. (1973) Geophys. Fluid Dyn. 4, 267

Malkus, W. V. R. and Proctor, H. R. E. (1975) Jour. Fluld Mech. 67, 417

Moffatt, H. K. (1968) Jour. Fluid Mech. 35, 117

224

Roberts, G. O. (1970)

Roberts, O. 0. (1972)

Roberts, P. H. (1967) Elsevier

Roberts, P° H. (1968)

Parker, E. N. (1955) Astrophys. Jour. 122~ 293

Robbins, K. (1975) Theslsp Dept. of Math., M.I.T.

Robberts, G. O. (1969) in The Application of Modern Physics to the Earth and Planetary Interiors, edited by S. K. Runcorn, Wiley, London, 603

Phil. Trans. Roy. Soc. Lond. Set. A 266, 535


Introduction to Magnetohydrodynamics, American


RoBerts, P. H. (1971) in Mathematical Problems in the Geophysical Sciences, Vol. 2, edited by W. H. Reid, A.M.S., Providence, 129

Roberts, P. H. and Soward, A. M. (1972) Ann. Rev. Fluid Mech. ~, 117

Roberts, P. H., and Stewartson, K. (1974). Phil. Trans. Roy. So¢. Lond. Set. A 277, 35

Weiss, N. O. (1966) Prec. R. So¢. Lend. 293, 310

Weiss, N. 0. (1971) Q. Jour. Roy. Astr. Soc. 12, 432

PENETRATIVE CONVECTION IN STARS

Jean-Paul ZAHN

Ohservatoire de Nice - FRANCE

I. Introduction

Penetrative convection occurs in a fluid whenever a conyectively unstable region

is bounded by a stable domain. This situation is encountered in many stars, and it is

also a very com~non circumstance on Earth: in the oceans and in the atmosphere. One

would therefore expect that the astrophysicists may largely benefit from the experience

accumulated on this subject by the geophysicists.

However, this is only partly the case. In the ocean, salinity plays a very

important role and especially so at the interface between a stable and an unstable

(mixed) region. In the atmosphere, the behavior of the convective planetary boundary

layer is dominated by the 24 hour thermal cycle, so that a steady state is never achie-

ved ~ as it is in a star (at least in one that is not pulsating). Furthermore, the ratio

between viscosity and conductivity, as measured by the Prandtl number, is of order unity

for water and air, but it drops to 10 -~ and less in a star. Finally, the effec~of stra-

tification are much stronger in stars where convective regions often span several density

scale heights.

For all these reasons, the astrophysicists have developed methods of their own

to describe stellar convection, even though some are widely inspired by those used by

the geophysicists. The same is true for convective penetration, whose study cannot he

separated from that of convection itself. The purpose of this review will he to recall

those methods, and to verify if they are suited to describe the penetration of convective

motions into stable surroundings.

~I. Phenomenological approaches

In those approaches, one hypothesizes a flow which is plausible in that it does

not seemingly contradict the laws of fluid dynamics and that it conserves heat and

kinetic energy. One then calculates the gross parameters that characterize this flow:

convective flux, mean temperature gradient. The most commonly used of such procedures

are based on the concept of mixing length, and have already been discussed in this

colloquium by D.O. Gough.

226

I. Non-local mixing-length treatment@

All mixing-length procedures applied to stellar convection are in fact based

on the two differential equations describing:

i) the variation with height z of the density excess 60 between a convective element

and the surrounding medium, in which the densities are respectively p* and P

~zz(~p ) d__e* - d_@_p ( ] ) dz dz ,

ii) the variation of the kinetic energy of that convective element

d l (2) d--z (5 P v2) = - 6P g ,

where g is the gravity.

The standard prescription (Vitense ]953) is to replace these equationsby

dO l ~ (3) ~P :k~ ~ ~ ~ ,

] v 2 ~Sp ~, (4) ~- = - C g P 2 ,

being the mixing length and C an efficiency factor which allows for the production

of turbulent energy. In this treatment, both the density excess and the convective

velocity are functions of local quantities only (the mixing length and the density

gradients); by construction the convective motions cannot penetrate into the stable

adjacent regions.

That constraint may however be relaxed by treating the original differential

equations in a less crude way. This was done by Shaviv and Salpeter (1973), Maeder (1975a)

and Cogan (1975), to be specifically applied to the overshooting from a convective stellar

core. The differential equations are integrated over one mixing length (or up to the

point where the velocity vanishes, whichever happens first):

d0 1 (" 1

1 v 2 6 p d z ( 6 ) y =-c g T ' • z-z.~£ l 1

227

(To formally recover certain results of the standard scheme, Maeder identifies the

integration distance with half the mixing length). The density stratification dp/dz

of the ambient medium is adjusted until the constancy of the total energy flux

(convective plus radiative) is realized.

This non-local mixing-length treatment permits the description of many ~eatures

of penetrative convection in the laboratory or in the Earth~atmosphere. A convective

element ceases to be buoyant at some distance from the unstable region, where also the

convective flux vanishes; from there on its momentum carries it still further into the

stable region, and since it is cooler than the surrounding medium, the convective flux

is of opposite sign. In a stellar core, the P~clet number is very high and thus the con-

vection is extremely efficient; it follows that the whole domain where the motions occur

is kept nearly adiabatic.

The main weakness of this approach, as one may expect, is that all quantitative

predictions depend on the assumption made for the mixing length. Another parameter plays

here also Some role, and it too can only be guessed: it serves to measure the fraction

of space filled by the convective elements. In the bulk of the unstable domain this

parameter is probably close to unity, but in the overshooting region, it drops to one

half and possibly much less, because it is unlikely that many downwards moving elements

are present there.

In a generalization of the mixing-length procedure proposed by Spiegel (]963),

the number of convective elements is not fixed a priori, but is governed by an equation

of conservation similar to the radiative transfer equation. Travis and Matsushima (]973)

have applied this non-local theory to the solar atmosphere, and they obtain an apprecia-

ble overshooting into the photosphere. In order to match the solar limb-darkening obser-

vations, they must choose a ratio of mixing length to pressure scale height of 0.35 or

less. Unfortunately, this value is too small to yield the correct solar radius, within

the assumptions that can be made for the chemical composition. Travis and Matsushima

suggest that this discrepancy be removed by allowing the above mentioned ratio, between

mixing length and scale height~ to vary with depth.

228

2. Other procedures

A different approach has been used by the meteorologists to model cloud

dynamics (Stommel 1947). It is based on the concept of thermals, and has since been

applied to a variety of other problems; it was Moore (1967) who brought it to the

attention of the astronomical community. A thermal is an organized cell which, llke the

eddy of the mixing-length treatment, exchanges heat and momentum with the surrounding

medium, but has also the property of gaining or loosing matter through entrainment or

turbulent surface erosion.

The only serious attempt to apply this concept to an astrophysical case was

made by Ulrich (]970 a, b), who used it to build a model of the solar atmosphere. He

had to overcome such difficulties as the absence of any ground level (from where the

thermals start on Earth), fragmentation (since the thermals are bound to move over

several scale heights) and radiative exchanges (the P~clet number becomes rather small

above a certain level). His model displays substantial overshooting well into the

photosphere, but one may wonder whether this is not due mainly to a simplifying assum-

tion he made for the correlation between the velocity of a thermal and its temperature

excess. Another consequence of this is that there is no sign change of the convective

flux in the stable region.

A similar treatment has been proposed recently by Nord!und (]976), in which the

medium is organized in two streams of rising and falling fluid. Those behave like the

thermals in the sense that they too exchange matter, heat and momentum, but here there

is no ambient medium. Dimensional arguments are invoked to write down the equations

governing the exchanges between the two streams. Solar models constructed with this

procedure are characterized by an appreciable penetration up to an optical depth of

T = 0.]; the quantitative predictior~of course depend on the choice of the dimen-

sionless parameters that occur in the equations.

229

III~ Direct approache s

In the past ten years a new approach has been explored thanks to the fast

computers with large memory storage that are now available: one can start directly

from the fluid dynamics equations, instead of replacing them by simpler ones that are

more tractable. Of course, it is not feasible yet to treat the most general problem:

as we will see, the solutions obtained to date all suffer from some kind of restriction.

But at least they help to build up an intuition which has been lacking so far. We

shall consider here only the nonlinear investigations; the main interest of the linear

studies has been to determine the critical conditions (Gribov and Gurevich ]957,

Stix |970, Whitehead 197]), but they cannot be used to predict the extent of penetration,

which is strongly related to the amplitude of the solution.

]. Bo_____ussinesq convection

The prototype of penetrative convection in the laboratory is the ice-water

experiment suggested by Malkus (]960) and performed among ethers by Townsend (1964) and

Myrup e~t ~. (|970). Water has the peculiar property of presenting a density maximum

at 4°C, so that a tank of water whose bottom is kept at 0°C will he conveetively

unstable up to the level of maximum density, and stable above. Veronis (1963) gave the

criterion for the onset of the instahility, which is of the finite amplitude type. There-

after Musman (]968) made the first quantitative predictions for the extent of penetra-

tion, using the so-called mean-field approximation (Herring 1963). The next improvemer

came from Moore and Weiss (]973), who solved the two-dimensional problem without furthe~

simplification.

A slightly different experiment is that of a fluid heated in its bulk by Joule

effect, in which the parabolic temperature profile creates two superposed domains of

respectively unstable and stable stratifications (Tritton and Zarraga 1967). This

experiment has been modelled by Strauss (|976)~ again with a two-dimensional code; his

results are similar to those of Moore and ~eiss (1973).

These two-dimensional studies are fairly ~ueeessful in predicting, at moderate

Rayleigh numbers, the mean temperature profile and thus the extent of penetration. But

it is doubtful that they can be extrapolated to the parameter range which is of astro-

physical interest (high Rayleigh numbers and low Prandtl numbers). Moreover, these

two-dimensional studies are unable to describe the time-dependent temperature fluctuations

which are observed at the boundary of the well-mixed region. These seem to be excited

randomly, and are essentially three-dimensional in their nature. The astrophysical

importance of these oscillations must not be underestimated: in the Sun, they would

occur just at the base of the photosphere and would generate gravity waves.

230

Another suggestion that the two-dimenslonal studies may be somewhat misleading

comes from the results obtained hy Latour ~t ~. (1977). They analyze the penetration

of convective motions from an unstable slab into the stable adjacent regions. The

solutions are expanded into orthogonal modes in the horizontal, and a finite differences

scheme is used in the vertical. In the special case of a single mode with a two-dimen-

sional planform, this procedure reduces to the mean-field approximation of Herring used

by Musman (]968). But one can also choose a three-dimensional planform representing, for

instance, prismatic cells of hexagonal base. The comparison of solutions derived with

the two types of planforms reveals that penetration is much stronger when the conveetlve

motions are allowed to be three-dimenslonal (Figure ]). In the simplest three-dimen-

sional case, where only a single planform is retained, the solutions are asyr~netrieal:

the overshooting occurs mainly on the side to which the centerline flow is directed in

the hexagonal cells. The mean temperature profile becomes symmetrical again when one

superposes two patterns of hexagonal cells with opposite centerline velocities;

remarkably enough, the total kinetic energy of the flow does not vary as one switches

from the one-mode solution to this two-mode solution. And the total extent of penetra-

tion too remains unchanged, if it is defined as the sum of the penetration depths at

either side of the unstable layer.

2. Convection in a stratified medium

In the laboratory (or Boussinesq) case, the extent of penetration is related to

the only natural length that characterizes the problem, namely the thickness of the

unstable layer. But what should one expect in a stratified medium, such as the solar

convection zone, where the unstable domain spans several density or pressure scale-

heights?

This question has not been answered yet. Toomre ~t ~. (]976) have studied the

penetration from the deeper convection zone of an A-type star; this zone is due to the

second ionization of helium, and it measures about one pressure scale height. Using the

technique mentioned above of truncated modal expansion, and retaining only one single

three-dimensional mode, they find that the motions penetrate up to one scale height

into the stable region below. More recently, they have established that the convective

motions penetrate also above, as far as to build a link between the deeper convection

zone and the upper one, which is caused by the ionization of hydrogen. But the situation

considered is admittedly not one of severe stratification, and these results cannot

be extrapolated to the Sun, for instance, Moreover, the solutions obtained so far are

all stationary, missing thereby the time-dependent character of penetrative convection

which may be of primordial importance.

Another difficulty with these drastically truncated modal calculations is that

they depend on the choice made for the horizontal wavelength of their single planform.

Fortunately, the results are not too sensitive to this parameter, which is felt mainly

in the horizontal heat exchanges; it does not play the dominant role of the mixing

length in the phenomenological approaches.

231

I

a

i I "~I # l

| I

,,f,, 'ii

-"t!

t - -

o i"'%

l ! 1 1 ! 1

V

b

: ' l . \ / - I / ! I l ~'.

. . . . . . . . '~ I / ! I t " - ~ - -

\ 1 ] \ II

, J ~f

c / ' ' k ~ W ; % 1

i*x e l

. . . . . . 4 . , I \ i l l ' , . . . . . . o . . . . : ' * / \ ~I

; I

0 ~ • " ° " "

i *

o 1 z

Figure I . Modal solutions for penetrative Boussinesq convection.

The unstable layer, which extends in depth from z = O to z = ], is imbedded in an infinite stable domain from which only a fraction of thickness ~z ~ 2 on each side is shown here. The same Raylelgh number R ~ |0 s characterizes the stability and the instability of the three superposed layers (it corresponds to about thousand times critical). The amplitudesof the vertical velocity, W, and of the temperature fluctuations,G, are displayed as functions of z. Figure la shows a single two-dimensional mode (which may be visualized as a horizontal roll), figure Ib a single three-dlmensional mode of hexagonal horizontal planform, and figure |c two non-interacting three-dimensional modes of that same geometry. In all cases, the value of the horizontal wavenumber is 2, and the Prandtl number is I. Notice that the overshooting into the stable surroundings is much more pronounced with the three- dimensional motions.

232

The only way to avoid any extra assumption would of course be to directly

integrate the basic equations in three-dimensional space. This has been done hy

Graham (]975), whose latest results are presented in this colloquium. But even the

most powerful computers which are presently available set a rather low limit on the

number of grldpoints that can be used. This in turn fixes the highest Rayleigh or

Reynolds numbers that can be reached: typically one hundred times critical. There is

thus still a very long road to go before meeting the numbers characterizing a stellar

convection zone, but in the meanwhile these numerical experiments are very useful as

a workbench to test the various approximations that have been proposed.

IV. Observational tests

It is relatively easy to confront theoretical predictions of Boussinesq

penetrative convection with laboratory experiments. But, as we were already reminded

by K.H. Bbhm, the comparison of astrophysical models with stellar or solar obser-

vations is more delicate, for the physical parameters that can he determined often

depend on other factors than just the properties of convection.

For the stars, one is forced to rely on the few gross parameters which can

be observed. In principle the classical tests for probing the internal structure

of a star may be used to determine the extent of the regions which are in nearly

adiabatic stratification, at least once their location is roughly known. These tests

can complement each other: the apsidal motion test (see Sehw arzschild ]958) is more

sensitive to the overall mass concentration in a star, whereas the pulsational period

of a variable star (see Ledoux and Walraven ]958) depends more on the stratification

of its envelope. There is even a s]ight hope to interpret the properties of the dynamical

tide in a close binary system, which are closely related to the size of the quasi-

adiabatic core of the two components (Zahn ]977).

But the most promising tests are probably those which sense the inhomogeneitles

in chemical composition. Prather and Demarque (1974) and Maeder (]975b, ]976) have

included some amount of overshooting in their calculations of evolutionary stellar

models. They find that the evolutionary tracks, lifetimes and cluster isochrones all

are appreciably modified by an increase of the convective core. Prather and Demarque

obtain the best fit between their theoretical isoehrones and the cluster diagram of M 67

for a penetration depth of about ]0% of the pressure scale height; Maeder's value is

slightly less and he uses it to calibrate his non-local mlxing-length procedure.

The thickness of a convective envelope (together with its penetrative extension)

may be inferred from the abundance of elements which undergo nuclear destruction at

moderate remperatures, such as llthium~ beryllium and boron. In the case of the Sun,

additional information can be gathered from the composition of the solar wind (Boehsler

and Geiss ]973). But when interpreting such observations, one must keep in mind that

other instabilities than convection may also lead to a thorough mixing of the stellar

material.

233

It looks at first sight as if the Sun should be the ideal object on which to

check the theories of penetrative convection. The solar atmosphere becomes eonvectively

unstable below optical depth T = 2, which means that the overshooting motions should

occur in the photosphere and thus be visible. The difficulty however is to distinguish

in the observations of Doppler-shifted lines what is due to waves or oscillations, and

what is due to genuine penetrative convection. The accuracy of correlation measurements

between velocities and temperature fluctuations is still not sufficient to permit the

separation of both types of motions (for a recent and complete review on such measure-

ments, see Beekers and Canfield ]976). And one encounters the same problem when it

comes to the interpretation of the non-thermal energy flux: the convective (enthalpy)

flux is blended with the flux of kinetic energy, which is carried by both convection

and waves. But the solar observations are rapidly progressing toward better precision

and spatial resolution, and one may hope that these questions will he settled in the

not too distant future.

284

Biblio~"~aphy

Beckers,J.M., CsLufield,R.C. 1976, Physique des Mouvements dans les Atmos- pheres Stellaires, R.Cayrel and M.Steinberg eds., CNRS, p.207

Bochsler,P., GeisstJ. 1973, Solar Phys. 32, 3

Cogan,B.C. 1975, Astrophys. J. 201, 637

Graham,E. 1975, J. Fluid Mech. 70, 689

Gribov,V.N., Gurevich,L.E. 1957, Soviet Phys. JETP 4, 720

Herring,J.R. 1983, J. Atmos. Sci. 20, 325

Latour,J., Toomre,J., Zahn,J.P. 1977 (in preparation)

Ledoux,P., Walraven,Th. 1958, Hsndhuch der Physik, t.51, p.353 (Springer)

Maeder, A. 1975a, Astron. & Astrophys. 40, 303

Maeder, A. 1975b, As,ton. & Astrophys. 43, 61

Maeder, A. 1976, As,men. & Astrophys, 47, 389

Malkus,W.V.Ro 1960, Aerodyn. phenomena in stellar atmosph., p.346 (Thomas edit)

Moore,D.W. 1967, Aerodyn. phenomena in stellar atmosph., p.a05 (Thomas edit.)

Moore,D.R., Weiss,N.O. 1973, J. Fluid Meqh. 61, 553

Musman,S. 1968, J0 Fluid Mec h, 31, 343

Nordlund,A. 1976, Astron. & Astrophys. 50, 23

Prather,M.J. Demarque,P. 197~, Astmophys. J. 193, 109

Sehwarzschild,M. 1958, Structure and evolution of stars, p.iA6 (Dover)

Shaviv,G., Salpeter,E.E. 1973, Astrophys. J. 184, 191

Spiegel,E.A. 1963, Astrophys. J. 138, 216

Stix,M. 1970, Tellus 22, 517

Stommel,H. 1947, J. Meteorol. 4., 91

8t~auss,J.M. 1976, Astrophys. J. 209, 179

Toomre,J., Zahn,J.P., Latour,J., Spiegel,E.A. 1976~ Astrop~s. J. 207, 545

Townsend,A.A. 1964, Quart. J. Ray. Meteorol. SOs. 90, 248

Tmavls,L°D~, Matsushima,S. 1973, Astrophys. J. 138, 216

Tritton,D.J., Zarraga,M.N. 1967, J. Flui d Mech. 30, 21

Ulrleh,R.K. 1970a, Astrophys. & Space Sci. 7, 71

Ulrlsh,R.K. 1970b, Astrophys. & Space Soi, 7, 183

Veronis,G. 1963, Ast~oph~s° J. 137, 641

Vitense,E. 1953, Z. fur Astroph. 3~2~ 135

9~itehead,J.A., Chen,M. 1970, J. Fluid Mech. 40, 549

9~itehead,J.A. 1971, Geophys. Fluid Dynamics 2, 289

Zahn,J.P. 1977, AstTon% & Astrophys. 57, 383

THE BOUNDARIES OF A CONVECTIVE ZONE

A. MAEDER

Geneva Observatory

It is worth noting that various definitions for the boundaries of a convective

Zone may be considered. Their importance for stellar evolution is very unequal.

I. A level r N is defined at the place where the Nusselt number N = I, the Nusselt

number being the ratio fo the total heat transfer in the turbulent state to that

in absence of turbulence (Spiegel, 1966). Thus, r N is the level reached when the

Contribution of convection to the energy transport changes of sign. If there is a

negligible transport by sound waves, the usual equation of energy transport in

stellar structure may be written :

T G Mr T I V t a d

M 4 = r ~ P N r

where N may be determined by an iterative process in a non-local form of the mixing-

length theory. For example, at the edge of a convective core, there are usually

2 levels rNi and rN2, the first one marks the transition from the convective zone to

the overshooting zone (convective motions with N < I), while the second one marks

the transition from the overshooting zone to the radiative zone (NE|). A frequent

but unsatisfactory treatment in stellar models is to consider rNl = rN2.

2. The level r T is defined at the place where the mean temperature excess AT of

a fluid element vanishes. Thus, at rT, the forces acting on the elements also vanish

and this level may be called the dynamical edge of the core. For subsonic convec-

tion, the levels r T and r N are evidently equal.

3. Following Shaviv and Salpeter (1973), a level r~ may be defined at the place

where ~ ffi O, where 6 is

(dT / dr )

236

~T/ ~r is the gradient in the surrounding medium, in the non-local formalism adopted

(Maeder and Bouvier, 1976) it is a non-local quantity. It was shown that the tempera-

ture fluctuations of the turbulent medium are able to make 6 ~ 0 at many places in

convective cores. So, this boundary r6 has no true meaning.

4. The level r is defined at the place given by Schwarzschild's criterion, i.e.j e

at the place where e = O, with

(dT / dr)ra d E = -- ]

(dT / dr)ad

Formally, r E and rN] do not coincide, e may be written

(dT / dr)ef f = N l ,

(dT / dr)ad

where (dT/dr)ef f is the fictious gradient, necessary if all the energy was carried by

radiation in the convective zone. In the calculated models, this gradient is slightly

subadiabatic for r + rN]. Thus, r E lies slightly below rN] , but due to the very small

deviations from adiabatieity, these 2 levels are essentially undiseernible at the

edge of a convective core.

5. A kinematical edge r v may be defined at the level, where the velocity of a

mean fluid element becomes zero. This level evidently coincides with the level rN2

defined before. It is this level which determines the extention of the zone of con-

vective mixing.

In a convective core, the significant levels are, in order of increasing dis-

tance from the centre, re < rN! = rT < rv = rN2. This order will be reversed at

the bottom of a convective zone, provided the convection is adiabatic there.

Numerical models show that the distance of overshooting (rN2 - rNi) /

expressed in terms of the mixing length is very insensitive of the various efficiency

parameters of convection. Comparisons with observations of open star clusters show

that an overshooting amounting to about 7 % of pressure scale height is likely to

occur in upper MS stars.

Bibliography

Maeder,A., ]975, Astron. Astrophys. 400, 303

Maeder,A., Bouvier, P., 1976, Astron ,. Astrophys. 50, 309

Shaviv,G., Salpeter, E.E., 1973, Astrophys.J. ]84, 19]

237

CONVECTIVE OVERSHOOTING IN THE SOLAR PHOTOSPHERE;

A MODEL GRANULAR VELOCITY FIELD

Ake Nordlund

NORDITA

Blegdamsvej 17, Copenhagen, Denmark

The solar granulation, with its horizontal temperature fluctuations,

and its associated velocity field, is a consequence of overshooting

convective motions. Theoretical estimates of the magnitude of the

temperature fluctuations and mass fluxes involved were obtained in a

recent paper (Nordlund, 1976, Astronomy & Astrophys. 50, 23). Here, a

simple model of the instantaneous granular velocity field is presented,

and the effects of this velocity field on photospheric spectral lines

are described.

The vertical velocity component is modelled by a simple, parameterized

expression:

pv z (x,y,z) = ~o(~2/2) sin (~x/d) sin (~y/d)e-Z/Zo/(l+e-Z/Zo).

The three parameters specify the amplitude (4o) of the vertical mass

flow, the horizontal size (d) of a model granule element, and a typical

scale (Zo) for the vertical variation of the mass flux.

The horizontal velocities are determfned from the vertical velocity by

the condition of continuity. Since the granular motions are slow and

approximately anelastic, the condition of continuity can be well approxi-

mated by

div (~v) = O.

This simple, quadratic pattern'of alternating vertical velocities, with

the corresponding horizontal velocities determined by the condition of

continuity, represents a crude model of the instantaneous granular

velocity field. Some important conclusions are possible, however, using

this model to fit the non-thermal broadening of photospheric spectral

lines:

The three parameters (~o' Zo" and d) can be used to fit the observed

238

half widths of a set of photospheric spectral lines, at two different

center to limb distances. The values 4o=0.35 kgm-2s -1, Zo=100 km,

d=1500 km produce a reasonably good fit. The velocity along a line of

sight varies along the llne of sight in the model granular velocity field.

When the line of sight velocity varies, the line absorption coefficient

is shifted in and out of the intensity profile of the line, and an

increased absorption results. With the given parameter values, this

effect is negligible for vertical sight lines. However, because of the

increase of typical optical scales for increasing angles of inclination,

the variation of the velocity along a line of sight becomes important

with growing angle of inclination.

Due to this effect, the observed center to limb behavior of the equiv-

alent widths of the spectral lines is also reproduced, without the need

for classical microturbulence. With a classical macro/microturbulence

model, the same behavior could have been achieved only by assuming a

depth-dependent and anisotropic macro- and micro-turbulence.

Turbulent motions on scales smaller than granular are certainly gener-

ated by the larger scale, granular motions. However, the (apparently

anisotropic) center to limb effects on the equivalent widths, which

correspond to classical microturbulent velocities of the order 1 - 2

kms -1, can be explained entirely as a consequence of the granular scale

velocity field.

In conclusion, this study shows that the observations of llne broadening

and line strength are consistent with a situation where the amplitude

of the velocity field is at maximum on granular scales, where the motion

is being driven by convection, and with the amplitudes of smaller scale

motion being progressively smaller and smaller.

THERMOSOLUTAL CONVECTION

Herbert E. Huppert

Department of Applied Mathematics and Theoretical Physics, Silver Street,

Cambridge CB3 9EW England

I. Introduction

The aim of this contribution is to survey a relatively new form of convection,

which is very easy to investigate in the laboratory, plays an important role in

the oceans and many chemical engineering situations and is likely to prove

essential in the understanding of some areas of stellar convection. Thermosolutal

convection (or double-diffusive convection as it is often called) owes its exis-

tence to the presence of two components of different molecular diffusivities which

contribute in an opposing sense to the locally vertical density gradient. The

different sets of components studied have covered a wide range including

a. heat and salt - two components relevant to the oceans and a number of

laboratory experiments;

b. heat and helium - two components relevant to certain stellar situations;

c. salt and sugar or two different solutes - components useful for laboratory

investigations; and

d. heat and angular momentum - components which are likely to be relevant to

some stellar situations. In each case, the most rapidly diffusing component has

been listed first. Thus, while in the paper the terminology of heat and salt

will be used, different components can be envisaged by reference to the above

examples.*

Aside from its many applications, thermosolutal convection has received

considerable attention because it can induce motions very different from those

predicted on the basis of purely thermal convection, that is, convection with

only one component. In particular, diffusion, which is known to have a stabil-

izing influence in thermal convection, acts in a destabilizing manner in thermo-

solutal convection. By the action of diffusion, instabilities can arise and

vigorous motion take place in situations where everywhere throughout the fluid

heavy fluid underlies relatively lighter fluid.

*Ed. Spiegel paraphrases this by the maxim: for salt, think helium. Is this his secret of gourmet cuisine?

240

An example of naturally occurring thermosolutal convection which highlights

its counter-lntuitive nature is afforded by Lake Vanda. Situated in Antarctica,

approximately 5 km long, 11 kmwlde and 65 m deep; Lake Vands has a permanent ice

cover of 3 - 4 m. Just below the ice the water temperature is 4.7°C and t~e

temperature increases with depth, often in a step-like fashion, until at the

bottom the temperature is 24.8°C (Figure l). There is a corresponding increase in

density, from 1.004 Em cm -3 just beneath the ice to a maximum of I.I0 Em cm -3

- - J

52

54~

-------~S-- --B 62--

TEMPERATURE °C

Figure 1. The temperature profile in Lake Vanda as a function of depth indicated in meters (taken from Huppert and Turner, 1972). Note the existence of a layer of uniform temperature (7.6°C) between 14.2 and 37.9 m which has been partially omitted from this figure.

at the bottom, due to the presence of salt. Vigorous convective motions take

place in the upper portions of the lake, maintaining the regions of uniform

properties, which are the hallmark of thermosolutal convection. Any model of the

lake based solely on considerations of temperature, or density, is doomed to

failure. Only by incorporating thermosolutal effects can a successful model be

derived (Huppert and Turner, 1972).

The plan of this survey is as follows. The two fundamental mechanisms of

thermosolutal convection are described physically in §2. These form the foundation

of the quantitative analysis of a suitable Rayleigh - Benard convection problem,

whose linear and nonlinear aspects are discussed in §3. The mechanism by which a

series of layers and interfaces can be maintained, as in Lake Vanda, is considered

in §4. In §5 a few ways in which a series of layers and interfaces can originate

are described. The structural stability of such a series is investigated in §6.

Conclusions are presented in §7.

241

2. The Fundamental Mechanisms

The first of the two fundamental mechanisms of thermosolutal convection occurs

in a fluid for which the temperature and salinity both decrease with depth, while

the (overall) density increases with depth, as indicated in figure 2a. In this

(a)

v

T S 0

?

(:b) T S p

Figure 2. Typical temperature, salinity and density profiles for: (a) the finger situation and (b) the diffusive situation, including a sketch of the motion of a disturbed parcel of fluid.

statically stable situation, the dynamic instability that arises can be examined

by considering a parcel of fluid displaced vertically downward. Initially warmer

and saltier than its surroundings, the parcel comes to thermal equilibriumbefore

its excess salinity can be diffused. It is thus heavier than its surroundings and

continues to descend. The ensuing motion consists of adjacently rising and

failing cells, interchanging their heat, and to a much smaller extent their salt,

much llke a heat exchanger. The kinetic energy of the motion is extracted from the

potential energy stored in the salt field. Experiments indicate that in typical

conditions, the plan form of the cells, called salt-fingers, is squarish with a

horizontal length scale of {(~g/KTV) (d~/dz)} -1/4, where ~ is the coefficient of

thermal expansion, g is the acceleration due to gravity, <T is the coefficient of

thermal diffusivity, v is the kinematic viscosity and ~d~/dz) is the mean

(positive) vertical temperature gradient. This length scale, discussed further

in the next section, represents a balance between dissipative effects acting pre-

ferentially on small scale motions and the increasing inefficiency of diffusing

heat over ever larger horizontal distances.

The second fundamental mechanism occurs in a fluid whose temperature,

salinity and (as before) overall density increases with depth, as indicated in

figure 2b. Displacement of the typical fluid particle vertically downwards now

places it in a warmer, saltier and more dense environment. As before, the thermal

field of the parcel begins to equilibrate with its surroundings more rapidly than

does the salt field. The parcel is then lighter than its surroundings and rises.

But due to the finite value of the thermal diffusion coefficient, the temperature

field of the parcel lags the displacement field and the parcel returns to its

242

original position lighter than it was at the outset. It thus rises through a

distance greater than the original displacement, whereupon the above process

continues and leads to a series of growing oscillations, or overstability, which is

resisted only by the effects of viscosity. This oscillatory form of motion has

been experimentally documented (Shlrtcllffe, 1969) and some of its characteristics

explored by Moore and Spiegel (1966) in an imaginative paper which develops an

analogy between this form of thermosolutal convection and the motion of a flaccid

balloon in a thermally stratified fluid. For sufficiently large temperature

gradients, steady motion can occur because a large temperature field can overcome

the restoring tendency of the salinity field. The criteria at which this first

occurs are discussed in the next section.

3. The RaTleish-B~nard problem

The fundamental mechanisms of the previous section form the basis of all

quantitative calculations. The most straightforward and hence frequently considered

calculation relates to the extension of the classical Rayleigh-B6nard problem:

what is the motion of a fluid confined between two horizontal planes across which

there is a temperature difference AT and a salinity difference AS? A major

motivation behind such studies is the expectation that just as the purely thermal

problem has successfully explained a variety of phenomena, as summarised by

Spiegel (1971), so also will the thermosolutal extension. And indeed this expect-

ation has already been partially fulfilled.

All calculations so far performed have essentially assumed two-dimensional

motion, dependent on one horizontal co-ordinate, x, and the vertical co-ordinate z.

Considering this restriction and non-dimensionalising all lengths with respect to

D, the separation between the planes, time with D2/KT and expressing the v e l o c i t y

q* in terms of a s t r e a ~ u n c t i o n ~ by

q* ~ (KT/D) (Bz~ , - Bx~), (3.1)

the temperature T* by

and the salinity S* by

T* m T O + AT (i - z + T) (3.2)

S* = S + AS (I - z + T) (3.3) o

where T and S are constant reference values, we can write the governing o o

Boussinesq equations of motion as

o-Iv2~t~-o-Ij(~,V2~) = -~3 x T+R s ~xS+V4~, (3.~)

~t T + ~x ~2-J(~,T) = V2T (3.5)

~t S + ~X~ -J(~,S) =TV2S (3.6)

where the Jacobian, J, is defined by

243

J(f,g) = ~xf~z g - ~zf~x g. (3.7)

We have also assumed the linear equation of state

P* = Po (I - aT* + 8S*), (3.8)

where ~ and 8 are taken to be constant, in the expression for the body-force term

in (3.4).

Four non-dimensional parameters appear in (3.4)-(3.6): the Prandtl number

~=V/KT; the ratio of the dlffuslvlties T=KS/K T, where K S is the saline dlffusivity,

which is less than KT; the thermal Rayleigh number ~ = ~gATD3/(KTV); and the

saline Rayleigh number R S = 8gASD3/(KTV).

To these equations must be added a series of boundary conditions. Mainly

because of their mathematical simplicity, the most frequently used conditions are

those obtained by assuming that both horizontal planes are stress free and per-

fectly conducting to both heat and salt. That such an assumption is a reasonable

one for a model which is to apply in the interior region of a star can be fairly

well defended (and often has been). One aspect of the defense incorporates the

belief that the use of other, possibly more realistic, conditions is likely to

lead to only slight quantitative differences. Indeed, Huppert and Matins (1973),

in a series of experiments described below, give an example of this. Mathematically,

free-free boundary conditions, as the above are often loosely called, are expressed

by ~=~2zz $ = T - S = 0 (z = O,I). (3.9)

a) Linear Disturbances

The equations governing infinitesimal motions are obtained by deleting the

nonlinear Jacobian terms of (3.4)-(3.6). The resulting differential system has

constant coefficients and a solution in terms of the lowest normal modes

~(x, z, t) = 9 ° slnwax ) (3.10a)

T(x, z, t) = T O cos~ax I ePtsin~z (3.10b)

S(x, z, t) = S o cos~ax (3.10c)

leads to the dispersion relationship

where

p3+(c+T+l)k2p2+ {(O+T+l)k 4 - ~2c~2k-2(RT-Rs)}p

+oTk6+~2~a2(Rs-T~) = O,

(3.11)

k 2 = ~ 2 ( 1 + a 2 ) . ( 3 . 1 2 )

S i n c e ( 3 . 1 I ) i s a c u b i c w i t h r e a l c o e f f i c i e n t s i t s z e r o s a r e e i t h e r a l l r e a l o r

c o n s i s t o f one r e a l r o o t and two compl ex c o n j u g a t e r o o t s . E x c h a n g e o f s t a b i l i t i e s ,

w h i c h a r i s e s when one o f t h e r o o t s e q u a l s z e r o , i . e . p~ O, o r e q u i v a l e n t l y

~t~0, occurs first for ~= 2 -1/2 and

244

- RS/T + 27~4]4 (3.13)

O v e r s t a b i l l t y , wh£ch a r i s e s w h e n the pa r r of complex-conjugate roo t s c rosses the

imaginary axis, that ~s Pr = O, occurs first for the same wavenumber, ~ ffi 2 -1/2

and ~ ffi (O+T)Rs/(~+I) + 27~4(l+r)(l+ro-l)/4. (3.14)

I0000

R S

5000

~" ~I~ (3.13)

(3.74)

I I | I I -I0000 -5000 ~ 5000 IOO00

-5000

-I0000

Figure 3. The linear stability results for 0 = i0 -I, T = I0 -I. Along (3.13) one of the temporal elgenvalues, p, is identically zero; along (3.14) two of the (complex conjugate) p are pure imaginary; and along C two complex conjugate elgenvalues coalesce on the real axis.

In the RS, ~plane the linear stability boundary is a combination of (3.13)

and (3.14), as depicted in figure 3, which presents a complete smmnary of the

linear results for ~ = 10 -I, T ffi I0 -I.

An investigation of the fastest growing mode evaluated by linear theory has

been presented by Balnes and Gill (1969), although owing to the linear constralnt~

the results are of at most academic interest. ~n agreement with the result

245

originally calculated by Stern (1960), Balnes & Gill find that in the salt finger

region, the unstable portion of the third quadrant in figure 3, the wavelength of

the disturbance of most rapid growth is much smaller than the marginal value,

23/2~, except very close to the marginal stability llne (3.13). This is in accord

with the physical description of §2, which indicates that a thinner mode acts as a

be~ter heat exchanger.

The following experiment is an example to which results based on linear theory

can be profitably applied. A uniform layer of hot, salty water is carefully placed

over a unlform layer of relatively colder, fresher water. The temperature and

salinity distributions across the initially paper-thln horizontal interface evolve

by diffusion, leading to a situation similar to that considered at the beginning

of this section. Equating the central gradient of the diffusing distribution to

the temperature and salinity gradients which appear in the marginal stability

criterion (3.12), Huppert & Martins (1973) calculate that under typical laboratory

conditions, specifically ~, T-IRs >> 27~4/4, salt-fingering should occur if

~S/C~T > T 3/2, (3.15)

where AT, AS are the initial temperature and salinity differences across the inter-

face. The results of a series of experiments, conducted with a variety of pairs of

solutes with different values of T, are in very good agreement with (3.15).

b) Nonlinear Disturbances

Fully nonlinear, but two-dimensional, investigations have been conducted by

Straus (1972) for ~, R S < 0 and by Huppert & Moore (1976) for ~, R S > 0.

The former reduces the complexity of the governing equations by assuming that

T ÷ O, R S ÷ 0 with Rs/T fixed.

In this limit, the inertial terms in the momentum equation and the adveetlon of

the disturbance temperature, but not the disturbance salinity are negligible.

Straus calculates the solutions for a variety of different values of u as

increases from the marginal stability value. + He also tests the linear stability

of these solutions. His principal conclusions are that as ~ increases, both

extremes of the range of stable wavenumbers increase significantly. Beyond a

specific ~, dependent upon Rs/T, the range of stable wavenumbers no longer

includes the wavenumber at marginal stability. The form of motion with the most

stable wavenumber corresponds to the long thin cells of the (somewhat different)

experiments and is close to that wavenumberwhieh leads to a maximum salt flux.

These results are interesting and suggestive, hut the two-dlmensional and small T

assumptions might limit the generality of some of the specific conclusions.

The calculations for ~, R S > 0 of Huppert &Moore aim to follow the form of

solutions as ~ increases for fixed RS, ~ and T. Drawing on the results of a

+The marginal stability point is supercrltical, that is, there is only the conductive solution for ~ less than the marglnal value.

246

number of numerical experiments, they put forward the following general con-

clusions. There are two rather different branches of solutions. Along one branch,

which may be initiated either subcritically or supercritically from the linear

oscillatory critical point given by (3.14), the solutlons are oscillatory. In

R~

4 ~

i

3(I0(I

10000

RT

90OO

8000

2"0 4.0 6 0 X 0

.Ms

, t i _ _ 1 21o ~ ' 4!o 6.0

M e

Monotonic

...! |0~0

Figure 4. The stabl~ solutlon branches in a thermalxRayleigh number, maximum Nusselt number at z 0 plane for (a) ~ = I, T = i0 -~, R s = I04 and (b) 0 = I, T = I0-I, R s = I07/2. Where relevant both local maxima are shown and the rapidly oscillating curve indicates that no definite maximum can be assigned to the aperiodic motion in this range. The dots indicate the transitions that can take place between the oscillatory and monotonic branches.

general, as R T increases, a transition point is attained at which the solution

changes from being relatlvely simple to being fundamentally more complicated, yet

247

still periodic. At a yet larger value of ~ another transition takes place, a

transition to aperiodic motion. Finally, beyond a still larger value, stable

aperiodic solutions cease to exist and only solutions which are ultimately steady,

and make up the second branch of solutions, can he found. The two branches for

particular values of o, T and R S are graphed in Figure 4. Steady motion exists

in a thermosolutal fluid because of the tendency of the temperature fleld to cause

an almost isosolutal core to be produced, confining all solute gradients to thin

boundary layers. The temperature, salinity and density fields for a typical steady

solution are presented in figure 5. For given o, T and RS, there is a minimum

T S p

Figure 5. The temperature, salinity and density fields for R T = 10 4 , O = I and T = 10-2.

= 10700,

value of ~ for which steady motion exists and one of the aims of the investigation

by Huppert & Moore is to calculate this minimum. For details of this result and

others the reader is referred to the original paper. The major finding is that

for sufficiently small T, steady convection can occur for values of ~ less than

that obtained from the llnear stability boundary (3.14) (and thus much less than

the value at which linear theory suggests non-oscillatory convection occurs).

The specific results obtained by Huppert & Moore, primarily by numerical

computation, were limited to 14 different values of o, T and ~. Guided by these

calculations, M. R. E. Proctor and independently Huppert & Gough are currently

attempting to obtain analytic expressions for various limiting cases, in

particular, the astrophyslcally relevant situation T ÷ O.

4. LaTers and Interfaces

As the experiment of Huppert and Manins, described in the previous section,

progresses, the fingers and the interface between the two layers extend in length.

Within the interface there is a strong background gradient of density~ and the

interface is hence an ideal site for internal waves, which are generated by dis-

turbances induced by the salt-finger motion. These internal waves cause the

fingers to sway back and forth, like a banner fluttering in the breeze. If the

fingers become too long, this motion causes them to loose their vertical coherence,

248

or break, much like a long strut subjected to an oscillatory transverse load.

Guided by the Navier-Stokes equations of motion rather than the analogies used

above, Stern (1969) argues that for very small T an established field of salt

fingers is limited in length by the requirement that

8FsI ( ~ j < C, (4.1)

where F S is the salt flux through the interface, to be discussed below, d~/dz is

the mean temperature gradient and C is a constant of order unity. Thus for a

fixed salt flux, the length of the salt fingers and the thickness of the inter-

face increase until equality in the constraint (4.1) is reached.

At the two edges of the interface the salt fingers impart an unstable

buoyancy flux on the adjacent layers, which causes the layers to convect.

Developing an analogy with purely thermal convection at high Rayleigh number,

Turner (1967) argues on the basis of dimensional analysis that the relationship

between the saline Nusselt number and the Rayleigh number is of the form

Nu$ ~ F S D/(KSAS) = fF(c~T/~AS, ~,T)R~/3 , (4.2a~b)

where D cancels in (4.2h) and fF is some function of its three argments. Also,

argues Turner, the resultant heat flux, FT, is related to F S by

C~FT/~F S = gF(~AT/BAS, ~, T) . (4.3)

Turner obtained experimentally the explicit form of fF and gF for heat and salt

in water and found that for the range of edT/BAS considered, 2 < ~AT/~AS < I0, fF

is such that as 0~T/~S ~ I the salt flux is approximately 50 times as large as

if the same salinity difference were maintained across a region hounded by two

solid boundaries, and fF decreases slowly with increasing ~AT/~AS. The constancy

of gF indicates that~ independent of o~T/~AS, a constant fraction of the potential

energy released by the salt field is supplied to the temperature field. Linden

(1973) experimentally evaluated fF and gF using a different technique and deter-

mined the same fF but a different, yet still constant, gF" Which result is in

error is still not known.

If a uniform layer of hot, salty water is placed below a uniform layer of

relatively colder, fresher water, heat and salt are transferred upwards through

the thin interface primarily by diffusion, with the resulting unstable buoyancy

flux driving convection in the layers as before. For this case, known as the

diffusive situation, the relationships equivalent tO (4.2) and (4.3) are

Nu T E F T D/(KTAT) = fD(6AS/~AT, O, T)~/3 (4.4a,b)

and

6FS/a~ T = gD(6~S/~T , ~, T) (4.5)

for some functions fD and gD" Using the results of another series of experiments

by Turner (1965) with heat and salt in water , Huppert (1971) suggests that for

249

thi8 partlcular case

Nu T = 3.8 (SAS/cU~T) -2 1~/3 (4.6)

=/l 85 0 85(B S/ T) I < S S/ T < 2 (4.7)

8Fs/~F T ] tO.15 2 < 8AS/~AT. (4.8)

A deductive model of the diffusive interface has not as yet been obtained, though

a number of ad hoc arguments, some of them described by Turner (1974~ lead to

formulae in soma agreement with equation (4,8). A few experiments with two solutes,

rather than heat and salt t have been performed. For both salt-fingering and dif-

fusive cases, all the experiments indicate a constant value of the flux ratios,

(4.3) or (4.5), for a large range of BAS/o~T. The deduction explanation for this

fact is awaited and is one of the major theoretlcal prizes still to be gained. To

be more general, a major advance in the subject would be achieved on building a

mathematical model which predicts the heat and salt fluxes for all values of

and T.

Notwithstanding our current lack of knowledge, the important conceptual

statement that can already be made is that the above mechanisms can be extended to

include a series of convectlng layers, separated by fingering or diffusive inter-

faces, as the situation demands. This is the explanation of the profiles of Lake

Vanda, those obtained under the drifting Arctic Island T3 displayed in figure 6,

and of many other oceanographic examples. The main aim of this review is to

support the suggestion that a process which occurs so readily on earth must also

play a fundamental role in stellar convection.

5. The Buildin~ $f Layers

As suggested in the previous section, a series of convecting layers separated

by thin interfaces can be easily constructed in the laboratory by carefully placlng

one layer on top of another. They arise in natural situations by a large number of

different mechanisms. A few situations have received a fair amount of quantitative

analysis and will be described here.

Consider a fluid with a uniform salinity gradient increasing with depth

subjected to a constant heat Tlux, F H, at its base. Initially a growing overstable

oscillation occurs. Shortly thereafter a convectlng layer develops adjacent to the

bottom because hot fluid rising from the base can penetrate only a finite height

into the stable salinity gradient. As time proceeds the height of this layer, h,

grows according to h = (2Bt)~/Ns, (5.1)

2 dS where B ffi - ~gFH/(Pc) and N S ffi - gS"~ • (5.2)

The relationship (5.1) is a consequence of the conservation of heat and salt and

the experimentally observed fact that the density (but not the temperature or

salinity) is continuous across the top of the layer (Turner, 1968).

This growth does not, however, continue indeflnltely. There is a thermal

250

3 0 0

3 2 0

E

3 3 0 - -

t

340 1

Temperature (degrees Celsius) - - 1.0" , ,,O"

(o) Typicol remperoture Prof/le Sect/on

- - ( b J

Section o.f Profile Recorded ot High Gain

0,01" C

l< - - O . I 'C - 4

150

E --250

C~

f~ I

500

350

Figure 6. The temperature profile under the Arctic Ice Island T-3 (taken from Neal e_.t_tal, 1969).

boundary layer ahead of the advancing front and when a critical Rayleigh number,

Rc, is reached the region above the first layer ceases to grow. This can be

calculated to o c c u r when h = ---(~<~RcB31K)I/41N ~ . (5 .4)

The second layer then grows, the thermal boundary layer ahead of its advancing

front becomes unstable~ and so in time a series of layers is built up. Heat and

salt are transferred across the interfaces~ in the manner of the last section~ and

in the course of time some of the lower interfaces disappear because the density

difference across them tends to zero. A combined theoretical and experimental

investigation of the depths of all the layers and time scales for their formation

is currently being undertaken by Huppert & Linden (1977).

This heating a salinity gradient from below mechanism, but acting in reverse,

that is, cooling a sallnity gradient from above, produces the layers under T3

shown in figure 6.

The above mechanism involes an entirely one-dimensional model. Many natural

phenomena can be expected to be two - or even three-dimensional. As yet such

extensions are only in the early stages of investigation.

251

In a series of qualitative experiments, Turner &Chen (1974) show that even a

relatively small disturbance applied to one side of a thermsolutal fluid, thereby

introducing horizontal inhomogeneities, can have significant effects. For example,

the raising of a small flap at the wall of a vessel containing a thermosolutal

fluid induces, in the salt-flnger situation, a rapidly propagating wave motion

which is accompanied by the initiation of convection over large horizontal distances.

In the diffusive situation, the disturbance propagates horizontally more slowly and

can cause local overturning which leads to the initiation Of salt fingers.

Another example concerns the introduction of a small source of warm, salty

water into a uniform layer of relatively colder, fresher water of exactly the same

density. Fluld which conwnences to fall diffuses its heat to the surroundings,

thereby beco~ng heavier. Nelghbourlng fluld, having been warmed, is relatively

lighter and rises. Large vertical motions, both upwards and downwards, in the

form of plumes result. As the motion proceeds, the density difference between

each pl-me and the surroundings increases. Thus, starting only with fluid of

uniform density, solely by diffusion both heavier and lighter fluid are formed.

If the plumes impinge on horizontal boundaries, they spread out and build a series

of layers and interfaces through the entire vertical extent of the fluid.

A final example is afforded by introducing an insulated sloping boundary. In

a stably stratified fluid, stratified with respect to only one component, such a

sloping boundary induces a thin slow steady upwards motion adjacent to the boundary.

The flow provides a convective density flux equal to the diffusive flux in the

interior and allows the isopycnals, horizontal in the interior, to bend near the

boundary and intersect it at right angles. In a fluid stratified with respect to

two components, the curves of constant T and constant S must intersect the boundary

at right angles and no steady boundary layer flow can accomplish this. Alternatively,

it is not posslble for a single boundary-layer to give rise to convective T and S

fluxes which balance the unequal diffusive T and S fluxes in the interior. Instead,

a series of layers and interfaces from throughout the fluld, as shown in figure 7

to build that characteristic structure of a thermosolutal fluld.

Figure 7. A series of layers and interfaces set up in a laboratory tank by introducing a solid sloping boundary (from Linden and Weber, 1977)o

252

The size of layers and the tlme-scale of their initiation is dependent upon

the angle of the sloping boundary, or more generally their existence is due to its

presence. However, this specific example presents another illustration of the

basic point: initially small horizontal inhomogeneities in a stably stratified

fluid leads to large scale layering with vertical transports considerably in excess

of those calculated on a molecular basis.

6. The Destruction of LeT era

In the experiment of heating a salinity gradient from below, discussed in

section 4, the tendency of the lower layers to merge as the density difference

across them tends to zero was mentioned. Such merging is due to the continually

imposed flux of heat from the bottom. Layers can merge, or be destroyed by more

natural, internally imposed conditions, which will be described in this section.

Consider a three-layer system consisting of two semi-infinite layers of

uniform T and S between which there is a finite layer of intermediate properties.

All layers are assumed to be convecting~ with temperature and salinity fluxes

across the interfaces in accord with (4.2) and (4.3) or (4.4) and (4.5). There

is then a single-valued relationship between the temperature and.salinity in the

intermediate layer for which the flux through the lower interface equals that

through the upper. The conditions under which such equilibrium situations are

stable is partially answered by Huppert (1971). Assuming that merging takes place

without any vertlcal migration of the interfaces~ he shows that only if the

conditions across each interface are in the 'constant regime', gF or gD equals

constant, will the layer system persist. Otherwise one or other of the interfaces

will disappear and two semi-lnflnite layers separated by one interface remain.

The analysis can be extended to any number of intermediate layers to yield the

same result. Thus the prediction is that no stable system of diffusive layers of

hot salty water exist if BAS/~AT < 2. A controlled laboratory experiment to test

this prediction has yet to be performed, although Turner and Chen (1974) and

Linden (1976) have observed merging which they believe to be due to the above

mechanism. Turning to large scale measurements, we can at present report that no

series of layers has been observed under T3~ in the Red-Sea or elsewhere with

BASIaAT < 2.

Experiments by Linden (1976) indicate that another form of instability is

possible, whereby merging occurs by the vertical movement of one interface to

coalesce with its neighbour. A quantitative analysis of this situation has not yet

been performed. It would clearly be interesting to know which instability is

favoured under specified conditions because layer merging will need to be accounted

for in any future quantitative model building.

7. Su~aar~ and Conclusions

This review has attempted to bring out the following salient points. Fluids

253

Stratified with respect to two (or more) components can exhibit motions very

different to slngly-stratlfied fluids. Instabilities can arise even when the

overall density is statically (very) stable by drawing on the potential energy

Stored in one particular component. The typically observed signature of a thermo-

solutal fluid is a series of convecting layers separated by thin interfaces through

which properties are transported by either diffusion or the action of fingers.

This transport is very much larger than one based on consideration of purely

molecular diffusion across a quiescent region.

Large stars have a heated hellum-rich core surrounded by lighter hydrogen.

The Composition gradient in the core/envelope regions is thus of the diffusive

type and it would be expected that convection of this form predominates. Some

attempt has been made to incorporate this process in a semiconvection zone,

although quantitative calculations would benefit from a precise description of

physics in this zone.

The salt-flnger type of instability has been hypothesised to occur in the

outer layers of differentially rotating stars (Goldreich and Schubert~ 1967),

where the two components with different diffusivities are heat and angular

momentum. Turner (1974) appears to be of the opinion, however, that such an

effect would be obliterated by baroclinic instabillties which occur on a much

larger scale. This concluslon should be questioned in view of the large obser-

vational evidence for the existence of salt-fingers in the ocean, also subject to

baroclinlc instability.

We conclude by suggesting that what is achieved so easily in the laboratory

and the oceans might also be attained by the stars~

This survey benefited from a careful reading of a first draft of the manu-

script by Dr N. O. Weiss.

References

Baines, P. G. & Gill, A. E. 1969 On thermohaline convection with linear gradients, J. Fluid Mech. 37, 289-306

Goldrelch, P. & Schubert, G. 1967 Differential rotation in stars, Astrophys. J. 150, 571-587

Huppert, H. E. 1971 On the stability of a series of double-dlffusive layers, Deep-Sea Res. 18, 1005-1021

Huppert, H. E. & Turner, J. S. 1972 Double-diffuslve convection and its implications for the temperature and salinity structure of the ocean and Lake Vanda, J. Phys. Oceano~. 2, 456-461

Huppert, H. E. &Manins, P. C. 1973 Limiting conditions for salt-fingerlng at an interfacej Deep-Sea Res. 20, 315-323

254

Ruppert, R° E. & Moore, D. R. 1976 Nonlinear doubte-diffusive convection, ~. Fluid Mech. 78, 821-855

Huppert, H. E. & Linden, P. F. 197 ? On heating a salinity gradient from below, (work in progress)

Linden, P. F. 1973 On the structure of salt fingersp Deep-Se.a Res. 20, 325-340

Linden, P. F. 1976 The formation and destruction of fine-Structure by double- diffusive processes, Deep-Sea Res. 23, 895-908

Linden, P. F. & Weber, J. E. The formation of layers in a double-dlffusive system with a sloping boundary, J. Fluid Mech. (to appear)

Moore, D. W. a Spiegel, E. A. 1966 A thermally excited nonlinear oscillator, Astrophys. J. 143, 871-887

Neal, V. T., Neshyba, S. & Denner, W. 1969 Thermal stratification in the Arctic Ocean, Science, 166, 373-374

Shlrtcllffe, T. G. L. 1969 An experimental investigation of thermosolutal convection at marginal stability, J. Fluid Mech. 35, 677-688

Spiegel, E. A. 1971 Convection in stars. I. Basic Boussinesq convection, Ann. Rev. Astron. and Astrophys. 9, 323-352

......

Stern, M. E. 1960 The 'salt-fountaln' and thermohallne convection, Tellus, 12, 172-175

Stern, M. E. 1969 Collective instability of salt fingers, J. Fluid Mech. 35, 209-218

Straus, J. M. 1972 Finite amplitude doubly diffusive convection, J. Fluid Mech. 56, 353-374

Turner, J. S. 1965 The coupled turbulent transports of salt and heat across a sharp density interface, Inst. J. Heat Mass Transfer. 8, 759-767

Turner, J. S. 1967 Salt fingers across a density interface, Deep-Sea Res. 14, 599-611

Turner, J. S. 1968 The behaviour of a stable salinity gradient heated from below, J. Fluldldech. 33, 183-2OO

Turner, J. S. 1974 Double-diffusive phenomena, Ann. Rev. of Fluid Mech. 6, 37-56

Turner, J. S. & Chen, C. F. 1974 Two-dimensional effects in double-diffuslve convection, J. Fluid Mech. 63, 577-592

THE URCA CONVECTION

Giora 8haviv

Departement of Physics and Astronomy

Tel-Aviv University

Ramat Aviv, Israel

The possible role that B-decays may play in stellar collapse was first discussed

by Gamow and Schoenberg (1940, 194]). The above authors proposed this mechanism a

mean of extracting quickly the energy content of the star and transporting it out-

side. In this way they hypothesized that stellar collapse may proceed.

The URCA process is composed of B-decay and inverse B-decay scouring inside the

Star. Let (A, Z-i) be a B-decay unstable nucleus in vacuum (i.e. on earth), namely

(A, Z-l) + (A, Z) + e- + U (I)

where (A, Z) is a nucleus with Z protons and A nucleons (protons plus neutrons) and

9 is the emitted antineutrino. The energetics of the process is seen in Fig. I. Note

that AQ must include the rest mass energy of the electron. The transformation of a

neutron into a proton leads to a lower energy configuration. The transition can go

from the ground state of the nucleus (A, Z-l) to the ground state of the nucleus (A, Z)

(arrow ]) or from the ground state to an excited state (arrow 2) if such exists and

if the transition is allowed. Only under high temperatures of the order of about

MeV is the (A, Z-l) nucleus excited and then transitions from excited states are pos-

sible. There is no basic difference between ground state and excited states trans-

itions. The most important properties for convection are: The energy difference

between the two states is shared by the two emitted particles, the electron (inclu-

ding its rest mass) and the neutrino. The electron is usually more energetic than

the surrounding and hence is slowed down (quickly) and deposits its energy in the

surrounding. The neutrino has an extremely small cross-sectlon for interaction with

matter (about 10 -44 2 cm ) and hence escapes from the star. The energy carried by the

neutrino is a net loss to the star. The rate of the decay depends on the density

(we shall return to this question) but is usually of order of minutes and longer.

T AQ

± ( A

,Z-I

)

(A,Z

)

Figure I.

The energetics of the E-decay.

inve

rse

~-de

cay

Ep s

hell

Figure 2.

The density profile in the star

and the URCA shell.

m/M

tot

Pshe

ll

257

The matter in the star is practically fully ionized at the relevant densities

(above 106 gm/am 3) and the electrons are degenerate. The fermi energy, which is the

average energy needed to add an electron at thermal equilibrium, is a monotonic

function of the density (at constant temperature ~$ep I/3 at the relevant densities).

At a density of about 5 x I05 gm/cm 3 the fermi energy is about I/2 MeV, namely it

equals the rest mass energy of the electron. As the density continues to increase

a moment will come at which the degenerate electrons, which are pushed to higher and

higher energy levels will be energetic enough to cause the inverse process, namely :

(A,Z) + e- ÷ (A,Z-I) + v , (2)

Here ~ is the antineutrino which has the same nuclear properties as the neutrino

(cross section ~ I0 -~# emZ)and escapes from the star, in general, so long as the

densities are below nuclear ones. At P=Pshell when the inverse process (2) occurs we

get the URCA shell. The name URCA was given by Gamow and Schoenberg after the famous

Casino in South America where the gambler was bound to lose his money. The URCA shell

is quite narrow, i.e. the process occurs for Ap << Pshell" At densities p < p shell

the dominant process is B-decay while for p > p shell inverse S-decay is dominant.

The location of the URCA shell is given by the condition

2 ef = AQ - meC (3)

where AQ is the energy difference between the two nuclei.

Tsuruta and Cameron (1969) considered the effect of density variations on the rate of

URCA losses. The picture of Gamow and Schoenberg is static. As the star contracts

the location P=Pshell advances outward and gives rise to neutrino (and energy) losses.

In the case of Tsuruta and Cameron the same nucleus can oscillate (by means of general

stellar vibrations) around P=Pshell" When the nucleus (A,Z) moves to a higher density

it absorbs an electron and emits an anti-neutrino, when the product nucleus (A,Z-I)

moves to the low density, it finds that the phase space of the sea of electrons around

it is free and it B-decays releasing the electron and neutrino. The sum of the two

processes can therefore be written schematically as:

(A,Z)-~ (A,Z) + v+ v (4)

Paczynski (J972) considered the effect of convection on the URCA process. The funda-

mental problem and motivation was the difficulty in the theory of neutron star form-

ation. There are some statistical arguments (Gunn and Ostriker 197]) that hint at

the fact that stars in the range 4-10 ~should be the progenitors of pulsars. However,

when models of such stars are calculated it is found that due to the particular

behaviour of stars with nuclear shell burnings, the carbon ignites at high densities

109 - 1010 gm/cm 3 and low temperatures. The ignition of the 2C ÷ Mg reaction under

258

these conditions is dominated by the corrections to reaction rates due to the high

density and gives rise to detonation. Numerical calculations byvarious people have

shown that no remnant is left. On the other hand, if carbon ignition is not achle~ed

with a violent reaction rate, e.g. if a fast and efficient cooling mechanism would be

available~ then the collapse could be delayed to still higher density and the outcome

of the explosion is a neutron star. The fast and concentrated energy production by

the carbon gives rise to convection. Paczynski considered the effect this convection

has on the URCA pairs. The estimates of Paczynski, based on mixing lengh theory

and some properties of stellar convective cores~ yielded the following result

L ~ T 170 (5) c

where L is the neutrino luminosity and T the central temperature. c

This very high temperature sensitivity follows from straightforward application of

the expressions derived by Tsuruta and Cameron (|970) for stellar vibrations to

convective cores. The oscillations considered are very fast compared to the typical

~-decay times and hence the nuclei will be out of equilibrium (and not in equilibrium

as assumed by Tsuruta and Cameron (]970) and Paczynski (1973)). Next, one has to

conserve the number of decaying nuclei. Suppose a convective core inside which an

URCA shell exis~ is given and a nuel~s inverse-8 decays far away from the URCA

shell on the high density side. This nucleus Cannot (practically) B decay before

it crosses the URCA shell to the low density side. Hence the energy loss must be

evaluated first per cycle and not immediately per unit time.

The extreme sensitivity of the neutrino lossesto temperature (much more than the

nuclear reactions energy production) led Paczynski to the conclusion that the UKCA

neutrinos can cool the star sufficiently fast and control earbon burning. Consequently

he assumed stable carbon burning which delayed the collapse of the star and

yielded the desired conditions for the formation of pulsars~aczynski 1973).

A new development came when Bruenn (1973) showed that an accurate calculation shows

that the final outcome of the URCA pair may be heating and not cooling. Consider

first the B-decay. Assume that convection turn-over time scale is short compared

to 8-decay rate (a good assuption). The B-unstable nucleus will therefore be

carried by the convection way past the URCA shell to regions of low density. The

escaping neutrino causes of oourse an energy loss but the emitted electron may

(since the electrons are emitted with a certain spectrum of energies) have energy

well above the average energy of the electrons in the medium (~f at that place). The

fast electron is slowed down and transfers its extra energy to the medium i.e. it

heats the medium.

The same situation occurs when the (A,Z) nucleus is transferred by the convection

to the high density region. Again~ the inverse B-decay process is slow compared to

conveetion velocities and the inverse process may oecur at densities for which ef

259

is much greater than the energy difference between the nuclei. Since the energy diffe-

rence is fixed, so will be the energy of the absorbed electron. Under these conditions

the absorbed electron will come from deep in the sea of electrons. Thus a "holeUis

created below the fermi level. The subsequent thermalization of the distribution, name-

ly the relaxation to a new thermodynamic equilibrium with a smaller number of electrons

will convert some of the energy difference~ef - Eth > 0 (the difference between the

average energy and the energy of the absorbed electron) into thermal energy of the whole

sea of electrons. Said in other words, a high energy electron may jump into the hole

and give its extra energy to the rest. This particular behaviour of 8 decays is well

known to nuclear physicists, namely if you put a G unstable nucleus in a container the

radioactive decay heats the medium in spite of the fact that the neutrino escapes.

The basic argument raised by Bruenn can be demonstrated in the following way. Let

E be the internal energy of the matter in which the species Ni 8 decays. We have from

the first law :

DE DE DE dE = (-~-) dV + (-~--) aT + E (-~) DN.

T'Ni V'Ni i l V,T,N i

= - <E > - pdV v (6)

where the heat lost from the unit mass considered is replaced by <E > the average

energy of the emitted neutrinos. Bruenn assumes the process to occur at constant

volume. This is not the case in stars. The fast pressure equilibrium will give

rise to compression in the case of electron capture (there are fewer particles) and

to expansion in electron emission. In spite of thls neglec?, the qualitative result

is correct. The change in temperature due to the electron capture is therefore given

hy

f ~E DE = L - - { + (

DE ~E + ( - 7 ) ] / ( ~ )

e V,T V,Ni (7)

The sum of the second and third terms on the right hand side

DE 3E ("ST') = (~-Tb"-') = Ef + %c 2 (8)

e V,T e V,S

is exactly AQ. Since we consider only very degenerate matter for which kT <<El, we can

approximate the thermodynamic derivatives with those evaluated at T=O. Hence one gets:

260

2 where m c

e is the rest mass energy of the electron. One finally gets:

~E ~E dT = ( - <e~> + Ae ) / (-~--) a"T" > 0 (9)

V,N i

Where Ae = ef - eth is the difference between the electron fermi energy and the 2

electron capture threshold and eth = -meC " AQ. Cooling will occur only if

<eg> > Ae. The calculations by Bruenn have shown that for T < Tneut (0) the e-capture

will result in heating and vice versa. The effect occurs at p ~ 109 gm cm -3, for T ~ 10 ~ °K, namely higher than the carbon ignition temperatures and hence Bruenn

concluded that the URCA process cannot stabilize the carbon burning and the story

was back the beginning. Figure 3, taken from Regev (1975) demonstrates the basic

result. Close to the URCA shell the electrons have very little extra energy and the

cooling dominates, however, outside a very narrow strip A0 << Pshell heating domina-

tes and if a convective core extends over a sufficiently large density gradient the

total heating may overcome the total cooling and convection may have the opposite

effect: heating instead of cooling.

The pendulum swung in the opposite direction after Couch and Arnett (1974) intro-

duced the idea of a cycle. Consider a given mass element moved up and down by the

convective currents. The energy balance at the high density side is

A E ec = ec - AQ E ee (10)

and at the low density:

aFi = aQ - E -~ (11)

Here ~ = ef + met2 is the chemical potential of the medium. The index ec denotes

that it must be evaluated at the point at which the e-capture takes place and vice

versa, ~- is the chemir.al potential at the place of e-emlssion. Consider now the

full cycle. The mass unit starts at the high density side by absorbing an electron.

It then moves upward where it decays as soon as the density falls. We find that

the downward moving mass unit has one extra electron compared to the upward moving

mass-unit. Consequently there is a net transfer of electrons downward. (The upward

moving electrons are "hidden" in the form of a neutron.) Couch and Arnett add therefore

the two energy equations to get:

ee - ec AE ec + AE = ~ - ~ - (e +av ) (12)

Theynow reason that the difference between the fermi energies of the two locations

must be invested in maintaining the convective flow since the electron must be

brought back to the original place. Note that the electrons move downward and hence

261

u

O~

%

O .

1.6

1,4

1.2

1.0

Heating

Cooling

Heat ing

I i I

2 3 ~; 5 6 T / 1 0 8 "K

Figure 3.

The regions in the O T plane for which heating occurs. The calculation is carried

for the Mg 25 - Na 25 URCA pair with uniform abundance (X = I0-4"I). The broken line 2

indicates the place where AQ =m e c - the URCA shell.

262

release potential energy rather than absorb. Moreover, in hydrostatic equilibrium

the total chemical potential must he constant, hence 9 mp ~ const or d~ mp

where # is the gravitational potential and m the mass of the proton. The gravita- P

tional potential is determined by the C and 0 nuclei and not by the electrons and

much less so by the URCA pair.

The fact that the change in chemical potential of the electrons must be equal to the

change in gravitational potential due to the nuclei gives rise to the high degeneracy

of the electrons. The difference ec _ P is therefore not equal to the difference in

potential energy of the electrons m e A~ but to the difference in potential energy Of

the nuclei - and this in turn is not so relevant (see later).

Couch and Arnett argue that the convective blob (of given mass) contracts upon

e-capture (pressure equilibrium - just the term ignored by Bruenn) and becomes denser

than the surroundinglwhile the downward moving blob emits an electron and becomes

lighter than the surrounding&. Thus the convective flow has to carry denser matter

upward and lighter downward. This difference costs the extra energy that appears

in the term d~ = ec - ~ . A similar argument to the previous one shows that this

is not the case. Consequently, the final result of Bruenn remains valid.

Regev and Shaviv (|975) considered the question of convective stability of the URCA

process. If heating is important,then convection may start before the Schwarzschild

criterion is violated because a small perturbation in the bubble may heat it. It was

found that convection may start earlier than assumed before (according to the Schwarz-

sehild criterion). However, the rise times are quite long and it is impossible to give

a final answer as to what will happen without a detailed stellar evolution calculation.

The analysis was a local one.

Lazareff (]975) considered the details of the convection process with URCA heating

and concluded that no stationary convective core can exist. This conclusion is

correct but for completely different reasons. Lazareff assumed mixing-length theory

in which only rising bubbles exist and considered the detai~ of this motion. He

assumed that during this motion the URCA process releases heat and concluded that if

this process is integrated over the whole convective zone the total entropy will

increase in time and hence no stationary state exists. The error in this kind of

treatment is readily seen in the case of no URCA pair. One finds that the entropy

density continues to increase even in the case of normal stellar convection. The

problem appears because (a) the downward bubbles are not included,

(h) the work done hy the buoyancy force (absorbed by the

bubble but lost by the surrounding) is not accounted for and

(c) the mixing-length theory discusses the perturbed quantities

and not the actual quantities and hence integral theorems are unavailable.

ShaviV and Regev (1976) proceeded in two steps. First the motion of a single bubble

was analyzed and then the global properties of the convective zone were discussed.

26,3

The change in temperature of a blob in surroundings

by (Regev (1975)) :

aT I aT ( r , g ) = v ( r , ~ ) ~ v ( r , ~ ) d--f

ad

containing an URCA pair is given

, T , X ) (13)

C P

when ~ is the place of formation, v the velocity, X the composition at time t.

Quantities with asterisk denote the values inside the blob. C is the specific heat P

at constant pressure. The composition of the blob changes in time according to

dX I ~ ( r , ~ ) • X v(r,E) dr = - X I X I (r,~) + ~2 (X - X I (r,g)) (14)

where X; ~ is the mass-fraction of the first species of the URCA pair and X is the

total mass-fractlon of the pair.

The equations of motion are then integrated in order to find the motion. A typical

result is shown in figure 4. We find that (a) for most cases of rising blobs the URCA

heating "helps" the buoyancy forces and the velocity is increased. (b) downward moving

blobs are somewhat disturbed by the heating (c) a subadiabatic gradient may lead to

unstable upward-~noving blobs (in agreement with the local stability analysis of Regev

and Shaviv (]975)) but prevents blobs moving downward.The most important result is:

(d) the URCA losses have a negligible effect on the motion of the rising bloh.Actually,

as the blob starts to move, its velocity is small and the effect of the URCA on acce-

leration very large, but as soon as the velocity becomes large, the time scale becomes

too short to have any effect on the motion. We find therefore that the URCA losses are

the result of spreading the URCA isotopes uniformly over the whole convective core.

Consider now a convective core as a zone with q~(r) losses and q(r) heating per unit

mass and time. Clearly, close to the URCA shell~q~ is greater than the heating but far

away q(r) l which includes the nuclear and the URCA heating;is the dominant factor.

The two contributions have different spatial behaviour.

The energy equation is :

P d-tdE + PV.~ = pq - pq~ + pqf - V.F (15)

where F is the radiative flux through radius r and qf the rate of heat generated by

dissipation. The equation of motion is given by

d~ Vp - ~ - pV~ (16) 0 dt

where f is the frictional force per unit volume and ~ the gravitational potential.

The integration over the whole convective zone down tb the place where ~=o yieldspafter

264

u

E u

>

0 M

7

6

3 w

5

iIi

×I

! I I I ........ I i

6 7 8 9 40 44 42

r/40 "4 R e

-6

-8

X

O

-10

" -12

Figure 4.

The velocity of a blob in the case of Na 23 URCA pair. Line ] is log v

for an adiabatle blob and line 2 is log v for a blob with the URCA pair.

Line a is the equilibrium abundance of Na 23 in the surrounding medium,

I is the distance of one mixing length,

265

Some manipulation ~the following result :

( ] / 2 pv 2 . . . . . . . . . . (17) -- + E ) = pq - pq~ + L~ - ~t In Lout

where Lin and Lou t are the radiative flux into and out of the convective zone respec-

tively. The bar denotes an integral over the whole convective zone. Consider first the

case of no URCA process, i.e. q = 0 and q is the energy generation due to nuclear reac-

tions. In a steady state, the time derivative must vanish and we find that the net out-

come of convection is to spread the nuclear energy generation over a large volume so

that radiative flux can carry the energy from the boundary. When the URCA pair is pre-

sent and the steady state is preserved, it follows that the total heating by the URCA

process (added to q) must be equal to the total neutrino losses, q . If this balance is

not maintained the convective core will not be stationery. A detailed balance can exist

only if the convective core has a definite extent. Moreover, even if such a steady state

convective core exists, it is• unstable. The analysis of the URCA losses shows that qv

dominates near the URCA shell but the heat gain dominates elsewhere. Thus if the nuclear

reactions increase their energy production and the convective core expands, the URCA

process will increase the heating even more unless the radiative losses increase faster,

which is not the case. We conclude therefore that steady state eonvectioD cannot con-

trol the carbon burning and the problem of the fate of these models and the progenitors

of pulsars remains.

A question of principle remains : how come that a process which conserves material has

as its outcome net heating ? The solution is that the URCA process is out of equilibrium.

The net heating is given by (Regev (1975))

q = A--N° meC2 ([AQ - meC2 _ e~ %2X2 - X2L 2 + [ef + meC2 - AWl]X ] - X]L! ) C]8)

where L l and L 2 are the neutrino loss rates by the e.c. and 6 decay rates per nucleus

respectively. N and A are the Avogadro number and atomic weight respectively. When the o

URCA pair is spread uniformly/q > 0 and we have heating, but at equilibrium X]Xl=%2L 2

and the expression for q becomes

No 2 q = - ~-- mec (X]L l + X2L 2) (19)

and we have cooling only.

We are led finally to the question of the distribution of the URCA pair. Two time-scales

affect the distribution : the convection mixing time rconv = £/Vconv and

TURCA = (%| + 12 ) -]j which is the decay time. Define a new parameter by

= -l

ami x Tconv TURCA = 3,5 x JO 7 (%! + Am)v -! z conv (20)

266

where the convective velocity is given in cm/sec. The limit of complete mixing is

obtained for s . << I while equilibrium is reached for a . >> I. In reality we find mlx mlx

a. = ]. Consequently, at the beginning of the convection the process is close to mix

equilibrium but as time goes on the URCA pair is driven away from equilibrium and the

heating appears. The entropy added into the convective zone is due, as pointed out by

Lazareff (75)) to the non-equilibrium state of the URCA pair.

Acknowledgement : It is a pleasure to thank Mr O. Regev for discussions that made this

analysis and presentation possible.

BIBLIOGRAPHY

I. Bruenn S. W. (1973) Ap J. Let~. 183, L125

2. Couch R. G. and Arnett W. D. (1974) Ap J. 194, 537

3. Gamow G. and Schoenberg M. (1940) Phys. Rev 58, ]I]7

4. Gamow G. and Schoenberg M. (1941) Phys. Rev 59, 539

5. Tsuruta S. and Cameron A. G. W. (1970) Astr & Space Sci. Z, 314

6. Gunn J. E. and Ostriker J. P. (1971) Ap J. ]60, 979

7. Lazareff B. (1975) Ast. & Astrophys. 45, 14]

g. Paezynski B. (]972) Astrophys. Left l_i], 53

(]973) ibid ]5, !47

(1973) Acta Astronomica 23, l

9. Regev O. (September 1975) On the Interaction Between Convection and the URCA

Process M.Sc. thesis Tel Aviv University

]0. Regev O. and Shaviv G. (1975) Ast. & Space Sci 37, [43

I]. Shaviv O. and Regev O. (I976) Ast. & Astrophys. (in press)

PHOTOCONVECTION

E.A. SPIEGEL Astronomy Department Columbia University

New York, New York 10027 U.S.A.

Convection under the influence of dynamically significant radiation fields

occurs routinely in hot stars (Underhill 1949 ab) and probably also in a variety of

other objects near the Eddington limit (Joss, Salpater, and Ostriker 1973), Yet

this topic, which is here called photoconvection, has not been actively investigated

prior to the present decade. Except for limiting cases, the stability condition

does not seem to have been worked out and only some preliminary notions exist about

the highly unstable case. This is somewhat surprising since it has long been sus-

pected that some of the vigorous dynamical activity observed in hot stars (Huang and

Struve 1960, Reimers 1976) is caused by radiative forces (Underhill 1949 ab). In

the hope that this neglect may be compensated for by the application of some of the

techniques described at this meeting, I shall sketch some of the main features of

this topic. Three aspects are considered. First, I list a set of approximate equa-

tions for plane-parallel photoconvection. Then I give a schematin treatment of the

onset of instability. And finally, I shall outline some of the arguments for be-

lieving that photon bubbles occur in the nonlinear regime.

I. EQUATIONS OF PHOTOHYDRODYNAMICS

The interaction of electromagnetic radiation with a plasma is a complicated

subject with a long and controversial history. However, many of the difficulties

are avoided if we consider densities and radiation frequencies that keep the index

of refraction of the medium quite close to unity. In that case, we can describe

the radiation field by transfer theory if we take due notice of the motion of the

material medium. The simplest description arises if we simply take the first two

moments of the transfer equation and supply a constitutive relation for the radia-

tive pressure tensor. For the matter, we shall adopt the model of a perfect gray

gas. Then the matter field is described by the velocity u, the density Q, and the

pressure p, while the radiation field is characterized by the flux~, the energy

density E, and the pressure tensor~.

These variables are expressed in the inertial frame of the system (star), in

which we will generally be working. It will be useful, however, to make use of the

expressions for radiative flux and energy density in the local rest frame of the

matter. These are

268

.÷ ~. 2 (l.la) E ffi E - zu°~/c

- P'u, (l.lb) F - ~- E~ +~÷

where c is the speed of light. These expressions are valid only to order lul/c,

which is the level of accuracy (at best) aimed for here. Nevertheless, in the

equations used below, we shall see some factors of c -2, because the radiation field

is relativistic. In particular, the quantity ~/E qualitatively plays the role of

a velocity for the radiation field and in the surface layers of stars the magnitude

of this velocity may be comparable with c.

In addition to the field variables, we have to specify certain quantities that

measure the effective interactions between the two fields. These interactions we

shall take to be Thomson scattering, absorption, and emission. We shall assume that

the Compton effect can be modeled by a suitable choice of absorption coefficient.

We shall call K the absorption coefficient and ~ the scatterlngcoefflcient (both

per unit mass); G will be constant and K may depend on density and temperature. The

source function (divided by c) is denoted by S and depends only on the matter's

temperature, as indicated below.

The equations describing the conservation of matter and the force balance of

the medlumare

~= -pvJ a t (1.2)

and

(1.3) d+

p~'~-= - W, - gp~-+' c F

where g~ is the acceleration of gravity, ~ is a unit vector, and

d d-[ = ~ + ~.V .

The last term on the right of (1.3) is the usual expression for the radiative force.

Analagous equations exist for the radiative fluid:

DE p(K+~) + +~ ~-[ + v-~ - p~c(S-~) - c u.F (1.4)

and

(1.5) -- -- - (S-E) u. 2 ~t c c

c

For the pressure tensor of the radiation field a standard form is

~-+ i ~++ ~÷ ÷÷ 2 - (uF + Fu)/c - ~+ (1.6) P ~ Z

where I is the idemtensor and T is a viscous tensor. In component form,

F' Bui ÷ (l.6a) TiJ = nL~ + ~ - 2 (V'u)6ij~3

where 6ij is the Kronecker symbol and the viscosity is approximated by

269

8E ( 1 . 6 b ) ~ . . . . . . 3p(10K + 9a)c "

Expression (1.6) arises when the radiation pressure tensor i s approximated

in the matter frame by the usual Eddington approximation plus a viscosity tensor.

For the constitutive relations for the matter we adopt

( 1 . 7 ) p = RpT and

(1.8) S = aT 4

where T is the temperature and R and a are constants. (We shall not specify

here.) The introduction of the temperature calls for another equation, as in

normal convection.

If • is the specific entropy of the matter, we may write

(l.9a) pT ~t = -pKc(S-E),

orj if we use the expression for the entropy of an ideal gas,

aT dp = -p~c(S-E), (1.9b) pCp d-~- dt

where C is the specific heat at constant pressure. P

These governing equations are consistent andmoderately accurate sets of

governing equations. I have said little about the basis of them (but see Simon 1963

or Hsieh and Spiegel 1976) since their physical content is reasonably clear. If

anything, these equations are, for present purposes, too complete. It appears that

there are a number of generally small terms which will hinder calculations and ob-

scure meanings. But many of these terms are unfamilia~and the challenge is to

discover when we can discard which terms. In what follows, I shall make a number

of guesses about this; I hope that these are not too misleading. In fact, much

of the discussion is just aimed at seeing what some of these terms do and in such a

schematic treatment you would not expect to see boundary conditions. I shall hardly

disappoint you. But before I comudt mayhem on the equations, let us modify the ap-

pearance of the last one by combining it with (1.4). We obtain, with the help of

(z.2),

PC dT dp + dE 4 E dP -V.(~ - 4 ÷ ( l . l O ) p d'-f - dt d-[ - ~ ~ d'-T" ~ Zu)

- - - u -F . 3 c

We may note that the left hand side of this equation is

the total (matter plus radiation) specific entropy. pTd6tot/dt where Ato t is

270

II. THE HYDROSTATIC STATE

As background to the problem of photoconvection it is useful to know the solu-

tion of the basic equations which describe the state in which the matter is static.

But note that this solution is not photostatic; the radiation is flowing through mat-

ter llke a fluid through a porous medium.

We consider stationary solutions whose properties are independent of horizon-

tal coordinate. If K# O, equations (1.9), (i.I), and (1.8) indicate that

(2.1) E = aT4;

if K = 0 this relation is not forced and T is an arbitrary function of t h e ver- +

ileal coordinate, z. In either case F is constant and is in the i-direction.

Now (1.2) is identlcally satisfied and (1.3) gives the hydrostatic equation

dz g*P ~ (2.2)

w h e r e

~c-I-O (2.3) g , = g - c F

is the effective gravity. (In the Eddington limit, g , = 0.) The radiative flow

equation (1.5) becomes

dE -3p ~ F, (2.4) d-~ = c

and (1.7) is unmodified. Thus all the governing equations are accounted for and we

have a simple system to solve once K is known. In general the problem is handled

numerically, but some analytically tractable cases exist. Let us look briefly at

the simplest: ~+u = constant.

We may introduce the total pressure

i (2.5) P = p + ~ E,

and combine (2.2) and (2.4). We find that

dP m (2.6) dz -go ,

and, on d i v i d i n g by ( 2 . 4 ) , t h a t

dP (2,7) 3 ~ = (K+~)F

The integral of this equation, after some rearrangement, may be written

g,c

(2.8) p = 3(~+o)F (E-E1) '

271

(2.12)

where

(2.13)

where E 1 is an arbitrary constant. It is often conveD/ent to choose E 1 as the

value of E at the top of the "atmosphere ".

We may now write a simple differential equation for E, or T, and find the

solution

i _ i +. (2.9) -z =~[7-~T Itan [ -yT ltanb -1 )] ,

where E 1 = aTe._ If T 1 - 0, this represents a complete polytropic atmosphere. In

any case, the medium is polytropic for z << 0 and T is proportional to -z

down there. For z >> 0, T - T 1 decays exponentially as we move upward and the

atmosphere extends to infinity for T 1 # 0.

In principle, all the other details could be worked out from this, but

nt~erlcal work is generally needed. However, some things are still simply expres-

sible in terms of the optical depth

(2.10) ~ = [-(K+c)pdz. #

z

In particular,

.ll) E = F(T+TI), (2

where T 1 is a constant of order unity.

Another quantity of interest in the static atmosphere is the temperature

gradient. In the present instance this is most simply expressed in the familiar

nondlmenslonal form

dlnT R dT _ 7 - 1 CD dT V = - d - ~ n P = gB dz ~' g8 dz

~=~.

For the atmosphere with K+o constant we find

(2.14)

where

(2.15)

1 l-s V = m ~ , 4 1-B

g, ~ i I - - +

g

tzt. THZ O+SST OF CONVECTZO~

The action of radiative forces under suitable conditions may promote wave

amplification (Hearn 1972, 1973; Berthomleu, Provost, and Rocca 1976) and posslbly

overstability (e.g., Spiegel 1976). The nature of this overstability seems to place

272

it more in the domain of stellar pulsation theory than convection theory, though the

two may become enmeshed in the nonlinear regime. On the other hand, monotonlc in-

stability, that is exponential growth without oscillation, is more clearly linked to

the development of convection when the time scales are dynamic, and I shall confine

myself here to discussing that topic.

The procedure for deciding whether convective instability arises is straight-

forward in princlple~ especially when we are not trying to study overstabillty or

flnite.amplitude instability. We decompose each dependent variable into a hydro-

static part and small perturbation.~ Here we shall indicate the latter type of

quantity by a prime, except in the case of velocity. We restrict ourselves to the

situation where ~/~t = 0. Then, on linearlzing in the usual way, we find from

(1.10) that

pCpblW = V'F , (3.1)

where

(3.2) 0Cpb I -(pCp dzd-~T-~+--'dS AS dO. = dz 3 ~dz ) "

But we may also proceed in this way on the basis of (l.9b) and in that case we oh-

rain the equation

(3.3)

where

(3.4)

pCpb2W = pKc(S'-E'),

oCph 2 = -(pCp d_.!_ dd_.~.z) dz

Now in a full treatment of the problem it would not matter which of these two

routes is taken since the final answer would be the same. But the stability criteria

that are normally used are obtained with approximations and the two approaches may

differ in that case since they have suggested different approximations to dif-

ferent people. In particular~ people have simply written down criteria for in-

stability with respect to adiabatic disturbances with differing notions of wh&t they

mean by adiabatic. Thus, the commonly encountered criterion results from equating

?.~' to zero. If we do this we find that hlW must vanish at marginal stability.

Since w in that case has small but arbitrary amplitude, we obtain the critical

condition b. = 0, which is the conventional one (Chandrasekhar 1939). On the other

hand, if we set the right hand side of (3.4) equal to zero (e.g., Wentzel 1970,

Spiegel 1976) we obtain b 2 = 0 as the condition for marginal stability. This cri-

terion holds strictly when absorption and Compton*scatterlng are omittedand its use

otherwise is dangerous.

The two criteria represent valid approximations under certain circumstances

and P. Vitello (private communication) has recently investigated what these are.

273

The discussion of this question shows that the conditions under which one or

other neutral stability criterion holds depends on the perturbation being made.

This is a common situation and we expect that the correct instability criterion is

to be found by choosing the most unstable mode.

To see how the problem goes let us begin to do the stability calculation.

From (1.5) we find for marginal linear perturbations that

= p(K~) p(m+o)

with

(3.6) ~ = "~ E'~+ ( ~ + ~) *F /c 2 - ~ ,

Also from (l.9b) we obtain

Cb (3.7) ~' = s' - P 2w.

KC

If we combine these results with (3.1), making use of other equations as needed, we

find an equation of the form

pCp[B2V2w + pKA ~z - 3p21<(K+O)BlW] = 3p2K(<+°)V'(KVT)'" (3.8)

where

(3.9)

and B I, B 2, and A

If n = 0 and C P

(3.10a)

(3.Z0b)

and

(3.10c)

4act 3 K = 3p(K~)

are quantities whose dimensions are temperature over length.

is constant,

81 =bl- 3p2(Kd~) dz 2 3P dz

Fd - ~T, [(r+~) ~]-I_ F Q(K+O)C dz 2 '

B 2 = b 2

Now consider the case in which the right hand side of (3.8) is set equal to

zero. That is, instead of trying to speak of an adiabatic disturbance, let us sim-

ply ask what happens to a perturbation when radiative conductivity is suppressed.

If geometrically small horizontal scales are the most unstable, as they are in

ordinary invlscid, non-conducting convection, we may replace V 2 by -k 2 where k

is the horizontal wave number. For qualitative purposes, we may also omit the term

274

with coefficient A since this is not important in the limiting cases we wish to

consider. Also, for the present argument, I shall set B I = bl, since this discus-

sion is merely schematic. The approximate condition then becomes

(3.11) k2b 2 + 3p2K(K+O)bl = O,

and in terms of the quantities defined in (2.12) and (2.15) we find the instability

criterion

(y-l) [~2 (4-,3~) ] V> ,

-- S[~{S+4(y-I) (1-6) ] + $2 [y62+4(y- l ) (1-6) (4+6) ] (3.12)

where

(3.13) ~2 = 3p2K(<+O)

k 2

This criterion holds approximately in the limit of zero viscosity and with the omis-

sion of radiative conduction terms as indicated. The dimensionless quantity ~2j

which arises in radiative cooling problems (Unno and Spiegel 1966), should be cho-

sen so as to minimize the right hand side of (3.12). The resulting value of ~ is

then inserted to give the local stability criterion. Of course, if we are led to

extreme values of ~ we should worry about the possible violation of physical con-

straints that have been removed in this simplified analysis. (I~nextremis, we could

Just solve the problem properly.)

To make the appropriate choice we observe that for y > 4/3 the right side

of (3.12) increases as ~ decreases. In that case, the instability criterion is

obtained with the largest possible values of ~, hence with modes of large horizon-

tel scale in the length unit [3p2K(K+O) ]-1/2. In stellar interlorsmost scales of

interest satisfy this condition and the conventional criterion would apply. In

transparent regions, however, it may be that geometrical constraints intervene and

large ~ cannot be achieved. In that case, the maxlmumvalues allowed for

should be taken, and here we should note that once ~ exceeds unity there is not a

large difference from the results at very large ~,

In cases where 7 < ~ the situation is reversed and the right hand side of 3'

(3.12) decreases as ~ decreases. The preferred value of ~ is now the smallest

one possible; that is we want the largest allowed value of k. If the particle mean

free path is much less than the photon mean free path we can choose small ~ with-

out worrying about the breakdown of fluid dynamics, But we do have to make sure

that we don't choose a k which is so large that diffusive effects wipe out the in-

stability. In fact this amounts tO finding the preferred mode in the usual way, but

here the choice determines not just diffusive corrections to the critical gradient,

but also the effective adiabatic gradient itself. Unfortunately, there is a compli-

cation that arises in this situation.

275

& The case y <~ will normally occur in ionization zo-~es and therefore has to

be treated with some care. In fact, Underhill (1949b) ~as evaluated the "adiabatic"

temperature gradient with partial ionization and in the presence of an important

radiation field. But that calculation was what I have been calling the conventional

criterion. That is, she has applied the condition of zero total (matter plus radia-

tion) entropy gradient which corresponds to the marginal stability conditions with

=m. Howeverj the possibility exists that finite values of ~ may be more correct

since the zones of partial ~onlzatlon tend to occur in stellar envelopes. We could

then ha~ a somewhat increased convective instability but the modes involved, being

radlatively leaky, might not carry heat effectively. It appears therefore that for

most purposes the standard convection criterion is good. However, it would be more

comfortable to have a detailed treatment of this problem, and I predict that there

soon will be one.

IV, PHOTON BUBBLES

In thinking about ordinary stellar convection we maybe guided by solar ob-

servations, but we have not such direct experience to guide us in photoconvection.

Instead, we may appeal to observations of a laboratory flow that is analogous to

photoconvection. We have already seen that the radiation in this problembehaves

(in the Eddington approximation) like a fluid flowing through a deformable porous

medium. This closely resembles the situation in a fluldlzed bad (Thorns 1973,

Prendergast and Spiegel 1973). Though the analogy is not a perfect one (Spiegel

1976), it can be used to suggest the qualitative nature of nonlinear photoconvection.

And one of the most striking implications of this analogy is that instead of convec-

tive thermals having relatively low densities, we should expect real bubbles

in photoconvectlon. These are filled with radiation and contain virtually no

matter. How this modification of the normal convective process may influence the

heat flux can only be crudely estimated (Thorns 1973), but there are also other

features of convection which are strongly affected. In partlcular~ bubbles feel

the full effect of gravity rather than the reduced gravity of ordinary eonvection~

hence large (that is, sonic) convective speeds may be anticipated.

In this section, I shall sketch an approach to the treatment of photon bubbles

borrowing heavily from the literature on fluidized beds (Jackson 1970, Rowe 1971).

In comparison to fluidlzation, this theory suffers from the disedvantage that we

have not yet seen a photon bubble. However, John Lin at Columbia is looking serious-

ly at the prospects for removing this drawback experimentally.

We wish then to study a photon bubble of radius r ° rising at speed V. We

shall assume that r O << H,~ where H, ~ R T/E~, and that the bubble may be taken to be

quasl-steady when described in its own reference frame. We may nevertheless intro-

duce the dynamical time scale r /V. Let us assume that this time is much shorter o

276

than the thermal time of e region of size r ° and then presume from this that

there is validity in neglecting thermal effects. Then we may tentatively set K = 0

in the basic equations.

Now let E and F be representative values for the ambient radiant energy

density and flux and let p, be a representative ambient matter density. The fol-

lowing dimensionless parameters are of interest:

V EV = V 2 (4.1) c =-f,~ =T,f2 .r,= p,cr o.

gr o

We assume that ~ << 1 and, from the analogy to fluidizatlon, we anticipate that

f2 is of order unity. If E and F may he estimated from their static values

(see (1.11)), we have E/F ~ e/T, for T >> i, where T is the optical depth. Hence

~ ~T. Then, when the bubble is only a few radii below the surface, T, ~ T and

we have that 8 ~ ET,, which is the regime we shall study here. A further restric-

tion to be used in the following analysis is ~ << I, but l shall mention at the

end what may happen at larger depths when 8 becomes of order unity.

If we nondimenslonalize the basic equations and make use of the foregoing

approximations, we obtain a greatly reduced set of equations. In dimensionful form

these are

(4.2) P ~t + u.Vu) - -Vp-gp~ + c F

(4.3) ~-E + V'(p~) = 0 ~t

(p/p~f) + u . V ( p / p "f) - o (4.4) 3-~

( 4 . s ) v.~" = o

(4.63 3 VE - ~c ~"

This description is about as primitive as it can be while still involving the ele-

ments of photohydrodynamice. Let us now seek approximate solutions for a bubble

rising at constant speed V. We presume that the medium is unstable, which is

true if = < 0.2.

Suppose, in first approximation, that the bubble is a spherical hole of

radius r • If the bubble does not greatly disturb the ambient density, we see that o

equations (4.5) and (4.6) are simply the transfer equations for a static medium with

a hole in it. This results because in t h e present approximation the radiation field

adjusts quickly to the state of the medium (~ << i); also the motion is so slow

that the difference between ~ and ~ may be neglected (~ << i)° We have also

assumed that the bubble radius is much less than the local scale height~ hence O

in (4.6) is approximately constant outside the hole. Equations (4.5) and (4..6) may

then he solved separately with D = 0 inside the spherical cavity.

277

Let us introduce a spherical coordinate system with origin at the center of

the hole and with % = 0 at the top of the hole. Far from the hole, the flux is

Fo~, where F ° is a constant given by the static solution and we have the condition

3p~F o ^ ( 4 . 7 ) VE ÷ - - - z a s r ÷ ~ .

c

M o r e o v e r , E and t h e component of ~ n o r m a l t o t h e b u b b l e s u r f a c e a r e c o n t i n u o u s

across the surface. Since (4.6) implies that E is constant inside thehole, we

have the boundary condition

0 1 1 r = r ( 4 . 8 ) E = E ° o

where E is the constant value of E inside the hole. Now (4.5) and (4.6) show o

that E is a harmonic function and, with conditions (4.7) and (4.8), we find

2 r o

oro - c o s e . (4.9) E - E ° - c o r

Since cE/(3p~) is a potentlal for ~, we have

(4.101 ~ ffi V-{Fo o r': [__.r. (~]2]coa e}. o

Alternatively, we can express F in terms o f a Stokes stream function:

1___!__ - 3T Fe i ~ (4.11) Fr = r2sin 8 ~8 ' = r sin 8 ~r '

w h e r e

( 4 . 1 2 ) r ~=_~1 For2[1 + 2(~)3]sln2e

The flux co, isis of ~e original unifo~ part plus a d~ole generated by the hole,

a reset familiar from analogue problems ~, for example, electroata~cs. ~e

radiative flow is shown in the neighborhood of ~e bubble in ~e figure on the

following page. Inside the bubble, the fl~ is 3Fo, if equation (4.6) ~y be wed.

~is last point is a ~llcate one as we have used ~e Eddin~on ~proxi~tion

for the transfer ~eo~. However, this approximation holds if the radiation is iso-

tropic ~d, when pot ° >> i, it probably is. ~e reason is that for T, >> 1 indi-

~dual photons will scatter off the b~ble surface (actually a l~er of ~ickness

(p~)-l) many times before escaping, hence the r~iation field inside ~e b~ble

should be reasonably isotroplc.

~e deformation in the radiation field produces an additional for~ on ~e

~tter. ~e total exte~al force density is 3

-gPz + 0~ F" -g*Pz " c ~pV cos 8

278

/

J

e %.. .'2

Streamlines for the radiative flux around a hole, according to (4.12). The solid circle ehc~s the original hole. ~ The dotted curve indicates the estimated deformation of the hole obtained by setting h = 0 in (4.22) with ~ = 0 and V chosen as in (4.24).

279

(4.13)

v-V(p/p 7) 0, (4,14) ÷ =

(4.1s) v.(~) = o,

where

roF r 3,

and

where g* " g-gR and gR ffi Fo/C" The additional dipole force produces a fluid

circulation which causes the hole to rise and, in general, to deform. Let us study

these effects.

If we work in the bubble's frame and assume a stationary situation we have

the equations

v'Vv ~ -Vh-g,~-V~ ,

I have not changed notation to indicate the coordinate transformation except to

call v = u - V z the new velocity. The correction to ~ due to the motion of

the bubble is of order 6 and is neglected.

First we shall determine V on the assumption that the bubhle remains

spherical. This we do with the approximation p = const, whence

(4.1s) v-~ = 0.

We may therefore take ~ to be the incompressible flow around a spherical ob-

stacle. Such a flow has a vanishing normal component on the bubble boundary and

-V ~ as r -~ ~. Solutlons of this problem are well known and if it approaches

we also set

(4.19)

we find

V x v = O ,

r 3

(4.2O) ~ffi-VV[r cos 811 +~r31 ].

Moreover, because of (4.19) we may rewrite (4.13) as

(4.21) V [h+#+g, zq~2/2 ] = 0;

hence for r ~ ro,

280

(4.22)

h = gro{l-[~ + (l-s) ] cos 8}

- ~V {i+ (1-3 cos 2 @) +~ (1+3 cos 2 @)},

where ~ = g,/g and an arbitrary constant has been chosen so that

On r = r we have o

9 2 (4.23) h = gro[l- cos 8 -~f (l-cos 2 8)].

h(ro,O,O) = O.

For p = const, h = p. Alternatively, the choice p/p7 = const, which satisfies

(4.14), gives h ~ p(7-1)/y. In either case we would like to have h = 0 on r = r o since p = 0 inside the bubble (and E is continuous across the interface). But

(4.23) shows this to be impossible with the present approximate treatment. However,

we do have the freedom to choose f2 to match the pressure boundary conditions as

well as possible. In fluidization theory the procedure used by Davies and Taylor

(1950) for ordinary bubbles is usually adopted. In the present instance this comes

down to setting ~2h/~@2 = 0 at r = r o, 8 = O, whence f2 = 4/g (see also

Batchelor 1967). The argument for this is that h and ~h/~8 are already zero at

r = to, 8 = 0, and we would like to extend the region where h is very small as

far as possible. Let us adopt this choice; (Any other choice of this type would

also give a value of f of order unity. For example we might mlni~zethe inte-

gral of h 2 over the surface r = r .) Thus we have an estimate of the speed of o

rise of the bubble which can also be used to see the magnitude of the distortion of

the spherical hole by the dipole force. For the latter purpose we may simply com-

pute the surface on which h = 0 with

= 2 (4.24) v ~(gro)l/2.

For ~ = 0 this surface is the dotted line indicated in the Figure above. The

distortion of the hole is caused by the need to balance the fluld-dynamical pressure

~2/2 and it represents a problem which is also encountered in the theory of ordi-

nary gas bubbles in liquids (Moore 1959). As long as appreciable speeds occur next

to the bubble this difficulty arises. In an actual fluidization bubble the problem

is resolved by the formation of an indentation at the rear of the bubble. The in-

dentation fills with particles which effectively move with the bubble. This feature

has to be built into the theory in a self-consistent way.

With the present estimates a second problem arises, namely that for r >> r o and @ > 0 we encounter a region of negative h when ~ > 0. This difficulty does

not arise in fluldization theory since that subject is confined to ~ = 0. We

therefore have no experimental guide to the meaning of this result. There are some

speculations that might be offered here but perhaps the message is simply that bub-

bles only occur when ~ is very close to zero.

281

Now it is evident that the foregoing discussion does not really provide an

acceptable theory. It might be different if photon bubbles were an observed phen-

omenon that we were trying to understand qualitatively. But the real question is

whether photon bubbles actually exist and the answer will almost surely have to be

given experimentally. In spite of these worries, I would llke to close this

theoretical discussion of bubbles with one further qualitative remark about what

may happen at very large optical depths.

The total radiant energy density includes the usual energy density plus the

pressure, hence it is ~E. The energy flux divided by this energy density gives

a speed to be associated with the radiant fluid.

F

3 When ~ exceeds some critical value %T , V exceeds v R, and the bubble is

moving faster than the radiative fluid. In that case, the radiation does not ad-

Just quickly to the matter. Rather, we may expect the radiation associated with

the bubble to be swept along with the bubble, much as in the corresponding case

Of fluldlzatlon where one sees a trapped cloud of fluid circulating in and around

the bubble. When this occurs, I expect that photoconvective transport should be-

come very efficient. The optical depth at which this occurs is given approximately

by

T ~ lO 24/9 I~eff] 4/~ • [i--~5 j

V. CONCtUSION

The main questions considered here have to do with the nature of photoconvec-

tlon and the conclusion which is tentatively adQpted is that the two-fluid nature of

the process may make for some qualitative differences from basic Boussinesq convec-

tion. I have tried to sketch how photon bubbles may behave in a~alogy with fluidi-

zation bubbles. The analysis is sufficiently simple that one can easily see what

is going on, but there is one point about the results that I want to emphasize.

The bubble is not simply held open by an excess of radiation pressure inside it.

The radiative force is vital to the process and this is proportional to the flux.

The figure in ¶IV is helpful in seeing how this works: flux converges onto the

bubble from below and diverges upward from the bubble. This forces the fluid

flowing by the bubble to go around it which in turn causes the hole which produced

the flux pattern in the first place. This seems to be a dynamically consistent

situation, whether or not the equations have been completely solved. Whether the

thermodynamics of radiation interacting with matter (and which has not been dis-

cussed at all properly here) can spell the picture, seams difficult to decide, and

that is, to me, the biggest question to be faced at present. But i£ we put doubts

282

aside for now we may imagine some astroDhysicall7 interesting aspects of photon

bubbles.

The generation of large amplitude, complicated velocity fields in hot

stellar atmosphere is one of these. Another is suggested by the rapid separation

of particles of differing properties in bubbling, fluidlzed beds. In this process,

called elutriation, particles of relatively large drag are carried up through the

bed by the bubbles. Similarly we can imagine that particles with large scattering

cross section may be carried swiftly through stellar material By photon bubbles.

Moreover, there are some interesting consequences involved when bubbles collapse

near a stellar surface. The heatlngmay cause hot bursts of radiation (as

J. Pringle has suggested) or radiation of acoustic and shock noise. Also, non-

spherical collapse could squirt matter off the stellar surface at high speed, as a

preliminary computation by J. Theys confirmS.

But these are presently speculative topics and more immediate aimS should

also command attention in this subject.

We need a more complete stability theory, a study of flnlte-amplltude

stability, and some numerical simulation. In this respect, we should be aware of

related work on high density plasmas (Estabrook, Valeo, and Kruer, 1975), though

much of what I have said here leaves out plasma kinetic effects and assumes rela-

tively low density s such as is encountered in stars.

I should like to conclude by acknowledging my indebtedness to the many

people whose remarks have influenced aspects of the presentation and to list Just

a few of them: S. Childress, L.B. Lucy, K.H. Prendergast, and J.C. Theys. I am

~rateful to G. Baran for running his contour routine. And flnslly, I thank the

National Science Foundation for supporting the work reported here under Grant

NSF PHY-7505660.

REFERENCES

Batchelor, G.K. 1967, An Intrqductign t O Fluid Dyqamlcs, Cambridge Univ. Press, p. 475

Berthomleu, J. Provost, A., and Rocca, A. 1976, Astron. and Astrophye., 47, 413 Chandrasekhar, S. 1939, "An Introduction to Stellar Structure " Davies, R.M. and Taylor, G.I. 1950, Prec. Roy. Soc. Lon. A, 200, 375 Estabrcok, K.G., Valeo, E.J. and Knuer, W.L. 1975, Phys. Fluids, 18, 1151 Hearn, A.G. 1972, As tron. and Astrophys., 19, 417 Ream, A.G. 1973, Astren. and Astrophys., 23, 97 Hsleh, S.-H. and Spiegel, E.A. 1976, Ap. J~., 207, 244 Huang, S.-S. and Struve, O. 1960, Stellar Atmospheres, J.L. Greenstein, ed.,

Univ. of Chicago Press, p. 300 Jackson, R. 1971, Fluidization, J.F. Davidson and D. Harrison, eds., Academic Press,

p. 65 Joss, P.C., Salpeter, E.g., and Ostrlber, J.P. 1973, Ap, J., 191, 429 Moore, D.W. 1959, J.F.M., ~, 113 Prendergast, K.H. and Spiegel, E.A. 1973, Co mmentsA p. Space Phys,, ~, 43

283

Reimers, M.D. 1976, Physique des Mouvements dans. les Atmospheres Stellalres, R. Cayrel and M. Steinberg, eds., C.N.R.S., p. 42]

Rowe, P.N. 1976, Fluidization, J.F. Davidson and D. Harrison, eds., Academic Press, p. 121

Simon, R. 1963~ 3. QuaBt ~ Spectr0sc. Radiat. Transfer, ~, 1 Spiegel, E.A. 1976, Phys~iquedes Mouvements dans les Atmospheres .Stellaires,

R. Cayrel and M. Steinberg, eds., C.N.R.S., p. 19 Thorns, V.A. 1973, Report RC59, Dept. of Computer Science, Univ. of Reading Wegener, p.p. and Parlange, J.-Y. 1973, Ann. Rev. Fluid Mech. , ~, 79 Wentzel, D.G. 1970, Ap. J., 160, 373

CONVECTION IN THE HELIUM FLASH

A. J. Wickett

Department of Astrophysics and Nuclear Physics Laboratory,

University of Oxford

Abstract

The evolution of a star through the helium flash depends upon uncertain aspects

of convection theory. Observations place some constraints on the theory of convection

in stellar cores.

Evolution of Stars in the Mass ranse 0.6 M@ to i~ M@

When stars finish burning hydrogen to helium in the core, the burning region

moves outwards to form a shell source. Matter moves inwards through the burning region

into the core which becomes progressively hotter and denser.

As the density of the core increases, the electron gas becomes degenerate. The

hydrogen outside the hydrogen-burning shell forms a diffuse convective envelope of

low density and great spatial extent (radius ~ 5xlO10m). The star is a red giant.

Figure 1 shows the structure just before the helium flash.

The site of the Helium Flash

In the eorejthermal conduction by degenerate electrons is efficient and there

are no heat sources before helium burning starts, so to a first approximation the core

is isothermal. However heat is lost from the densest part of the core by the plasmon

neutrino process

Y plasmon ~ ~ + 9"

A slight temperature inversion therefore appears, which causes helium burning to ignite

some distance (- D.1 M@ ) from the centre. The peak temperature T m " 108K.

285

9.0

8,0

7.0 log p

6.0

(kg m-3

5.0

~1.0

3,0

2.0

1.O

0.0

-1.O

-2.0

-3.0

-4.0

6.0

7.o

log T

6.O

5.O

4.O

m

m

m

m

m

m

m

I .... i i i I ! | ....

- j

, , , I I ...... I I I I I 0.i 0.2 0.3 0.4 0.5 M 0.6 0.7

Me Figure !. Temperature and density profiles for star of 0.75 M@ and zero-age composition X = 0.7, Z = 0.0004 at start of helium flash

286

The rate of the reaction

34He ~ 12 C + Y

(the 3~ reaction) is proportional to about the 40th power of the temperature. Since the

degenerate electron gas can take up much heat without a large pressure increase, a

thermal runaway can occur.

Quasistatic evolution through the Helium Flash

The energy generation rate in the hellnm-burning shell is great enough to require

a convective zone outside it to transport the heat away. The peak temperature rises

a~d the convective zone extends; the rest of the star adjusts its structure adiabat-

ically. When the electron degeneracy is lifted in the burning region~expansion takes

over and the peak temperature falls. Figure 2 shows how the structure of the centre of

the core changes through the flash.

Calculations with an accurate equation of state (Thomas 1970, Demarque and Mengel

1971, Zimmermann 1970, Wickett 1976) show that if, in the convection zone, the temper-

ature gradient is adiabatic or slightly superadiabatic as prescribed by the standard

mixing-length theory and if the convective zone does not reach further towards the

centre than the initial helium ignition zone, the burning timesoale at the peak of the

flash is large compared with the convective timescale. Thus the traditional formulation

of convection theory leads to a quasistatic helium flash.

Convective Uncertainties

Edwards (1975) shows however that the traditional convective stability criterion

3tar < adiabatic ~> unstable

is modified in the presence of a very temperature-dependent energy generation rate.

Even a positive gradient (temperature rising with radial coordinate) is unstable if,

as is the case here, the reaction rats depends sufficiently strongly on the temperature.

It is not clear exactly how this oonclusion affects the evolution; however a calc-

ulation was performed in which the convective zone was extended to the centre and the

temperature gradient was the adiabatic one. The formulation is more fully described in

the author's thesis (1976). At the peak of the flash, the burning time

8,.2

log

T

8.1

8.o

7.9

7.8

7.7

7.6

6.5

Figure 2.

...... I .

....

....

....

....

..

t ......

.....

I i

~odel

.....

/Will

'

I .

..

.

I I

7.0

7.5

8.0

log~

(kg ~3) 8.5

Temperature-density plots for core of star of figure i

during helium flash

r

9.o

288

T T n=~

for constant density and no heat flow at the hottest point was equal to the convective

timescale

i c

T C U " C

U and 1 are the convective speed and mixing length evaluated by the usual theory. o c

The time for thermal runaway is considerably less than T n as the burning rate rises

sharply with temperature. So convection cannot keep up with the rate of increase of

heat production. Thus if this picture of the evolution is correct a thermal runaway

and explosion must occur.

D~namicCalculation

A hydrodynamic code including gravity and nuclear reactions for a spherically

symmetric model was used to follow the explosion.

A spherical detonation wave with peak temperatures ~ 2.5 x 109K and propagation

speed l0 times the sound speed in front and twice that behind passes through the core.

Half of the helium burns to neon (or species of similar mass number) releasing

~ 5 x 1043j. The entire star is disrupted with energy typical of a supernova of type I.

Observational Constraints

We assume that the 108 Galactic globular sluster stars (Allen 1973) have masses

distributed according to Salpeter's (1955) mass function

dN = M. -1"35 d in M.

with a lower limit of 0°l M@. The variation of age at helium flash, T hf" with mass

din T hf - - - -5.31 dlnM,

was calculated using Eggletonls (1971, 1972) code. For age i0 i0 years we are interested

in N, = 0.92 M® and find that Galactic globular cluster stars now undergo 5.5 x l0 "13

helium flashes per star per year or, 5 x lO -5 per year for the whole Galaxy.

289

Since one would expect to see a supernova remnant for l06 years (Woltjer 1972)

and no supernovae remnants or gas of any sort is seen in globular clusters, we con-

clude that the supernova rate is less than l0 -l# per star per year. So there is an

Upper limit of one supernova for 50 helium flashes.

Conclusions

It is not clear how to calculate the temperature gradient or the convective time-

Scale when the energy generation rate is very temperature-dependent. It is however

clear that the traditional prescription does not work.

As to the helium flash supernova model, the observations allow 2% of helium flashes

to be that violent. If this is the case then the small fraction might arise either from

rather narrow ranges of mass, rotation etc. being faveurable or, in some ways more

appealingly, from fluctuations in the convection. This implies that the evolution of

two stars with the same gross properties may be substantially different because of the

random nature of the convective process.

Acknowledgements

This work was supported by the United Kingdom Atomic Energy Authority and the

Science Research Council.

References

ALLEN, C. W., 1973. Astrophysical Quantities, 3rd ed., Athlone Press, London

DKMARQUE, P. and MENGEL, J. G., 1971. Ap. J. 164, 317

EDWARDS, A. C., 1975. M. N. R. A. S. 173, 207

EGGLETON, P. P., 1971. M. N. R. A. S. 151, 351

EGGLETON, P. P., 1972. M. N. R. A. S. 156, 361

SALPETER, E. E., 1955. Ap J. 121, 161

THOMAS, H.-C., 1970. Ap Space Sci. 6, 400

WICKETT, A. J., 1976. The Hydrodynamics of the Helium Flash, D. Phil. thesis,

University of Oxford

WOLTJER, L., 1972. Ann. Rev. Astr. Ap 10, 129

ZIMMERMANN, R. E., 1970. .The Hydrodynamics of a HeliumShell Flash in a Star of one

Solar Mass, Ph. Do thesis, University of California, Los Angeles

WAVE TRANSPORT IN STRATIFIED, ROTATING FLUIDS

M. E. McIntyre

Department of Applied Mathematics and Theoretical Physics,


SUMMARY

Momentum and energy transport by buoyancy-Coriolis waves is illustrated Dy means of a simple model example. The need for careful conslderation of a complete problem for mean-flow evolution is emphasised, especially when moving media are involved. Then a recent generallsation of the wave-action and pseudomomentum concepts is introduced, and used to exhibit in a very general way the roles of wave dissipation, forcing, or transience in the mean flow problem, for a certain class of "nearly-unidirectional" mean flows. This class includes differentially-rotating stellar interiors, which could well be systematically changed by wave transport of angular momentum. Similar results hold £or MHDand self-gravitating fluids. Finally the physical distinction between momentum and pseudomomentum is discussed.

i. INTRODUCTION

Some of the most spectacular natural manifestations of wave transport

effects are those believed on the basis of recent evidence to Occur in the

stratospheres of Earth and Venus I-4. Closely analogous effects appear

likely to influence the evolution of the rotation of stellar interiors 5,

and to be important in other astrophysical contexts 6. They are often

associated with rather complicated kinds of low-frequency fluid-dynamical

waves, in which buoyancy and Coriolis forces are essential. The waves set

up a "radiation stress" whereby the mean azimuthal velocity at one height

and latitude can undergo systematic acceleration at the expense of a

corresponding deceleration at a more or less distant location. Thus

transport of angular momentum by the waves is involved. This transport can

result in drastic changes to the pattern of differential rotation (which

in turn can drastically affect the wave propagation and lead to some

interesting feedback effects~.

291

An idealisation illustrating this kind of "radiation stress" phenomenon

is the model problem suggested in figure i. Inertio-gravity waves, which

are the simplest type of buoyancy-Coriolis wave, are being generated by a

slippery, corrugated boundary moving parallel to itself with constant

velocity c:

C

FIGURE I. Inertio-gravity waves being generated in a stably-stratified fluid (specific entropy increasing upward) by a rigidly-moving boundary. The frame of reference is rotating with constant angular velocity~= (O,O,A) and the effective gravity g = (O,O,g) = constant. (It can be shown that the waves have their crests or'lines of constant phase sloping forward, unlike the sound waves which might be generated if the boundary moved much faster.)

Here the Cartesian x direction plays the role of the azimuthal direction,

and the mean state is assumed independent of x. The mean pressure gradient

has no x-component; thus the fluid is free to accelerate in the x direction

in response to the radiation stress.

If the waves are being dissipated in some layer L at the top of the

picture, there is a systematic tendency for the mean flow to accelerate

there. So the wave-drag force which the boundary exerts on the fluid is not

felt at the boundary, as far as the mean flow is concerned; it is felt at

L. This is a typical radiation-stress effect.

If the waves were generated not at a boundary but by a moving system

of heat sources and sinks in some layer in the interior of the fluid, then

total momentum would be constant, and the mean acceleration at levels where

the waves are dissipated will be accompanied by a corresponding deceleration

where they are generated 3'4. The close connection between mean flow changes

and wave dissipation or forcing can be verified by detailed solution of

the appropriate sets of equations, but is not usually obvious from the

equations themselves.

Where the waves are dissipated will depend not only on the physics of

292

the dissipative processes but also upon the solution of the wave propagation

problem for the particular mean-flow profile involved. Other things being

equal, we usually get enhanced dissipation of the waves in places, if any,

where their intrinsic frequency (i.e. frequency in a frame of reference

moving with the local mean flow H ) is Doppler shifted towards zero - that

is, we tend to get enhanced dissipation near an actual or virtual "critical line "8 5(y,z) = c.

In this review I shall pay particular but not exclusive attention to

the class of problems exemplified by figure i. Their characteristic feature

is the existence of a coordinate x (cartesian or curvilinear) such that mean

quantities are independent of x, and "mean" can be defined as an average

with respect to x. Such problems, which I shall call "longitudinally

symmetric" happen to comprise an area of recent advances, and also serve

to illustrate some of the subtleties and pitfalls which can arise in thinking

in general terms about the transport of conservable quantities such as

energy and momentum by waves in material media,and most particularly waves

in moving media. (We are, of course, dealing with moving media par excellence

as soon as Coriolis forces are relevant.) In section 2 some of these points

are illustrated by describing in more detail what happens in the problem

of figure i. Most of the phenomena encountered can be found in one or other

of several related problems which have been discussed in the literature by Eliassen 9, Phillips IO, Matsuno II, Uryu 12, Grimshaw 13, and others.

In sections 3 and 4 I turn from illustrative example to general theory,

and survey some very recent developments which appear to be of quite wide significance, but which have proved to be especially powerful for longitud-

inally symmetric problems. A simple yet very general version of the

"wave-action" concept is involved, resulting from a synthesis and extension

of ideas from "classical field theory "14 and the more recent work of Eckart 15, Hayes 16, Dewar 17, and Bretherton 18. Equally relevant is the

pioneering work of Eliassen & Palm 19 and Charney & Drazin20; and a related

but not identical line of development is contained in the work of Soward 21 on the Braginskii dynamo problem. A remarkable feature of the general

results is that they enable useful statements to be made without requiring

validity of approximations of the "slowly-varying wavetrain" type and attendant concepts like "group velocity'. Also, they can be developed for

finite-amplitude waves 22. Their special value in longitudinally-symmetric

problems is that they lead to ways of expressing the problem for the

mean-flow changes which do directly exhibit the abovementioned general connection between those changes and wave dissipation or forcing 22-25.

Finally (section 5) I shall make some remarks about that elusive entity,

or rather, nonentity, wave "momentum'.

293

2. MORE ABOUT THE PROBLEM OF FIGURE 1

2.1 Equations

The simplest relevant set of model equations is the usual set for a

Boussinesq, incompressible, stratified fluid in a rotating £rame of

reference, with constant angular velocity JL :

~,t + ~'~ + 2J~^B + polyp - e~ = -X (2.1a)

8,t + U.V8 = -Q (2.1b)

V.u = O . (2.1c)

Here u = (u,v,w) is velocity, ( ),t stands for 6( )/6t, ~ is the unit vector

(O,O,i), e is the buoyancy acceleration given by minus the effective

gravity-plus-centrifugal acceleration (assumed constant in this model)

times the fractional departure of the density from its constant reference

value Po" For descriptive purposes we shall think of 8 as a measure of

temperature or potential temperature. The departure of the pressure from

the hydrostatic value associated with Po is denoted by p . The terms X =

(X,Y,Z) and Q may be thought of as representing arbitrary body forces and

~eating, which may or may not be functionally related to the fields of motion

but which in any case will be zero if the waves are neither dissipated nor

generated internally.

2.2 Excess momentum flux, and the mean-flow problem

Let an average with respect to x be denoted by an overbar: for instance

the mean velocity in the x-direction in figure 1 is 5(y,z). If we average

(2.1a) and make use of (2.1c) the result may be written in suffix notation

(i,j=i,2,3) as

ui,t + {uiuj + ~ij},j ÷ (2~A~)i - ezi = -{u[u~},j - xi " (2.2a) Po

(It will be convenient in what follows to use (x,y,z) and (u,v,w)

interch___angeably with (Xl,X2,X 3) and (Ul,U2,U3).) Here ( )" is defined as ( ) - ( ) , the departure from the mean, and (),j means d( )/6xj . Eq.(2.2a)

contains mean-flow quantities only, except for the term involving the

Reynolds stress uiu j . The equation tells us that uiu j is the excess mean ...... > i momentum flux due to the waves. Note that uiu j is a wave property, by which

I mean something which can be self'consistently evaluated as soon as you know the linear wave solution, i.e. when you know the fluctuating quantities ( )" to leading order.

294

It might be tempting to conclude that nothing more need be said: Eq.

(2.2a) states that the momentum transport by the waves is equal to uiu j ;

so "obviously" -ui~ is the stress whose divergence will give the mean

acceleration u,t, or at least the contribution to this acceleration

attributable to the waves. The average of Eq.(2.1b), namely

8,t + {ujS},j = -{u~8"},j - Q , (2.2b)

is irrelevant, one might think, because how, after all, can the excess heat

flux u{8" due to the waves affect momentum transport?

This conclusion would, however, be wrong (for reasons to appear

shortly), and the fact that it has appeared in the past literature

illustrates the dangers of "incomplete reasoning" about wave transport

effects on the basis of superficial consideration of a relevant-looking wave

property - in this case the excess momentum flux u~u~. Another illustration

will be encountered in section 2.6. In fact the only safe general recipe

for getting a self-consistent picture is to include a conslderation of the

complete problem for the mean flow and its solution correct to second order

in the wave amplitude a. In the present example, the wave properties u[u~

and ~ appear as forcing terms in the mean-flow problem; and both turn

out to play essential roles.

The result o£ averaging (2.1c) is

~.~ = 0 , (2.2c)

and this completes the set of equations, (2.2), for the mean quantities

and 8 . To obtain a well-determined model problem it is simplest to suppose

that the flow is bounded laterally by a pair of vertical walls y = O, b

on which the normal component of velocity vanishes, implying that

= O on y = O, b . (2.3)

We must beware, however, of assuming that ~ vanishes at z = O; in fact for

a rigidly-translating, corrugated boundary whose shape is described by a

given function h ,

z = h(x-ct, y) , (2.4)

where h=O(a), h=O, and c is a (real) constant, it can be shown that

+ O(a 3) at z = 0 . (2.5) = (v'h),y

This illustrates the fact that ~ , which is an average along a horizontal

line such as ~ in figure I, can represent a vertical mass flux, into or out

of the thin region betweeen ~ and the actual boundary, which is continuous

with a horizontal, O(a 2) mass flux within that region, associated with any

tendency for the disturbance velocity to be one way along troughs and the

other way along ridges in the boundary.

295

In fact, such a tendency turns out to be the rule rather than the

exception when Coriolis effects matter; for instance if h is of the form

a sin k(x-ct) then v" for conservative, plane inertio-gravity waves on a

uniformly stratified basic state of rest turns out to be exactly in

quadrature with w" and therefore exactly in phase with h at z=O. This can

easily be verified by setting 8,z = constant, 5 = 8,y = 0 , and X=O, Q=0,

and calculating the elementary plane-wave solutions ~ exp i(kx + mz -~t)

of the linearised disturbance equatlons derived from (2.1) (namely (3.2)

below). Other pertinent features of such plane-wave solutions are that 8",

being proportional to the vertical displacement through the basic stable

stratification 8 is (like h at z=O) in quadrature with the vertical ,Z '

velocity w'; also incompressibility dlctates that u" is in phase with w',

since (2.1c) implies iku" + imw" = O . Thus u'w', v'8" are nonzero, and v'w',

zero, in a plane inertio-gravity wave. The frequency of such a wave ~,

(= kc), satisfies the dispersion relation

~2 = (~ zk2 + 40~2m2)/(k2 + m 2) (2.6)

when H = O. (It should be noted that this implies that c 2 must lie between

4jl2/k 2 and O,z/k 2 for the inertio-gravity waves to be generated.)

2.3 Solution

I shall now describe, for the simplest relevant example, the result of solving the O(a 2) mean flow problem; ~ and Q will be set to zero, so that

we are talking about the effect on the mean flow of the waves alone. The

waves are supposed to have propagated upwards as far as L either because

they are being dissipated there or because a finite time has elapsed since

the bottom boundary started moving. Well below L we can take the waves to

have reached a steady state and the moticn to be conservative - we assume

that X'and Q'are zero there as well as ~ and Q. To keep life as slmple as

possible we shall assume that H = 0 initially, and follow its evolution as long as it can be considered to be O(a2). We also take e,z = constant +O(a 2)

for the moment.

The simplest kind of mathematical analysis for the waves (we omit the

details, since the results of section 4 will supersede them) makes the usual

kind of "slowly-varying" approximation, in which the plane wave solution

is locally valid. This involves inter alia an assumption that the layer L

is sufficiently deep compared with a vertical wavelength. We also take h

to be of the form a.f(y).sin k(x-ct) , where f(y) is a sufficiently slowly-varying function (which we assume vanishes at y=O,b). Then by the

properties of plane inertio-gravity waves previously mentioned, the important term on the right of the x-component of (2.2a) is -(u'w') and

Fz that on the right of (2.2b) is -~SVT,y . The remaining terms are not of

course exactly zero, because plane waves represent only the leading

approximation; but in fact it is consistent to neglect them. The response

of the mean flow to the forcing -(v'8"),y together with the forcing

represented by the inhomogeneous boundary condition (2.5) involves a mean

"secondary circulation" indicated schematically by the arrows in figure 2.

296

The picture assumes that the wave amplitude is a maximum near y=~band falls

monotonically to zero on either side, so that (~) changes sign once, ,Y

near y=~b. The mean flow feels an apparent "heating" on one side of the

channel, and "cooling" on the other (about which more will be said in section

2.5). This gives rise to an O(a 2) mean vertical velocity~which beautifully

satisfies the boundary condition (2.5) and, by Eq.(2.2c) , demands a mean

motion across the channel, i.e. a contribution to V, in the vicinity of the

layer L where the wave amplitude goes to zero with height.

"Z- f

1 WAVES

i

I

!

t

I

!

w

!

t

t

FIGURE 2. Left: end view (looking along the x axis) of the problem of figure i. Right: typical profile of the mean acceleration in the longitudinal or x direction. The left-hand picture indicates how the secondary circulation ~, ~ is closed by a mass flux "in the bottom boundary', associated with a positive correlation between the disturbance y-velocity, v', and the depth -h of the corrugations in the boundary.

The Coriolis force associated with this O(a 2) contribution to ~ accounts

for a contribution to H t which is generally comparable with that from the

Reynolds stress dzvergence-(u w ),z zn the x component of Eq.(2.2a). In fact

the two contributions, in the present simple problem, can be shown to stand

approximately in the ratio

-6s 2 Reynolds stress divergence _~ Coriolis force associated with wave heat flux 4_Q- 2

. (2.7)

The two contributions are comparable in magnitudewnenever the Coriolis term

is significant in the dispersion relation (2.6); indeed if k2<<m 2 in (2.6),

~2 e 4~2 and the two contributions are almost equal and opposite. In that

297

case an estimate of effective momentum transport from the Reynolds stress

-~-~walone could be too large by an order of magnitude. (It is always wrong

in order of magnitude in another example, namely that of quasi-geostrophic,

vertically propagating Rossby waves II'12 - in which case it is too small,

by a factor of order the Rossby number.) Only when both contributions are

accounted for will the calculated total rate of change of mean momentum

u,tdydz agree, as it must, with minus the horizontal wave-drag force F

exerted by the fluid on the lower boundary. F is defined correct to second

order as the integral with respect to y of

= P'h,xlz= O + O(a3) . (2.8)

This agreement (the detailed verification of which is omitted here) provides

a useful check on the correctness of the overall picture.

2.4 Lagrangian-mean flow and "radiation stress"

There are crucial differences between the foregoing picture, which is

based on Eulerian averaging, that is averaging at fixed values of (y,z),

and the same problem solved using a Lagrangianmean (definable approximately

as the mean following a fluid particle). It turns out that the Lagrangian-

mean secondary circulation is negligible sufficiently far below L. In a

region of steady waves, when Q=O, the fluid partlcles merely oscillate about

a constant mean level, and have no systematic tendency to migrate up or down.

This is no more than mlght be expected for adiabatic motion in stable

stratification. So in a Lagrangian-mean description there is no secondary

circulation linking the regions of wave generation anddlsslpation, and thus

no "Coriolis" contribution to the effective transport of momentum by the

waves. The analogue, in the Lagrangian-mean momentum equation, of the

Reynolds stress - ~ in the Eulerian-mean momentum equation (2o2a), thus

gives a more direct description of the momentum transport, as was recognised

by Bretherton 18,26, who suggested that it be identified as the radiation

stress. Its xz component, taking the place of -u'w r in the present,

Eulerian-mean description, is equal to the "wave-drag" force in the x

direction per unit area across amaterial surface whose undisturbed position

is a plane z = constant - which force is evidently the same as (2.8) when

z=O. These ideas have been further developed by Grimshaw 13 for the

slowly-varying case, and an exact "generalised Lagranglan-mean" descrip-

tion for arbitrary, finite-amplitude waves has been developed by Andrews

& McIntyre 22. The reader is referred to those papers for more discussion

of the differences between the Lagrangian and Eulerian-mean descriptions,

and to Bretherton 26 for a sufficient physical explanation, in terms of the

average Coriolis force on a thin piece of fluid bounded above by a corrugated

material surface and below by a flat, "Eulerian" control surface, as to why

the corresponding momentum fluxes differ in general; see also (4.7) and

(4.9) below.

2.5 More details about the Eulerian-mean secondary circulation

Returning to the example of figure 2, we describe in a little more detail

298

how, in the present description, the forcing term - ~ , y "gives rise"

to the Eulerian-mean vertical velocity W, since this will help motivate the

more general analysis of section 4. In the region below L, where we may

suppose the waves to be in a steady state and not dlsslpating (X'=O, Q'=O),

the forcing term -(v'e~,y is in fact balanced mainly by ~ 8,z on the left

of (2.2b), when we rewrite that equation with the aid of (2.2c) as

8,t + ~ 8,y + ~ 8,z = -{v'S'),y . (2.9)

The term ~ e ~, is O(a 4) and therefore negligible, because ~ is O(a2), and

e,y is also O(=a 2) for the following reason. We have taken ~=O(a2), which

implies that 8,y=O(a 2) because the x component of the curl of (2.2a) gives

(when X=O and ~,~ are O(a2))

~,y = -231U,z + O(a 2) . (2.10)

(This is known in geophysical fluid dynamics as the "thermal wind

equation" ) The other term 8 t on the left of (2.9) turns out to be negligible

unless we are within a distance of order the "Rossby height"

H R = 2~b/(e,z)%

£rom the layer L where the waves are unsteady or dlssipating 9. This point

will be further explained in section 4.6 below. Thus, sufficiently far below

L we have a balance between W~,z and -(~,y , which implies that

~ - ~ y , (2 ii)

where we have defined

= v'8"/(O,z) , (2.12)

again using (2.10) and our temporary assumption that H=O(a 2) in order to

neglect e,y z .

There is another didactic point to be made here, incidentally: it has

o£ten been assumed in the literature, for instance in connection with

thermodynamic arguments, that the nonzero value of ~-~ signifies a tendency

for the waves to transport heat across the channel. Even more than with , this is true but misleading. There is no tendency at all for the

mean temperature to rise on one side and fall on the other, if we are

sufficiently far below L. The adiabatic heating or cooling associated with closely compensates the divergence of ~ This compensation is

intrinsic to the nature of the wave motion, as is underlined by the already-mentioned consideration that indlvidual fluid particles are not

being heated or cooled below L, because the motion is adiabatic there.

The right-hand half of figure 2 schematically indicates the profile

o£ the mean acceleration H If the layer L is shallower than the Rossby It" height H R, then additional contributions to ~ and ~arise in a layer of depth

H R centred on L. These adjust the values of e,t and 5,t in such a way as

299

to keep the thermal-wind equation (2.10) satisfied; there is

a circulation only in a layer of depth H R (see Eq.(4.12) 'room" for such below).

It turns out that Eq.(2.11) still holds for mean flow profiles H(y,z)

and 8(y,z) which vary sufficiently slowly and are such that the Richardson number ~,z/(U,z )2 is large. Then

V e ~,z (2.13)

can be significant below L. The associated Coriolis force does not, however,

lead to an acceleration of the mean flow well below L, because it turns out

that it is always cancelled by an equal and opposite Reynolds stress

divergence if the waves are steady and conservative. The fact that this

cancellation must take place will be seen as a corollary of the much more

general results to be described in section 4.

2.6 Energetics

Our simple example is also quite instructive as regards questions of

energy transport. Suppose that the waves are dzsslpating in the layer L and

heating the fluid there. (The amount of heat involved does not change

significantly within the Boussinesq approximation, but that is beside the

point.) What is the source of this energy? Obviously, the work done by the

agency moving the bottom boundary. How does the energy get from the boundary

up to L ? Answer: there is a vertical energy flux p'w" due to the waves. Indeed, the rate of working by the boundary

-~C = -cp h,x z=O = P w z=O ' (2.14)

in virtue of (2.8) and the fact that w'= -ch

here are wave properties. ,x" All the quantities involved

What we must not forget, however, is that this simple picture, while

correct to O(a2), depends crucially on the circumstance that 5 is zero, apart

from the O(a 2) contribution indicated in figure 2. If we look at the

similar problem in which the boundary is brought to rest

and the fluid is moving past it with velocity H = -c + O(a2), the picture

is quite different. Clearly the boundary can now do no work. The source of

energy is now the kinetic energy of the mean flow near L, whose density is changing at a rate Po times

(~u2),t = UU,t =-cu,t + O(a4) • (2.15)

The integral of (2.15) over the yz domain is, indeed, equa I to Fc, by the remarks above Eq.(2.8).

I n t h i s p r o b ] e m there is no need for the waves to transport

any energy into L at all; and indeed it does turn out that there is no net transport, despite the fact that p'w" is still the same as before. Note first

that in the region of steady, conservative waves below L, the work done

300

across a material surface corrugated by the waves is obviously zero, because

in the present p r o b ] e m such a surface is immobile. Alternatively,

in an Eulerian-mean description of the energy budget correct to O(a2), based

on the model equations (2.1) and (2.2), the total energy tlux across a

horizontal control surface contains two terms which combine to cancel the

contribution Ip'w'dy . The first comes from the O(a 2) part of the advection

by w" of the leading contribution ~po(U + u') 2 to the total kinetic energy:

PoW'(~52 + 5u" + ~u "2) = po H u'w" + O(a 2) . (2.16)

(This contribution to the total Eulerian energy £1ux has been drawn

attention to in the literature just about as often as it has been forgotten

aboutL) The second contribution is the mean pressure-working f~dy

associated with the y-dependent part of the mean pressure whose gradient

balances the Coriolis force associated with H .

The general conclusion to be drawn is that, whenever moving media are

involved, we must expect that a solution to the O(a 2) mean flow problem will

be essential to a self-consistent picture of the way in which waves

contribute to the energy budget. We must also remember that, as always, use

of the energy concept requires us to pay attention to £rames of re£erence!

There is no such thing as "the" net energy transport due to the waves; and

the transport can be identified with p'u" only if the medium is everywhere

at rest.

None of this affects the quite separate fact that the wave property

p'~" is the quantity usually related to the group velocity (when that concept

is applicable). It usually turns out that

p'u" = Ex(group velocity relative to the local mean flow) (2.17)

for plane waves, where E is intrinsic wave-energy density, a wave property

which in the present problem takes the form

E =]~po ( u.2 + S.2/~,z) (2.18)

(The second term is the "available potential energy "27 associated with

vertical displacements of particles in the stable stratification; there is

no internal energy because our model assumes incompressible flow. Brether-

ton & Garrett 28 have analysed the idea of "wave-energy" as a physical concept

in some depth, and in particular have established condltions under which

E can be uniquely defined in a general manner independent of the mathematical

formulation of the wave problem. I shall not go into that here except to

say that, roughly speaking, E is the work you would do in setting up the

disturbance, in a frame of reference in which the mean flow is at rest -

which clearly makes approximate sense in problems of moving media only when

the mean state varies sufficiently little over a wavelength.) I said that (2.17) "usually" holds, by the way, because there are some exotic cases such

as Rossby waves where the two sides differ by an identlcally nondivergent

term. However, in the case of our plane inertio-gravity waves the two sides

301

can be verified to be equal.

3. THE GENERALISED WAVE-ACTION PRINCIPLE

I deliberately avoided writing down the full conservation equations

for the energy budget, partly because the details get quite complicated for

all but the very simplest mean flow structures. Besides, I want to leave

space to introduce another kind of conservable quantity, generalised

wave-action, which apart from its wider significance will lead to a more

powerful approach to problems of the kind just dlscussed. This approach will

depend in no essential way upon any "slowly-varying" approximations.

Wave-action is an O(a 2) wave property which, in its most general form,

satisfies a conservation relation, apart from source terms in X" and Q',

for any mean-flow structure whatever. This remarkable property is to be

contrasted with the equation for wave-energy, whose "right-hand side"

contains a complex of terms representing exchange of energy with the mean

flow:

~E + ~. (~E + p'u') = 5t

. . . . . • - -

= po[-5i,juiuj - (~ij-zizj)e,iuj e - 2 , zt+UjS,zj )]

-Po[~'.~" + 8"Q'/8,z] • (3.1)

wave-energy fails to be conserved as soon as you have a moving medium. Eq.

(3.1) is just for the Boussinesq case, and is derived by dotting the

linearised versions of (2.1a) with PoU" and of (2.1b) with poe'/e,z and adding. We should always keep in mind that (3.1), as implied by the

discussion just given in section 2.6, represents only a part of the whole

energy budget. The linearised equations corresponding to (2.1) are

Dtu + u'.~7u + 2 NAu ÷ pj " - 6)" = " (3.2a)

DtS'+ u'.~z~ = -Q" (3.2b)

V.~" = O , (3.2c)

where the linearised material derivative

Dt( ) = ( ),t + ~.V( ) .

Here we shall define the generalised wave-action correct to O(a 2) only.

(It can be defined exactly, for finite-amplitude disturbances, once one has the generalised Lagrangian-mean description 22, but that is beyond the scope

of this review.) Two preliminaries are needed. The first is to introduce

the O(a) particle displacement field ~(x,t), which is defined to satisfy

302

~7.~ : O, ~ = O, and

Dt~ = u i , (3.3a)

where u ~ is the Lagrangian disturbance velocity

u" u = ~ + [ . V ~ ( 3 . 3 b )

For the problem of figure i, (3.3) simplifies, correct to O(a2),to

Ot~ = v', Ot~ = w', Ot~ = u" + ~u y+ ~U,z , (3.4)

where ~, ~, and ~ are the components of ~.

The second preliminary is a formal device (see Hayes 16 and Bretherton 25)

which is introduced for the sake of the greatest possible generality: we

reinterpret the Eulerian averaging operator ( ) as an ensemble average, over

an ensemble of wave solutions distinguished by a single, smoothly varying

parameter o~ . In a stochastic problem o~ would range over a "sample space';

but random waves are merely one possible case. In the determznlstic problem

of figure i, for example, we can generate a suitable ensemble just by

translating the boundary and the wave pattern a distance ~ in the x

direction. Then (--~ may be trivially re-defined in terms of an integral over

o~ rather than over x. For the axisymmetric mean flows important in

astrogeophysical applications the principle is the same but the details less

trivial 22'25. Quite generally, we nave the basic property

{( ),~} = {( )},~= 0 , (3.5)

whenever the ensemble of disturbance fields depends differentiably upon the

parameter c~ , which we shall take to be the case.

Instead of dotting the linear ised momentum equation (3.2a) with PoU ",

we now take its dot product with the derivative Po~,o~ and average. After some manipulation there results 22'23

~_ +()A ~7.B = -po( ~,o~ .X" + ~,o~ q') + O(a4) (3.6)

where

and

s so that q = O and

A =

B=~A+ t~p"

q" = - e" - ~.7g + 0(a3),

3.7a)

3.7b)

3.8a)

303

(3.8b)

Thus q" = O when Q" = O, i.e. q" = O for adiabatic motion. So A is conserved,

with flux ~ , whenever X" and Q" are zero.

In the derivation of (3.6), the property (3.5), and its corollary that

if f'(~,t,~) and g'(x,t,~) are any two disturbance fields then

f',~ g" = -f'g',~ , (3.9)

and ~ =O(a2). are needed a number of times. Also (2.2) are used, with

Eq. (3.6) corresponds when X" and Q" are zero to one of a class of exact

conservation relations pointed out by Hayes 16, which arise from replacing

certain space or time derivatives by ( ),~ in the classical %nergy- m omentum-tensor formalism 14. The relationship between these exact conserv-

ation laws and the adiabatic conservation laws discovered by Whitham 29 is

discussed in some detail in Hayes" paper. Hayes took the range of ~ to be

(0,27), which makes A unique and is convenient for applications to periodic

or almost periodic waveS, since ~may then be interpreted as phase. However,

it is convenient here not to fix the range of ~ , in order to leave a little

more flexibility in applications. Then A is defined only up to a

multiplicative constant.

Hayes called his conserved quantity the "action" irrespective of what

variational principle was used as the starting point. It can be shown 25 that

A (with O<~<2~) is equal to Hayes" invariant when the governing variational

principle is Hamilton's principle, expressed in its classical sense in terms

of the particle displacements I. A is to be carefully distinguished from

the other conservable quantities to which other varlational principles may

lead, via Hayes" modification of the energy-momentum-tensor formalism, and

which may or may not be wave properties. For example, Hayes" invariant is

not a wave property when the Clebsch-Herivel-Lin variational principle 30

is used as the starting point.

A, or more properly its generalisation to finite amplitude 22, repre-

sents the fundamental, exactly conservable wave propertywnich, in problems

of slowly-varying, conservative waves (X, Q both zero), reduces to the adiabatically-conserved wave-action whose physical meaning and precise

relation to Whitham's adiabatic invariants was elucidated by Bretherton &

Garret, 28. The connection between A and Bretnerton & Garrett's wave-action

can be made via the scalar virial theorem 22, i.e. the result of dotting the

momentum equation with ~ rather than with ~ (This reduces to

"equipartition of energy" in the non-rotating case.) Bretherton & Garrett's

wave-action is defined in those "slowly-varying" circumstances in which the

wave-energy E is uniquely defined, and is then equal to E divided by the

intrinsic frequency, or frequency in a frame of reference moving with the local mean flow, ~÷.

The result (3.6), or more generally (3.11) below, appears to have two

distinct types of appl~cation. One is the same as that envisaged by

304

Bretherton & Garrett, namely to computing the spatial and temporal

dependence of wave amplitude. The second application is to the calculation

of mean-flow evolution. Both applications depend on the fact that A and

are wave propertles, and it has been found in both appllcations that the required information is obtained, in at least some cases 23, from far less

computation than would otherwise be needed.

For reference we quote the corresponding result for a compressible fluid of density p and general equation of state

p = S(8,p) , (3.10)

where p is pressure and 8 potential temperature or specific entropy; 8still

satisfies Eq. (2.1c) in either case. The result is almost as simple as before

(again quite unlike the corresponding wave-energy equation):

[ .... 1 %y + v.~ = -P!,~ .x" - ~ ~ q" p,~ + O(a 3) • (3.11)

where A and B are still given by (3.7) with~) O replaced by E , the mean density, and 32

§ = ~-iOs(~,~)/6~ .

(3.12)

(3.13)

The definitions of ~ and q" are the same as before, except that because of

compressibility we have ~.~ = - p~/p + O(a 2) (pL= p-. ~.~).

4. THE GENERALISED ELIASSEN-PALM RELATIONS, AND THE CONNECTION BETWEEN

MEAN-FLOW ACCELERATION AND WAVE GENERATION OR DISSIPATION IN LONGITUD-

INALLY-SYMMETRIC PROBLEMS

4.1 Conservation of pseudomomentum

For simplicity we now revert to the assumption that the mean flow is

independent of x i, as in the example of figure 1 (where i=l). That is, the

mean flow is invariant to translations in the x i dlrection. Associated with

this invariance is a conservable wave property

Pi = -Po~,i "(ul +~^~ ) (4.1a)

because in this case we may replace ( ),~ by -( ) ,i in the generalised

wave-action principle (3.6). The associated flux is

Qij = 5jPi - ~j,i p" ' (4.1b)

and (3.6) is replaced by

305

Pi,t + Qij,j = Po(~,i "~" + {i q') + O(a4) " (4.2)

(Of course I could have derived (4.2) in the £irst place by dotting (3.2a)

with -~i rather than ~,~. But I wanted to go via the wave-action principle

because of its importance as the starting point for applications in various

other contexts. For instance it is not quite so obvious how to find the

analogue of (4.2) in the more general Kind of longitudinally-Symmetric

problem where the mean flow is rotationally invariant, until we take (3.6)

or (3.11) as starting point and apply it to the ensemble generated by

rotating the disturbance pattern22'25.) For reasons to be explained in

section 5 I shall call Pi the density of pseudomomentum.

There is still no obvious connection between (4.2) and the mean-flow

equations (2.2) - although there would have been at this stage had we been working with the generalised Lagrangian-mean description 22. Because we are

using the Eulerian-mean description, some more analysis will be needed.

First, we use the remarks in section 2.5 to motivate a simple transformation 23,24,32 of the mean-flow equations, which will take them one step

closer to the connection with (4.2). The idea is to subtract out the

contribution to the O(a 2) Eulerian-mean secondary circulation expressed by the stream function ~ of (2.12). The second and £inal step 3,4,23,

24,26,32-35 will involve manipulation of the linearised equations in a way

foreshadowed by the celebrated work of Eliassen & Palm 19 and recently brought to a very general form by Andrews & McIntyre 23'32.

4.2 Preliminary transformation of the mean-flow problem (2.2)

Now take x=x I as the direction of symmetry. Define v and by

v= ~ +v , w=- ~ +w (4.3) ,z y ,

_, _, a 2) so that v , w represent the "residual" O( mean secondary circulation

left over after subtracting out the part corresponding to (2.12). Then a

small amount o£ manipulation converts the mean-flow problem (2.2) into the form

u,t + Uy~* + UzW* = -Sxy,y - Sxz,z -

2Jlu,t + P,ty + v,tt = -Y,t

-8,t + P,tz + w,tt = -Z,t

O,t + ~ yV* + e --* , , z w = -G - rz

_, w,

V~y + W,z = O

(4.4a)

(4.4b)

(4.4c)

(4.4d)

(4.4e)

where

Uy = H,y - 237_ , Uz = H,z (4.5a)

308

Sxy = ~ - UzV~e~/e,z (4.5b)

SXZ = u'w" + UyV'8"/8,z (4.5c)

O = w'e ''~ + v'e' e,y/e,z (4.5d)

-- (v "e"/~, z) = (v ~" ) ,y + (v'w'),z + ,zt + Y + O(a4) (4.5e)

= (v'w'),y + (w---~") ,z - (v'e'/e,z) ,yt + ~ + °(a4) • (4.5f)

Correct to O(a2), (4.4) can be regarded as a set of equations for the five unknowns

{~,t , V*, ~*, §,t ' P,t} , (4.6)

since Uy, U z, ~,y and e,z can be regarded as known "coefficients" apart from contributions O(a2). We are of course thinking of the linearised wave problem as having been solved, so that the wave properties on the right are

known forcing terms.

The transformation (4.3) is dependent on our choice of coordinates; a coordinate-independent preliminary transformation may be used instead, at the cost of getting more complicated-looking versions of the results 32.

4.3 Excess momentum fluxes and generalised Eliassen-Palm relations

We now multiply the x component of (3.2a) by ~ and then by ~ 26, and average. The resulting pair of relations reduces, after a little manipulation in which we use (3.4), (3.9) (with ~ replaced by x) , and our

assumption that ~ = (~,O,O) + O(aZ), to

u'v" + pjl~7, x p- _ Uz ~ = ~ + _IUy(-~),t + (~u'),t + O(a4) (4.7a)

u w + po -I ~,x p" - Uy~-~ = ~ + ½Uz(~),t + (~u'),t + O(a4)" (4.7b)

The second term in (4.7b) is to be compared with (2.8), the wave-drag force - are the excess momentum fluxes per unit area, and indeed ~,xp" and - p;

(or minus the "radiation stress" components) in one of the forms in which they appear in Lagrangian-mean analogues of (2.2a) 13'18'22. Eqs.(4.7)

relate these to the "Eulerian" excess momentum fluxes u'v" and ~-~. Note that -~,x p~ and - ,x~ are also equal to the y and z components of the nonadvective part of the flux of the pseudomomentum component PI; see

(4. ib).

Next we multiply (3.2b) by ~ and average to get

-v'8" + (~e'),t + ½e,y(~),t + e,z~ = _-~v + o(a4 )

and ~ times the first plus ~ times the second of (3.4) gives

, (4.8)

307

i ~ i , t = ~ + ~"w"~ (4.9)

These may he used to eliminate ~v? and ~w" from (4.7). The result is a pair

of relations like (4.7) except that Sxy and Sxz appear in place of u'v" and ~ , and the remaining terms, apart from ~xp-~ and ~ , are either of the form [--5, t or contain a factor X" or Q'. If the first relation is differentiated with respect to y and the second with respect to z, and (4.2) used to eliminate the two terms in p" , there results

Sxy,y + Sxz,z -- (~X~,y + (~---fS.z +

+ .&x~ + ~,x=V uz(~l~,z),y + UyC~V/~,zl,z

+ + - pol l - Oz C o' ÷ -

+{Uy[(~-~ + ~Oy ~)/[9,Z] + Uy~ + ~Uz~},z ] + O(a 4) . (4.10a

By multiplying (3.2b) by 0", averaging, and differentiating with respect to z, we get

G,z = I~/~,z),~ + ~-~/~,z],z + o(a4) 1 (4.10b)

4.4 Deductions

The results (4.10) imply that the O(a 2) forcing terms on the right of the mean-flow problem (4.4) can be expressed as a sum of terms each of which falls into one of three categories:

(i) the Eulerian-mean external forcing terms ~ and Q (which we shall choose to regard as unconnected with the waves),

(ii) wave terms of the time-differentiated form ( ),t ,and

(iii) wave terms all involving X', Q', or q" , that is, all depending explicitl Z on the forcing or dissipation of the waves.

This immediately shows that in the proDlem of figure l, for instance, the forcing of mean-flow changes vanishes below the layer L, where the waves are steady and conservative, for any initial pro£iles of mean flow and stratification, no matter how complicated, and no matter whether or not approximate, "slowly-varying" descriptions of the waves are valid.

It also carries the implication that although strictly conservative waves or instabilities (X', Q" and q" zero) can change the mean flow if they themselves are growing or decaying in amplitude, such changes are temporary

in that no net change to the mean flow persists if the waves propagate cut of the region of interest. This is almost obvious 23 from the fact that all the O(a 2) wave terms on the right of (4.4) can then be written in the form (-~,t ,with the aid of (4.10). However, if avery small amount of dissipation

308

is present, its effects can be greatly enhanced by the occurrence of

mean-flow changes that would otherwise be temporary. For a striking example

of this, see reference ii. A change in the mean-flow profile due to wave

transience can bring about an approach to "critical-line" conditions

somewhere, giving the dissipation terms a chance to take over locally, in

turn causing further and more permanent mean-flow changes. Theoretical work

on such highly nonlinear feedback effects, which could easily be important

in the evolution of stellar differential rotation, for example, is still in its infancy 7'8.

4.5 Extensions

Precisely analogous results, enabling the same qualitative conclusions

about mean-flow evolution to be drawn without solving wave problems in

detail, have been derived for:

I) rotationally as well as translationally invariant mean flows, with

mean velocity predominantly in the longitudinal (azimuthal) direction 22-2~ 32

2) a fluid with a general equation of state p = S(8,p); the Boussinesq

approximation is not essential 22'25'32

3) a self-gravitating fluid 22

4) a conducting fluid, with mean magnetic field as well as mean velocity

predominantly longitudinal 31.

4.6 Simplifications for the problem of section 2

It is of interest in connection with the previous discussion to exhibit

the approximate form taken by the transformed mean-flow problem for the

almost-plane inertio-gravity waves of figure i. A self-consistent set of

approximations requires that the Richardson number ~,z/(H,z)2 be large, and

the depth of the layer L, and other scales of mean variation in the vertical,

including the Rossbyheight H R= 2_O.b/(~,z )i , large compared to the vertical

radian wavelength m -I. The horizontal scale b is then large also, compared with k -1, assuming that both terms in the dispersion relation (2.6) are not

too different in magnitude. The upshot of such approximations is that all

the terms of the form (-~,y or (--) ~ in (4.10) become negligible, and (4.4)

can be shown to simplify, when X and ~ are zero, to

U,t - 2/i~* = - ~,x.X" - ~,X q" + poIPl,t (4.11a)

2]lu,t + P,ty = O (4.11b)

-8,t + P,tz = O (4.11c)

- -* (4. lld) ~,t + ~9,zW = O

V,y + w, z = O • (4.11e)

808

According to the second and third equations the mean flow stays in

geost~ophic and hydrostatic balance as it changes: knowledge of P,t implies

knowledge of ~,t and 8,t " (The thermal-wind equation (2.10) holds, with

its O(a 2) contribution negligible). If we eliminate 5 t and 8 t in favour of P,t in (4.11a) and (4.11d), and cross-differentiate to eliminate ~* and m* w via (4.11e), there results

{--~(~yT. + ~Z--~ -~-HR~ ~I P,t- = 2J9-{ ~,x.X" + ~,X q; - polPl,t} y, • (4.12)

The form of the elliptic operator on the left shows why the Rossby height

= 2J~b/(8 z)% takes over as the vertical scale of the response of the H R mean flow, whenever the forcing on the right of (4.12) is confined to a layer

L of depth smaller than H R .

5. PSEUDOMOMENTUM IS NOT MOMENTUM

It is surprising how often one seems to encounter the conceptual mistake

that since waves can transport momentum, they must possess it. No less than

Lord Rayleigh 36 appears to have been under this impression when he wrote

that "if the reflexion of a train of waves exercises a pressure upon the

reflector, it can only be because the train of waves itself involves

momentum'. Nowadays one often reads about "the momentum of the waves" and

abo~t waves "exchanging'(their) momentum with the mean flow, and suchlike;

and there appears to be a tendency to assume that this momentum which the

waves are supposed to have is to be identified with the wave property Pi defined in (4.1a), or rather the wave property

Pi = Eki/~+ , (5.1)

to which Pi can be shown to reduce in those "slowly-varying" circumstances where Bretherton & Garrett's 28 arguments apply.

On the other hand, as Brillouin 37 pointed out in 1925, Rayleigh's

statement is a non sequitur because in a material medium you can perfectly well have a nonzero flux of momentum unaccompanied by any momentum density

- try leaning against a brick wall. Two specific counterexamples to

Rayleigh's statement which I happen to know are the obvious one of waves in solids (phonons) 37'38 , and a simple fluid-dynamical example Ipublished

in 197340 . In the latter example, apacket of "inertia" (pure Coriolis) waves

propagates along a waveguide comprising an incompressible , homogeneous

liquid between rigid, parallel boundaries in a rotating frame of reference.

The mean momentum is zer____~o, for reasons of mass continuity. Nevertheless

310

there is a non-zero recoil force when the wave packet is reflected from an immersed obstacle.

That problem has the further interesting feature that the wave packet

does have a well-defined "fluid impulse', I, which gives the recoil force.

is not equal to the integral of P; indeed it can have the opposite sense]

(In fact the momentum flux due to the mean pressure ~ plays a leading role in this particular problem.)

An example of a somewhat different kind is the celebrated problem of

a packet of electromagnetic radiation in a refractive medium. This problem

has long been controversial, but has been convincingly clarified in recent years by Penfield & Haus 42, Gordon 39, and Peierls 38, to whose papers,

together with the review by Robinson 41, the reader is referred for some very

interesting history. In this problem, provided we neglect dispersion, there

is a definite, non-zero total momentum M i which travels with the waves.

Again, this is not generally equal to the integral of Pi (or rather its

electromagnetic counterpart, the "Minkowski quantity'), nor is it equal to

the electromagnetic part of the total momentum (the "Abraham quantity').

On the other hand recoil forces, are, this time, simply related to Pi (but not to Mi!) in at least some circumstances 39.

Brillouin's point is that waves don't have to possess momentum; the

examples show that in fact they sometimes do and sometimes don't - and that

when they do, the momentum is not necessarily related to recoil forces. The

wave property Pi may or may not be closely related to either; and whether

it is depends in fact on 91obal considerations - on the full O(a 2) mean

problem and its boundary conditions. So we must either say that Pi may

sometimes "be interpreted" as momentum, and sometimes not, depending on the

global problem - surely a most unsatisfactory conceptual structure - or we

must decide that Pi is simply an entity in its own right, not necessarily related to any momentum, whereupon the conceptual problems disappear. This

is why I like to have a separate name "pseudomomentum" for Pi, just as one

likes to have separate names for other pairs of quantities, like energy and

torque, which have the same dimensions but different physical natures. I

want to use the term "momentum" in its ordinary, elementary sense, of course

(which we use when thinking intuitively about forces and accelerations).

The terminology follows the usage of workers in solid-state physics, to whom all this has naturally been more obvious than to most. Blount and Gordon 39

have carried the terminology, and the distinction between momentum and

pseudomomentum, into classical electrodynamics, and shown how it helps

clarify the issues of the so-called Abraham-Minkowski "controversy" over

electromagnetic waves referred to above.

"But, the reader may say, you said in section 3 that Hayes

conservation relation results from replacing certain space or time

derivatives by ( ),~ in the variational derivation of the conservation

relation for the energy-momentum tensor. So the result of going in the

reverse direction, which is just (4.2) without the "dissipative" terms, is

nothing but the conservation law satisfied by the "momentum" part of the

311

energy-momentum tensor itself. So surely Pi i_ss a momentum."

My reply is that that would follow if the same mathematical formalism

always represented the same physical entity. Here, however, the blanket

term "energy-momentum tensor" tends to obscure the fact that different

variational principles, springing from different basic formulations of the

physical problem, can be used as starting point. Different basic formulations carry different implications about the kinds of invariance

properties associated with conservation laws. Certainly Eq. (4.2) is

associated with invariance under translations in space, just as is

conservation of momentum. But the translational invariances referred to

are in fact quite different, a point well made by Peierls 38. Conservation

of the i-component of momentum, for instance, depends on invariance of the

basic physical problem, including force potentials (e.g.gravitational),

under translations in the x i direction. Conservation of Pi depends on

translational invariance of the mean flow, insofar as it enters into the disturbance problem. This condition does not necessarily involve external

force potentials. Other necessary conditions for the two kinds of conserv-

ation law to hold are also quite different. For instance, whether or not

momentum is conserved has nothing to do with whether or not the motion is

adiabatic, while conservation of Pi does depend on the motion being

adiabatic since it requires, inter alia, that q" = O in (4.2).

It is the description in terms of particle displacements (and Hamilton's

principle in the classical sense) that lies behind the appearance of

pseudomomentum rather than momentum in the "energy-momentum" tensor. By

contrast, if we form components of the Eulerian-mean energy-momentum tensor

from the usual pure-Eulerian variational principle in fluid dynamics, namely the Clebsch-Herivel-Lin principle 30, then in place of Pi and its flux

Qij we get the Eulerian-mean density and flux of momentum ,

pu i and P6"ij - Puiuj , (5.2)

to within an identically nondivergent contribution. This should be no

surprise in view of the foregoing remarks on translational invariance.

Conservation of pseudomomentum, as distinct from momentum, is connected

with invariance to a displacement of the disturbance pattern while mean

particle positions are kept fixed, as distinct from a displacement of the whole system, particles as well as disturbance pattern 38. The idea of "fixed

mean particle positions" cannot be directly expressed within a purely

field-theoretic or Eulerian description, which does not keep track of where

fluid particles are. But it is implicit in a description of the disturbance

in terms of particle displacements (from "mean particle positions'). (What

this means for finite-amplitude disturbances is dealt with in reference 22.)

One possible reason why momentum and pseudomomentumhave sometimes been

mistaken for one another may be that in certain examples, even~ore idealised

than those already cited, not only are both quantities conserved (requiring

X'and Q~to be zero in the case of pseudomomentum), but also their conservation

relations reduce to the same form. If in these examples we generate the waves

312

starting from an initial state in which momentum and pseudomomentumare both

zero, it can happen that they evolve in parallel and remain equal. The

simplest example is the trivial one of an electromagnetic wave in vacuo.

Here there is no medium present to make the translational symmetry

operations different. But there are also examples involving waves in media,

all of them longitudinally-symmetric problems, in which the lon@itudinal

components of pseudomomentum and mean momentum evolve in parallel. Perhaps

the most celebrated example is that of Stokes" periodic waves on the surface of an infinite, inviscid ocean. The initial conditions of no motion are

hidden in the assumption of irrotational motion~ Approximate longitudinal

symmetry would be enough for approximate conservation of pseudomomentum;

but we need also that there be no mean horizontal pressure gradient and that,

concomitantly, the mean mass continuity equation (which constrains the

distribution of mean momentum but not that of pseudomomentum) plays no

significant role. Exactly the same considerations explain why a further such

example is provided by the problem of section 2, in the case when H R << H, the scale of the layer L. Eqs. (4.11) then imply that, for conservative

waves,

poU,t = Pl,t (X=O, Q=O, HR<<H). (5.3)

These examples are very special, and in any case the question of whether

or when there happens to be a momentum density equal to Pi is not the most

relevant one in practice. Statements which are more useful and general can

be made about the fluxes of momentum and pseudomomentum. Especially when a Lagrangian-mean description is used, the excess momentum flux due to the

waves is often simply related, although not usually equal, to the flux of pseudomomentum. It is basically this fact which accounts for examples of

the kind just mentioned. It is also why (4.2) could be used to eliminate

the terms in p" during the derivation of (4.10a) from (4.7). The reasons

for the existence of such relations are hinted at by Eqs. (2.14), (2.17),

(5.1), and the well-known argument about the relation between wave-drag,

phase speed, and the rate of working across a material surface.

More explicitly, in many "slowly-varying'situations it turns out that

the mean flow can be defined in such a way that (i) the excess momentum flux is the only wave term in the leading

approximation to the O(a 2) mean-flow problem (Eqs.(4.11) provide an example

of this), and (2) the excess momentum flux is then either equal to the pseudomomentum

flux, or differs from it by a contribution Cij which in some cases does not cause systematic mean-flow changes because it can be balanced quasi-

statically by the reaction of the medium. When MHD effects are not involved, and the fluid is compressible, Cij is

an isotropic~ pressure-like contribution which can be thought of as a kind

of acoustic "hard-spring" effect coming from the nonlinearity of the equation of state Ig'37'44. Analytically this results from redefining the

mean pressure in such a way as to avoid having a wave term in the equation

of state for the mean flow. For electromagnetic waves in refractive media Cij is again isotropic 42'39 and comes from "electrostrictive" and "mag-

The same remark applies in the classical theory of sound waves.

313

netostrictive effects 41. In MHD problems Cij is not, however, isotropic 17.

Whether or not such statements are helpful or misleading depends on whether or not, in the problem in question, the difference Cij between the pseudomomentum and excess momentum fluxes has time to be balanced quasi- statically by the mean stress in the medium (it usually does have enough time in "slowly-varying; situations), as well as on whether that mean stress happens to affect the answer to the particular question being posed. (To take a classical example, Ci~ is relevant to the force exerted by the absorption of acoustic waves znto the end wall of a closed container, but not into an absorber immersed within a larger volume of fluid 43,44. It is

presumably the former situation more than the latter, incidentally, to which the problem of solar wind acceleration by Alfv4n radiation pressure is analogous.) When (5.1) holds, and "group velocity" is meaningful, it is often true that the analogue of (2.17) holds also, namely that Qij equals Pi times the jth component of the group velocity. So there are some slowly-varying situations (those in which the questions being asked permit Cij to be ignored in some sense) where one can say mnemonically that the

waves transport momentum as if a local momentum density equal to Pi were being carried along through a vacuum at the group velocity. But the difficulty of saying in general terms when this mnemonic is applicable, and

when it isn't, brings us back in the end to the point made earlier: the only safe and completely general recipe for studying wave transport effects is to consider not only the "wave properties~ which can be evaluated from the O(a), linearised problem, but also a self-consistent analysis, correct to O(a2), of whatever global mean-flow problem is relevant.

REFERENCES

i. Lindzen,R.S. 1973 Boundary-layer Meteorol.4, 327-43 2. Holton,J.R. 1975 The dynamic meteorology of the stratosphere and

mesosphere. Boston, Amer.Met. Soc., 218pp. 3. Fels,S.B. & Lindzen,R.S. 1974 Geophys. Fluid Dyn.6, 149-91 4. Plumb,R.A. 1975 Q.J.Roy.Meteorol.Soc.lOl, 763-76 5. Spiegel,E.A., Gough,D.O., personal communication 6. E.g. Sakurai,T. 1976 Astrophys.& Space Sci.41, 15-25 7. Holton,J.R. & Mass,C. 1976 J.Atmos. Sci.33, 2218-25 8. E.g. Grimshaw,R. 1975 J.Atmos.Sci.32, 1779-93 9. Eliassen,A. 1952 Astrophysica Norvegica 5, 19-60 iO.Phillips,N.A. 1954 Tellus 6, 273-86 ll.Matsuno,T. 1971 J.Atmos.Sci. 28, 1479-94 12.Uryu,M. 1974 J.Meteorol.Soc.Japan 52,481-90 13.Grimshaw,R. 1975 J.Fluid Mech.71, 497-512 14.Landau,L.D. & Lifshitz,E.M. 1975 The classical theory of fields, 4th

English edition. Pergamon, 402 pp. See also Dougherty,J.P. 1970 J. Plasma Phys. 4, 761-85

15.Eckart,C. 1963 Phys.Fluids 6, 1037-41 16.Hayes,W°D. 1970 Proc.Roy. Soc.A 320, 187-208 17.Dewar,R.L. 1970 Phys.Fiuids 13, 2710-20 18.Bretherton,F.P. 1971 Lectures in Appl. Math. 13, 61-102 (Amer.Math. Soc.)

314

19.Eliassen,A. & Palm,E. 1961 Geofys. Publ.,22#3, 1-23 20.Charney,J.G. & Drazin,P.G. 1961 J.Geophys.Res.66, 83-109 21.Moffatt,H.K. 1977 Magnetic field generation. Cambridge Univ. Press (to

appear) 22.Andrews,D.G. & McIntyre,M.E. 1977 Submitted to J.Fluid Mech. 23.Andrews,D.G. & McIntyre,M.E. 1976 J.Atmos. Sci.33, 2031-48 24.Boyd,J. 1976 J.Atmos.Sci. 33, 2285-91 25.Bretherton,F.P. 1977 Submitted to J.Fluid Mech. 26.Bretherton,F.P. 1969 Q.J.Roy.Meteorol.Soc.95, 213-43 27.Lorenz,E.N. 1955 Tellus 7, 157-67 28.Bretherton,F.P. & Garrett.C.J.R. 1968 Proc.Roy. Soc.A 302, 529-54 29.Whitham,G.B. 1974 Linear and nonlinear waves. Wiley. 30.Bretherton,F.P. 1970 J.Fluid Mech 44, 19-31 31.Andrews,D.G., personal communication 32.Andrews,D.G. & McIntyre,M.E. 1977 To appear in J.Atmos. Sci. 33.Eliassen,A. 1968 Geofys. Publ.27#6, 1-15 34.Dickinson,R.E. 1969 J.Atmos.Sci.26, 73-81 35.Uryu,M. 1973 J.Meteorol.Soc.Japan 51,86-92 36.Rayleigh,Lord 1905 Phil.Mag.lO, 364-74. (Sci.Papers,5, 262-71) 37.Brillouin,L. 1925 Annales de Physique 4, 528-86 38.Peierls,R. 1976 Proc.Roy. Soc.A 347, 475-91 39.Gordon,J.P. 1973 Phys. Rev.A 8, 14-21 40.McIntyre,M.E. 1973 J.Fluid Mech 60, 801-11 41.Robinson,F.N.H. 1975 Phys.Reports (See.C.of Physics Letters) 16, 314-54 42.Penfield,P. & Haus,H.A. 1966 Phys,Fluids 9, 1195-1204 43.Brillouin,L. 1936 Revue d'Acoustique 5, 99-111 44.Rooney,J.A. & Nyborg,W.L. 1972 Amer.J.Phys. 40, 1825-30

WAVE GENERATION AND PULSATION IN STARS

WITH CONVECTIVE ZONES

Wasaburo Unno Department of Astronomy, University of Tokyo

Bunkyo-ku, Tokyo, JAPAN

SUmmary. Wave generation processes are classified in (i) strong and (2) weak, and (a)

Spontaneous and (b) stimulated processes. Then, the case (2b) operating in convective

Zones is discussed in detail. Both the dynamical and the thermodynamical coupling be-

tween pulsation and convection are formulated by use of the diffusion approximation for

the turbulent convection. A mixing length variable with time is thereby introduced.

The work integral is transformed so that each of its terms can reveal the mechanism re-

sponsible for the stellar stability. An important destabilizing mechanism associated

with the convective flux is found to exist among other known mechanisms. The mechani-

cal work is shown to be rather important.

Wave generation processes. The genuine hydrodynamleal generation of waves is due to

the nonlinear Reynolds stresses (Lighthill 1952, Unno 1964). The adiabatic wave gene-

ration in an isothermal atmosphere was thoroughly studied by Stein (1967). In this

Case, the waves are generated spontaneously. The propagation is not isotropic, but the

anisotropy is not very strong because of dominating quadrupole emissions. The wave am-

plitude in situ is small in subsonic turbulence, bu tthe effect can be appreciable after

the waves propagate in the outer layers. On the other hand, if the medium is made strong-

ly anisotropic by the presence of magnetic field or rotation, the monopole and dipole can

be very important, and the wave generation can be strong. The generation of Alfv~n

waves from turbulence under the presence of a strong magnetic field was found to be very

effective (Kato ]968, Roberts ;976). A change in the basic structure of the medium is

then expected. Spiral arms in galaxies, spicules and sunspots (Parker 1974) may be

the manifestation of such cases.

For waves that are trapped in some region of a star, the stimulated emission should be

Considered. The emitted wave and the underlying oscillation have a phase relation so

that the whole process forms a self-exciting system. Therefore, in principle, the

strong stimulated generation of waves may not be an inaccurate concept. Osaki (1974),

however, considered that the resonance between the nonradial oscillation and the over-

stable convection in a fast rotating core could be the cause of the 8 Cephei variabil-

ity. In such a case, the theory remains qualitative.

For trapped waves or pulsations, the weak stimulated emission can be accumulated and

become important. The thermodynamieal excitation of pulsation has been worked out by

many authors (Zhevakin 1953, Baker and Kippenhahn 1962, Cox 1963, Christy 1964) as the

316

cause of the variabilities of Cepheids and RR Lyr stars. Less investigation has been

done for stars having deep convective envelopes because of the theoretical difficulties

i~ the treatment of a convective zone.

Modulation of convection by pulsation can be calculated on the basis of the mixing-

length theory (Vitense 1953) slightly generalized to include the time dependence (Unno

1967). Recently, Gabriel, Scuflaire, Noels and Boury (1975) calculated the thermody-

namlca! coupling of the convection with the nonradial pulsation, and they demonstrated

an appreciable effect of the convective flux perturbation on the stability coefficient.

The dynamical coupling has been neglected so far. But, it is caused by the 2erturba-

tions in the turbulent pressure, visocsity and conductivity, and its effect on the sta-

bility is not negligible as shown later.

Table i. Classification of wave generation mechanisms

I. STRONG

(monopole) dipole anisotropic

2. WEAK

(quadrupole)

isotropic

A. SPONTANEOUS B. STIMULATED

noise {phase relation propagating "trapped

STRUCTURE CHANGE

8-Ceph spicules spiral arms sunspots (Osaki) (Kato, Parker)

Effects on Outer Layers Excitation of Pulsation

homog. (Lighthill) thermal & dynamical 5mn Oscill. pulsating stars

isothermal (Stein) ~stability

Table 1 summarizes the classification of the wave generation mechanisms discussed above.

The excitation of spiral arms in galaxies (Mark 1976) is considered as an example of a

strong emission mechanism. The exeitation of the solar 5 min. oscillation studied by

Ando and Osaki (1975) belongs to the weak stimulated emission mechanism. In these two

examples~ however, the coherence in spatial and temporal wave patterns is not completely

perfect, and the emission mechanism may better be considered as partially spontaneous

and partially stimulated. The solar stability (Dilke and Gough 1972, Boury, Gabriel,

Noels, Scuflalre and Ledoux 1975, Shibahashi, Osaki and Unno 1975) is an interesting

example involving the weak stimulated emission mechanism. The convection-pulsatlon

coupling is important. A qualitative change in the underlying solar structure discussed

in the present Colloquium may not take place, since the emission mechanism is weak.

Basic equations °f the pulsation-convection coupling. We shall hereafter restrict our-

selves to study the mechanisms of pulsational stability operating in the convective zone.

At present, no complete description of the compressible inhomogeneous turbulence is a-

317

vailable. We shall, therefore, approximate the nonlinear effects of the turbulent con-

vection by the eddy dlffusivltles. Then, the conservation equations of mass, momentum

and thermal energy are described by (Unno 1969)

d~ 1 dt ~- pV-m+~V-(<v>V~) %- ~V.~ , (I)

d~ 1 i d---{ = - ~ V(P+Pt) - V~ + ~ [V(<~>V-u) + (V.<~>V)n] , (2)

d! = ! (~N + ~V 1 1 dt T - ~ V'FR) + ~ V. (<%>VS) , (3)

where d/dt is the Lagrangian time differentiation, P, T, P, S and ~ denote density,

temperature, pressure, specific entropy, and the velocity, eN, EV, ~R and Pt denote the

nuclear energy generation rate, the turbulent viscous dissipation, the radiative flux,

and the turbulent pressure, respectively, and <l>, <~> and <v> describe the coeffi-

cients of turbulent conductivity, viscosity and diffusion that are approximated by

<pu'%>, Z being the mixing length and the prime indicating the convective fluctuation.

The small scale fluctuations inside a representative convective element have been smooth-

ed out so that the Pt(=<pu'2>), <l>, <~> and <~> appear from their nonlinear effects.

Since the turbulence has a continuous energy spectrum, the magnitude of these terms

should depend also on the scale of motion under consideration (Nakano 197~), but this

dependence will not be explicitly described in this paper. We shall also neglect <~>,

since mixing is efficient in the convection zone. The viscous dissipation ~V which is

dimensionally given by

C V ~ u'S/E (4)

in accordance with equation (2) should not be disregarded, since for the adiabatic con-

vection the following relations,

~V0 ~ (llP0)<%>oVTo'VS0 ' (5)

~co ~ - <X>o TorSo • (6)

reduce essentially to the original Biermann formalism (1932) and then the energy con-

servation in steady state,

P0~N0 = V'(FR0 + ~CO ) , (7)

is ensured by equation (3). Here FC denotes the convective flux and the subscript 0

indicates the statistically steady undisturbed state. The equilibrium structure is

determined by

(I/o0)V(P 0 + Pt0 ) + V{ 0 = 0 (8)

318

in addition to equation (7). Subtraction of these equilibrium equations from the basic

equations (2) and (3) yields the equations for perturbations. The difference in spatial

and temporal spectra between pulsation and convection can be used to separate the system

of equations governing pulsation and convection from each other.

The work in te$ral of the nonradial pulsation.

- im (~p /p 0 + ¥ ' 6 r ) = 0 ,

- m26t + (I/P0)WPI - (Pl/p02)¥P0 = ~i '

- i~6S = ~ ,

The equations of pulsation are given by

(9)

( i 0 )

( 1 i )

where ~q and qlrepresent the Lagranglan and Eulerian perturbations of any variable q, t

the time differentiation d/dt0=8/~t+u 0 .V 0 of the Lagrangian perturbations are replaced

by -I~ , and

1 Pl ~0 3 1 : - v¢1 - ~o v P t l + TjZo v e t ° - t v ( < ~ > ° " ~ ) + ( v ' < ~ > ° v ) ~ 1 ' ( n )

~ = ~[T-I (EN + eV - O-I¥'FR ) + P-I¥'<%>¥S] (13)

The effects of the modulated convection enter through Ptl in 71 and ~gV and ~<l> in

~. The calculation of these terms will be made later.

Now we can define the pulsation energy Ep per unit volume by

2 + ~ dPo ) tdlnP0 _ ~ ) - i P1 - p9~01 ) 2] I (14) Ep= ½Po [ml -c2po 2 (- ~ "71dr (71P0

where c2=YiP0/Po and 71 =(~inP/81nQ) S. It consists of the kinetic energy and the poten-

tial energies of acoustic and gravity waves. From equations (9), (i0) and (ii), after

some manupilations, we obtain

~Ep ~t + ¥" (PlUl + EpU0) = P0(~I" ~I + ~T6~ ) (15)

Integrating this equation over the whole volume of a star and assuming the boundary !

conditions of PI~I=0 and 40=0 at the surface, we obtain

~d I EpdV = W = WM + WT ~ Re I d t O0(~I*'~I+6T* ~ )dV" (16)

The result is independent of the quasi-adiabatic approximation and the Cowling approxi-

mation used above.

The mechanical work W M is the sum of the turbulent pressure work Wp and the turbulent

viscous stress work WS;

3 1 9

Wp = Re -i~r*.V~PtdV = 2m f Ptolm[(~p*/00)(~u'/u'0)]dV ,

W S = Re Ul*- (-~m) [V(<P>0V-~r) + (V-<~>0V)~r]dV

-- -c°~ f<~>0 (16P/Po 12 + IV~f2)dV

The calculation of Im[(Sp*/p 0)(6u'/u0)] will be given later.

W T consists of the nuclear energy work WN, the viscous dissipation work WV, the radia-

tive work WR, and the convective work W C. After some manlp~%lations, we obtain

WN=Re f p0(~T*/T0)6eNdV = f P0ENO(gN)T,adI~T/T0'2dV , (19)

W R = Re f p0(dT*/T0)6(O-IV-k'VinT)dV

- []-~'(~'0 ) + ~ I~01']dV + I 2V°]F"^Re[~T* ~6r,.. ~0 ~u ~0 @T'~v f (~26r 2 ~6r 2-£ (£+i)

+ Re FRO ~0" ~ - r ~r + r ~ 6r)dV , (21)

W C = Re I P0 (6T~/T0)6(Tp-IV'<X>VS)dV

= Y3-1 - F'~3-i- - u ~0 z0

where

+ Re f FC0 ~6T, 6u' ~0 ~ f F 2T~V.-_ -f~Q Re[~* ~r- T O Dr (u~ ~ + )dV + u i 0 -0 ~-~--]dv

f dT*f82~r 2 8~r 2-Z(£+I) 6r)dV + Re FC0 TO0 " ~ r Dr + r 2

.81nT. =d_~_~ H0 dr Y3-1 = [~-~no)S , V 0 dlnP0 ' dlnP 0 '

4acT ~ (81nK. .~l..~..~ I- K = 3<p ' ~ = t~--~TnT)S ' (gN)T,ad = t~inT )S

(17)

418)

The thermodynamieal work

(22)

L denotes the surface luminosity, and equation (6) has been used. The radiative con-

ductivity K appears in the expression of the radiative flux FR, ~R=-I<VInT. In calculat-

ing W R and WC, the following identity has been used,

~Bo (~2~r 2 ~r 2-£(I+i) dr) - 2V.(AoVB 0) ~r ~(V'AoVB0) = - A0 ~-r-- ~ r 8r + r Dr '

320

where the spherical harmonics yim(@,~) for the angular dependence of a normal mode has

been assumed. The work integrals will be transformed to more convenient forms for re-

vealing physical processes, after the convection modulated by the pulsation is discuss-

ed in the following.

Time dependent convection.

out from the basic equations (i), (2) and (3) as follows,

Equations of the time dependent convection can be separated

dp'/dt I + V.(pu') % V.(pu') % 0 , (23)

1 p' 1 , p' (d~l +~)~' +~VP'- ~r YP =- ¥~ ' - ~'Pt +~ YPt (24)

dS'/dt I +~"¥S = [T-1(e N + e V - p-IV'~R) + p-l¥'<X>VS]' , (25)

when d/dtl=~/~t+~l.¥ , TV~Z/u', the primed quantities are the Eulerian perturbations due

to convection, and the coordinate and the quantities without prime include the Lagrang-

Jan perturbations due to pulsation in addition to the equilibrium values.

Now we neglect the right hand side of equation (24) and use the Boussinesq approximation

instead of equation (23). Then, using the lateral component of equation (24) to elimi-

nate P' from the radial component, we obtain

(d+! , k~ 2 p' BP k~ 2 T' ~P Tv)U r = k - - ~ ~ k--- f- IpTl ~- (- ~) , (26)

where pT=(alnp/alnT)p and a spatial dependence of exp(i~-~) has been assumed for the

convection variables. Also, equation (25) can be rewritten as

!

(d~1+!+!-!)S' +u r ~s°=0 TC TR ~N Dr

(27)

where TC, TN, and T R denotes the time scales of the turbulent conduction, the heating

by nuclear burning, and the radiative cooling of a convective element. We shall here-

after write u' for u r' and T for ~V and T C without distinguishing the different numeri-

cal factors of the order of unity that are unimportant in the stability analysis.

Equations (26) and (27) have to be supplemented by the definition of the mixing length v

A which is modulated by pulsation. We assume that a convective element born at time t

has a mixing length equal to the instantaneous scale height H (=-~r/~inP) initially and

evolves according to the law 0A3=const. during its llfe time T. Then, the relative ex-

is given by ([6H]/H0)e-i~' at its birth and will be further increased by cess

(-i/3)([~p]/p0)(e-i~t-e-i~t') during the time interval from t' to t, the time dependence

in ~H and ~p being expressed explicitly and [~H] and [~p] being the compressible ampli-

tudes. For the average convective element, we obtain

321

[%~_~0 e-~0t = r[[6H___~] e_i~t' _ 1 [~P] (e_i~t _gi~t')]exp( - ~') d(t~t') ~0 H0 3 Po

or

~ 1 (~H iw'c 6p) o+ -Wo '

(28)

assuming a constant birth rate and constant life time for the convective elements. A

more precise theory for the time-dependent mixing length has been developed by Gough

(1976).

For equilibrium (d/dtl=0), equations (26) and (27) reduce essentially to the Vltense

(1953) formalism. The modulation of convection by pulsation can be calculated from

the Lagranglan variations of equations (26) and (27), (Unno 1967, Gabriel, Scuflaire,

Noels and Boury 1975). For qualitative study, we now consider an envelope (TN/TR -~°)

in which convection takes place almost adiabatically (TR/~-~) . The result comes out

to be :

BT' 1 (B% D6r) T O ' - l-i~T ~0 - Dr (29)

6u' i (~£ D6r) .H 0 D (6P) , u o' - l-i~ i~0- ar + 2-imT a-~ PO (30)

where the variations of PT and of the mean molecular weight have been neglected. The

value of B%/~0 is calculated from equation (28) in which by definition (H=-dr/dlnP),

~H/H 0 = D~rlSr + H0(D/Dr)(SP/P 0) (31)

Thus, the modulation of convection to be used for the calculation of the work integral

is now expressed in terms of (aT/T0) and (D~r/Dr)~ on account that Vad(~P/Po)=(Y3-1)

(6P/Po)=BT/To •

Some useful formulae. Readers can skip this section, unless mathematical details are

of interest.

The work integrals W R and W C should be simplified further by use of the equations of

adiabatic oscillations. Neglecting ~i and ~ , we obtain from equations (9), (i0)

and (ii) after some manipulations,

l(l+7)GM --~ar ( r 2 6 r ) - ~zy2 "r~r = - r2 (1 - ~2r2 " 0 p -

D GM= 1 D~P GMr 6_~ a--{ [--~Br) - ~2Br 00 Dr rZ P0

, ( 3 2 )

(33)

322

Multiplying equation (33) by r ~, differentiating, and using equation (32), we obtain

r 2 9,(9,+i)c~ ~t3 ~i r~ ~P ~-P0 ~r~ _ r2 ~r + 2-£(bl-l)r2 ~r = ---GMr (°~2 - ~ ~0 - ~r [G---M~.rp 0 ~r + -] , (34)

where the variation of Mr has been neglected as in the Cowling approximation ($i=0).

Thus, we have a formula appearing in equations (21) and (22) :

6T__*+82~r 2 8~r 2-£(£+i) Re[T 0 (-~fr - ~-r + r 2 6r)]

where

÷ - - I~TI~I (Y3_l)~og 2 Va d t 1T~ TO) I

- ~ [~-:+'ad i+Tl:' {"~ - ~ (+-~+)> i~oi+T +,j

+ + 0',=!- ~o + #-:.)' I~1: , [3 -' l ad

~g2 _= GMr/r 3

(35)

(36)

Next we shall derive the expression of 36r/~r. Differentiating equation (32) with r

and using equation (33), we obtain

32(r2~r)/Sr2_£(E+l)~r = _(8/~r)(r260/pO)-£(Z+l)~-2A~P/00 , (37)

where A=(dlnOo/dr)-(i/Yl)(dlnPo/dr). The operation : [(37)/r2-(34)] gives the expres-

sion of ~r/~r. Unfortunately, the resulting expression is not so simple. We shall

be satisfied with a crude approximation in which we introduce an effective number of

nodes n such that ~2(r2~r)/Sr2~-n26r and ~2(rZ6P/p0)3r2~-n2~P/P O. If we neglect A, we

obtain from equation (37),

~r/~r ~ - {n2/[n2+E(£+l)]} (6P/PO) (38)

Physical processes of stability. With the help of the results in the preceding two

sections, all the integrands in the work integrals given by equations (17)-(22) can be

transformed into the forms proportional to the power of pulsation.

After a number of partial integrations, we obtain finally,

WM= ~I ~t0'[2u}- r2Pt ,Z y3-± ~ ~-~ -

W T = (i/2)L(Vad -I - K T) I~T/T012r= R

- I m2)V' IK~R°+{%-I ) - ' - 2+u~R+'~TR}FC011+T/~0t~dV

323

+ I POeN0[ (~N)T, ad + (Y3-1)-1+2aT-Vad-r(l-Vad,P+eN,p/2) ]IST/ToI 2dV

+I I'dv ~0eV0[2(UT -£T ~'6T )-i-{~ In(r2p0gV0)+

+ I [FR0(Vad-1-V0-1 )+FcOVad-I ]H0[[ a (~T/T0)/ar] 2+{% (%+1)/r~} 16T/ToI 2 ]dV

d I-V ST 2 ,R ,R 9 aT 2 (4o)

where approximations (5) and (6) have been used, aT, ~ etc. are defined by, [equations

(28)-(31)],

~Sr ST nZ/(y3,11 ~T ' a r = ~T ~ 0 % - n2+£(%+I) T~ '

SH ST , 8 .ST. V_a~)~T H_O - 8 (ST) --H0 = HT TO0 + HT ~LTo0) = (aT + qad TO ÷ Vad ~TO0 '

, ~ ,~T,.

~0 ~0 +~T~'~o ~ =ra~H~ ~rl% HT

' ~ 8 .ST, ~u I aT , 8 caT. V~d-IVad.P) aT ( £T u0 To + = + + + "rr%' = l-iLoT 2-i~T ~0 l-im~

and the other symbols have been defined before except that

~N,P = d(IngN0)/dlnP0 " and Vad,P = d(inVad)/dlnP0

The turbulent pressure work represented by the first integral in equation (39) can be

positive or negative depending upon the mode (% and n) and the relative convection time

mT, while the turbulent stress work represented by the second integral is always nega-

tive. The ratio between them is of the order of (~T) -l which can be larger or smaller

than unity depending upon the mode and the stellar structure. If the integrands of the

thermodynamical work W T are estimated to be of the order of (Fco/~0) I6T/T012 and FCO ~

PoU0 in convection zones, the ratio WM/W T becomes to be of the order of ~T or (~T) 2

The mechanical work should, therefore, be taken into account in the stellar stability

calculation.

In equation (40), the first llne on the right hand side shows the spherical effect and

the <-mechanism. If K increases strongly with increasing temperature and density (nega-

tive KT) , larger radiation loss results at lower density higher entropy state, causing

the destabillzation of the oscillation. The sphericity has a similar effect, though

not so pronounced. The second llne shows the correction to the K and other effects. The

third llne shows primarily the e mechanism which is the destabilizing effect due to high

324

temperature sensitivity of the nuclear burning process. The fourth line shows the simi-

lar effect by the viscous dissipation, but its work like the turbulent pressure work can

be positive or negative. The term proportional to FRO in the fifth line describes the

Cowling-Souffrin-Spiegel (or radiative Cowling) mechanism (Graff 1976), and the term

proportional to FC0 describes the turbulent conduction mechanism (or convective Cowlin~

mechanism) which seems to be found by the present analysis. The sixth line shows the

corrections to these two mechanisms. Because of the latter corrections, the radiative

Cowlin~ mechanism may not work for high frequency modes, while the convective Cowling

mechanism may remain effective. If a fluid element moves up (or down) slowly (small ~) in a suDeradiahatic layer, the

temperature inside becomes higher (or lower) than outside, and the density inside be-

comes lower (or higher) than outside because of the pressure balance. Both the radia-

tion and the turbulent conduction decrease (or increase) the thermal energy inside

where the entropy is higher (or lower). This explains the radiative and convective

Cowling mechanisms. Both mechanisms are due to the superadiabaticity, but they differ

greatly in efficiency, because the gain (or loss) in thermal energy is of the same or-

der as the convective flux for the convective Cowling mechanism while it is smaller by

a factor of (?0-Vad) for the radiative Cowling mechanism. Although none of the mecha-

nisms can be neglected, the K mechanism and the convective Cowling mechanism are the

main destabilizing mechanisms of pulsation in a convective star. Various approximations have been employed in the derivation of equations (39) and (40).

Many of them should be easily avoided in numerical work of stellar stability.

The author thanks Dr Y. 0saki for discussion.

References. Ando, H., Osaki, Y. 1975, Publ. Astron. Soe.Japan, 27, 581 Baker, N., Kippenhahn, R. 1962, Z. Astrophys., 54, ll4 Biermann, L. 1932, Z. Astrophys., 5, 117 Boury, A., Gabriel, M°, Noels, A.,--Scuflaire, R.,Ledoux, P. 1975,Astron.Astrophys.

41, 279

Christy, R.F. 1964, Rev.Mod.Phys., 36, 555 Cox, J.P. 1963, Astrophys.J., 138, 487 Dilke, F.W.W., Gough, D.O. 1972, Nature, __240, 262 Dziembowski, W., Sienklewicz, R. 1973, Acta Astron.,23, 273 Gabriel, M., Scuflaire, R., Noels,A., Boury,A. 1975, Astron. Astrophys.,40, 33 Gough, D.0. 1976, Astrophys. J., 2i].4, 196 Graff, Ph. 1976) Astron. Astrophys., 49,299 Kato) S. ]968) Publ. Astron. Soc. Japan, __20, 59 Lighthill) M.J. 1952, Proc. Roy. Soc., A 211,564 Mark, J.W.-K. ]976, Astrophys. J., 206,]-~-~-- Nakano, T. 1972, Ann. Phys., 73, 326 Osaki, Y. 1974, Astrophys.J., ]89, 469 Parker, E.N. ]974, Solar Phys., 36,249 Roberts, B. ]976, Astrophys. J., 204, 268 Shibahashi, H., Osaki, Y., Unno)W', Puhl Astron. Soc. Sapan, 2__7, 40J Stein, R.F. 1967, Solar Phys., 2, 385 Unno, W. 1964, Transaction IAU XilB, Academic Press, New York, p.555. Unno, W. ]967, Publ. Astron. Soc. Japan, 19, ]40(ef.Proc.Astron.Soc.Australia,!,379 ,

1970) Unno, W. |969, Publ. Astron. Soc. Japan, 2[I, 240 Vitense, E0 1953, Z. Astrophys., 32, ]35 Zhevakln, S.A. ]953, Russian Astro-n.J., 30, ]6]

FULLY DEVELOPED TURBULENCE, !NTERMITTENCY AND MAGNETIC FIELDS

Uriel Friseh

C,N,R,S,

Observatoire de Nice, France

I. INTRODUCTION

Turbulence is usually associated with the idea of chaos, i.e. erratic behaviour of

some observable quantity. Let me stress that there are at least two different kinds

of chaos.

Temporal chaos is known to appear in certain systems having only a few degrees of

freedom. Take for example the Lorenz model (Lorenz, ]963) which has three degrees of

freedom. It is a crude truncation of the Rayleigh-B~nard problem with only one Fourier

component in the X- and the Z- directions so that the motion can in no way be consi-

dered as spatially chaotic. Nevertheless, there is a strong numerical evidence that

the temporal spectrum becomes continuous when the Rayleigh number crosses a certain

threshold, an indication that temporal chaos has developed. This kind of chaos can

appear on the largest scales of the system which makes it easy to observe. Possible

candidates for such theories are irregular variable stars, the geodynamo, etc.,...

Very different is the problem of fully developed turbulence which is essentially a

spatio-temporal chaos : when the Reynolds number goes to infinity all space and time

scales, down to infinitesimal are excited. Such chaos may develop in a finite time

and has universal sealing properties (e.g. a power-law energy spectrum). In the

astrophysical context fully developed turbulence may not always be directly observable

because of lack of resolution of the small scales. But it will always manifest itself

indirectly through a drastic modification of the transport properties.

In the next two sections we try to give simple phenomenologieal insight into univer-

sal properties of fully developed turbulence, particularly the question of intermit-

tency, or in other words spottiness of the small scales. Intermittency is very much

at the center of present theoretical studies (Kraichnan, |974, Frisch, Lesieur and

Sulem, 1976, Frlsch, Sulem and Nelkin, ]977). Experimentally, it is rather difficult

to observe because the small scales carry very little energy. However , magnetic fields

which are very sensitive to small-scale velocity gradients can he used as tracers of

the small scales (in the ~D case). It is therefore of great interest to note that re-

cent high resolution observations of the small-scale solar magnetic field indicate a

very intermittent structure (Stenflo, ]975). Non magnetic intermittent turbulence being

rather poorly understood it seems premature to consider the MHD case in detail. However,

many overall features are probably common to both cases in particular the steepening of

326

the spectrum. The reader interested in the non intermittent aspects of MHD turbulence

such as the non linear dynamo effect is referred to Pouqnet et al. (1976),

2, KOLMOGOROV 1941 REVISITED

Big whorls have little whorls

Which feed on their velocity

And little whorls have lesser whorls

And so on to viscosity.

L.F. Richardson, ]922

By the Kolmogorov 1941 (in short K41) theory, we mean the general class of arguments

developed by Kolmogorov, Obukhov, Onsager and others which has led in particular to

the 5/3 law (see Batchelor, 1953, and Monin and Yaglom, ]975, for review). The 5/3

law may be derived from dimension analysis, but more insight is gained from a simple

dynamical argument borrowed from Kraichnan (1972, page 213). We define the energy

spectrum E(k) as the kinetic energy per unit mass and unit wavenumber k. It is a

convenient simplification, with no significant loss of generality~ to consider a dis-

crete sequency of scales or "eddies"

= £ 2 -n n = O, |, 2, ... (2.1) n o

and of wavenumbers k = Z -I . The kinetic energy per unit mass in scales ~ £ is n n n

defined as

I kn + 1 E(k) dk- (2.2) E n = kn

Let us assume that we have a state of statistically stationary turbulence where ener-

gy is introduced into the fluid at scales % Zo, and is then transferred successively

to scales ~ ~|, ~ £2' ..., until some scale %d is reached where dissipation is able

to compete with non linear transfer (Fig. I). If we now make the essential assumption

that eddies of any ~eneration are space fillinb, as indicated in Fig. |, we may write

E ~ v 2 (2.3) n n •

where v is a velocity characteristic of n-th generation eddies (in short, n-eddles). n

In Eq. (2.3) and subsequentl~ factors of the order of unity will be systematically

dropped except when such factors would cumulate multiplicatively in successive cas-

cade steps. Note that v n is not the velocity with which n-eddies move with respect

to the reference frame of the mean flow, this being mostly due to advection by the

327

largest eddies. It is rather a typical velocity difference across a distance ~ £ n

the latter being the only dynamically significant quantity. (In this respect the

"velocity" in the second line of Richardson's poem is misleading). We now define

the eddy turnover time

t n % Zn/Vn . (2.4)

The quantity tn-! may be considered as the typical shear in scales ~ in' and there-

fore defines the characteristic rate at which excitation at scales % £ is fed into n

scales % in+l"

There are, however, at least two important exceptions to this statement. First we may

define a viscous dissipation time

t diss % i 2/v , (2.5) n n

where ~ is the kinematic viscosity of the fluid. If

t diss << t (2.6) n n'

then transfer is no longer able to compete with dissipation, and most of the excita-

tion in scales ~ £ is lost to viscosity. Second, if n

>> t = £o/Vo , (2.7) tn o

then the shear acting on scales % £ comes mostly from scales ~ £o' n

used instead of t as a dynamical time. n

and t o should be

Assuming that neither of these two exceptions applies, (this may be checked a poste-

riori) we make the fundamental assumption that in a time of the order of t a sizeable n

fraction of the energy in scales ~ Zn is transferred to scales % in+l" The rate of

transfer of energy per unit mass from n-eddies to (n+l)-eddies is then given by

E n ~ En/t n ~ Vn3/£n . (2.8)

Since we assume a stationary process in which energy is introduced at scales ~ Z o

and removed at scales ~ £d' conservation of energy requires that

en - ~ , £d < £n -< £o " (2.9)

328

NotiCe that ~ can be thought of as a rate of energy injection, a rate of energy trans"

fer or a rate of energy dissipation, From the point of view of inertial range dynamics,

the second of these three definitions iS the most relevant, Using (2.8) and (2.9) we

solve fO~Vn and En~

v n ~ ( ~ £n )I/3, mn ~ (~ £n )2/3 ' (2.10)

which by Fourier transformation yields the K41 spectrum

E(k) ~ ~ 213 k-513 (2.11)

The eddy turnover time of Eq. (2.4) is given by

t ~ ~ -I/3 I 2 / 3 ( 2 . t 2 ) n n

Equating (2.12) to the viscous diffusion time (2.5) determines the Kolmogorov micro-

scale

£d = (v3/;)114 (2.13)

gq. (2.13) gives the length scale at which the cascade is terminated by viscous

dissipation.

Injectio

Transfer

O0@C O© O00000000OOOO O0000C O00000000C O00

Dissipation Fig. I. The energy cascade according to the Kolmogorov 194] theory, Notice that

at each step the eddies are space-filling.

329

3. INTERMITTENCY : THE 8- MODEL

Big whorls have little whorls

Which feed on their vorticity

And little whorls have lesser whorls

Which hardly ever can you see.

Since the first experiments of Batchelor and Townsend (1949) there has been strong

evidence that the small scale structures of turbulence become less and less space

filling as scale size decreases. (Kuo and Corrsin, 1972; see also Monin and Yaglom,

1975, for review). Dynamically this spottiness of the small scales can be made

plausible by a simple vortex stretching argument. Consider the point M within a large

scale structure which at the initial time T o has the largest vorticity amplitude I~I.

This point is also likely to have a large velocity gradient IVy[ ~ [~I- The strai-

ning action of the velocity gradient on the vortieity may then be described by a

crude form of the vorticity equation

Dt ; (3.1)

hence it is expected that the vorticity downstream of M will rise to very large

values (possibly infinite at zero viscosity) in a time of the order of the large eddy

turnover time t ~ I~l -I. o

Even if the vortieity at time T has a very flat spatial distribution, the non- o

linearity of Eq. (3.]) will lead to a very narrowly peaked spatial distribution at

+ t . So we see that small scale structures may be generated in a very loca- time T o o

lized fashion. This argument can be made fully rigorous fur the Burgers equation, but

not for the Navier-Stokes equation (L~orat, 1975). For the Navier-Stokes equation

there is the important complication that the velocity gradient at a point ~ is not

related in any simple way to the vorticity at ~; instead it is given by a Poisson

integral with a fairly substantial local contribution, but also with some coupling

to nearby points. This could smooth out the vorticity peak, but the smallest scale

structures will still have some tendency not to occur uniformly.

Assuming that the small eddies do indeed become less and less space filling, let us

now define the 8-model. At each step of the cascade process any n-eddy of size

= 2 -n produces on the average N (n+])-eddies. If the largest eddies are space £n £o filling, after n generations only a fraction

330

B n = B n (B ffi N /2 3 - < ]) ( 3 . 2 )

of the space will be occupied by active fluid (see Fig. 2). Furthermore we assume

that (n+1)-eddies are positionally correlated with n-eddies by embedding or attach-

ment. (For the sake of pictorial clarity this feature is not included in Fig. 2).

I°

Injection/-

@C) C) ©©©

0 0 0 0

• 6 m

TransFer © o

Dissipation

Fig. 2. The energy cascade for intemittent turbulence. Notice that the eddies become

less and less space filling.

It is straightforward to work out the modification to K4! in the B-model. Let v n now

denote a typical velocity difference over a distance ~ ~n in an active re$ion. The

kinetic energy per unit mass on scales ~ £n is then given by

2 ( 3 . 3 ) E n ~ B n v n

331

The characteristic dynamical time for transfer of energy from active n-eddies to

smaller scales is still given by the turnover time t n = Zn/Vn as in K4I : the

generation of (n+])-eddies arises from the internal dynamics of the n-eddy in which

it is embedded. We can express the rate of energy transfer from n-eddies to

(n+1)-eddles exactly as in K4], and as in K4! this quantity must be independent of

n in the inertial range:

Vn3/£ n - en % En/tn ~ 8n ~ E (3.4)

Defining

= - log 2 B ,

we combine Eqs. (3.2 - 3.4) to obtain

-e I /3 £ 1/3 (Zn/£o)-~13 (3.5) Vn n *

e - - I / 3 £ 2/3 (£n/£o)U/3 (3.6) tn n I

En ~ ~ 213 Zn2/3 (Zn/Zo)+U/3, (3.7)

and

E(k) % ~ 213 k-5/3 (k ~o )-u/3. (3.8)

All the intermitteney corrections may be expressed in terms of the self-similarity

dimension D = 3 -~ , a special case of Mandelbrot's (1975) fractal dimension, which

is related to the number of offspring by

N = 2 D (3.9)

That D can suitably be called a dimension is made clear by Fig. 3 which shows three

very familiar objects : a unit interval, a square and a cube which have dimensions

D equal to I, 2 and 3 respectively. If we reduce the linear dimensions of these

objects by a factor of 2, as in the cascade process, the number of offspring needed

to reconstruct the original object is 2 D. For mere complicated self-similar objects

a natural interpolation is N = 2 D, where D need no longer take only integer values.

(Some rather exotic examples can be found in Mandelbrot (1975).) It has been shown

by Mandelbrot (1974) that D is also the Hansdorff dimension of the dissipative struc-

tures in the limit of zero viscosity. D = 2 would correspond to sheet-like structures,

but in view of the experimental value of the exponent for the dissipation correlation

function a more likely value is D ~ 2.5 (See Frisch et al 1977).

332

-~ ................ ' . . . . I

/ ~ I ; , i I ,

/ i ' 4

I I, s~ --]~--7

Fig. 3 When the linear dimensions of a D-dimensional object are reduced

by a factor I (here 2), ID pieces are needed to reconstruct the original.

Moze exotic examples with non integer D, such as probably oceurin turbulence,

may be found in Mandelbrot (1975).

333

Remark 3.1

Remark 3.2

Remark 3.3

Equation (3.8) relating the correction to the 5/5 exponent of the K4I

theory and the fraetal dimension was first derived by Mandelbrot (]976)

using the Novikov-Stewart (1964) model.

Since 0 < D < 3 the corrections to K4] can not make the spectral expo-

nent larger than 8/3. This same upper bound can be derived rigorously

from the Navier-Stokes equation for finite energy turbulence (Sulem and

Frisch, 1975). This reference also gives a heuristic argument to show

that D > 2.

When (3.4) is used for the largest scales we obtain

Vo3!~ ° , 7 (3 . J O)

the same result as in the non-intermittent case. This is important since

(3.10) is frequen~lyused in practical calculations. Intermittency may,

however, be of practical importance in other ways, particularly when

chemical reactions are involved (Herring, 1973).

Dissipation sea!e

Equating the turnover time (3.6) to the viscous diffusion time £n2/~ we obtain the

dissipation scale

R -31(4-p) (3.11)

where we have introduced the Reynolds number

R = £ v /v ~ e 1/3 £ 4/3 -I (3.12) o o o

Singularities

Both the K41 and the ~-model imply that the three dimensional Euler equation (Navier-

Stokes with zero viscosity) leads to a singularity in a finite time. Indeed, if we

start with very smooth initial conditions, say only large eddies, then the complete

hierarchy of eddies, down to infinitesimal scales should be established in a time

t± ~ E t ~ ~olVo (3.13) n=0 n

334

Since there is now no viscous cutoff the enstrophy given by

f° fl = k 2 E(k) dk (3.14) 0

will become infinite at time tx. There is in fact some numerial evidence that such

singularities exist (Orszag, 1976a, b)° There are also a few known exact solutions

which display singularities at a finite time, but these solutions are badly behaved

at large distances (Childress and Spiegel, 1976). Finally various stochastic models

or low order closures of the statistical Euler equation can be shown to produce sin-

gularities in a finite time (Lesieur and Sulem, 1976; Andr~ and Lesieur, 1977).

NOTE ON THE M H D CASE

The K4| theory can be easily modified to account for the effect of Alfv~n waves. It

then yields a k -3/2 spectrum (Kraiehnan, 1965; Pouquet et al., 1976). How intermitten-

cy can be handled in the MHD case is not yet clear, but it is again likely to steepen

the spectrum. That the spectral exponent can become as large as 2.5 as suggested by

certain solar observations (Harvey, 1976) is a possibility which cannot be ruled out.

Finally we note that singularities should appear in the MHD case as well as in the

nonmagnetic ease. There are even some indications that they are present in two-

dimensional MHD flows (Pouquet, 1976) although they are known to be absent in the

non magnetic two-dimensional case (Wolibner, ]933). The presence of singularities at

a finite time in the MHD equation implies that magnetic field line reconnection

at high kinetic and magnetic Reynolds number occurs in a time which does not depend

on the magnetic diffusivity : it is essentially the large eddy turnover time.

335

REFERENCE S

Andre, J.C. & Lesieur, M. 1977, Evolution of high Reynolds number turbulence,

J° Fluid Mech.,to appear

Batchelor, G.K. & Townsend, A.A. 1949, Proe. Roy. Soc. A 199, 238

Batchelor, G.K. 1953, Theory of homogeneous turbulence, Cambridge U. Press

Childress, S. & Spiegel, E. 1976, Private communication

Prisch, U., Lesieur, M. & Sulem, P.L. 1976, Phys. Rev. Lett. 37, 895

Frlsch, U., Sulem, P.L. & Nelkin, M. 1977, A simple dynamical model of intermittent

fully developed turbulence, submitted J. Fluid Mech.

Harvey, J.W. 1976, Private communication

Herring, J. 1973, Private cormnunieation

Kolmogorov, A.N. 1941, C. R. Aead. Sci. USSR 30, 301, 538

Kraichnan, R.H. [965, Phys. Fluids 8, 1385

Kraichnan, R.H. 1972 in "Statistical Mechanics : New concepts, New problems, New

applications", Rice, S.A., Fried, K.F. & Light, J.C. Eds., University

of Chicago Press, Chicago

Kraichnan, R.H. 1974, J. Fluid Mech. 6_~2, 305

Kraichnan, R.H. 1975, J. Fluid Mech. 67, 155

Kuo, A.Y. & Corrsin, S. 1971, J. Fluid Mech. 50, 285

L~orat, J. 1975, Thesis, Observatoire de Meudon, Meudon, France

Lesieur, M. & Sulem, P.L. 1976, "Les ~quations spectrales en turbulence homog~ne et

isotrope : quelques r~sultats th~oriques et num~riques" in "Proc.

Journ~es Math~matiques sur la Turbulence", Temam, R. ed., Springer Lecture

Notes in Math., to appear

Lorenz, E.N. 1963, J. Atmos. Sci. 20, 130

Mandelbrot, B. 1974, J. Fluid Mech. 62, 33]

Mandelbrot, B. 1975, "Les Objets Fractals : Forme, Hasard et Dimension", Flarmnarion,

Paris

Mandelbrot, B. 1976, "Intermittent turbulence and fractal dimension : kurtosis and

the spectral exponent 5/3 + B" in "Proc. Journ~es Math~matiques sur la

Turbulence", Orsay~ June 1974, Temam, R. ed., Springer Lecture Notes in

Mathematics, to appear

Monin, A.S. & Yaglom, A.M. ]975, Statistical Fluid Mechanics, vol. 2 , MIT Press

Novikov, E.A. & Stewart, R.W. 1964, Isvestia Akad. Nauk USSR, Ser. Geophys. n = 3,

p. 408

Orszag, S. |976a, '~tatistical Theory of Turbulence" in "Proc. Les Houches 1973",

Balian, R. ed., North Holland, to appear

Orszag, S. [976b, Private communication

336

Pouquet, A. 1976, "Remarks on two dimensional MHD turbulence", preprint

Pouquet, A., Frisch, U. and L~orat, J. 1976, J. Fluid Mech. 77, 32!

Stenflo, J.O. ]976, "Influence of magnetic fields on solar hydrodynamics :

experimental results", to appear in "Proc. of IAU Coll. n ° 36" held at

Nice, September 1976

Sulem, P.L. & Frisch, U. 1975, J. Fluid Mech. 72, 4i7

Wolibner, W. 1933, Math. Zeitsehr. 37, 698

TURBULENCE : DETERMINISM AND CHAOS

Y. POMEAU

CEA~ DPhT, Gif sur Yvette, FRANCE

After the article of Ruelle and Takens (I), there has been recently much interest

in the problem of the "onset of turbulence". That is, instead of trying to understand

the structure of a well established turbulence flow, one studies the way in which a

flow "jumps" from a quiet stable laminar state to a turbulent state whenits Reynolds

(or Rayleigh) number increases.

The idea of turbulence is connected with the one of "chaos". The ergodic theory (2)

allows one to give a precise content to this last notion (Take care that this differs

from the one given in the review paper by May (3) , see also (4)) . One first considers

a time dependent quantity, say u(t), as, for instance, the fluid velocity at a given

point in a turbulent flow of fluid under constant (or periodic, or eventually station-

nary "in average") external conditions, in such a way that one may define a gliding

average as | [t+T

<~[u(t)]> = lim ~ dr' ~(u(t')), T ~ ~ Jt

where ~ is any smooth function.

We assume that these averaged quantities are independent of the initial conditions,

at least for "almost" any choice of them and that they do not depend on t. The "signal"

u(.) has the property of mi~ng, that we shall consider as defining the chaos if :

<~[u(t)]~[u(t+t')] - <~><~>> + o

tw~ for any smooth ~ and ~ . This property expresses the idea that, after a sufficiently

long interval of time, say t', the system "forgets" the detail of the initial condi-

tions (= the fluctuations of u at two very distant times are uncorrelated).

A good example of such a "chaotic" signal, with astrophysical implications, is

provided by the time dependence of the magnetic field of earth (5) . The geological data

show that the earth'smagnetic dipole has reversed a large number of times. It is of

interest to know whether these reversals occur "regularly" or at random. For that pur-

pose, let us consider the autocorrelation function which is built up from the data (6)

as follows : ~ and ~ represent the same function. This function is equal to +I when

the polarity is the same as now, and to (-I) in the reversed case. The autocorrela-

tion function of this random signal is given in Fig. I. It shows rather clearly that,

from this point of view, the reversals are at random, and the reversals follow appro-

ximately a Poisson law.

t 338

dipo]e data from Heirzler et al. J of geoph.reso 73,2119 (1968)

There is often a misunderstanding about this idea of chaos, which is implicitly

connected with the one of "noise". At least, at the level of the equations of the

motion, it is often thought that chaos mus~ be introduced by some noise source and

that chaos may exist in non-deterministic systems only (as, say, a damped harmonic

oscillator in contact with a heat bath). In order to Understand how chaos may arise

very simply from a deterministic process, let us consider a "discrete" dynamical

system. This dynamical system mimics a system depending continuously on time, wherein

measurements are made at discrete instants, say t| , tl+T, tl+2T, .o., tl+nT, .... •

Thus we shall define for this dynamical system a "variable" and a time translation

operator (that is an operator which allows one to jump from the value of the variable

at any time t to its value at time t+O . This is a dync~nieaZ system if the transfor-

mation acts continuously and is inversible, in mathematical terms it is a homeomor-

phi8m) (= time can he reversed to get the initial data from the final state). To

define the variable of our dynamical system, we consider a set K with a finite num-

berp say k, of elements and the doubly infinite sequences of elements of K :

{ .... } u t = ...i_n , .... i_l~Zo~Zl~...~in, ln+l,...

where ij (j = 1,2 .... ,k) 6 K . Thus, giving the initial data u(t I) , means that a

particular sequence is known. The transformation that allows one to find ut+ T , once

u t is given, is just the shift of the sequence ; by definition :

in(Ut+ T) = in_1(ut)

339

If we consider now the class of functions of u defined by

m=-Oo

with +co

I max ~0m(i m) I 2 < oo , m =-o= i 6K

m

it is easy to see that, if the im'S are taken a t random in K w i t h the p r o b a b i l i t i e s

Pl'''''Pk (k= cardinality of K) such as jE=l pj = 1 , then

+oo K

<~(u) > = I ~ Pn t~m (Yn) ' m=-co n = ]

where Yn is the n th element of K , and

<[~(ut) - <~>] [@(Ut+N) - <@>]> --+ 0 ,

which proves the property of mixing for our discrete (and deterministic) system.

Actually this last property expresses a very simple fact : ~ and @ depend on terms

of the infinite sequence {i m} which are located in a fixed part of this sequence.

Shifting at each step this sequence on the left, one "loses " part of the knowledge

of the values of the {i } in this region, as new i's come from the right which are m m

uncorrelated with the already known i's . Obviously this double infinite sequence m

looks very much as the k-ary expansion of a real number (except that it is doubly

- instead of singly - infinite). This helps to understand that, in a mixing dynamic-

al system, the noise source might just be the infinite (say decimal) expansion of

the real numbers defining the initial datas.

Of course this notion of ten.oral chaos is not sufficient to define

turbulence, as another typical feature of turbulence in unbounded flows is the

absence of spatial correlation with an infinite range. There is an obvious extension

of the mixing property to the 8pat~al ease, that is a position dependent function

u(~) has this mixing property iff

<:~[u(~)] ¢ [u (~+~ ) ] - <~> <@>> , o ,

where the averages are now to be understood as gliding space averages. This definition

implies, of course, that the flow is unbounded in some direction and that the turbu-

lent state is invariant under the translations along this direction. Although there

is clear evidence (7) that the turbulent flows have the mixing property both in

time and space, we shall only consider the time dependent properties, as the connec-

tion between spatial chaos and the original non linear Navier-Stokes equation is

rather unclear at the present time.

340

On the contrary, if one neglects completely this question of the spatial struc-

ture, one is led to consider the fluid motion as described by the solution of a sys-

tem of ordinary differential equations (O.D.E.). Of course, one is mainly interested

in the qualitative properties of these O.D.E., as the fluid equations cannot be

replaced by a fully equivalent system of a finite number of O.D.E. with an explicit

form. This qualitative theory of the O.D.E. has been the subject of detailed investi-

gations (8'9), in particular one may understand quite well how the solutions of such

a system may have the property of mixing.

Recently Ruelle and Takens (I) have drawn attention to the possible connection

between the onset of turbulence in flows and some bifurcation properties of O.D.E. •

Without going into too many details, I will just explain what is presently known

about the onset of turbulence. Following Martin and McLaughlin (]0), one may consider

three different ways for the occurrence of turbulence .

I. The onset o~ turbulence in the Lorenz s~stem

By a drastic reduction of the Oberheck-Boussinesq equations for a flow convect-

ing in a horizontal layer, Lorenz (ll) has obtained the following system of O.D.E. :

|,a

l.b

i.c

d~ d-~ = o (y-x)

dY = - •z + rx - y dt

dz d-~ = xy - bz

where ~ , r and b are numerical parameters. By numerical computations he has shown

that in some range of values of these parameters the motion described by these equa-

tions is chaotic and that the trajectory, after some transient, reaches very rapidly

an "attractor",which does not depend on the initial conditions. This attractor is

very interesting, as it is presumably structurally stable, that is it exists (and

remains attracting) for values of the parameters in open intervals, The idea of

structural stability is actually much more general(12),for O.D.E.,likethe system (1),

it just means (|3) that one may add a small perturbation depending on x , y , z on the

right hand side, without changing the topology of the velocity field defined by this

system .This means that by a homeomorphic mapping of space (i.e. a change of varia-

÷ f(~) that is both continuous and inversible) one may change the trajecto- bles x'=

ries of the perturbed system into the ones of the unperturbed system. The structural

stability is essential, as it means that the properties of the system under considera-

tion do not depend on the detai~ of the equations, and that they remain essentially

the same under any kind of (small enough, but finite) perturbation (actually one

thinks of perturbations arising from a lack of knowledge of the exact form of the

equations).

341

The structure of the Lorenz attractor has been recently studied by a number of

authors (]4), and I will just give a brief (and I hope clear enough) account of

these works. A simple way to describe it is the one of Williams (14). He first consi-

ders a "semi-flow", that is a flow on a surface where two sheets may collapse. This

cannot really represent the solution of O.D.E., as the motion cannot be traced back

unambiguously, as it should be allowed for O.D.E. But this helps to understand the

structure of the Lorenz attractor. I have tried to draw as clearly as possible this

surface on Fig.2 . It has two holes and the line along which the two sheets collapse

is the dashed line. On the right part of the figure is the section of the surface by

the mid -vertical plane.

\/

The trajectories run on this surface approximately as follows :

They revolve around each hole by diverging slowly and if at one of these revolutions

the trajectory cut the dashed line beyond the middle point , it is inserted at the

next turn close to the other hole and revolves by diverging slowly around this "new"

hole. Finally it jumps at random from a hole to the other. To understand why this

motion is non periodic, one considers the so-called Poincar~ transform on the shaded

segment where the two sheets merge. Let us define on this segment a coordinate~ say

x which varies between -] and +] . The Poincar~ transform defines a function f(x) ,

-I ~ f(x) ~ +] if -] ~x~+] : if the trajectory cresses the shaded line at x , its

next crossing will be at f(x). The function f(x) hasa discontinuity at x= 0 , and

looks approximately as represented in Fig.3 .

/ / I

342

If one assumes that its derivative (when it exists) is everywhere larger than I,

then it is clear that the application x ÷ f(x) cannot have any stable ~ixed point

or even any stable period . It is important to understand that this is possible only

because f(x) has a discontinuity, otherwise if f and its derivative were continuous

If'(x)Icould be not everywhere larger than l, if f['l,|] ~[-l,l].

Now the question of the mixing character of the motion is turned into the ques-

tion of the mixing character of the application x ÷ f(x). This mixing property is

rather obvious (if one does not want to get a rigorous proof) : consider two

points{x l, x2}very close to each other, and theirsuccessive transform :{f(xl),f(x2)}:

{f[f(xl)], f[f(x2)] } ..... ; {f(n)(xl) , f(n)(x2) } ....

(By definition f(n)(x) = f[f(n-])(x)] and f(]) E f) . The distance between the tans-

forms~f== x I and x 2 is multiplied after each application of f by a quantity larger than

min l~x 1 that is larger than I, thus it increases at least exponentially. This means

that, at some time, the two image points will be separated by the discontinuity of

f, and" the subsequent trajectories starting from xl or x 2 will be completely diffe-

rent from each other. This is a version of the mixing property : a small fluctuation

in the initial conditions yields,after some time, a huge difference in the arrival

points ; in other terms unless one knows the initial conditions with an infinite

accuracy, the motion is unpredictible after some ~n~te time.

Let us come back now to a more realistic description of the Lorenz attractor

I have already noted that it cannot be considered as a surface in the usual sense,

since two sheets cannot merge owing to the deterministic character of the equations

of the motion. To understand what really happens, it is only necessary to replace the

shaded line where the two sheets collapse by a small surface parallel to this line

(as a thin stick).Now the Poincar~ transform is no longer given by a function of one

variable, but by a plane transform: that is,given a starting point inside the stick,

one wonders what is the next crossing point of the trajectory inside this stick.Essen-

tially (although things are a little bit more complicated), the Poincar~ transform

looks very much like a Baker's transform (15) : the stick is first cut in two

pieces, (this cutting remembers of course very much the discontinuity in f), each

piece is stretched and the two resultant pieces are put together inside the stick

(see fig. 4). The transform of the coordinate parallel to the stick is very similar

to the one dimensional transform represented in fig. 3, but now the Poincar6 trans-

form has been made invertible as the coordinate perpendicular to x allows one to

~Actually Lasota and Yorke (Trans. of the A.M.S. 186, 481 (1973)) show that there

is an absolutely continuous invarlant measure for such f, and f is ergodic for this

measure.

343

distinguish between the arrival points with the same x. By a repeated action of the

transform pictured on Fig. 4, one obtains a section of the attractor, which is the "ob-

ject" stable under an infinite number of applications of Poincar~ transform. If one

cuts this section by a line perpendicular to the axis of the stick, one easily sees

that, after each application of the transform, the central part of the stick is de-

leted, then the central part of the remaining segments is deleted, and so on. This

is precisely the way in which one generates Cantor sets. Accordingly the Lorenz attrac-

tor has the structure of a Cat, tot set perpendicular to its "surface". It is a surface

with a number of sheets which has the power of a continuum.

A

E ~ I F ~ .......... I

Let us notice that these properties of the Lorenz attractor have actually not

been proved from a rigorous study of the system (1), although it is a very reasonable

extrapolation from the computer studies.

Another important feature of the Lorenz attractor is that it appears by an inver-

ted bifurcation from a pair of stable fixed points : in a domain of values of the

parameters (r,~,b), one reaches~from some initial conditlons,one of the stable fixed

points, or/and some other Lorenz attractor.This manner of occurrence of turbulence is

well known, for instance in Poiseuille flow(16); in a range of values of the

Reynolds number the laminar flow is linearly stable, but unstable against perturbations

with a finite amplitude, and at the upper limit of this domain the laminar flow

becomes linearly unstable. It must be stressed that the stability of convection

flows is much less well known than the one of the Poiseuille flow, so that it is not

$44

~lemr if the Lorenz system describes, even in a rough way, the bifurcation toward

turbulence of convective flows. From this point of view the existence (or the non

existence) of two sorts of flows for the same value of the Rayleigh number would be

an important test.

2. The theories of ,La~dau-HoFf and of Ruelle-Takens

Landau (17) and Hopf have given a quite convincing picture of the onset of turbu-

lence. In order to understand their idea, it is necessary to introduce the notion

of quasiperiodic funation. Let us consider a dynamical system with a periodic limit

cycle. The existence of such an oscillatory behaviour is known to occur (18) in con-

vective flows.

The theory (19) shows that, by a certain type of bifurcation (called the Hopf

bifurcation), this limit cycle may give rise to amotionwith two ~ncommensurate frequen-

cies, say ~| and ~2 (these frequencies are incommensurate if no non zero integers P and q

exist such thatp~| = q~2). Then any function of time in this flow should be ~asi-

periodic. To define such a function, let us consider a function of two variables,

say t I and t2, which is periodic of period 27 with respect to each of the~ variables:

f(t I + 2~n, t 2+ 2~m) = f(tl,t2)

whatever the integers n and m are.

From this function we may build the quasiperiodic function ~(t) = f(mlt,~2t). II

ml and ~2 are incommensurate, this function will appear (at least at first sight) as

completely choatic, although it is not chaotic in the sense of the mixing property.

Its frequency spectrum is concentrated at the frequencies m1' ~2' and

more generally at any linear combination p~! + q~2 with integer coefficients.

The extension of this construction to a function that depends periodically with

the period 2~ on any set of variables, say t!, t2,...,tn, allows one to define the

most general quasiperiodic function which has not the mixing property.

One must take care that such a quasiperiodic behaviour is not structurally stable

(except for the case of a single period). If a parameter as, say, the Rayleigh number

varies in the domain of quasiperiodic behaviour, then a periodic (and structurally

stable) limit cycle should be reached every time when ~I and ~2 (which both depend

on the Rayleigh number) are commensurate. The non periodic behaviour is reached for

~solated values of the parameter R only (20). The idea of Landau and Hopf is the a following: when the Rayleigh number increases many new bifurcations occur, which

always correspond to frequencies 1~commensurate with the already existing ones. Then,

one can show that, if the quasiperiodic function depends actually on all these fre-

quencies, then it tends toward a chaotic signal (in the sense of the mixing @roperty)

after an infinite number of frequencies have appeared.

345

One may wonder first about the validity of this theory, as it is unclear whether

an infinite or finite Rayleigh (or Reynolds) number is required for the occurrence

of an infinite number of bifurcations.

Landau and Lifshitz (2|) relate the number of "degre~of freedom" of a turbulent

motion with its Reynolds number R e . Asymptotically this number should increase as

(Re)9/4 , so that the existence of an infinite number of degrees of freedom requires

an infinite Reynolds number. They assume that each "degree of freedom" is actually

connected with the freedom in the choice of the phase at a bifurcation where a new

frequency appears. Although this notion of "degree of freedom" is widely used in

theoretical physics, the way in which the (R)9/4 formula of Landau Lifshitz counts the e

number of bifurcations between R = 0 and a large value of R is rather unclear. e e

On the other hand followlngRuelle and Takens (l) , the Onset of turbulenc% as dese;ibed

by Landau and Lifshitz, cannot be a '~eneric" phenomenon. They show that, after the

occurrence of a few non cormnensurate frequencies, the next bifurcation is toward a

non periodic attractor. This non-perlodic attractor is built as follows : let us

consider the case of a quasiperiodic motion with four ~ncommensurate frequencies, if

one represents the trajectory by the motion of a point in a four dimensional space,

it intersects a 3-dimensional hyperplane (i.e. a usual 3-d space) following a full

torus. Again one considers that the motion describes a one to one application of

this torus into itself (that is, each point inside the torus has an image which is

the next crossing of the trajectory with the hyperplane). It is possible (22) to

find a transformation of the torus into itself that is continuous, invertible , and

which transforms the torus into a strange attractor after an infinite number of

applications (see fig, 5). This attractor is also structurally stable.

Therehave been attempt~ 23) to find if this picture of the onset of turbulence is

valid for Taylor instabilities, it is not completely clear whether the experimental

findings are or not in agreement with this theory.

346

3. Approach of chaos by succesivebifurcationfis

For some values of the parameters, the Lorenz system (I) has one (or two) perio-

dic limit cycle(24~ When a parameter such as r varies in this domain, this limit cycle

becomes very rapidly much more complicated by a mechanism of "cascading bifurcations'~

These bifurcations are rather striking as they describe a smooth transition from a

periodic limit cyclewith a single period toward a strange (i.e. non-periodic) attractor.

For r ~ rl, the period of the limit cycle is, say T ; for r just below rl(bi-

fureation point)the period is 2T ; but as the motion is anharmonic, the amplitude of

the Fourier component of frequency |/2T start from a zero value at rl, then increases

continuously as r becomes smaller than r I . In order to make clear possibility of this

mechanism of frequency division, it is enough to draw a closed trajectory (= the

limit cycle) in the space of thevariables x,y,z (Fig. 6), when r becomes just a little

smaller than rl, this closed curve with singl~ orb becomes a closed curve with two

orbs, as drawn (approximately)in fig. 7 .

i

J

/

347

There is a series of such splitting of the limit cycle when r decreases, so

that the period of the limit cycle becomes 2T, 4T, ... 2nT. There is an infinite num-

ber of such bifurcations when r decreases from r I to, say r . The limit r~ is appro-

ached in a geometric fashion : let r be the value of r which the limit cycle of 2 n

period 2nT becomes unstable giving birth to a limit cycle of period 2n+]T, then the

quantity (r2n - r )/(r2n+l-r~itends to a limit as n increases, which is an universal

number (= independent of the detail of the equations) approximately equal to

0.214I 693. At the end of these bifurcations (i.e. at r , and for lower values ofr )

the period of the motion is infinite,which means that this motion is chaotic. Actually,

the recurrence time for a given point on the limit cycle is the period. It is easy to

see that the autocorrelation function for such a periodic system musttake the

same finite value at separation time of one period, two periods,... N periods, which

forbids it to tend to zero at infinite separation times (so that a periodic system

is obviously not ahaot~c). On the contrary, if the period of the motion is infinite,

the autoeorrelation function may tend to zero for large separation times, and the

motion may be chaotic.

At the present time,these are only theoretical examples for this occurrence of

chaos by an infinite number of bifurcations in a finite domain of variation of the

parameters. However this is probably (25) the best understood case.

CONCLUSION

Even if the proofs of theorems are quite remote, there is some hope at the pre-

sent time for understanding the way in which turbulence may occur in flows (convective

or not). However let us emphasize again that this approach leaves aside the question

of spatial chaos. Further studies in this domain are needed in order to know

whether the spatial chaos Dccurs or not in the infinite Rayleigh (or Reynolds) number

limit,as required by the Landau theory of turbulence.

REFERENCES

I. D. Ruelle, F. Takens, Comm. Math. Phys. 2iO , 167 (I971)

2. P.R. Halmos, Ergodic theory, Chelsea Pub. Comp. New York (1956)

3. R. May, Nature 261, 459 (1976)

4. T.Y. Li and J.A. Yorke, Am. Math. Month. 8_~2, 985 (1975)

5. T. Rikitake, "Electromagnetism and the earth interior" Elsevier (1968)

6. C. Laj, Y. Pomeau, in preparation

7. Monin, Yaglom, "Statistical Fluid Mechanics'~ Vol. I-2, MIT Press (1971)

348

8. I. Kubo, notes from the Nagoya Univ.,

E. Hopf, "Ergodentheorie", Splnger Verlag, Berlin (1937),

M. Smorodinsky , "Ergodic Theory,Entropy", Springer Verlag Lectures Notes in

math. 2 14, Berlin (1971)

9. S. Smale, Bull. AMS 73, 747 (|967)

I0. P.C. Martin and J.B. Me Laughlin, Phys. Rev. Left. 33, 1189 (1974), Phys. Rev°

At_!2 , 186 (1975)

II. E.N. Lorenz, J. Atmo. Sci. 20, 130 (1963)

12. R° Thom "ModUle math~matique de la morphoggn:se", 10-18, Paris (1974)

13. A. Andronov and L. Pontryagin, Dokl. Akad. Nauk. SSSR 14, 247 (1937)

14. D. Ruelle, in "Turbulence and Navier-Stokes equation" Lecture Notes in Math,

565, Springer Verlag, Berlin (]975),

O. Lanford, III lectures at IHES (1975)~

R.B. Williams "On the Structure of the Lorenz Attractors", preprint

15. V. Arnold, A. Avez, "Ergodic Problems of Classical Mechanics", Benjamin, New York

(1969)

]6. R. Betchov and W.O. Criminale Jr. "Stability of Parallel Flows", Acad. Press,

New York (]966)

17. Landau et Lifehitz, "M:canique des Fluides" chap. III, §27, ed. Mir. (Moscou)

(1971)

18. D.R. Caldwell. J. of Fluid Mech. 64, 347 (1974)

]9. N.N. Bogoliubov, J.A. Mitropolskii, A.M. Samo~lenko, "Methods of accelerated

mechanxcs convergence in non linear " " Spinger Verlag, Berlin (1976)

20. V.I. Arnold, Small divisors I, Izv. Akad. Nauk. SSSR Ser. Mat. 25 (1), 21 (J961),

Small divisors II, Usp. Mat. Nauk I8 (5), 13 (]963) ; 18 (6), 9I (]963),

M.R. Hermann, Thesis, Orsay (1976)

21. Reference 19, p. 154

22. Reference 9, p. 788t

M. Shuh, Thesis, Univ. of CaLif. Berkeley (]967)

23° J.P. Gollub, S.L. Hulbert, G.M. Dolny and H.L. Swinnay, to appear in "Photon

correlation, spectroscopy and velocimetry " Ed. E.R. Pyke and H.Z. Cun~ins,

Plenum Press (1976)

24. J.L. Ibanez, Y. Pomeau, to be published

25. Ref. 3, and P. Stefan, preprint IHES (Bures-sur-Yvette) Dec. ;976~

A.N. ~arkovskiy, Urk. Math. t. 16, I (1964),

B. Derrida and Y. Pomeau, to be published

STELLAR CONVECTION

D.O. Gough

Institute of Astronomy, Cambridge, England

The most important function of a convection theory for stellar model building is

to determine the temperature stratification in terms of the heat flux. Apart from in

some recent work by Latour et al. (2), mixing length theory, in one or other of its

guises, still provides the only prescription that is used. Unfortunately this is not

a reliable procedure because of the crude way in which the dynamics is treated. Fur-

thermore, the resulting formulae depend on the mixing length £ which occurs in the

theory as an undetermined function. Work on convection that may one day lead to a more

satisfactory theory is taking place, but none of it has yet reached the point to

warrant displacing the methods currently practised in stellar evolution computations.

The reader is referred to the reviews by Spiegel for a discussion of the astrophysi-

cally relevant work on convection up to 1972 (2,3).

A.ATTEMPTS TO MODEL THERMAL CONVECTION

One of the principal factors inhibiting progress in stellar convection theory is

that conditions in stars are very different from those in the laboratory. Stellar con-

vection is characterized by high values (1020 ) of the Rayleigh number R, which is a

dimensionless measure of the temperature gradient, and low values (10 -9 ) of the Prandtl

number ~, which is the ratio of kinematic viscosity to thermal diffusivity. On the

other hand in the laboratory R is quite low (< 1011 ) by astrophysical standards, and

is of the order or greater than unity. Moreover, stellar convection zones extend typi-

cally over many scale heights of pressure and density, leading to compressible motions,

whereas in the laboratory the depth of a convecting layer is always a minute fraction

of a scale height: the motion is essentially incompressible and can be described by

the Boussinesq approximation (4).

Most of the theoretical work is aimed at mimicking laboratory conditions. A thin

layer of fluid bounded by twQ isothermal planes, the lower boundary being at the higher

temperature, is usually considered. The equations of motion are solved, usually in the

Boussinesq approximation, either numerically at moderate R and ~ (5-]0) or analytically

at low R close to the critical value R c at which a static fluid layer becomes unstable

to convection (;I). At present the computational difficulties are too severe to extend

these calculations to values of R and ~ of astrophysical interest. The main objective

Reprinted from: Trans. (Reports), 16A, Pt II, 169 (1976)

350

is to determine the dependence of the Nusselt number N, a dimensionless measure of

the heat flux, on ~ and R. A plausible extrapolation procedure might then lead to

a better prescription for stellar convection.

The most obvious procedure, one might think, would be to apply mixing-length theo-

ry to laboratory convection and attempt to determine I experimentally. However it is

the astronomer's belief that this is of no use, for whereas in the laboratory eddies

extend across the whole of the convecting layer, in a compressible fluid many scale

heights deep the shear produced by differential expansion and contraction of verti-

cally moving fluid is thought to disrupt the convective motion in about a scale height

(Schwarzschild, (12)). This is a nonlinear argument, and so is not contradicted by the

fact that the most unstable linear modes extend across the entire convectively unstable

region (Spiegel, B~hm, (13-16)). Accordingly, I is presumed to be proportional to a

density or pressure scale height (Opik, Vitense, ]7,18)), usually the latter, the con-

stant of proportionality being determined astronomically, and contact with terrestrial

experience is lost.

In its most usual form the mixlng-length theory provides a local relationship be-

tween the heat flux F and the snperadiabatlc temperature gradient D = V - Vad" There

are many uncertainties in the theory, and consequently there is opportunity to incor-

porate into it several adjustable parameters, though only two are of immediate inte-

rest (]9): ~ ~ 1/H, where H is an appropriate scale height, and a measure 7 of the

radiative losses. The theories are calibrated either by constructing a solar model

and adjusting it to have the correct luminosity and effective temperature at an age

of about 4.7 x ]09 yr (Schwarzschild et el; Sears (20,2])), or by fitting a theoreti-

cal sequence to a young cluster diagram (Demarque and Larson; Copeland, Jensen and

J4rgensen (22,23)). Both methods yield ~ = |, the precise value depending on the de-

tails of the theory adopted, but leave y undetermined. Some comfort is derived

from the observation that this implies an eddy size at the top of the solar convection

zone comparable with the length scale of the granulation (Schwarzschild (12)). The

gross structure of a main sequence stellar model is insensitive to ~, which matters

only near the outer edge of the envelope convection zone. The parameter ~ does affect

the convective envelopes of red giants, however, which have large nonadiabatic regions

(Henyey et al (24); Schwarzschild (25)). Red giant models are probably more sensitive

to other details of how convection is treated, too, especially in the surface layers

where fluctuations are large in magnitude and horizontal extent.

It should be noticed that the mixing-length formalisms used in stellar structure

computations are based on the Boussinesq approximation to incompressible flow. One

would have expected (4) this to have been valid had Z been much less than H, but in-

dications are that it is not a good approximation otherwise (Graham (26), Deupree (27)).

Thus the calibration 1 = H exposes an inconsistency in the theory. However, the evi-

dence for the functional dependence 1 = H is hardly overwhelming, and adopting it no

doubt introduces errors that are just as great. The apparent success of the mixing-

351

length formalism lies in the fact that the gross structure of a main sequence stellar

model is almost independent of the functional form of the relationship between F and D

(Gough and Weiss (19)).

Stellar mlxlng-length theory ignores viscosity. This sounds plausible since the

Reynolds number of the heat-transporting flow is large and ~ is small. It implies

that N depends on ~ and R only in the combination ~R which is independent of visco-

sity. Furthermore, since N increases with R at fixed ~, it must therefore increase

with ~ at fixed R, provided ~ remains small.

The beginnings of an attempt to bridge the gap between laboratory and stellar

conditions, using a trO~cated modal expansion, has recently been reported (Gough,

Spiegel, Toomre (28,29)). Although the analysis is in the Boussinesq approximation,

it can treat with the same assumptlonsthe extreme values of R and ~ typical of stars

and the more moderate values encountered in the laboratory. The procedure can be made

to reproduce some of the features of laboratory convection, but its most obvious draw-

back is that it contains several undetermined parameters. In its simplest form there

are just two such parameters, characterizing the horizontal scale and shape of the

convective eddies. Although this is perhaps an improvement over mixing-length theory,

which depends on an undetermined function Z, an unambiguous calibration by comparison

with laboratory convection has not been possible. The theory has the property that for

Y<< I, N is a function of ~R, provided the convection is three-dlmensional, which

accords with astronomers' prejudices.

It should be pointed out, however, that the ~ dependence of N is not universally

believed. This arises partly because almost all laboratory experience is with fluids

that have ~ ~ 1, and for these both theory and experiment show that N is almost inde-

pendent of ~ at fixed R. Furthermore, numerical solutions of the Boussinesq equations

at moderate R, which until recentlyhave always constrained the flow to be two-dimen-

sional, have predicted almost no ~ dependence~ and even a slight increase of N as

is decreased below unity (Veronis (5); Quon (8); Moore and Weiss (9)), though it has

been argued that this may be a result of constraining the horizontal length scale of

the motion (Lipps and Somerville (6); Willis, Deardorff and Somerville (7)).The sim-

plest modal analysis predicts that N is independent of Y when the motion is two dimen-

sional (28). Analytical expansions of the full Boussinesq equations for R near R c re-

veal only a weak dependence on ~ in that case too (Sehluter~ Lortz and Busse (30)) •

but, like the modal result~ suggest a strong decrease in N at low ~ when the motion is

three-dimenslonal. This led Jones, Moore and Weiss (31) to investigate numerically

axlsyrmmetrieal convection in a cylinder which, though mathematically dependent on only

two space variables, is geometrically three-dimensional. They reproduced the analyti-

cal results for R just above Re, but showed that at moderate R the flow readjusted it-

self to resemble the two-dimensional flows, and produced an N that is independent of

~at high and low ~, and slightly decreasing with ~ in the neighbourhood of ~ = I.

352

The issue is unresolved. Jones et al. suggest that their flow is unstable and that

at sufficiently Nigh R it would become turbulent with N independent of viscosity at

low ~. Moreover, three-dimensional calculations reported recently by Veltishchev and

Zelnin (10) at ~ = 0.7 and ~ = ! suggest that the flow does not adjust itself to the

kind of structure that is preferred when axisynm~etry is imposed, and that N is less

when ~ = 0.7 than it is when o = I. This concurs with the evidence from laboratory

experiments (29), though this is admittedly weak. Finally, if a convection theory

based on eddies of scale I = H and governed by dynamics similar to that exhibited by

the two-dimensional and axisymmetrical Boussinesq computations (and therefore implying

N is independent of ~) were subject to the usual astronomical calibration, the result

would be that I would be but a very small faction of H, which most astronomers would

find unpalatable.

Little study has been made of fully developed convection in a layer of compressible

fluid many scale heights deep. Graham (26) has made some two-dimensional computations

for a perfect gas at moderate R and ~.The property exhibited by similar Boussinesq

calculations that N is a decreasing function of ~ when ~ = | is accentuated as the

layer depth, and the effects of compressibility, are increased. Moreover, no tendency

for eddies to break up on a scale of H was found. Graham's more recent three-dimensio-

nal compressible calculations yield similar results (32). Compressible modal calcula-

tions have also been performed in the anelastic approximation; by van der Borght (33)

with ~ = 1 and by Latour et al. (]) under more realistic stellar conditions modelling

an A star envelope. As with the Boussinesq calculations there are undetermined para-

meters which can be chosen to produce plausible results. Once again, time-dependent

calculations (]) show no tendency for the motion to break up into eddies on the scale

of H.

B. PENETRATION AND OVERSHOOTING

The edges of stellar convection zones are not rigid inpenetrable boundaries as they

are in most laboratory and theoretical investigations. The density stratification chan-

ges from being conveetively unstable to convectively stable, from the point of view of

linear stability analysis, and fluid accelerated in the convectively unstable region

c~n penetrate, Qr overshoot, into the adjacent stable regions.

This phenomenon has been of interest particularly to meteorologists interested in

mixing at the atmospheric inversion (34). D.W. Moore (35) gives a brief account of the

relevant physics. A convective element, or thermal, on reaching the top (or bottom) of

the conveetively unstable region, still has a temperature excess (or deficiency) rela-

tive to its i~mediate surroundings and continues to experience a buoyancy force. If

the element were to maintain its identity and move adiabatically in pressure equili-

brium with its surroundings, buoyancy would not disappear until the level z = z s were

353

reached at which the specific entropy were the same as at the level at which the ele-

ment originated. This point does not necessarily mark the edge of the zone of penetra-

tion, however, because the element still has momentum to carry it on yet further. En-

traiament of stable fluid, on the other hand, retards the motion of the element. Thus

Z = z~ may either overestimate or underestimate the extent of penetration. Observa-

tions of the motions of cloud tops suggest that it is usually an overestimate

though in some circumstances, such as in tropical storms, large plume - like

structures penetrate well above the tropopause (36). An additional complication, usually

ignored by meteorologists in this context, is radiative diffusion, which tends to re-

duce both the buoyant acceleration and retardation by reducing temperature fluctuations.

Buoyant thermals penetrating into the stable layer advect heat upwards, though this is

offset by the induced return flow. Near the outer edge of the penetrated region both

upward and downward moving fluid presumably transport heat counter to the net flux.

Some aspects of the situation can be modelled with the ice-water experiment. This

consists of a layer of water cooled from below at 0=C and with its upper boundary main-

tained above 40C, the temperature of the density maximum. Laboratory experiments show

that the unstable layer extends beyond the limits it would have occupied had there been

no motion (Townsend (37), Adrian (38)), and in addition plume-like motions in the un-

stable region penetrate into the stable layer above. Adjacent layers of convectively

stable and unstable fluid have also been created by inducing spatially varying tempe-

rature gradients in water near room temperature, either by imposing time varying boun-

dary conditions (Krishnamurti (39); Deardorff, Willis and Lilly (40)) or by internal

heating or cooling (Failer and Kaylor (4|); Whitehead and Chen (42)). The nature of

the motions in the stable layer is not entirely clear, but the temperature fluctuations

observed by Townsend (37) seem to be the product of trapped gravity waves. Theoretical

numerical experiments in two-dimensions by Moore and Weiss (43) also exhibit the en-

croachment of the unstable region into the region that would have been stable in the

absence of motion, and the excitation of gravity waves. They also predict weak vis-

cously driven countereells which are not seen in the laboratory, and little evidence

of plumes. Earlier steady one-mode mean-field calculations, which in some sense repre-

sent two-dimensional motion, yielded similar results, without the gravity waves (Mus-

man (44)). Thus some of the observed features of laboratory experiments are reproduced

theoretically; the differences, as Spiegel (3) has pointed out, might result from the

two-dimenslonal constraint imposed on the numerical computations.

Although the ice-water experiment sheds some light on the mechanism of penetration

it seems difficult to generalize to stellar conditions. There is some evidence from the

two-dimensional numerical experiments that penetration increases as Prandtl number de-

creases (D.R. Moore (45)). Modal calculations by Latour (46) et al.(]) modelling three-

dimensional convection in A star envelopes predict greater penetration by the almost

plume-like columns in the eentres of hexagonal cells than by two-dlmensional rolls.

However, this analysis has not been applied to the ice-water problem; there is yet no

354

bridge between stars and laboratory experience.

The various theoretical prescriptions that are usually employed to describe over-

shooting from stellar convection zones are all essentially based on mixing-length theo-

ry (Spiegel (47), Parsons (48), Ulrich (49), Scalo (50), Shaviv and Salpeter (51)). Of

necessity they are nonlocal theories, though they all rely on the Boussinesq approxi-

mation. They have been used, in particular, to model the solar atmosphere which is per-

haps the most sensitive astrophysical testing ground at present, because quite detailed

comparison of theoretical predictions with observations can in principle be made. It

is not easy to deduce the height dependence of the solar atmospheric velocity fluctua-

tions~ nor is it easy to disentangle convective motion from waves, though an attempt -I

has been made (Frazier (52)). It seems likely, however, that velocities of about 2 Pan s

extend well above the photosphere (de Jager (53)), which agrees with a model computed

by Ulrich (54), though the theory does appear to overestimate the overshoot. Travis and

Matsushima (55), using a theory of Spiegel (47), compare their models with limb darke-

ning measurements and conclude also that too great an overshoot is predicted if a mixing

length to scale height ratio ~ of about unity is adopted; they favour ~ ~ 0.35, in

contradiction to the usual calibration. A subsequent investigation by Travis and Matsu-

shlma (56) of the eolours of cool main sequence stars and metal-deficient subdwarfs also

suggested a low value for ~. Nordlund (57), using Ulrich's approach, found overshoot to

a lesser degree for a given ~, and produced a model in better agreement with the Harvard-

Smithsonian Reference Atmosphere (58) and similar to an earlier model built by Parsons

(48) using a convective heat flux calculated from a nonlocal estimate of vertical velo-

city and a local estimate of temperature fluctuations. In contradiction, Edmonds's ana-

lysis (59) of the photospheric velocity and brightness fluctuations favours a greater

degree of overshoot, so the matter seems unresolved. One thing that does seem clear is

that at their present stage of sophistication nonlocal convection theories should not

be relied upon to explain fine details, especially in regions in which the assumptions

on which they are based are not satisfied. All the theories have adjustable parameters

and can no doubt be tuned to rationalize the limb darkening; adjusting the radiative

loss coefficient in Spiegel's theory, for example, could probably lead to an atmosphere

hardly distinguishable from Nordlund's with an ~ consistent with the evolutionary ca-

libration. Indeed Spruit (60) has produced a model with the correct centre to limb

flux variation using a local mixing-length theory with no overshoot at all, though

presumably this does not reproduce the fluctuation measurements discussed by Edmonds

(59). Although there seems to be too much uncertainty in the theories at present to

apply such subtle tests, detailed analyses of inhomogeneous atmospheres must eventually

be undertaken both for theoretical model building and for analysing observational data.

Horizontal temperature fluctuations increase the horizontally averaged opacity, for

example, since opacity is a steeply increasing function of temperature, which leads to

an increase in the actual mean temperature gradient. Furthermore, since the magnitude

355

of the fluctuations decrease with height, the temperature gradient currently inferred

from limb darkening observations (55,57) is overestimated when fluctuations are ignored.

Abundance measurements may be affected. Turbulent Reynolds stresses generated by both

the convection and the gravity waves in the photospheric regions also influence the

stratification.

Another important consequence of overshooting is material mixing, particularly

at the edges of convective cores. Early estimates (Roxhurgh (6]); Saslaw and Schwarz-

schild (62)) which ignored the influence of the convective energy flux on the tempe-

rature stratification, implied negligible mixing rates. But recently Shaviv and Sal-

peter (51) pointed out that the modification to the stratification increases the pe-

netration of the motion into the stable envelope, just as in the case of the ice-water

experiment. Maeder (63) and Cogan (64) independently confirmed this conclusion with

more detailed calculations. The influence of the overshooting on colour-magnitude dia-

grams for old open clusters was subsequently investigated. Of particular interest is

the position and magnitude of the gap at the top of the main sequence, which can be

more accurately reproduced theoretically if an appropriate degree of mixing at the

core boundary is assumed (Maeder (65), Prather and Demarque (66)). Using Shaviv and

Salpeter's prescription for overshooting, Maeder (67) found once again that a value

of ~ somewhat less than unity gives the best results. This too should not be regar-

ded as contradicting the usual calibration, partly because the chemical composition

adopted for the models may not have been appropriate, partly because there lles buried

in the mixing-length formalism an undetermined parameter in the relation between ve-

locity and temperature fluctuations that does not appear in the formula for the heat

flux (|9), partly because the geometry of the core has not been taken into account,

and partly because the ratio of the mixing length to pressure scale height can hardly

be a universal constant.

Calculations by Sugimoto and Nomoto (68) and Iben (69) suggest that theoretical

predictions of nucleosynthesis in post main sequence stars would be significantly

affected by overshooting beneath convective envelopes. It would also have some bea-

ring on the observed lithium abundance in the sun (Spiegel (70)).

C. SUBCRITICAL CONVECTION

In the relatively straightforward case of ordinary convection discussed in ~A,

N is an increasing function of R at fixed ~. That is not necessarily the case when

agents such as rotation, magnetic fields or nonuniformities in composition are pre-

sent to inhibit the motion. The minimum Rayleigh number R o above which convection can

exist is modified by the presence of the stabilizing agent, but it is not always

possible to determine its value by linear stability analysis. It is often the case

that direct convective motion of finite amplitude can adjust itself to reduce the

efficacy of the stabilizing forces~ and so exist at a Rayleigh number below the

356

value R C predicted by linear theory (Veronis (71)). This is called suhcritical con-

vection. Of course such a state can be achieved only if it were approached hy lowering

R from a value greater than R~ or if a metastable state, with R < R < R~, were appro- o

priately perturbed by a finite amount. It seems that the latter is often achieved spon-

taneously because in many circumstances there is a range of R below R e within which

the fluid is overstable, that is to say unstable to infinitesimal oscillations

(Veronis (71), Weiss (72)). Weak experimental evidence exists to support the idea

that such motion might grow to an amplitude great enough to trigger subcritical con-

vection (Turner (73), Shirtcliffe (74), Rossby (75)).

The most widely studied problem of this type, and perhaps the easiest to understand,

is thermohaline convection. Turner (34) summarizes well the present state of knowledge.

The case of interest here is when salt stabilizes a layer of water heated from below.

Veronis (7] , 76) studied the overstability and suhcritical direct convection and gave

a simple physical explanation of why they should occur (71)~ Laboratory experiments

reveal that instability does occur first as a growing oscillation (74), and that con-

vection subsequently organizes itself into a series of superposed shallow layers sepa-

rated by diffusive interfaces (Turner and Stormnel (77)), a configuration that has

been observed to occur naturally (Hoare (78), Neal, Neshyba and banner (79)). The

fluxes of heat and salt appear to be controlled by the diffusive interfaces, and Tur-

ner (80) has observed that their ratio ~, When measured in units of the fluxes that

would have occured had motion been absent, appears to be independent of the ratio X of

the jumps in salinity and temperature across the layer, over a wide range of ~. Indeed,

it has been suggested (Turner (34)) that this value of # depends only on the diffusion

coefficients of the fluid, though recent experimental work indicates that it depends

also on R (Marmorino and Caldwell (81)).

The astrophysical relevance of thermohaline convection is to the edges of convective

cores of stars (Spiegel (82)),where the products of nuclear reactions~ usually helium,

take place of salt. When the usual criterion for convective instability is employed in

a massive stellar model evolving off the main sequence, it is found that once a suffi-

cient, stable discontinuity of composition is built up at the edge of the convective

core, the envelope immediately outside it is also convectively unstable. This has been

considered unacceptable by many astrophysicists, and it is assumed that the disconti-

nuity is somehow smoothed out, usually to precisely the degree that results in no more

than marginal stability immediately beyond the truly convective core (Ledoux (83),

Tayler (84), Schwarzschild and Harm (85)), though other amounts of mixing have been

proposed (Gabriel (86), Sale (87)). Different criteria are used to define marginal

stability, which lead to rather different results, but it does not seem possible to

choose between them by astronomical means (i e,g, Ch~osl and Summa (88) ;

Robertson (89); Swefgart and Demarque (90); Z~6~kows~i (91); Varshavskfi (92);

Sreenivasan and Ziebarth (93); Stothers and Chin (.@4)).

This situation has some similarftles to thermohaline convection set up by heating from

357

below an initially isothermal layer of water stably stratified with salt, but the ana-

logy is not perfect. Gabriel (86) has argued against generalizing from laboratory expe-

rience in this case. However, the idea that at least one shell of ordinary convection

is created outside the convective core does not seem implauslhle, though it is not ob-

vious whether the interface separating the two convecting regions would be stable enough

to survive the disrupting forces of the turbulence. Such a possibility has been pointed

out by Tayler (84, 95) as a mathematically consistent alternative to the conventional

procedure. To determine the structure of the region an understanding of the diffusive

interfaces is required. Had ¢ been independent of % and ~ one might have had some con-

fidence in extrapolating laboratory measurements, especially sinee~ if thin convective

layers are formed, this is one place where the Boussinesq approximation might be valid ;

but it appears that the answer is still out of reach.

Subcritical convection may also be relevant to solar type stars. It has been pointed

out that the stability characteristics of the solar core are potentially similar to

those of the thermohaline situation : overstable to infinitesimal perturbations and

able to sustain direct convection of finite amplitude (Dilke and Gough (96)). An

important difference~ however~ is that whereas the usual saline layer derives its ener-

gy from an externally imposed heat source and will convect so long as that source is

maintained, the sun must derive its extra energy from burning a supply of 3He which is

mixed from the edge of the core. The amount of SHe available is finite and after it

is burnt convection is presumed to cease, and the solar core becomes quiescent again

until a new supply of fuel has accumulated near its edge. If it occurs, this process

may have some bearing on the solar neutrino problem and the occurrence of terrestrial

ice ages. Subsequent more detailed analysis has supported the overstability postulate

(e.g. Noels et al., (98) Unno (99, 97, I00) ), though some computations have cast

doubt on it (Christensen-Dalsgaard and Gough (10])). The likelihood of subcritical

convection is questionable too (Ezer and Cameron (102) ; Ulrich (]03)), though some

evidence for it has been found (Rood (I04)).

D. ROTATION AND MAGNETIC FIELDS

Uniform rotation inhibits convective motion and so increases the critical Rayleigh

number above which convection can take place. At finite amplitude the motion can redis-

tribute the angular momentum so that subcriticai convection can occur (Veronis (IO5)).

Typically the constraint cannot be cancelled entirely and the rotation reduces the heat

flux. This is not always true, however. Rossby found in the laboratory that rotation

sometimes increases N at fixed R, a behaviour seen also in three-dimensional numerical

experiments (Somerville and Lipps (]06)) and a modal analysis (Baker and Spiegel (]07).

It gives fair warning to those who argue that factors inhibiting linear instability

necessarily inhibit subsequent nonlinear development.

358

Astrophysical interest in the interaction between convection and rotation has

been concerned in recent years with the structure of the convection zone in the solar

envelope and the maintenance of the differential rotation. Of particular interest are

the numerical experiments by Gilman (108). The subject has been reviewed recently by

Gilman (]09), Durney (]]0) and Weiss (]]J). Tayler (I12) has discussed convection in

rotating stellar cores.

The solar convection zone will not be understood until it is known how convection

interacts with magnetic fields. It is hard to infer the field strengths beneath the

surface, especially since the topology of the convective motion is such as to submerge

the field (Drobyshevskiand Yuferev (I13)). The formation and decay of magnetic field

concentrations are of obvious interest, and are reviewed in the proceedings of IAU

Symposium n ° 7]. Like uniform rotation, a uniform magnetic field tends to inhibit con-

vective motion. The linear stability characteristics of a plane Boussinesq fluid layer

heated from below are similar to the rotating case with no magnetic field. But Weiss

(114) has pointed out that there are fundamental differences between Lorentz and Coriolis

forces and that care must be taken when comparing the two cases. Weiss found that the

nonlinear development of both overstable and direct infinitesimal motions can be oscil-

latory, provided the magnetic field is not too weak. The final state is not necessarily

one in which there is equipartition between kinetic and magnetic energies (Peckover and

Weiss (I15)). Modal calculations (van der Borght, Murphy and ~piegel (I16)) have

revealed only a decrease in N at fixed R as the magnetic field increases, hut a magnetic

field appears to be able to interact with a rotating fluid in such a way that the

resulting Nusselt number is greater than it would have been in the absence of the field

(van der Borght and Murphy (I]7)).

Of interest recently has been the question of whether convection can be the source

of dynamo action. Childress and Soward (118) demonstrated that the kind of flow encoun-

tered in a rotating convecting fluid is suitable for amplifying magnetic fields, as has

been noticed also by Spiegel (3). Perturbation expansions about the marginal state

(Soward (119) ; Roberts and Stewartson (]20)) and a modal analysis (Baker (121)),

both of which incorporate the forces on the fluid arising from the perturbed magnetic

field, indicate that a convectlng fluid can indeed sustain a magnetic field by induction.

E. TIME-DEPENDENT CONVECTION

New difficulties are encountered when a star is varying globally on a time-scale

comparable with the convective turnover time. This may occur when the star is not in

hydrostatic equilibrium : during gravitational collapse, a nova or supernova explosion,

a flare or envelope ejection, or whenever a star pulsates. It is perhaps for pulsating

stars that an understanding of the time dependence of convection is most urgently needed

because both theory and observations have progressed further than in the studies of other

359

classes of intrinsically variable stars. Many of the gross features of the observations

have been explained, but the position of the red edge of the Cepheid strip, for example,

remains unsolved. This can probably be blamed on an inadequate treatment of convection

in the theoretical models.

Most computations of stellar pulsations have either ignored convection entirely,

ignored perturbations (either Lagrangian or Eulerian) in the convective heat flux indu-

ced by the pulsations, assumed the convection to adjust instantaneously to its changing

environment, or assumed it to relax towards the state given by the usual mixing-length

formulae at a rate proportional to the amount by which it deviates from that state and

inversely proportional to the eddy lifetime. The last of these prescriptions is perhaps

the most credible, and was first used by Cox et al. (]22) to compute model Cepheids.

However, its obvious deficiency is that it contains a free parameter : a constant of

proportionality that determines the rate at which convection readjusts to the pulsation.

This in turn determines the phase difference between the convection and pulsation, upon

which the pulsational stability of the star directly depends.

Attempts have b~en made by Cough (123) and Unno (324) to generalize the mixing-

length theory. Unfortunately there are different ways of formulating the fundamental

postulates. The resulting formulae are essentially the same for hydrostatic stars but

differ when the star is presumed to pulsate. M~reove~ there appear to be no relevant

laboratory experiments with which to compare the various possibilities. Despite these

uncertainties it would be interesting to know how sensitive pulsating stellar models

are to the assumptions behind the convection formalism, and Whether a possible choice

of the uncertain parameters exists that rationalizes the observations. Computations in

the quasiadiabatic approximation suggest this may be so, but apart from misrepresent-

ing nonadiabatic effects these computations are deficient in an important respect :

they do not take due account of the turbulent Reynolds stress.

It is usual to ignore the Reynolds stress when computing stellar models, partly

because ~ pO~t~O~ mixing-length estimates are less than the gas pressure gradient in

all but a thin region at the top of the hydrogen ionization zone. Attempts to include

this stress in nonpulsating stars have been made, notably by Henyey et al. (24) and

Parsons (48) but the formulation adopted is not entirely consistent. The mixing-length

formula for the Reynolds stress adds second derivatives of temperature and pressure to

the hydrostatic equation, raising its order and introducing singular points at the edges

of the convection zones. This has led to numerical instabilities (24) which have been

removed by judiciously ignoring high derivatives. It is claimed that this should not

alter the results substantially. A consistent computation should be done to check.

It is even more important to include the Reynolds stress in pulsating models. The

motion of most of the star is almost adiabatic ; density and pressure perturbations are

almost in phase and the work done in a single cycle is much less than the energy ex-

changed between thermal, gravitational and kinetic forms. There is no reason to suppose

that the turbulent stress is in phase with the density, however, so even though its

360

magnitude may be much less than the gas pressure gradient the work it does might not

be negligible.

The modal approach adopted by Latour et al. (1) includes the Reynolds stress.

Because viscosity is include the equations are not singular, but the low Prandtl num-

ber gives such severe numerical trouble that it has not yet been possible to compute

stellar models with deep convection zones. In principle this method can be used for

pulsating stars, but certain aspects of the turbulent energy transfer are lost in the

modal truncation and once again the results would have to be treated with some caution)

A recently discovered pulsating star of some interest is the sun (Hill and Stebbins

(125) ; Fossat and Ricort (]26) ; Severny, Kotov and Tsap (I27) ; Brookes, Isaak and

van der Raay (128)), which is pulsating in many modes simultaneously. The pulsations

are of too low an amplitude to have a noticeable influence on the structure of the

star, hut they could provide a powerful diagnostic tool. The oscillation periods are

in satisfactory agreement with theoretical estimates (Christensen-Dalsgaard and Gough

(130)) Scnflaire, Gabriel, Noels and Boury (]29)), but how the oscillations are dri-

ven is not yet known. It is unlikely that convection is unimportant. Differences bet-

ween linear analyses which have either ignored convective flux perturbations (e.g.

(97) Shibahashi et al. (100, ]Ol) ) or have taken them into account (Noels (98, 131)

et al.) using Unno's (124) approach suggest that the stability of the modes of oscil-

lations are rather sensitive to the assumptions adopted. In the light of experience

with stars in hydrostatic equilibrium (Gough and Weiss (]9)) perhaps it is too opti-

mistic to hope for an unambiguous solar calibration of time-dependent convection in

the near future.

REFERENCES

I. Latour, J., Spiegel, E. A., Toomre, J. and Zahn, J.-P., (]976), Ap J., 207, 233

and 545

2. Spiegel, E. A., (197|), A~n. Rev. A. Ap , 9, 323

3. Spiegel, E. A., (]972), Ann. Rev. A. AP , ]O) 26I

4. Spiegel, E. A.) and Veronis, G., (]960), Ap J:., ]31, 442 ; (]962) 135, 655

5. Veronis, G., (]966), JFM, 26, 49

6. Lipps, F. B. and Sommerville, R. C. J., (1971), Phys. Fluid s, 14, 759

7. Willis, G. E.) Deardorff, J. W. and Sommerville, R. C. J., (1972), JFM) 54, 35]

8. Quon, C., (1972), Phys. Fluids, ]5, J2

9. Moore, D. R. and Weiss) N. O.) (1973), JFM, 58, 289

]O. Veltishchev, N. F. and Zelnin, A. A., (1975), JFM, 68, 353

]]. Busse, F. H., (197]), Proc. IUTAM S~p., Herrena!b ~ 4]

]2. Schwarzschild) M., (|96|), Ap J., 134, I

13. Spiegel, E. A. and Unno, W., (]962), PASJ, 14, 28

361

]4. Spiegel, E. A., (1964), Ap J., 139, 959

I5. Spiegel, E. A., (1965), Ap J., 141, 1068

16. B~hm, K. -H., (1967), IAU Syrup. NO. 28, p. 366

|7. Dpik, E. J., (]938), Publ. Obs. Astr. U. Tartu, 3__O0, No. 3

18. Vitense, E., (1953), Z. Ap, 32, 135

]9. Gough, D. O. and Weiss, N. O., (1976), MN, ]76, 589

20. Sehwarzschild, M., ~oward, R. and H~rm, R., (1957), Ap J., ]25, 233

21. Sears, R. L., (]959), Ap J., 129, 489

22. Demarque, P. R. and Larson, R. B., (]964), Ap J., 140, 544

23. Copeland, H., Jensen, J. 0. and JMrgensen, H. E., (]970), A. and. Ap, ~, 12

24. Henyey, L,, Vardya, M. S. and Bodenheimer, P., (1965), Ap J., ]42, 84]

25. Sehwarzschild, M., (1975), Ap J., ]9_55, 137

26. Graham, E., (1975), JFM, 70, 689

27. Deupree, R. G., (]975), Ap J., 201, 183

28. Gough, D. 0., Spiegel, E. A. and Toomre, J., (1975), JFM, 68, 695

29. Toomre, J., Gough, D. O. and Spiegel, E. A., (]977), JFM, 79, ]

30. SchSlter, A., Lortz, D. and Busse, F. H., (]965) JFM, 23, ]29

31. Jones, C. A., Moore, D. R. and Weiss, N. O., (]976), JFM, 73, 353

32. Graham, E., (]975) private communication

33. Van der Borght, R., (1975), MN, ]73, 85

34. Turner, J. S., (Cambridge Univ. Press, ]973), Buoyancy effects in fluids

35. Moore, D. W., (]967), IAU Symp. No. 28, 405

36. Saunders, P. M., (1962), Tellus, J4, ]77

37. Townsend, A. A., (]964), Q. J. roy. Met. Soe=, 90, 248

38. Adrian, R. J., (1975), JFM, 69, 735

39. Krishnamurti, R., (1968), JFM, 33, 445 and 457

40. Deardorff, J. W., Willis, G. E. and Lilly, D. K., (|969), JFM, 35, 7

4]. Failer, A. J. and Kaylor, R., (]970), JGR, 75, 52|

42. Whitehead, J. A. and Chen, M. M., (1970), JFM, 40, 549

43. Moore, D. R. and Weiss, N. 0., (]973), JFM, 6_!I , 553

44. Musman, S., (1968), JFM, 3], 343

45. Moore, D, R., (]975), private communication

46. Latour, J., (1973), Thesis, Univ. Nice

47. Spiegel, E. A., (]963), Ap J., 138, 2]6

48. Parsons, S. B., (1969), Ap J. Suppl., ]88, 127

49. Ulrich, R. K., (1970), ~ Sp. Sci., !, 7]

50. Scalo, J. M. and Ulrich, R. K., (1973), Ap.~J. , 183, 151

51. Shaviv, G. and Salpeter, E. E., (1973), Ap J., 184, ]9]

52. Frazier, E. N., (]968), Ap J., 152, 557

53. de Jager, C., (1972) Solar Phys., 25, 71

54. Ulrich, R. K., (1970), Ap Sp. Sci., !, 183

362

55. Travis, L. D. and Matsushima, S., (1973), Ap J., J80, 975

56. Travis, L. D. and Matsushima, S., (1973), Ap J., ]82, 189

57. Nordlund, ~., (|974), A. and Ap, 32, 407

58. Gingerich, O., Noyes, R. W., Kalkofen, W. and Cuny, Y., (J971), Solar Phys., 18, 347

59. Edmonds, F. N., Jr, (1974), Solar Phys., 38, 33

60. Spruit, H. C., (1974), Solar Phys., 34, 277

6]. Roxburgh, I. W., (1965), MN, 130, 223

62. Saslaw, W. C. and Schwarzschild, M., (|965), Ap J., 142, 1468

63. Maeder, A., (]975), A. and Ap, 40, 303

64. Cogan, B. C., (1975), Ap J., 20], 637

65. Maeder, A., (1974), A. and Ap, 3~4, 409

66. Prather, M. J. and Demarque, P. R., (1974), Ap J., 193, 109

67. Maeder, A., (1975), A" and Ap, 43, 61

68. Sugimoto, D. and Nomoto, K., (1974), IAU S ymp.: No. 66, 105

69. Iben, I., Jr~, (1975), Ap J., 196, 525

70. Spiegel, E. A., (1967), Highlights. in Astr. (IAU), ~6l

71. Veronis, G., (1965), J. Marine Res., 23, ]

72. Weiss, N. O., (1964), Phil. Trans., A 2____~56, 99

73. Turner, J. S., (1968), JFM, 33, 183

74. Shlrtcliffe, T. G. L., (1969), JFM, 35, 677

75. Rossby, H. T., (1969), JFM, 36, 309

76. Veronis, G., (]968), JFM, 34, 315

77. Turner, J. S. and Storm~el, H., (1964), Proc. Nat. Acad. Sei., 52, 49

78. Hoare, R. A., (}966), Nature, 210, 787

79. Neal, V. T., Neshyba, S. and Denner, W., (]969), Science, 166, 373

80. Turner, J. S., (]965), Int. J. Heat and Mass Trans., 8, 759

8]. Marmorino, G. O. and Caldwell, D. R., (]976), Deep-Sea Res., 23, 59

82. Spiegel, E. A., (}969), Comm. Ap Sp. Scl., i, 57

83. Ledoux, P., (]947), Ap J., IO_~5, 305

84. Tayler, R. J., (]954), Ph.D. The£i£, Univ. Cambridge

85. Schwarzschild, M. and Harm, R., (1958), Ap J., 128, 348

86. Gabriel, M., (1970), A. and Ap, ~, ]24

87. Saio, H., (]974), Ap Sp. Sc~., 29, 4J

88. Chiosi, C. and Sun~ma, C., (]970), Ap Sp. Sci., 8, 478

89. Robertson, J. W., (]972), Ap J., 177, 473

90. Sweigart, A. V. and Demarque, P., (]972), A. and AP, 20, 445

91. Zi61kowski, J., (1972), Aeta Ast., 22, 327

92. Varshavskii, V. I., (]973), Soy. Ast. AJ, 16, 86]

93. Sreenivasan, S. R. and Ziebarth, K. E., (]974), Ap Sp. Sei., 30, 57

94. Stothers, R. and Chin, C., (1975), Ap J., ]98, 407

95. Tayler, R. J., (]969), FIN, ]44, 231

363

96. Dilke, F. W. W. and Gough, D. O., (1972), Nature, 240, 262

97. Christensen-Dalsgaard, J., Dilke, F. W. W. and Gough, D. O., (]974), FIN, I69, 429

98. Noels, A., Gabriel, M., Boury, A., Scuflaire, R. and Ledoux, P., (1975), M~m. Soc.

Roy..Sci. L~ge, # Ser., 8, 317

99. Unno, W., (1975), PASJ, 27, 81

]00. Shibahashi, H., Osaki, Y. and Unno, W., (]975), PASJ, 27, 4Ol

lO]. Christensen-Dalsgaard, J. and Gough, D. 0., (]975), Mfim. Soc. Roy..Sci. Liage,

6 Set., 8, 309

102. Ezer, D. and Cameron, A. G. W., (1972), Nature phys. Sci. , 240, 180

]03. Ulrich, R. K., (1974), Ap J., 188, 369

]04. Rood, R. J., (1972), Nature Phys. Sci. , 240, 178

]05. Veronis, G.) (1959), JFM, ~, 401 ; (I965), 24, 545

IO6. Sommerville, R. C. and Lipps, F. B., (]973), J. Atmos. Sci.) 30, 590

107. Baker, L. and Spiegel, E. A., (]975), J. Atmos. Sci., 32, ]909

]08. Gilman, P. A., (]976), IAU Symp. No 7], 207

]09. Gilman, P. A., (1974), Ann. Rev. A. Ap, 12) 47

]IO. Durney, B. R., (]976), IAU Symp. No 71, 243

Ill. Weiss, N. 0., (1976), .IAU Symp. No 7|) 229

ll2. Tayler, R. J., (1973), MN, ]65) 39

If3. Drobyshevski, E. M. and Yuferev, V. S., (1974), JFM, 65, 33

114. Weiss, N. O., (1975), Adv. Chem. Phys., 32? ]01

]15. Peckover, R. S. and Weiss, N. 0., (1972), Comp. Phys* Comm., ~, 339

116. Van der Borght, R., Murphy, J. O. and Spiegel, E. A., (]972), Aust. j. Phys.,

25, 703

]17. Van der Borght, R. and Murphy, J. O., (1973), Austr. J. phys., 26, 617

]]8. Childress, S. and Soward, A. M., (1972), Phys. Rev. Left., 29, 837

If9. Soward, A. M., (]974), Phil. Trans., 275, 61!

120. Roberts, P. E. and Stewartson, K., (1975), JFM, 68, 447

]21. Baker, L., (]975), private con~nunication

122. Cox, J. P., Cox, A. N., Olsen, K. H., King, D. S. and Eilers, D. D., (1966), Ap J,,

:,,44, ]o3s

123. Gough, D. 0., (1965), GFD, II (Woods Hole Ocean. Inst.) p. 49

I24. Unno, W., (1967), PASJ, ]9, 140

|25. Hill, H. A. and Stebbins, R. T., (]975), Ann. New York Acad. Sci., 262, 472

126. Fossat, E. and Ricort, G., (]975), A. and Ap, 43, 243

127. Severny, A. B., Kotov, V. A. and Tsap, T. T., (1976), Nature, 259, 87

128. Brookes, J. R., Isaak) G. R. and van der Raay, H. B., (]976), Nature, 259, 92

129. Scuflaire, R., Gabriel, M., Noels, A. and Boury, A., (]975), A. and Ap, 45, ]5

130. Christensen-Dalsgaard) J. and Gough, D. 0., (]976), Nature, 259, 89

131. Gabriel, M., Scuflalre, R., Noels, A. and Boury) A., (1975)) A. and Ap, 40, 33

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