Cosmic Gas

Dynamics

Prof. Izmodenov

Lomomnsov Moscow State University Moscow,Russia

Praperaed By: Patel Mehulkumar L

C 1I :lp1.(' )' I

Basic Fluid Equations

Fluid dYllallJ ics is 1,] 1<' ruul.iuu um dl 'snip l i" l1 01' III(' flo w 01' ;1 lal'/-',I' 1I11 l1l1 lf'r 01'

par{'id( ~s , Such a d ( ~scriplioll is wid ('],\' a pp l ical,I (' ill ;ls l l'op llys ic;ll ]l l'ol>k lJIs .

anr l flu id dy lla i llic;d P]'l l(' { 'SS I ~S pl ay n l« :v roh ' i II Ilia 11\ ' ; \1'(,< IS "I' ;ls1rophysi cs .

IJI t.hi» h ook 11](' JI11 id undor co usi(kr a l ioJJ wil l l',ell('I';ll ly 1H' a I',as, th ought.J w equations of Iiuid clyu .uuics cau a lso IH ' app l i f ~d 10 (1r:sni],ill l', 1.11<' motion

of a «ollocl.ion of s t a rs , 01' eV('JJ ga lax ies . provicl.«! t.luu 11111' is iU{.I ·]'('s !.ed ill

t.h« co llpdivp hohuviour Oil su flicie ld ly I:I I'/-',e sca les.

T he ten u fl uid ill g<:JJera J n: l'ers {." g.; ISI 'S a lld liq ni.]«. FJllids n]'(' Ilis

j,jul',l1ished fro m so lids ill t.hat . so lids have' ri gidi ty. llol.l! ,~o l i d s alld llu idsdoIorru w hen a s tress is »pp liod t o them: hut. unlil« - a so lid . a sim p le fluid

has no lelJ<!eucy t o ret 11 I'U t o it s orig in n) stnte whr-» the ilpplier] s1n'ss is

roiuovod .

T he continniun description is fuud mueut.al !.o the Ji u id approach to r1p­

scribing t.ho dynamics of a coll ection o f p.nti clcs . 'I'ho domai n of vali d itv of

t he con t.iuunm d esITip1.ioJ] is d l'1,f'n JJiJH'd l,y cOlllp;n'i llg 111<' "ollisioll al nu -au

Jiu ' paUl I of 1.1 ](' pa r1.i f·]r,s w it h 1:he m acroscopic ]('Jlg !.!t SC; I I" l . 0 1' ijd (']'es t

in t.hc pruhlcui , If I « L t.hon i t is rensoJlah lf' t o in t.rod ucx - 1,11( ' cOJlcppl of

a fluid volume element. whoso lin ear siz e i,s much large1' th an 11m! mu ch

sm all er than L . The number of particles inside a Iluid clem ent is ]a1'g(·.

a nd we call associa te wit.li the Iln id elouieut a hul k velocity 11. Ind ividua l

p ar ti cle veloc ities have a random co mponent in add ition (.o 11 but" 1)1'('a 11SI '

th« moan frce pnt .h is smal l. t he r.uu kun motion d( H's ]JO( illllllPdiaf ely t.al«:th p particl e 1'<11' from its ncig hbonring part icles lH'(',1l1,SC' th e p ;lrlil'1l ' 11 :nC'1s

on ly a clis tnuco of ordr-r I ]JI'fo]'(' 11lI(krgoiu/-', a colli si OJJ a nd ch;ul ,l~ iug il s

di rect ion . \Ve «an a lso assrwiaj·f' wit h t I)(' flu id d (·]J)(,JJj. ot.hr-r 111 ;HTOSI'0l'i l'

proper ti es such as a d ensity fI (to tal mass of OJ(' particles inside 1:1](' olcmont.

di v id ed 1Iy it s VOIlllJJC ). Over a shor t t ime inte-rval lro in time I to time I +iSt

A siroplujsicu! Fluid Dynll1Hics Basic Fluid Equations

1.3 T he Momentum Equation

1. 2 T he Cont.in u ity Equation

(1.2)

(1.3)

(1.4)

Iim (f(1' -+- 11, r5t , t +<5I) -- [ (1',1,) )1m .St,- .o M

Df

D I

the fl uid , over a shor t. peri od of time ot , in (.]I e limi t ilS ill 1ends to zer o.Since (con ed to Iirs t order in M) the fluid elcme ut will have moved fromr at time I t o r + u M al; t ime t. + r5t , tI)() material derivative is

Consider a volume V , which is fixed ill space, enclosed by a surface S' onwhich n is th o ou tward-poiut.iug )J()l'IlJ al vecto r (Fi g. ! .J). The to t.al mass

of fluid in \I is .f~! pdV , whe n : p(1', t) is th o de nsity of the fluid. The ti mederivat ive of the mass ill V is t he mass flux int o V across its su rface 8 , i.e.

d / ' j'-I . P (111 = - (pu ) . n <IS .(t.. F S

Since V is a volume fixed in sp ace, t he time derivat ive on the left of Eq . (1.2)can be t aken inside the integral and becom es a derivative at. fixed positi onin space. The surface term on the right-hand side of the equati on can beIT-expressed as a volume integr al using the divergen ce t heorem . Hence weobtain

This is t he continui ty equation (or m ass conse rvation equat ion). Com biningEqs. (1.1) and (1.3) t he cont inuity equat ion can also be expressed as

DpDt + pV . u = O.

18 j'rPdV = - \7 . (p11,) dV .

v Dt . v

Since this ho lds for any ar bit rary volum e \7 in the fluid , it follows that

8pDt + V · (p11,) = o.

T he fluid properties , such as its density p and velocity u., will in general hefunctions of positi on r and of t ime I . We shall always use DI Dt to denotet he rate of change of some qu ant ity with resp ect to time at a fixed posit ionin space. In describing fluids it is also very useful to define t he material

derinaiioe, which will be denoted D IDt: t his is th e rate of change of somequantity with respect to ti me but travelling a long wit h t he fluid .

Let tir ,/.) be any quanti ty, for example, temperature of t he flu id . Itmay happen that the temperature of all indi vidual parcels of fluid is notchang ing with time, so t he material derivat ive D f l Dt is zero; but if somefluid is hotter t han other fluid t he n t he tem perature at a fixed point inspace may st ill change wit h time as fluid of different temperature passest he po int at wh ich the temperature is measur ed . In fact , in that case ,Df! fJt = - 11,' V f where u ir , t ) is t he velocity of the fluid. More generally,

the material derivative is relat ed to the rate of change at a fixed poin t in

space as

1.1 T h e Material D eriva t ive

we may define the fluid eleme nt to transfo rm by transla t ing ea ch point of

t he element by an amount u (r-, t )M, wh ere u tr, t ) is t he local mean velocityat the position r of t he elem ent. By virtue of th e above considerat ions, thefluid clement will st ill cont ain essenti ally th e same number of part icles at/. + 15t as it did at Lim e i; an d mo reover they will b e almost a ll t he samepa rt icles asI JCli)]'( ~ . Heucc Ow iuacro sco p!« properties of tJw fluid clem ent

will evolve only slowly, ami by a diffu si ve process.For fur ther discussion of the cont inuum descripti on a nd t he fluid ap­

proach , see e.g . Batchelor (19G7) aud Shu (1992) .If the mean free path of t he particles is not. much smaller t.han the macro­

scopic scale of interest, then the appropriate description of t he collective

properties of th e particles is kinetic: t heo ry. T he equat ions of fluid dynamicscan indeed lw derived from t.he mic roscopi c basis of kinetic theory. .For apresentation of thi s approach , ::;ee Shu (1992). Here we shall ins t ead assu met he conti nu um descr iption from the outset an d see how simp le considera­tions of the motion of the fluid , and the forces acting on it , lead to the fluid

dynamica l equations .

(1.1)Df = 8.f + u . \7f .Dt fJt

Eq uation (1.1) can be deri ved by cons ider ing t he change in f when following

One ca n simi larly deri ve a momentum equation, or eq uation of moti on , forthe fluid by consider ing t he rate of change of the total momentum of thetluid insid e a volume V . It t urns out to be easiest t o consider a volume

4 A sl.TOl' hysical F lui d D yn nU/.u ;s

( I.( i )

]\lasS of a fluid element and is invariant followin g tho ruotion so it s 11l;11 t'rinl

dcrj vnt.i v( ~ is zero. 'Thus

d /. / . /h ,-- flU d lI ;co [' -- - d I'd! , \ ' ,\ l rt

vnnrl 11( 'II Cf\ apPh'in g; t.hr: di \'C'l" .l',c'IW( ' t.!W()]'('lll jo tl« : s1lr fac I' iIILq l:r:d ill

Eq, (1 .5) , w r: ol n.ain

/

. f)7J / .p- - dV = (-VII -I ['f )dV.

, Ii])( , v

Since this holds for :I1lY a rbitary m ntcri nl volu me II , it. follows 1.1];11.

(1. 7 )

(1.5 )

( U I)

(J .8)

o« ( ,) 1 ( \Pm = - -Vp -+ pf + II V - l1 + ~V V' 1/,)) .

With viscosity included , E q. (J.9) is cal led tl io Navicr-Stokos equation. Its

inviscid form , Eq . (1.7) . is cal led tJIC Euler equation. T hroug ho ut most. of

this book we sh all neglect visc osi ty : the justification of this app roxim at ion

j' . /. D 1

(J i j H j ciS = ' 1 > (J ij d \',,<; • F u :J.' j

(d iverg enc e theorem), a nd so if 1'. is a cons tant it follows t haI. t ho equation

of mo tion for a viscous flu id is

where p, is t he so-cal led dyna mica l viscos ity: sec ('. .g . E ntr-lrolor (l 9(j7) .

Landau & Lifshitz (J95D). Also Il i i is the Kron ecker i1e11a: sel' Ap]l(:ndix A,

Now

T his is i.lw m omentum ('cj1 w t io ll for a ll i\lviscid lIuid.In a gClwra l viscous fluid (i t docsn 't IlcC'd to I lf ' as c'x t.]'CllW all oxmnpl«

as treaclel) t.he ith C'.()]llj HJl1 ent of 1.Iw fo]'( '( ~ ex ( 'l' t.c ~d Oil ~·mrfa('" S' II.\' 111C'surround ing fluid is no t. just f <; - 1) 'lJ i d8 ]mi. is .fs (Tij 'IJj d8. whore (Tij is

t.ho 8 1.1'1.:88 I.I.:n80 1'. [No t.o her e thai. the su m m nl.iou (,Oll VC ~ ll t.io ll is w )('( L so

tlmt, if a ll index is re]l C'aU~d it should lw sum me d over. A lIo11-]'C']lc'ntC'd

ind ex d enotes a COlll]ion ent of a vec t.or or te-nso r. SC·t' A ppC'1 \C1ix A . AIso

note that , t h roug hou t. th is book we shall use r or :r. t,o dl' no j.r· vl'c: tor position :

l)11 t for it s ith com p on ent form wr: always wri te :1 :i . ) For g;asl's and sin rplo

liquids it is foun d th at

F ig.1.J An arbit rnrv volume of Ilnid \I , wit h s llrface,.') au d ou t.ward-point.ing noruinl n .

where f is the body force per unit mass , (Note that for ce per uni t mass

h as dimension s of ac ce leration .) For ex ample, f could he the gravit at iona l

acce leration g , The second kino of for ces acting a rc sur face forces - forc es

exerted on the surfac e S of F by the surround ing fluid . In a n ituns cul fluid ,

suc h as we shall mo stl y he cons ide r ing. th e surface force acts norm all y t o

the surface and it s net effec t is

'moving with the [iuid , so that no fluid is flowin g ac ro ss it s surface into or

out of F . T he momentum of the fluid ill F is I, .p7J d F , and 1.Iw rate of

change of t his momentum if; eq ual 1.0 t.he net Iorc« ac ti ng on ti le fluid ill

volnnw V. These .a re of two kinds. F irst there arc h od y forces , such as

gravity, wh ich act on the particles inside F : their net, effec t is a force

j' - pndS ,,S

p being t he pressure. Ther e is no flux of momentum across the surface

ca r ried by flui d par cels m oving , since by defin iti on none crosses the surface

of a material volu me . Eq uating force to chan ge of momentum we obtain

Ie! rpudV = r- pn dS + / . pf dF .

C t i F .Is ' F

Sinc e V is a llmt.el'ial volu me, whe n t he t ime d eri va t ive is t aken inside

t he in tegral it becomes a materi al deri vative; but t he prod uct pdF is the

(i A stropliusical Fluid DynamicsBasic Flui d Equations 7

ill as trophysica l contexts will be seen in Section 2.8. However , viscosityplays a key role ill some applica t ions , notably in astrophysica l accretiondisks which we discuss in Chap ter 9.

(b being th e Dirac delt a function in ;~ -D space), Eq. (] .1:-1 ) cau I)(~ rew rittenas a part ial differential equation, Poisson's cquaiiou:

(1.15)

1A Newtonian Gravity

A muss '11/ at. position 1" exerts OI l any other mass In a t posit ion I ' auat.tructivo force proportional to th e pr od uct of the two masses and inverselyproportional to th e square of UIC dist an ce betwee n them , directed towardsiu uss 111,':

F = '/ng (1') == Om rtf' (1' - r' )II' -- 1" 12 [r- _.- '1"1 -

Grn 'In' (1' - '1" )11' _. 1" 13 (L/ O)

1.5 The Mech a n ic a l a n d Thermal E nergy E quat io ns

If one takos Newton's third law, F = uui = In (tlv /rl/. ) and multiplies hyvelocity '0 , OJl( ~ ol.taius tltil l. rate of work of the forces, F u, is equal to thorate of chau gc of kinetic energy , tl ( ~ '11 I.'IJ'2) /d/.. Simil arl y, taking th e do tproduct of th e equation or iuot.iou for a fluid , (J.7), wit h tho fluid velocity'U yields

Note that (1' - - 1")/11' - 1" 1is th e unit vector along the line of action of thoforce , Now

D (1 2)'u -Dt. '2 -1-- u · \7]) -+- 'U ' f .(i

(1.16)

(t he derivatives are with res pect to T': they treat r ' as a cons tant vector) ,so the grav ita tional acceleration g(I' ) can be written as th e gradient of apo tential function 'I/{ r ):

Similarly, the gra vita t ional field due to a fluid can be written as a po­teu tial, namely the SU Ill of the pot entials du e to all the fluid elements . Themass of a Huid eleme nt of volume dV' at posit ion 1" is p('r ' )dV', so thetota l gravitational pot ential is

Equation (1.16) says that th e rate of change of th e kine tic energy of a uni tmass of fluid is equal to the rate at which work is done on the fluid bypressure aw l body forces. This is somet imes called th e mechan ical energyequation.

An equ ation for t he total energy --- kinet ic and intern al t hermal energy_._- can be derived in the same manner as was the momentum equ ation inSection 1.3. Let the internal energy per un it mass of fluid be U. Thenth e rate of change of kin etic plus internal ene rgy of a material volume (i.e.one moving with the fluid ) must be equ al to the rate of work dono on thefluid by surfac e and body forces , plus the rate at which heat is added tothe fluid . Heat can be added in two ways: one is by it s being generated ata ra te E per unit mass wit hin t he fluid volume (e.g. by nuclear reactions) ,while the second is by the heat flux F across the sur face S (e.g. rad iativeheat flux). Thus

(1.11)

(1.12)

- (1' - 1.1)IT- 1" 13

- Gm''1/' =

IT' - 1" 19 = - \7'(/" where

where the int egration is over the whole volume of the fluid .tional acce leration is - \74' .

Using the result

In the same way as for th e momentum equa t ion, one rewrites all the surfaceint egrals in thi s equat ion as volume integrals, using the divergence th eorem.The resulting equation holds for an arbitrary volum e V and so one deduces

(1.17)

1;7fpd V - .f~ F . nelS' .

dj' (1 2 )- - '/L -+- U pdVdz V 2

= r 'U ' (-pn) dS -+- .r;7'U· f p dV -+­is

(1.13)

(1.14)

T he gravita-

- 41Tb(1' - 1" )'J ( 1 )\7- IT' - 1" 1

8 A st rophy si cal F luid J)'i/narnir." B o.si» Fluid Equ ation s

One ca n deri ve all equa tio n for the thermal ene rgy alone by dividingE q. (1. ]8) by the density and then subtracti ng t he kin eti c ener gy equation

(1. lG) to obtain

T he d ivergence of 11. has hoon rep laced by -- fl '" J Dp! D/. using th e «on t.inuity

equat ion (1.4).

No t ing th at the volume p er unit mass is just t he reciprocal of tho den ­sity, i.e . 11 = p--- l , we recognise t he therma l ener gy equat ion (J .19) a :-.; a

statement of th o first law of th ermodynamics :

J)U)+ .Dt = - V . (pH) + pu. r] + pe - V · F .

t ha t

( D (1 ")P Dt 211,-

DUDI

]1 DfI- ._- -+ I -fI'2 Di

1- V · P .p

(J .] 8)

(L ID)

wlWl"e 11 is a fixed volum e onclos ing th o wh ole flui d : c.g. C o x ( I! lk ll ). III

deri vin g t.ho ahovo cql1atiml it is helpful first t,o l:s1.nhlish (1"(1111 I'q. ( 1. 1:l)

(.hat Iv(i)fI! (}f) I!,dV = .f~ · (P(N,! ()t)d V where 11 is l.h« WII O]I' Il'giou IJ{"I'llpi f'r]

by t.lic fluid .. T ho VO] U Il H' int.cgrnls of di v{']"!'/'lice te n us ill Eq . (J .2 1) ('an 0 [" C(lm s('

1)(' j'( ~-( ~XPl"css( ~d 11.''; surfar«- in t.q~rals . If t.l io flu x in :-.;quan ' ] )]";\('kd.:-.; ill 1.1 ]('sccond t erm on tho ]d'i of Eq. (J .2 1) vanish e-s at the sur[";)l'( ' (If F. whichmigh t. for ex,w Jph: ]"('»]"(':-';CIl1. l.hr- interior 0 [" a star. 1.]Il '1I t.h« 1.o1.n] ( ']\(, I"g ., ' ill

\i ('an ollly challg(' tJl1'IllIgll iuu-rual he.u :-';(llIrcc:-.;jsiJl]':-'; (I ) or ]wnJ flu x ( F )

a(TOS:-'; thc surface.

1.6 A Little J\tIore Thermodynamics

T IH' :-.;eu llld law of U\('rlUOd.I'IJallli cs :-.;l a (.(':-.; t. ltat .

ellJ = (- ]I)elV + 5Q , (1.2()) M:2 = 'I' <I S' , ( 1.22 )

dU = TdS - plW .

From this various relations hot.woon 1hormodvnam ic dorivativcs can hI' d ( ~­

.lucod. For example, it follows immodiatcly fro m Eq , (1.2:{) 1lJ:11

whore S is a thormodvnaiuic s(.al,e vari ab le, t.h« 8]iN4il: cn l'l'OJiY (i .1 '. 1.hc

ent ropy per unit mass). Comb ining th is with Ow firs tlaw, Eq. ( 1.20), vio kls

( i.2G)

(1.2:;)

( 1.24)

D'2.f!DyD:):,

of !uy ) ~ =: _ ((~J;) .(Of ! a:c)y uy .r

(DT ) ( a]l )all s = - as ,, '

Another usoful mauipula tion th a t is a general property of part.i n1.lorivat .ivr-»

is 1.l1il1.

T = ( ~~~) v and - 11 = C~~ :)5' :

but a prop er ty of par t ial differentiation is that [j'2 .f ! (hUll

so we find that

This follows by rearran gin g d f = (Df ! ch:)y dx + (Df !uy)", dy to make (h:t he subj ect of t he formula, and identi fying t.he resulting codtlcjen1. of el y as

th e deri vative (D:l:!ay).r . Variou s t he rn )(J(lynHm ie rolat ion s th at ar e useful

~ ( ( ~11,'2 + U + ~ 7/') pdV + ( v· [(~n'2 + U + !.!. + '1/') Pu] avdt iF 2 2 iF 2 P

= [" i p« - V' . F )dl! , (1.21)iF

t hat is, the change in the) int er nal energy is equal (.0 the work (- p)d V don o(on t he fluid ) plus t ho heat added . Note th at \I , U , ]I are pro perties ofthe fluid (in fad t hey ar c t hermod yn amic st ate vari ables ) am] we denote

changes in them wit h 1;]18 symbol "d" . In contrast, there is no su ch pl'Op­

erty as the heat content a nd so we can not speak of t he change of he at.

content . Instead , we can only speak of t he heat added, a nd we t hereforeuse a different. no tation , i.e. 5Q .

Equations (1.16)- (1.19) can be generalise d t o include a viscous xtrcsst er m . In th at cas e, _ p- l u . V]i in E q . (1.16) and 11 . (- ]in) in equati on

(1. ]8) are repl aced by p- l 7/.iU(Ji:i ! D:rj and lI ;(Ji.i n .; resp ecti vely, where rTij isthe st ress tensor as ill Eq. (1.8). T he consequence for the t herm a l en er gy

equat ion (1.19) is that. kinetic energy is conv erted to heat by viscosi ty, sothat one obta ins an add it ional heating term sim ila r to f . This is discussed

in more detail in Chapt er 9.

W ith some effort, one can use t he above eq uations to derive an integralequation (sometimes also referred to as the total energy equation) for t.liora te of ch an ge of t he t otal ener gy (kinet ic plus intern al plus grav it ationa l

poten tial energy) for t he whole fluid volume:

10 Astropluisical Flu id D yn am ics

f~.

..~

Basi c Fluid E ou ai icnis 11

ill s tellar physics a nd ast ro phys ica l fluid dyn amics can be found in e.g.J\"ippcnhalm & 'vVeiger t (l9f)()) .

We defille th e udiabuiic exponen ts 1 1, 12, 7 J by

1. 7 P er fect Gases

A perfect gas is ow: for wh ich

For example, the first equat ion can be der ived by noting

Not« that all thoso partial derivatives ar c at constant specifi c (mj,m py: 'adi­alJiltic' hen : me ans without exchange of heat, so MJ = () = dS , i.e, 5' is

COllSeall/" 'J'h o quanti ty ((2 -- l ) /~f'2 == (lJl n T /D hlp)", is oft en referred toas '\7".1 '

We define (;1" the sp ecific heat at constant pressure, to I)(~ L1w am ount ofheat required to make' a unit increase ill tmnperatllre, without th e pressurr:

changing: tlm s, from E q. (1.22) , (; 1' = T (US/DT)p. Similarly we define ell,tho specific hea t at constant volume, to be t he amount of beat required toinako a unit increase in temperature at cons tant V. 'The f()]]owing three

useful res nlts re late I.lw aruoun! of heat. added to the clJanges in pairs oft lwfl llo(lyna mic va ri ablns:

(1.2!J)

(J.:m)

( 1.:31)

(1.;)2)

(1.3:3)

(1.34)

dT

R'J'

U(T) .

dV

II

Jill

U

I d U d'J' avo = dB = T (dU + pdV ) = dT T -+ R 17- .

(R being ::;Ol1l C constant ) , and

Now for all adi ab atic chaugo orS1l( ;]1 a gas ,

It follows from Eq . (1.29) t ha t.

dp+

From Eq. (1 .:)2) awl th e dofiuif .ion of ~h it follo ws that

nI :J = 1 + dU/dT .

Elim inating dT between Eqs . (1.31) and (1.32) gives that '/'] is given by t hesame expression (1.3:3), and likewise for 12 (eliminat ing dV). Thus for aperfect gaH, the three adiabatic: exponents arc equal.

Hencefor ward in this b ook , since for a perfect gas a ll t hree adiabati cexponent s are equal , we shall usc 1 t o denote all of them when no confusioncan ar ise .

In fact , for a uionatomic gas (in which the molecules are simply pointInasses) one can show that 1 is equal to 5/3, as follows . For a monatomicgas, the internal ene rgy is just the t ranslational kinetic energy of all themolecules. Assuming the gas to be isot ropic (all directions equivalent ) a ndall t he molecul es identical , t he total internal energy of the gas in volume Vis

(1.28)

(Uln T )DIll P .'; .

(1.27)

1:1 -- ](DhI T )

= Dhl]J 8'

( OS) (DS)T -D dp + T -::>. . dV]J v oV l'

T ( op) - 1 rdP + T(DS/cW)l' d V]DS v (oS/op) Ii

~('!. - ]

/ 2

fJ(J - TdS

~ . _ (DIIlP)Y1 - 7- ,.d lll (J S.

fJ C)

and using Eq. (1.25) on t he factor outs ide th e square brackets and Eq. (1.26)to m anipulate the last t erm, together with the defini ti ons of 1 1 and 13. Notethat In V = - In p. The other two above expressions for fJC) arc der ivedsimilarly,

where m is the m ass of a molecu le, N is the nu mber of molecules in V , and(for exa mple) v; is th e mean squa red velocity in t he x direction . Supposethe volume V is enclosed by a rigid rect angular box of length I in the x­

direction (a nd of cross-sec t ional area A = V/ I ). The force on t he end ofth e box is pA. Cons ider a sing le mo lecule. It has some z- velocity Vx ' In

12 A strophysical Fluid Dyn nmicsB asic Fluid E qnntim :»

1.8 The Virial Theorem

If a gali iii l)J)(!ergoiJJg ionizat.iou , dUj rlT if: grC'a 1r ~r th an it wou ld ot lwrwif:C'

bC' , becallf:C C'1lr'rgy gor ~f: into io nizing the gm-; ; so Iron: I'.g. E q , (I,:l:3) 1Jwarliabaii e I'Xj!ollent s .uo red uced ill value.

ti me t:lt == 21 j v,,. it bOIIll(,('S off that end of tho box OJJce. III brJlln r~i!l g: its

:r:- lllOl IW l l t lll Jl ch angr-s Ily an muount 2111 1':,. (ass llllJing all cl astic «ollision ).

Thus, s ince lorco (= pressmr: x area) is equal 1.0 t he rate of challgr ~ ofm U]JWIl(,1II 11, s \Illll lling 0 \'('1' all molecules gi vr ~s

( 1.·11)

U sing 1.1)(: r1 i v r ' I'),!, ( ~n r: ( ' t.lu-orern and ill(' i. lou t.i t.v ,\7 , t : iI:I' ; /( ):r ,

pressu]'c term ill Eq . ( I .:l! }) ('all 1)(' rr-wtit .t.c-u ;IS

/. r . V7)(W = - 1',711" n el S I ;{ / ' J!C 1I ' .

. v '!'s ' \\ :\7C' supp ose tll at 1.1 )(: prossn r« van ishes at 1.11(' houudarv of 1]](' Ilui .l volu me­

(i,his «a u 1)(' a !-'.()oel ap]']'l)xillla l,joll lor ;I st.u. for I lX:IJllp1 (' ) so 1,h:II , fl \l'

f: nrf:lul tr-rrn is ;1,( ' ]'( '.

FiJiall y,

/. /. j' (/1(1'/)) , II ; 11 1''17 , / 111 .-- ( ' T: '17 _.' ",..__ /1(1' ) ( . ( .- " 1" v ( , (I( " -' , ' v I 'I

' \ , ' , I ' , 1' .. '" 1' ,

/' / ' "'. (1' '1" ) ., ; ,I I '

-"(/ ,------------" -C- (1(1 )el\ (1(7 )el l' 11' 1" I'~, 1.' , 1"

' I' , (. ,I") 1'" (r' 1') ') , 1 '1- - -~ (/ / , ' , ( .~ -~ -c1 - --..2. 1 _;_I '\" " " " " (1(1') 111' ji(r )(11:2 ' , 1' , ' " 11' - ri ll I" 1"

1 / ' I' I '\ ' ( ' ) 1\ "-- -"C; , ,------ --(! (1,)d, (! l ' I '2 . v . 1" 11' -- 1" I

(J.:lG)

( U ri)

JV 'III,--;­__ '/) 2

I :r '

:\ GU :=.: 2RT and ')' =: :l

L 2 1/1.1':1 L 11/, :2J)/I = -- = - 11t:ll 1 C/:

Nun ;,'; and 1i0 . from E (jli. (J .2!J) , ( I.a:n and (I .:H ),

T ile velocity 11, is the rat.e of (:!lange of position [()llowillg f,l lc fluid :

Taking the dot product. with r am] integrating over tho whole volumo ofth e fluid gives

Hence Eq. (1.7), wit h f n~p]acrxl by gravi tati onal acceleration and llsingEq, (1.12), can be rewr itten

])1'U = - .

Dt CU 7)

(1.:J8)

(th o lat er steps exploit th e synnnetry IH'twcCJI l ' al II I 1") where

I}J == ~ I' 1/' (I(T )elV2, "

is the total grav itat.ional energy . P utting all thi s togd ,]](']' violds

2 I'1 d I _ ?T + '3 IV + IJI- - ') - - • 71 ( •2 dt - . v

(1.12)

( I. I:n

(1.-1-1)

wh ere T == ~ Iv P 11,2 elF is t he total kinetic cncrgy of the flu id. (No te t hathere and in simi lar exp ressions we write 1)2 when what is meant is In12 , i.e.1/, . U -- t ho quantity is a scalar. )

(1 4 1) ' 1J I t nil of th e uiriolwhere I == '/;1/W2rlF . Equation . ' 1 . IS ' Je sea a]' 0theorem.

One can also deri ve a tI'JlSO], vi ri al t!]('o r I' JII. liv iaking ill!' i t h ('lllllj )(I ] WI J1

of E q , (1.:38) and mult.iplyiug by t.h« jib coiu p onout of 1' :

/

' ])21' / . / .

,T: > - -peW = - . r . vp dV - T' . v 1/' peW .

. F Dt2

, 1 , "

T he left-hand side of Eq, (1. :J9) call be rewr it t en as

d / Dr / (Dr)2 T 1 d2

/ 2- r' -pdV - -- ()( 11· = - ----:z . 11'1prlV --2T ,elt . F Dt ,F])t 2 dt ' \ '

(1.39)

(lAO)

11 can the n be shown that

1 d2 Ii j

:2 dt :!-n: + s.. I' ]lrlJ! + IIl i l, .- 7:J · 7J .

. l '

(J .iJ [))

( 1.,Hi)

15

err)B ----... · ~v, ( r

. ~

u (r)

Busic Fl uid Equation»

A

Fig. 1.2 '1'1 1/, m ot.ir» of J",iglt i>o llrillg p"i llt.~ A (at. p osition r ) and B (al. p ost ion r ·j aT'),wltid . leads 1.0 t.h« «volut.iou of ,.1 ", lJ1a1.erial Jill" .,j"lJ1ell t (}.,. with t.iru «.

(J.47)

A s t lVp h ysi cal F lu i d D ynami cs

'r.,

14

where

/. p:c;:cj d If ,

· 17

1 / .,ceo ','; . PU;Ui <1 l! ,

L., l'

1 , / . / . (:1:; - (:Cj - .J./ ) . .= -' ~- C - - - ; :j_Lp(r} W p(1")<117'.

2 . v. \/ ' Ir' - r INote tl mt if Eq. (l Au) is c()lIt. racted over i and j (i.e. multiplied by Ii ; i a ndsun uuod over i aw l .i) th en the :-;(;ala1' vir ial theorem (] .114) is recovered .

Dori var ious of L!w virial thoorciu ill d iJf(:rmlt forms call be found illCllilJldrasddwJ' ( l UW ) aw l Tassoul (l!J7I)).

1. 9 Vortici t y

Au important. deri ved quant ity Ior a ilnid How is the vorticity The fluid velocity at A is ·LI C,.) and al. 13 it is '11, ('1' + or); therefore af ter ashort, tiuic 51. t ho separation of 13 from A has changed to

For il Il uid ro ta t illg rigidly with angular veloc ity 0 , for exam ple, 11. = O X 'I'awl

using a stalldard vect or identi ty (cf. Appendix A). Generally, in a fluid

flow 11. the vorticit y at any location is equal to twice the local ro t at ioll ra teof a fluid line elem ent at that location, as is proved below. This do es 110 1,

mean however t.hat st reamlilies have to be cur ved for t he fl u id to possessllOIl-:6ero vort.ici ty, For examp le, conside r the shea r How 1l = Cue; wher e Cis a 1I011-:6Cro constant : t.liis is a ullidirecti onal shear How in the a- directionwit h magnit.ude pro por t.ionn] to y . Here as elsewhere we use e e e to

x, y , zdenote unit vectors in th e X-, y- and z-direc t ions, It. is a st raightforwardexe rc ise to show t hat the vor ticity of such a How is w = - Ce

z, whi ch is

non-zero al though the streamlines (lines everywhere parallel to t he flow)arc st raight lines.

To see t he rel ationship bet ween vorticity and local rotation of t he fluid ,we :::;lwll uow analyse the relati ve motion of a fluid in the vicinity of a point.Let A be a poi nt moving with the fluid , which a t a n ini tial time i is atp ositi on 1' ; and let. B be ano th er poin t which a t t ime t is a t nearby positi onr + 0'1' (F ig. 1.2). At t ime t the refore the positio n of B re lat ive to A is Sr,

(1.fiO)

(1.51)

DOr' .- - = Or, · V 'UDI

correct. to O (lit.). T he las t. term call be simplified by expanding 11, (1' + 61')in a Taylor series about r and keeping on ly tenus up to li1'. Treating Sr]l OW as a func ti on of t ime, a n equat ion for t he rate of change of lir' with

time can be ca n be ob tained from here by dividing the difference bet weenthe new separation and the old one by Of; and taking th e lim it as ot -> 0:

where, since we are following the separa tion bet ween materi al poin ts , wewri te t he derivative as a material derivat ive. This equat ion therefore de­scribes the evolution of a material elem ent 01'. The right-hand side ca n

be expressed ill index notation as (<5r') j ehtdoxj . T he te nsor \7u , like anyother second-r ank tensor , can b e split into a sy miuetr ic part and an ant i­syu un otric part:

OUi _ 1 ( OUi Dai ) 1 ( OUi U U) )= - -,- + - + - -,- - -, -OXj 2 u:c; o:r ; 2 ch j 0 1:;

The second term on the right of Eq . (1.51) is anti -symmet ric: usin g the def­

iuit ion (1.48) of vort icity, it ca n be written as ~~EijkWk (d. Ap pendix A) .Hence, subs ti t uting into Eq . (1.50) it makes a contribution to t he right-

(J,4i))

(1.49 )w = 20 ,

16 A stro plujsi col Fluid DinuimicsI3lLsic Fluid R'Iuo ti"" ,, 17

Finally, using the cont inuity equation (1.3) to elimina te \7 . 'lJ" we obtai n

(1.59)

(U j8)

(1.:>7)r = i u ' rlr .. C

df f' ])11, i' D- = ) - . dr -I- 11, • - dr .dl . c Dt . c Di.

df j' ( 1 )-1-' = z \7p x \7p -I- \7xf . ndS .d . s fJ

If we cho ose C to he a material curve , moving wit.h the Iluid. I h('11

The last term can be re wri tten as the integral a round C of \7 ( f;.U2)using

Eq. (1.50) , a nd for an inviscid fluid We' can replac e Dul Di in th; first term

on t.he ri ght of E q. (1.58) using t he m omfm1.mll eqnat ion (1.7). \\Ie consider

only :~-D fiui d do mains for whic h curve C can he sp <II llH'd hy a snrfal'(' Swholly co]]t ained in the fluid domain; so St okes's t.lJPorC]l] ca ll he applied,

Henc e Eq . (1.58) b ecolJlcs

We ca n immediately deduce from this that for a ba rotropic flu id wit h on ly

slIr faccs of const.aut density and of cons t.a nt ])J'f'SSl1r(' "oillCid(' . l1 11 d it is

possible 1,0 write oithor variahlr- as a fuu rt .ion sllkl.\, "f 1.111' " fl ll'r v.ui nl r]«.

c,g. p = p(y ). Cou vorsoly, if .lonsitv .u u] pn ~sslll'I ' a]'(' just l'lllJ('liilllS (I I I( ' Id'

t.J j(' ot.J wr, thcn tho fluid is barotropic. If l.! ll' fluid is h:tl"ot.m pi" .u ul a llY

hody Iorco f that is present. is COll sI'l'vaj,ivI' (i.e, V xf CC' (l, ill' ('quiv;I1('lItly

.f U IIl be written as th « gradicll1. of a sca l.u pll1.l'llti a l ). Sll ill pn rt.ir.ul.u' \,l ll'j'( '

ar c lI O vi scous forces , th en t.ho last. two terms ill Eq. (I .fiG) vanish and 11]('vOl'Licity oqu nt.iou (J .f-,ri) is of ti ll) sa il II , form as till' e<j naLio ll ( I.fi ll) for IIll'evo]nJ,ioll of a lIl:l1r'ria] line olom ont. V/e d( ~dnr:r ' 111:11. ill t.lii« (':IS,' vortex

lilies iuo vo wit h I.lw Ilu id . (/\ vo rtr -x lin« is n linr: ov r-rvwh orr: pl1 r:dkl 1. 0 1.11( '

vor1.icity. ) YVI ' C:III ddill e a '1101 11':1' 1'111)(' 1.(J 1)(" ]oosldy sjH':l,kil lg. a I1I1JJd]e

of vorl.cx lines or IliOn ) pre'cisc'ly a tu l«: w1IOSf' sllr f:wr' is no wh cr« ('f'oss(·d

by vorI.ox lilies and whos(' Sl1I'fa(' (' is it.sl,]f I:OlllpOSl'd of vor tex lines. III 1.1]('p]'( 'sel lt, cnse , then , tl w wal ls of a vo rtex I n l)( ~ Iorru n nl:!I,('r ia l Sll l' ['a ('(~ m o ving

with the Iluicl. YVe definc t.li« s L ]'( ~ n gth of a vortex 1,111)( , to Ill' Is' w· ndS,wlwre oS' is any cross-section an)a c111, a(TOSS 1lJ(' 1. 111)(' a ll,] n is a vector

lIoru IaI to that arr :a. Sin('(' lIO vortex liw 's <TOSS 1.l 1<' wa lls ',(';1 l '01'1.('x t u1H '.aw l vorti city is di ver gc'u CI '-free. it follo ws from 1.111' di vergc!w(' 1.1 J('Ol'l'lI I t.h.u .

11](' s trengt h of n vor te x tube is a we ll-dcfi]J('d qu a.nt.it.v. i.o. ilis iJldl 'lH)ndl ~J1t

of which cross-sec ti on a rea. we ch oose OJ ] which t.o eva lua te it.

Generall y in a flu id we can define t he circuuuuni ahout. a. d osed curve

C containe d wit hin t he fluid to b e

(L!j 2)

(1.55)

(1.53)]~ \7]J -I- f .p

1-I- u·\7w = w·\7u - (\7 ·u)w -I- z \7px \7p -I- V xf ·

p

hand side which is oqual to - ~ Jr x w == ~w x Sr. T hus, comparing this

with the velocity due to a solid-bo dy rot.at.ion , wo deduce t hat the antisviu­

metric part of \7tt. conti butos to tho motion of point 13 relative 1:0 point 1\a motion wh ich is a rotation wit h angul ar veloc ity ~w . Equival ently, 1lw

vor t ici ty is given by Eq . (1.49) where n is interpreted as the local rotation

rate.The symmetric part of du, ID:l:j , i.o. th e first. term on th e right of

Eq . (1.51), is called the rate of strain te nsor Cij :

, _ 1 (DU j ()'/J,:J,' )<',.1 -- ;- -, - -I- -, -

2 ch::i d;r i

It s traer) ekk is equal t o \7. 'lJ, and t.he isostropic part of Cij, namely j Ck/;,r)ij,

rep resents an ex pa ns ion or compression of the fluid in th e region of point.

A. The remai nder of c i:i , n am ely ei,j - *ekk0 i.i' has ze ro trac e and ]'()prelsents

a loc al shear of the fluid ,An evo lution equati on for vo rtici ty ca n I lC deri ved by t aking the curl of

t he m omentum oquation. In iti all y we shall consider the inv iscid case . First

we rewrite t he 11' \71), term in Eq , (1.7) us ing a vector iden tity t o obtain

since t he curl of a grad ient vector is zero . The \7 x ('lJ, x w ) can be ex pande d

using identity (A A) from Appendix A. Noting that \7 . w vanishes by its

d efinition (1.48), Eq. (1.54) becomes

awat

Dtt. (1 2)-a = l/, XW - \7 -1),t 2

Taking the curl of this equa t ion gives

aw 1Dt = \7 x ('lJ, x w ) -I- p2 \7Px \7p -I- \7 x f

D (w) _ a (w) (w) (w) 1 1Dt P = at p -I- 11·\7 P = p ,Vu -I- p3\7p xVp -I-p\7xf·

(1.5G)Equat ion (1.5G) is ca lled the vort.icity equat.ion. It describes how vort icity

evolves in a flu id.

A flu id for which Vp x \lp = 0 ever ywher e is called barot.mp ic: sinc e

the vector gr adients of density and p ressure arc everywhere parallel , the

;..

iI,! ,

18 Astrophysical Fluid Dynamics

couscrv.u.] vo body forces I; the circulation around a material curve is in­variant with time. 'I'liis is a statement of Kelvin's circulatian tlicori-m.

Moreover the strength of a vortex tube: can b<:: expressed, using Stokes'sLhooreiu, as a flow around a matciia] curve embedded in the walls of tIwtube awl encircling the tube's axis. Hellce the strength of a vortex tube insuch a flow is invariant also: if the ilnid motion is such as to cause the vortex

tube to become W1ITOwm' (knowll as vortex stretching), the ruagnitude of

W llJust increase so Lhat I w . n dS over the cross-section of the tube isconstant.

A fluid for which Vf/XVp cJ 0 is called baroclinic. This means thatsurfaces of constant density are inclined to the surfaces of constaut pressure.

C hapter 2

Sirrrple Models of AstrophysicalFluids and Their Motions

In t.he first chapter we est.a]ilish ed the mo monturu equat ion (1.7), th e (;011­

tinuil,y cq naLioll (JA) , POiSSOll'S eq uat.iou (1. JfJ) aIHI t.lio energy equation(1.19). Assuming t hat th e onl y hod y forces arc (hw to self-gravity, so that

f = ~ \74' ill Eq, (1.7), th es« equations are:

Dup- = - \7p" p\7 I/' , (2. 1)

tn

Dp+ p divu 0 , (2.2)

DL

\7 '2 'lf' 47fG p, (2.::\)

DU p Dp J(2.4)- - = E - - \7 · F .

Dt p2 Dt p

Note that these cont ain seve n dependent variables, namely p, t he t hreecomponents of u , p , Vi an d U. The three components of Eq . (2.1) , to­get her wit h Eqs . (2.2) -(2.4) , pro vide six equat ions, and a seve nth is theequat ion of state (e.g. that for a perfect gas) which provides a relationbetween an y three thermo dynamic state var iables , so that (for example)the internal energy U and temperature T can be wr itten in terms of p an dp. (It is assu med t ha t, E and F are know n functions of t he other variables.)T hus one migh t hope in principle to solve these equations , given suitableboundary conditions . In practice this set of equat ions is intractable to ex­act solut ion , and one must res ort to nu merical solutions. Even these can beextremely problematic so that, for example, understand ing t ur bulent flowsis st ill a very challenging research area. Moreover , an analytic solution toIt somewhat ideali zed problem may teach one mu ch more than a nu merical

19

2.1 H ydrostatic Equilibrium for a Self-gravitat ing Body

solution, OlJ(~ useful idoa lization is wh ere we assume that th o fluid velocityaw] all tiin c derivatives are zero. T hese are cal led equilibrium solut ionsand describe a steady sta te . Alt ho ugh a t rue steady state iuay be rare in

reali ty, t li« t ime-sca le over whi ch all as trophys ical sys tem evo lves may bevery lon g, so that at an y particular t ime th o st ate of many ast rophysicalfluid bodies sucl: as st an; may he well represented by all equi librium mod el.E V( )JJ when t.li« dynamical belravi our of the body is im portant, it call oft.eu

IH: descri bed iu terms of sma ll departures Iro m an equilibrium st a te . Hencein thi s c:lJ ilptcr W(~ st.a rt Ly looking at some equ ilibrium mod els awl thendcri Ill' equ ation s describing small perturbations about all equilibriu m st ate .

21

(2.8)

(2.9)

(2.10)

Grn(T)

Si m ple AIodels

Int egrating on ce gives

where

(Note that rn( r) is the mass inside a sphere of radius '1', centred on theorigin .) Now \7'1/) = (ch/)/d1') e ,. if 'II> is only a func tion of 1', er being a unitvector in the radial direction . Hence Eq. (2.8) implies

. Gm.9 == - \7 '1jJ = - - ,- e ,. ..,. '2

A stroplnjsicu i Pl1Lid Dyn am ics20

If we suppose that u = 0 everywhere, and tha t all qu antities are indcpen­den t of tiuio, th en Eq, (2.1) becomes

(2.;» )

Equat ion (2.10 ) st ate s that in the spherically symmetric case, the gravita­tiona l accelerat ion at posit ion r is du e only to the mass interior to r andindependent of the density dist rubt ion outs ide r : this is kn own as Newto n 'ssphere t heorem . Also , hy Eq. (2.5),

which implies that

(2.11)

(2.14)

" _ Grnpvp - - - -2 - e ,. .r

a2

p = ')""f.!• •2 '

The vector \7p poin ts towards the or igin, so the pressure decreases as T

increases.

One can only make further progress by ass uming some rela t ion be tweenpressur e and density. Suppose then t hat the fluid is a perfect gas, so

R pTp = - - == a2p ; (2.12)

/.L

a is known as t he isothermal sound spe ed. Suppose fur t her t hat t he tem­perature T and mean molecular weight /.L are both constants throughoutthe fluid , so a is also a constant . Then Eq. (2.11) becomes

Gmp- 72

d~' ('r2

pa'2 ~~ ) = -4nG1'2 p . (2.13)

Seeking a solut ion of the form p = A1·n , where A and n are constants , gives

(2.7)

Let us seek a solut ion where everything is independent of () and ¢ (andhence dependent only all t he radial variable 1' ) . T hen Eq. (2.3) becomes

'2 , I, a( ')eN) 1 8 (. (} 8'1/!) 1 8 '2 '1/' (2 6)\7 , = -- 1'-- + - - -- sm - + " .'1/ '1' '2 8T 8T 1''2 sin () 8(} EJ(} '1''2 sin2 () EJ¢'2

2.1.1 Spher-ically symmetr i c ca se

In spherical polar coo rd inates (1',0, ¢) (see Appendix B),

the cont inuity equat ion becomes trivial ; and Eq. (2.:3) is unchanged. Afiuid sa t isfying Eq. (2.5) i::; said to be in hydrostati c equilibrium . If it isself-gr av it ating (so that '¢' is determined by t he density distribution wit hinthe fluid), then Eq. (2.:3) must also be satisfied .

Putting 'U = 0 and a/ot = 0 in Eq. (2.4), we obtain that the heat sourcesgiven by f must he exactly balanced by the heat flux term p - 1 \7 . F. Ift his IJOhl::; , t hen t.he fluid is a lso said to be in t hermal equilibrium . Sincewe have not yet considered what. t he heat sources might be, nor th e de tailsof the heat fiux , we shall neglect cons ide ra t ions of t hermal equilibrium atthis po int .

22 Astropliusico l Fluid Dyn amics S im ple Models

polyt rope of index n = J.5. Note that the isotherm a l l a'y( ~r is ob t ai JH'd illth e limit. n. --7 00 .

Subst.it.uti us; Eq. (2 .1R) into Eq. (2. 1G) giws

This is the sing ular self-grav itating isoth ermal sp here solution . It is not

physically realist ic at r = 0, where p an d (I are singula r , but noneth elessit is a useful an alytical mod el solution. Of course , in a real non degen erat est ar , for example, the inte rior is not isot hermal : the t emperat ure increaseswit h depth , which in turn means t hat t he pressure inc reases and the star ispr evented from collapsing in uJlon itself wit hout. recourse to infinit e pressurean d de nsity at t he centre.

/, (n -l- J) 1 __ 1dp

\ p "11 d z

and this integrates to give

- y (2. 1!I)

1[1.1)(' <lonsil .y van i shes at. .: C7 (J (wh ich cou ld 1)( ' a rcn sonahlr - app ro x imnl.ion

if z = 0 wore tho snrlaco of il s t.ilr) 1.lH' /l I.h(' COJlst.anf. of inU'gra t. ioJl illEq. (2.20) is zero. Hence fOJ' a plaue-parallol p nly t.rop« of fini te ill<lex "II,

( Z)71 - I ) ),,-tI . ] - ' I' / JP ex: - " all(]! ex: -- z , a .so J ex: p p ex: - -;; , so t Ie t.el11]>cratun:increases linearly wit h depth .

2.1.2 Plane-parallel layer under constant gravity

In modelling t he atmosphere and ou ter layers of a star, Ute spherica l g(~­

omctry can often he ignor ed , so that such a regio n can he approximated

as a plane-parallel layer. Moreover , in the rarified oute r layers of a starthe gr avi t ational acceleration 9 may he ap proximated as a con stant vect or.T hus , in Cartesian coor d inates (:J: , y , z) we have a region in which every­

thing is a function of z alo ne and 9 = -ge z , where 9 is constant . Note thatwe t ake z t o be height , so e z points upward s. Hence Eq. (2.5) becomes

- y zpI / 1I = -I- constant. .

(n-l- l )K (2.20)

Hence, in t his case, H = 0.2I9 and is constan t . T hus p = Poexp(- zIH ).

A useful fam ily of solutions is that of plan e-parallel ]!olytropes, where

(2.21)

(2.22)

G'mp== -- - -1'2

dm

d r

dp

(17 ·

2.2 Equations of Stellar Structure

Alt hough it if) an aside, it. m ay be instruct ive t.o point out hr-ro the rc la­

tionship between the fluid equations that we have der ived t hus far an d t heequa t ions of stellar st ruct ur e describi ng a stat ic, spher ically synnnct.ri r:star.

These ar e commonly used in studies of stellar str uct ure and evolution.

T he equation of hydr ostat ic equilibrium is j ust t he morneutum eq uationin the sta t.ir: case: wit h sphe ric al sy mmetry, so that. quantities are onlyfunctions of radial var iable 1' , th is is given by Eq . (2.11):

A differential equation for variation of mass m (1') containe d within a sphereof radius r is j ust the deri va t ive of Eq . (2.9):

In a spherica lly sy mme tric star t he hea t flux F is purely radial: F =F(T)e,.. The flux F(1') is rela te d to t he total luminosity th rough a sphereof radius T by L (r ) = 41r1'2 F (T). An equa tion describing t he rad ial variationof luminosity follows from Eq. (2.4) . Setting the derivati ves Oil th e left of

tha t equat ion to zero. and using the expression for d ivergence in sp her ical

(2.15)

(2 .17)

(2 .18)

(2.1G)

]J = J(p I -l- l / n

dp- = - g p(z ) .dz

1

1 dp 1-1H = - ­p dz

Since self-gravity is being igno red , Eq. (2.3) is not used .

In the isothermal case (pi p = 0.2 constant) , Eq. (2.15) ca n he integr a ted

to give

where the constant Po is the de nsity a t z = o.The density scale height H is defined by

(wit h bot h J( and n constant); n is ca lled the polytropic index . For exam­ple, in the ad iabatically st rati fied convect ion zone of the Sun the pressur e­

densi ty rela ti on is well described by p = K p" where I = 5/ 3 (except inregions of partia l ioni zati on ) and hence , comparing with E q . (2.18) by a

I!I

iI ,r .

rII . ,!

'. :yi :;.~

Ii

24 Astrophysical Fluid Dy n am icsSimple Mo dels 25

where D is the coefficient of diffusion. Typically D can be related to t hemean free path l amd mean speed v of particl es carrying the fiux, by

A fourth and final differenti al equat ion describes how heat is transported inthe star. In the bulk of a st ar like t he Sun this is by radiation. To combineradiative t ransfer with fluid dynamics in general is a substant ial topic init self, and is excellently expounded by Mihalas & Mihalas (1984). Here weonly consider a stat ic case and moreover , in the interior of a st ar the rad ia­t ive transport is well described by a diffusion equat ion . The prototypicaldiffusive transport equat ion for the fiux j q of some qu antity q with density(I " per unit volume is

polar coordinates (Appendix B), it follows that

dL 2- = 4 7fT pf. .dl"

(2.23)

(2.24)

More details of the derivation and use of these equat ions, and of st ellarstructure awl evolution in general, may be found in e.g. the book by Kip­penha hn & Weiger t (1990). It. should be evident from the above discussionof the origins of the st andard equat ions of ste llar structure that, if the staris not stat ic or not spher ically symmetric, it is ap pro priate to ret urn to thefull fiuid dynamical equations to ob tain equations ap propriate for modellingthe star.

2.3 Small Perturbations about Equilibrium

In many interesti ng instances, the motion of a fluid hod y may be consideredt.o be a small disturbance about an equilibrium state. Suppose that inequilibr ium the pressure, density and gravitat ional potent ial are given by

P = Po , P = Po , 'l/J = 'l/Jo (all possibly functions of position , but independentof time) and u = O. Using Eqs. (2.5) and (2.3), the equilibrium quant it iessat isfy

In the pr esent case of radiative diffusion of heat , the particle velocity isthe speed of light (which in this subsectio n only we denote by c), andast rophysicists describe the mean free path in te rms of a mean opacityto radiatio n (1'), thus l = (I'p)- I . The dens ity of thermal energy in theradiation is U = aT4

, where a is t he radiation constant . T hus finally weob tain the four th different ial equat ion aft er some rearrangement to makedT/ dr the subj ect of the formula as

Suppose now that t he system undergoes small mot ions about the equ ilib­rium state, so

so for example p'(r , t) == p(r , t) - Po(r ) is the difference between the act ualpressure at t ime t and posit ion r and its equilibrium value t here. Subs t i­tuting these expressions into Eqs. (2.1) - (2.3) yields

(2.28)

(2.29)p = Po + p' , P = Po + p' , 'l/J = 'l/Jo+ 'l/J' ,

(2.25)

(2.26)- 3 1'p L

167facr2T 3

1D = "3v l .

dT

dT

in regions of the star where heat is transported solely by radiation. Inconvectively unst able regions the heat transport is by convect ion (or somecomb ination of convection and rad iation) . If convection is very efficient , thistyp ically leads t o a strat ificat ion that is very close to marginal st ability (seeSection 4.1.1) in which the temperature grad ient is instead given to a verygood approximation by

We suppose that the pert urbations (the primed quantities and the velocity)are small; hence we neglect the products of two or 1110re small quant it ies,since these will be even smaller . This is known as lineari zation, because weonly retain equilibrium terms and terms that ar e linear in small quantities.

- \l(po+ p' ) - (Po+ P')\I ('l/Jo + 'l/J') ,

(2.30)- \I . ((po + P')u )

4nG(po + P') .

(Po + p' )(~~ + u . \lu) =

~(po + p') =at\l2('l/Jo + 'l/J ' ) =

(2.27)dT

dr

26 A strophusical Fluid Dynam icsSi m ple M odd s 27

This simplifies the above equat ions to

auPo Df; =

ap'Dt

2 ,\7 (1jJo +'l/J ) =

- \7(Po + p')

- \7 . (Po1L ) ,

41TG(PII -I- p') .

(pO + p') \7'1/Jo - Po\7'1}/ ,

(2.:n)

T he linearized form of this equat ion is

iJ,l,/ " Po (()(I' )- -I- u , v 1Jo = - - + 1/. ' \7(111 .iJf PII Uf

(2.:17)

Subtracting equilibrium Eqs. (2.28) leaves a set of equat ions all the termsof which are linear ill smal l qu antities:

Equations (2.:32) (2.34) give five eq uations [couutiug the vector oqun­t ion as three) for six unknowus (1/" v; p' , 'ljl' ). We need anoth er equationto close the sys te m: t hat equation com es fro m energy considerations. Ingenerality, we sho uld perturb the ene rgy equation (2.4) in the same manneras Eqs. (2.1) --(2.3). But there ar e two limit ing cas es, isot hermal perturba­t ions and adi ab atic pert ur bations, which are sufficiently common t o be veryuseful ami are simpler than using the full pertur bed equation (2 .4) becausethey don 't involve a det ailed descript ion of how E an d F are perturbed .

T ho converse si t.nat.ion is whe n ' thr: ti ll!eseak lor heat. exd lHl lp;e hotwocn

neighho uri ng matnrial is mu ch Jonger t.han t he t.imescalo of the port .url ia­

t.ious. 'I'hcn we can Ray that. over a t.i moscak- T 1.h(: heat gai lled or lost by

a flu id eICIIH'llt is zero: (KJ = O. By Eq . ( 1.2H) th is iUlplies t.hai

(2 .:11-\)

(2 .:39)

(2.40 )

dp1"-- .

P

, ]1 Dpfl i»

d ]J

]1

1)]1

Dt

D,p,' n , Po (D f/ )- + 1l. . v Po = - -.- + 11 · \7Po .of. Po Dt

2.3.2 Adiabatic [iuctuaiion»

or in terms of materia l derivati vos

The lineari zed for m of t his equation is

(2.32)

(2.:1:3)

(2.:34)41TGp' .

= - \7 . (Po1/.) ,

Dnp075t

Dp'at

\72 '1j/

2.3.1 Isothermal flu ctuations

Let t he typical time scale and length scale on whi ch the perturbations varybe T and A, resp ectively. Suppose that the timescale on which heat can beexchanged over a dist ance A if> mu ch shorter than T. Since heat tends toflow from hotter regions to cooler ones, efficient heat exchange will eliminateany tempera t ure fluctua tions. Assuming a. per fect gas , pert ur bing equation(2.12) gives

dp

p

dp dTp -I- T (2.35)

(In the last equation, is also an equilibrium quant ity because we havelinearized , bu t for clarity the zero su bscript has heen omittod.) We seethat t his is of t he same form as Eq. (2.37) hut wit h an additional facto r At.

The adi abatic approximation will generally be a good one when con_sidcring dynamical motions of e.g. t he deep in teriors of sta rs , whor« thedyn amical t imesca le is much shorter th an the t hermal timoscal« . In j.]ml

case, different iating Eq. (2.32) (t he linear ized equat ion of motion) wit hrespect to ti me, an d using Eq .(2.40) t o eliminate p' and Eq. (2.:33) (t helinearized continuity equat ion) /.0 eliminate p' ; yields

For iso thermal fluctuations , dT = O. Hence dp jp

mater ial deriva t ives,d pj p. In terms of

D p

DtpDp

pDf(2.36) In the penul timate term, the equilibrium Eq s. (2.28) have been used to

eliminate \7'1/Jn in favour of \7vn.

28 Astrophysical Fluid Dynamics Simple Models 29

2.4 Lagrangian Perturbations

We have previously considered perturbations evaluated at a fixed pointin space, so for example p' = p(r, t) - Po (1') is the difference between theactual pressure and the value it would take in equilibrium at that same pointin space. One can also evaluate perturbations as seen by a fluid element(d. the material derivative). Such a perturbation will be denoted lip, forexample. Now lir is the displacement of a fluid element from the positionit would have been at in equilibrium. So

so taking the divergence of the first of these equations and substituting for\7 . u from the second gives

(2.46)8u ,

Po-.- = -\7p8t

shall do so in the remainder of this chapter. The ./// term is of course alsoabsent in problems where self-gravitation is ignored altogether. However itis crucially important in the Jeans instability (see Chapter 10).

Suppose now that we have a homogeneous medium, so that equilibriumquantities are independent of position (and hence in particular \7po = () =

\71/!O)' Equations (2.32) and (2.33) can then he rewritten

(2.42)lip == p(ro + 61') - po(ro) = p(ro) + lir· \7po - po(ro) ,

2.5 Sound Waves

Perturbations such as p' at a fixed point in space are called Eulerian;perturbations such as op following the fluid are called Lagrangian.

where 1'0 is the equilibrium position of the fluid element; in the secondequation, the first two terms of a Taylor expansion of p(1'0 + lir) have beentaken: strictly we should have lir . \7]), but or· \7po is correct up to termslinear in small quantities. Equation (2.42) can be written

(2.47)

(2.48)

(2.49)

,)2 ,U (I 2 ,-a·~ = \7 p .t

where c6 == ,PolPo is a constant. Integrating with respect to time givesp' = c6P', which can be used to eliminate p' from Eq. (2.47):

82 p' 2 2 ,8t2 = Co \7 p .

Suppose further that the perturbations are adiabatic. Now Eq. (2.40) for ahomogeneous medium becomes

This is a wave equation (d. the I-D analogue 8 2p'18t2 = c6 82p'18 x2)and describes sound waves propagating with speed co. In fact, Co is calledthe adiabatic sound speed. If we had instead assumed isothermal fluctua­tions, we would have obtained a wave equation with Co replaced by a, theisothermal sound speed; d. Section 2.1.2.

One can seek plane wave solutions of Eq. (2.49):

(2.43)

(2.44)

lip(ro) = p'(ro) + or· \7po

where the argument on the left is written 1'0 (rather than 1') and this isagain correct in linear theory. Of course, Eq. (2.43) holds for any quantity,not just pressure. We note that, in linear theory, 81iII8t = DoI IDt (whereI is any quantity). The material rate of change of the displacement or ofa fluid element from its equilibrium position is equal to its velocity. Hence

Dol' 801'U = Dt = 8t .

Just as the Poisson equation (1.15) has integral solution (1.13), so Eq. (2.34)has solution

p' = Aexp(ik· T' - iwt) , (2.50)

the integration being over the whole volume of the fluid. In the integralon the right-hand side of (2.45) the positive and negative fluctuations inp' tend to cancel out, so that it is often a reasonable approximation to saythat ib' s::;; O. 1'h11s W8 will fneollfmt.lv nron ih' ill Eo (?411 Tnnpen we

'1//(1') = j' -Gp'(r) dV ,11' - 1'1

(2.45)where the amplitude A, frequency wand wavenumber k are constants.(Here and elsewhere, it should be understood when writing complex quan­tities that the real part should be taken to get a physically meaningfulsolution.) Substituting Eq. (2.50) into (2.49), one finds that a non-trivialsolution (A i= 0) requires

A strophiisical Fluid Dynam ics S im ple M"dds :{ I

This is known as th e dispersion relation for the waves. It. specifies therelation t hat must. hold betw een the frcq oncy and wavenumber for the waveto he a solution of t he given wave equat ion . W ith a sui table choice of phaso,

one can deduce from (2.50) that

whence

f( z ) = Aexp(kz ) +- l3ex p( :-k:: ) .

,P

]I'

r'i r

(l cos(k . r - wi ) ,

n c~ eos(k . T: - wi.) ,L

- (]~ k sin (k . l ' - w I,) ,f! IlW L

(2.fJ2)

(2 . fJ:~ )

The fluid is infinit ely deep , and t he solution sho uld not l)(~(:()]lJI' infinitr- ;IF:z ----> - 00; honc« 13 = () .

The boundary cond it.iou at. tlH' frr ~e :-;urfal'f' is Ihat 1,111' pn 'sslIn ' a i, 1,1\1'cdp/, of th r: flu id should he' ('ollst ,llJl,: hl'])(,(, (51) ~ 0 (.]1('1"1 ' . ' I'lms, at, 111('

sur face,

for some const ant a mplit ude rr. Not.e that th e adia batic pressure and den­

sity fluctuations a re in phase, where as th e di spl ace ment r if> 7r / 2 out ofph ase . A sound wave is call ed longitudinal , b ecause the fluid displacement

is parallel to the wave number I.~ .

011 t he ot her hand, taking; Lllf' dot product of Eq. (2.rd ) wit h c;, and using;Eqs. (2.56 ) and (2.G8) with B = I), yi<'1 dF:

2. 6 S u r face G ravity W aves (2.(jO)

We seek a solution wit h sinusoidal horizont al variat ion in the x direction :

and taking the divergence of t h is gives

(2.n] )

P- l 1CT7/whcr c. Hen ce the boundary condit ion (2.59) can only 1)(' sarisfiod if wand k satisfy t he dispersion relat ion

It is clear that these ar e surface waves: for t.hc p ortu rhorl quant.iti cs a lldecrease exponentially with dep th . In rea IiLy, of course, th e f uid canu: 1j, 1)('

infinitely deep , so B is not identica lly zero. Instead, A and 13 will have {,o

he chosen such that sonic boundary cond it ion is sa t isfied a t the bottom of

the fluid layer. However , provided the dept h of t he layer is much greaterthan k - 1

, it will generally he the case that B has to be much less tha n A.If the layer has depth h , and t he conditi on at z = - h is that the ver tica l

disp lacement is zero, B =I 0 a nd by the first. pari of Eq. (2.GO) th e lower

boundary condition amounts to requir ing that Dp' / G.: = 0 t here. All theabove equat ions hold , except the last part of Eq. (2.60). It follows that13/ A = exp( - kh) and t he dispers ion relation is

(2 .54)

(2 .55)

(2.56 )p'(x , z , t ) = j( z )cos(k.r - wt)

As a second example of a simple wave solution of the linearized p erturbed

fluid equat ions, cons ider incompressible motions (V . u. = 0) of a fluid ofconst ant density Po which occupies the region z < 0 be low t he free surface

z = 0 (so p is cons t a nt at the sur face) . Suppose also that gravity 9 =

-ge z is uni form and points downwards , and that self-gravity is negli gible.

This is a reasonable model for ocean waves on deep water, for example.Equation ( 2 . :~ 3) implies t hat p' = O. Hence Eq. (2.32) be comes

In the regime kh. » ] ( "deep layer" ) t hen t his is approximated byEq . (2.61) . In the opposite limi t of kh «: 1 ("shallow layer" ) the disp ers ionrelation annroxirnatns t.n I,J 2 = (nhlk,2 i P I . ' = , r;;r; L·

(wit h k > 0 for definiten ess), where f is an as yet. unknown fun ction; andwit hout loss of generality k > O. Sub stituting this into (2.55) gives

(2.57)

w2= gkt.anh kh . (2 .62)

2.8 .2 Importance of viscosity

This is the typical time scale for oscillations of e.g. Cepheid variable stars.tdYll is called the dynamical t imescale.

Molecular viscosity, which provides tangent ial forces in fluids, comes aboutmicroscopically because molecules from faster-flowing fluid diffuse intoslower-flowing fluid , and vice versa . As can be seen from e.g. Eq. (1.9) , theviscosity 11 has dimensions M L -1 T - 1 where ~M, L, T denote mass, lengthand time respectively. Let the molecules have mean velocity v and meanfree path l . Then on dimensional grounds,

33

(2.63)

(2.64)fi ~ pvl .

(R:3 ) 1/ 2 ,

T ~ t dYlI = GIll ex (mean density) - I / 2 .

2.8 Order-of-magnitude Estimates forA strophysical Fluids

S imple M odels

2.8.1 Typical scales

A Elystem call often be characterized by a typic al length scale L , time scale'T and velocity U. T hese are usu ally related by U = £ /T.

The appropriate length scale E for a part icular motion may be differentfrom the size of the whole syste m - e.g . for sound waves, L might be thewavelength, T the per iod and U the Hound spee d.

For exa mple, for motion in a gravit.at ional field , wit h length scale E, thetime scale is T ~ (£/g)I / '2 . For mot ion in a star's gravitatio nal Held withg = G111/ R'2 , where R = £: is the radius of t he star an d III it s mass,

Often people work with the kinematic viscosit y v == 11/ p; t hus u ~ ul. Equa­tion (2.64) can also instructively be ded uced by cons idering the tangent ialforce at a plane int erface between two fluids moving at different speeds,assuming that such force comes about by molecu les diffusing a distance oforder l across the boundary at a sp eed v and deposit ing their momentumin the new environment , noting the relation t hat force is equal to rate ofchange of momentum.

To make further progress, we need to relate v and l to macroscopicproperties of the fluid . If t he collisional r.rnss-spl"t,inll for t hp mn!p('ll !p<: j c

f'W', ,

i,.~-~.

{~

~.

32 Astrophysical Flu id Dynamics

2.7 Phase Speed and Group Velocity

Before leaving the to pic of waves it is worth noting two different conceptsregarding the speed at which waves prop agate. Consider a wave which islocally a plane wave , prop agating with wavenumber k and with frequ ency w.

These two quanti t ies are related IJy a disp ersion relation , so w = w(k ). Sucha wave is proportional to eik .x - iw (k )i . T he ph ase of the wave is k ·x - w(k )t ,and wave fronts are surfaces of constant phase. One concept of the speedat which a wave propagates is the pha se speed. Compare the wave at somelocat ion x and t ime t with the wave at a location x + n6.x and slightlydifferent t ime t + /st , where n is a unit vector in any chosen dir ection.The ph ase at t he second location and t ime will be the same as at t he firstlocation and time if 6. :,{; = (w/k . n )6.t . Hence we refer to w/(k . n ) asthe phase speed in dir ect ion n. In par ticular, the ph ase speed in t he x­direct ion is Vp h x = w / k x , an d likewise for the y- and z-direct ions, providedthe waven umber has a non-zero compo nent in that direction . Sometimesthe phase velocity is defined to be a quantity with the direction of k andmagnitude equal to the pha se speed in the direction of k ; but it should benoted th at the x-, y- and a-components of this 'vector' are not in generalthe same as the phase speeds in the x- , y- and z-direct ions . Note also thatin directions almost perpendicular to k (so k . n almost zero) the phasespeed can beco me arbitrarily lar ge; but this does not correspond to anyphysical transport at that speed.

A second concept of the speed of a wave is the group speed or group

velocity. A packet of waves of different wavenumbers but similar to k osay propagates physically at a veloc ity vg given by Vg i = Dw/ oki , or inshorthand vg = ow/Dk , evaluated at k = ko. This is the group velocity .We can also speak of the magnitude of t his vector as the gro up speed.For a proof that t his is indeed the velocity at which a wave packet wouldpropagate, see for example t he book by Lighthill (1978) . This is t he velocityat which wave energy propagates.

In the case of pure sound waves, it is st raight forward to show from theirdispersion re lat ion (2.51) that the gro up speed is Co and that the phasespeed normal to the wave fronts is also Co . Hence in t his case these twoare equal. In t he case of sur face gravity waves considered in Section 2.6,the phase speed normal to the wave fronts is w/k, but differentiating thedispersion re lation (2.61) gives that the group sp eed is on ly half of this ,so in this case the two speeds are not equal. Although in the case of puresound waves the group velocity is in the direction of the wavenumber, t his

I ,

ii

I!!fi

. :, ,j

,

II

II'j 't

I

where T is in Kelvin and {J is in kg m -:i . (See Appendix A for the valuesof physical rous t.an ts. ]

Now the left-hand side of t he Navier-Stokcs equat ion (U J) is pDul Dt .while a typi cal viscous te rm is ILy 2U . If viscosity were dominan t , then t hetimesca le of moti on wo uld be determined by its effect:

(2.70)

HI' .II

rtf:(11.1"2 / 1:

jillI £"2

Ilip 1---

1

H I' = ]1 dz .

I!m . Y u l

IIJV2ul

2.8 .3 The adiabatic opproximaiion.

Suppose Tp is the ti mescale Ior the tra nsfer of heat. (11.Y flux F ). if thisis much greater th an t he timescale of th e mot ion U W Il (1 )(' G I ll t.n'at. t.hemotion as H.dial>at ic (oQ = 0). This is th e adinbati« npproximatiou . Feu Ll H'SIIIl , 7/.. ~ 107 years ill d IP. in l.erior, and abou t ono dav noar t.li« snrfnr« .Th e fundamental period of oscillat ion of the S IlIl is ah'ntl OJl( ' hom (sef'Cha pter 11 ). so fur most Jlurposes t he ad iabn t.i« upproxnu ation is cxrol lc-ntfor describing oscillat ions of the SII11 . Til th e solar at.mosphore , however, T»c all he much shorte r , in fact so sho rt that there are Rome circ um st.au ro» inwhich one can treat t he mot ion as isothermal (Section 2.3).

871H1,1" M ond s

the left-h and sidp. of S q. (1.9) to tho viscous term 0 11 t.lIl ' righ t- hand sidl':

ViscoUS dfed,s an' imp ortnnt. if H(, ::; 1. For str-llnr s('a1l:s, lor sj"'l' d:- "1,,,;(­to the sound sjJl'ed inside the S1I1 1 (~ ](I " IUS-- I) . Hi ' CV HI I

" , :\lJd ;;0 rv cu

for subst.ant ia lly subsonic Hj H'(,ds tl w H.eYllolds number is g('w' n illy verymuch greal.!~r t.luu: unity. T his shows once again (.]mt n rolocu lnr V iS( T O IlS

effects are gencrally negligib le i ll the std lm «on l.cx t. However, smnll-scal«tnr lmj(>nt. flows can have an dfcd . on the mean lar gc-scale mot.ion sim ilarto t.ha! of viscosity: th is is known as t.nrhul cnt viscosi ty . Jt mnv woll 1)('the source of "v isr.osi tv" i ll JlII l.lIY a.c;1.rop liysi, ·;I! \'is l"l ' lI:< lWITl'i. io Jl d isks. IiI]·

exa mp)('.

2.8.4 The approximation of incompreseilnliu]

Th e flow is incom pre ssibl e if Dpl Dt = (J fur th en the density of a fluidelement docs not change with t ime. By the conti nuity equa tio n (2.2) th isis equivalent to Y . u. = O. (Some authors prefer to take Y . u = 0 as I.hedefinit ion of incompressibi lity, and lJ[IIDt. = 0 as the conseq uence of that. ]Roughl y speaking , the cond it ions for t his to hold a re that U is mu ch lessth an the sound sp eed and that I: is much smaller than tho pr essure scaleheight.

(2.(i[) )

(2.GG)

(2.G7)

(2.68)

(kHT)1/ 2

'V rv - -

In

1/2(kJ3 m) J/ 2p. ~ T.(Y

A st roph ys ical Fluid D yna.m ics34

Thus

For a crude est im ate, o ~ (radius of a!.olll? , so about 10 -- 21J1ll2 . (Thisundercst.imate s (Y in an ionized gas, where electromagnetic inte rac tions arcim portant .] T he mean mass t n. = ii.m,,, where ji here denotes the moanmolecular weight and m il is the atomic ma ss un it . Assu ming reas onabl yt hat a ll the constants implied in the ,~, relations above are of order unity,this gives

(Y (not to be confused wit h tho stress to nso rl}, and the nu mber density ofparticl es is 11 (so that. (J = 11m , where In is the mean molecul ar mass), t hen0 11 average hot WCUll collisions a particle sweeps out a cylinder of vol um« a]

and thus such a cylinder must. contain 011 aver age one particle: nlo ~ l .Hence I ~ tn.](J (Y . TI l<) mean kinetic energy of a molecule ~ knT, when,k n is Boltzmann 's cons t.ant., so

or, rearranging, Tv ~ 1:2 / ,1. For typi cal ste llar values (1' = lOGK , (I =

lkgm- 3 , ji. = 1, I: = 108m ) we ded uce using (2.67) that t he viscoustimescale is of ord er 10 2 1 s ~ 3 X 1013 years. Even for a star t his is avery long time , so mo lecular viscosity is unlikely to be important on st ellarscales. T his will generally be true for ast ro physical fluids, t hough someform of viscosity is impor tant in e.g. accret ion disks (see Ch apter 9) .

A commonly used measure of th e importan ce of viscous effects is t heReynolds number R.e, which is the rat io oft-he advect ion te rm (implicit) on

Astrophysical Flui d Dynamics

For exam ple , for the Earth's atmosphere Hp ~ 10km. Of course, co m ­

pressibilit.y cannot. be ignored for modelling sound waves!

Chapter 3

Theory of Rotating Bodies

Most if not all objects in t he universe rotate , and the effect s of rotation areimpor tant to an understanding of the s tru cture and dyn am ics of many as­trophysical systems. Rotation is indeed suliiciont.ly iiuportun t to the subjectof astrophysk al tluid dynamics th at we ret ur n to it several times in ad dit ionto the present chapter: in Chapt er 4 on fluid instabiliti es, in Chapter 7 onthe dyn amics of planet ary atmospheres, in Cha pter D on accretion disks,and elsewhere . T he present chapter est ablishes the equations of motion ina rota ting fram e of reference a nd considers the equilibr ium st ruct ure andshape of a slowly and uniformly rot a t ing st ar (or gas eous plane t). Vo/ e shallalso consider briefly the int ern al dynamics of a rot at ing star, and someconseq uences of orbital rotation of stars in a binary sys tem.

A great deal of research has been made into equilibria of rotating bodies,particularly in t he case of bodies with unifo rm density, by such illustriousnames as Lap lace, Jacobi, Liouville, Riemann , Poincare , Lord Kelvin andJeans. Much interesting deta il of the results and his tory can be found ine.g. Lyttleto n (1953) , Lebovitz (1967), Chandras ekhar (1969) and Tassoul(1978). Just to give some brief hist oric al context, we mention that in theease of bodies of uni form rota ti on there are two families of equilibriumconfigurations. One consists of the Maclaur in spheroids : these are axisy rn­

metric configur ations . T he second family consists of the Jacobi ellipsoids,which are non- axisymmetric. When the rotation is sufficient ly fast , as mea­sured by t he quanti ty n2 /27fGp where p is the density, then t he Maclaurinsequence terminates and for faster rotation the only equilibr ium configura­tions for homogeneous bodies ar e the t ri-ax ial J acobi ellipsoids.

;17

3.1 Equation of Motion in a Rotating Frame

where n is the angular velor-itv of the rotating frame relative to the inertialone. Applying the same rule a second time givps

When considering rotating systems, it is usually most convenient to workin a rotating frame of reference, The fluid velocity u is the ratcl of chang(~of a fluid element's position r with time, If we use D / Dt to denote rate ofchange as measured in an inertial (nonrotating) fi'aI1W, and d/dt to denotprate of change as measured in the rotating frame, then

Theory of Rotating Bodies

mmetry. For slow rotation, the distorted body is axisymmetric aboutsy t t' axis as one would expect. Although we shall not COIISHlprthe I'O a ,Ion '," ,. . . " ..

f, "or rotation can give rise to SOIlIn surprrscs, notahly th« Jacobiit here, dS",. , . ..., " .., • ,"

. . '1' hich are triaxial figures of oquilll niurn. For a h.Jllu (XjH)Slj]OJ!elhps()]C S w. , '. . ,

~ 1 1· . .ct C'('(' 'J'J(' classic texts hv CJmudras('khar (I%(J) and Lvtt.lr-t.ouof the S11 ).1C .L, ,0. ,I, .' ,e", ". ..

(195:3). , " .' ',' '" 'I" ._vVe work in a frame rotating WIth the body: III that h eUI]( 1, H. ,c(jm

.' . , described lrv u = 0 and CJ/Dt = 0, Since we arr: llJoddlmg ahbnuIll IS.,o .. .1 • •

1 mhorical configuration, we use spherical polar coordinates (,., (),4)).ncar y sp . . '., " ,I . ordinates WI'l' ting 0 as i le., (where e z IS a unit vector alollg theIn t rese co ) e, , " , t. ., • " •

polar axis,

(3,] )dr- + Oxrdt

Astrophysical Fluid Dynamics

DrDt

38

The last term is the centrifugal acceleration; the penultimate term is theCoriolis acceleration, which is zero if u = 0 and is perpendicular to thevelocity otherwise.

where in the last step we have now assumed that the rotation rate n does

not vary with time. There are some subtleties to VActors in rotating frarnesand calculating their rates of change, and the reader who would like moredetails is referred to Chapter 3 of Jeffreys & .JefFreys (1956).

Now in the inertial framA, Eq. (2.1) is the equation of motion; so substi­tuting for Du/Dt =:: D

2r/Dt2from Eq, (3,2), and identi(ying dr/ell. as the

velocity as measured in the rotating frame, gives the following equation ofmotion in the rotating frame:

(~ + nx)2 rell.

(:1 01)

(3.G)

(3.5)

(1 2 2)- Ox(Oxr) = \7 2'0 tv .

- Ox(Oxr)

1 · ' . 'j the velocit.v of tli« fiuid as seen from the nonrotating Iraino, ifTns IS Jusr . ", , ',I

, j rc..st in the rotating' frame. J'vlon~(Jvcr,it IS a'.

n27' sin2

() e, + 0 2, sin () cos () eo

\7 (~n2r2 sin ' ()) ,

So the centrifugal acceleration can he written as the gradient of a potential.Note that T sin () is simply the distance from the rot.ar.ion axis, Indeed theabove would he simpler in cylindrical polar coordinates (tv, ¢, ;;) since co­

ordinate tv (pronounced "pomoga") is the distance from the axis: Eq. (:3,5)would become

We are interested in finding equilibrium solutions of Eq, (a.3), Settingu = 0 gives

(:3,2)

(3.3)

d2r dr+ 2fh- + nX(Oxr),dt 2 dt

1--\7p - \7'I/J - 20xu - OX(Oxr) .p

du

dt

3.2 Equilibrium Equations for a Slowly Rotating Body \7p - p\71> , (3.7)

In this chapter we shall consider how to calculate the shape of a fluid body

that is rotating slowly with a uniform rotation rate. \Ve shall consider inparticular the case of a slowly rotating star; but the equations apply equallywell to, for example, a slowly rotating gaseous planet, It will be assumed

that in the absence of rotation the body would be spherically symmetric,and that rotation induces a weak distortion of the shape from spherical

where

1 2 2 ' 28 (3 8)1> = 1/) - 2'n r sin .

is the total effective gravitational potential (gravitational plus centrifugal).

vVe can argue qualitatively from Eq, (3.5) what the effect of rotationon the equilibrium shape of the body will be. At the poles (8 = 0,71') the

centrifugal acceleration -0 x (0 xr) is zero, and it is radially outwards at

t~lC equa tor ((j = 1T/ 2). It thus reduces the effective gravitational accelera_t ion at th e e(~uator, i.e. t he centreward pull is not so st rong there as at t hepole~ and so .mstead of be ing sp herical the bo dy "bulges" at the equa tor.

C ' . ~ ~le gra(~lCnt vector \7f of a ny sca la r is perpend icul a r to surfaces of_on,st ant f ,. so a normal n to the surface of constan t .f satisfies n x \7f = 0

It follows from Eq. (3.7) that surfac es of constant p are also surfaces ofcon stant !P, and vice versa, T hus we can write p = p(p ), and so

is constant (i.e. indep endent of B). The rotation is slow and t he distortionweak , so rfl and e ar e small and we neg lect pro du cts of smal l qua nt iti es .

Then (:3.13) implies that

41

(:U:3)R(l + f(B))

Theory of Rotuting B odi es

q)S l lrfac(~

where f(O) is a function of B. T hen

OM

A st rophysical Fluid Dynamics40

dp\lp = - \lp

d1'

Substituting t his into Eq. (:-~ .7) yields

(3.9)OM

- -(1- f (O) )R

is indep endent of 0, i.e.

1 .. 2 "-n"'R sin- (j2

(:3 .14)

so (J is abo 11 Iuuct.ion of q" i.e, p = p(q,).

, , ~:Iencef~)rward , for definiteness, we shall speak of the body as being astar , but, It could equally be a gaseous pl anet , for example. T he outersurf~ce ~)f the star is a surface of constant pressure (b ecause t he pressureout side IS const ant, say zero) and so <l? is const ant on the surface.

\~~e consider the case where if t he star wer e not rotating it wou ld be~phencally symmetric, and rotat ion induces a weak distor t ion fro m spheric~Ity, VVe SUppo se that t he star has mass M and (I'll t he t ti ). nonro a mg' caseradi us R .

Note th at n2R is t he equatorial acc eleration due to centrifugal forces ; an dGMI R'2 is the gravitational accelerat.ion. So, the dime nsio nless quantityn2R3I (OM ) is the ratio of centrifugal acceleration to gravitational accel­

era t.ion .T he radii at the pole and at. t he equator are obtained from Eq. (3 .12)

by pntting e = 0 and 0 = 1T12 respectively. T hus t he relative diffe rencebetween equatorial and polar radii is

R( l + f(1T12)) - R (l + f(O))R

dp(J = - ­d p , (:3.10)

f(fJ )1 n2 j { 3 . 22" G !YT sin 0 + constant.

] n2R3

2" OM .

(3.15)

(3 .16)

(3.11)

3.3 The Roche Model

On the surface P - " I, 1 n2 .2 " 2 () ." ' - 'J-' - 2' 1 sm IS constant. Let us approximate the

gravitat IOnal potential t/J by what it wou ld be in the nonrotating cas e:

1/-' = _ OMT

Thus th e relative difference in radi i, which is a measure of the shape dis­tortion, is n 2 R'J I (0fl.i) ti mes a coefficient of order unity.

The only thing wro ng wit h t h is ar gument is the use of Eq. (3.11) to de­scribe the gravitational potential. \Ve should properly use the gravitationalpotential appropriate to the distorted st ar . We proceed to do t his now.

3.4 Chandrasekhar-Milne Expansion

at t~~e, s~rface and o~ts ide . the star . This is equivalent to approximating thegraxitational potential as If all t he mass were at t he centre and is called t he

R oche model. T l: is is ~ reasonable approximation in the case of a centrallycondensed st ar , III which mos t of the m ass is concentrated near t he centre .

\Ve suppose t hat t he surface of the rotating st ar is described by

r = R(l + f (B)) , (3 .12)

The missing ingredient in t he previous sect ion was a proper treatment of thegravita t ional potenti al of the dis to rted star. In the Chandrasekhar-M ilneexpansion, one cons iders the O(n 2 ) pert urbation not onl y to the shape ofthe st ar but also to it s gravitational potenti al. T he procedure is des crib edin more detail in Tassoul (1978) .

We know that on the surface <l? is constant; also p = p(<p ) everywhere.Now the gravit ational potential satisfies P oisson 's equation (1.15) ; and it

is straightforward to demonst.rate t hat

(:\.2(;)

.1 Co whi le t he first-orrler torins1~ 7f .i ]. u ,'

Theoru of Rotating Bodies

Ell ('{ ] 8) becomesBence " . . .

1 ( " (lp I ) - :m:l "LJ7rG Pu + <I rl (I) .1l'1I

. 2The zero-order terms give \7 1/)"

yield

A stmphysical Fluid Dy nam ics42

It follows t hen from t he defin ition (3.8) of q> that

The problem involves th e \72 operator , so it is mo re natural to write j,]1(.

B-d ep eJl(lence not as F>i1l2

() but ill te rms of Legendre pol Yllomials of cos 0:P" (cos 0) , since

Matching zero -order t erms gives Pu = p('1/),,) , and first-order terms give

are solut ions of Laplace 's equa t ion , \72V = O. The firs t three Legen drepolynomials are

wher e 'lj)" and p" are the gravit a t ional potential and deusity in t he spher­ically symmetric, nonrot:ating star , and the primed quantities are smal lperturbations, of order [2 2R 3j (G.M ), induced by the rotation. As before,

we express the ste llar sur face as 7' = R(l +E(B)), where Eis the same order as

the other small perturbations; and we neglect product s of small quantities .Recalling that p = p( (j)),

(3 .20)

(:\.27)

00 ()n+1GM + " A ~ P,, (cos(1)'1/' - - -- L..-J n-' - T 11.=0 T

Also, (j) is constant Oil the surface, so

1/)'1 (R(1 -I- r )) -I- ij>1(7 ',0 )

To allow complet e generality, we would now wr it e

, ,'jS,, the firs t two j.('nus of al rt l' O' whic h after expanding l j! ."is illdepml( ell ,0 , . . '

Taylor series expan sion , gJvos

_ ~ (~h/'" I ) I (l/ (R J i) ( -j- r:o lls !: 1l11) "dO) = R dr R

" f ij>' once (1) '. (3 26) is an lnliomogencous differe ntial equation or :Equat.lOn " ,. f E ( '~ 28)

I ' f the surface follows rom ' q. . . .is found, the s, iape o . . " ' I , I t i I1111F> t he regul ar in the interior ,rt " q)' is that t i e so u ,IOn " .

One cone I ,IOn on " f 1 ,I cond it ion , we must ensure th at. ti lar at l' - 0 To llll a seconu c ' .and III par ,I ClI . ( - . . I " tllv o n t o t he oxtcrual gravi!<1 -I . "J' t'at,)'011 '11 po tential 1/' mate ics smo o ' ) ..the gra" , ., ,. ' .

tional fiel d.Ou tsid e the star,

A . f riel' [22fl3 /GMsince \72'1/' = 0 there by Eq . (1.15), where " IS 0 0 ( . .

Insid e the st ar ,

1 2 2 {I P. ( os(O)} = 1/.! (1') -I- q/ + ~n21'2 {l - P2(COS(0)}1/' = (j) + -[2 r -' 2 e 11 .3 )3 (3.30

(3.] 9)

(,3 .2(J)

(3. 21)

(3 .22)

(3.23)

] c 2= 2" (.h: - 1) .

p = p,,(7') + P'(1', B) ,

:J: ,

v = l'''P', (cosO) and V = 1" - (n+JJp,, (cosO)

1 ( ') () 1 dp IPu -+ P = P 1/'" -+ <I> = P 1/)u + q) dq). "lj,u

Po (.1:) = 1 ,

(j) = '1/;,, (1") + (j)1 (1',O) ,

Thus (j) may be rewritten as

Let us write

(3 .24 )

00

<]/ (1', B) = L qJn(1' )Pn (cosB) ,n=()

00

E(1',B) = L E,,(1')P,, (eosB)11 = 0

(3.31)

45

(:3.39)

Theory of R otatin g Bodies

ef 'l/Ju 2 d'ljJu _ 1 G__,_ + --- - L7r P'/l 'dr '2 r dr

at 'I' = R.Equatio

lJ(3.39) pro vides the surface boundary condit ion that must be

applied to the differential equat ion for P z· Taking just the P2 (cos 0) t erms

froUl Eq. (3.26) , this differential equat ion is

<Ip \47rG - <I' '2 P2 'dl!>

'II.

and using Eq. (3.28) with (3.:12) to eliminate E'2 , Eq. (:1.:17) ]leconwH

dP'2 3 ;r tl7rR'2;r 5 n '2R__ + -~z --pu~'2 - H .

dr R M 3

where all functions are evaluated at r = R . Now recalling that 1/.'u obeys

poisson 'S equation so

1. "

~..,'1/)'11 ( R (l + EO + E2P2)) +

- GAd

A st rophysical Fluid Dyn am ics

(see Tassoul 1978). However , to avoid needless algebra, we note that theproblem for uniform rotation has only P'2 (cosB) and Po(cosB) angular de-pendence "-- see Eel" (oJ ')6) ' d (3 30) .d . .u . v.. an . -- an so we anticipate the solut ionto be

qi' (r,O) = Po(-r ) + P '2 (r)P2(cos B) , E( r, B) = EO(r ) + E2(r )P'2 (cos B) ,

' . . . . (:3.:32)and similarly for th e external field (3.29) . The surface boundar y condition

b~com~s t hat of requiring 'l/J and o~jJ/or to be cont inuous there: for other­w~se , since 4' satisfies th e Poisson equat ion (1.15), a discontinuity in ouoof th ese quantities would imply that there was an infinite density at th,

surface, Continuity of 'ljJ means, equat ing (:3.29) and (3.:30), that . '

~ \ = clp'/l / d'ljJud<D dr dr

1L

(3.'11)

(3.42)

'ljJ ,, (R)

(3.34)

so Eq. (3.40) gives the following ordinary differ ential equat ion for <1> 2:

~i. (rzd<l>z) _ .Q.<I> z = _ 47r.,.z dPu<l> '2rZ dr dr ".2 rn(r ) dr '

Similarly, cont inuity of a ljJ / ar implies

To zero-order , these two equat ions give simply

d'l/Ju (R)dr

GMR 2 . (3.36)

with boundar y condi tion (3.39) at r = R and <l>z regular at r = O.In t he general case it would be necessary to solve the ab ove equa­

tion numeri cally. However , in the spec ial case where Pu is cons t ant, it

is straig ht forward to find t he solut ions of Eq. (3.42) in the form <l> z = ArP

for constants A and p, and after applying boundary cond it ions to de­duce t hat <1>2 = (5/ 6)Ozr-2. Further , it follows from Eq. (3.28) that02 = - (5/6)0 2 R 3 / GM. Hen ce the difference between the equator ial and

polar radii , divid ed by R, is (5/4)0 2R:3/ GM in the case of a homogeneous

stellar model.

The first- order terms propo rtional to P2 (cos B) give two equations whichafter A 2 has been eliminated between them , yields '

(3.37)

3.5 Dynamics of Rotating Stellar Models

We have not so far considered how energy is transp orted in the ro tat ing star.A well-known result , which is discussed at leng th by Tassoul (1978), is thatone cannot have a uniformly rotating star in strict radiative equ ilibrium .

3.6 Solar Rotation

Deta iled observat ion s have been ca rr ied out of t he Sun' s su rface rot.atioll

rate over many years, by t ra cking surface featllreH such as sunspots and.more recentl y, using Doppler velocity measurellleu t s . The rotation ra l:cvaries wit h latitude, t.he equato ria l regions h avin g a rota t ion peri od of a ho ut25 days while high la ti tudes rotate 1110re slowly, wi th peri ods in excess of olle

mont h. T he surface rotation rat e is comnHm ly expressed in an expanHionjn cos '2 e, wher e e is co-lat it ude, e.g . Snod gr ass (J98:3). The Doppler rnte.

·/l ·

Assuming the contrary leads to what is kn own as von Zcipel '» po'/'aclox,

The same is t rue if the rot.atiou rate is a Iunc t.ion onl y of distance Iron, therotation axis. \Vc conclude ther efore th at tlw ro t ation rate must hHV( ~ a

more gene ral form, dep ending on cy lind rica l polar coordinate z aR well asdi stance fWIII the axis, or that stiict radiative oqui libri mu (IoCR 1101. hold .

\<\10. cons ider now the la tter possibili ty. The VOIl Zeipol paradox in dl'c'etsays that t.ho radiative Hux ca nnot he bal anced everywhere by th e ( ~Jl( 'rgy

generat ion . Som e regions have n net influx of heat: these will Iwat Ill) aile!tend 1.0 rise under 1moyancy, O th ers will cool a IHI sink, This t end s 10 setup motions in m cridiounl planos: thi s is ca lled morid ionnl cir oulat.iou . Itcall he S b OW Il , e .g . Kippcnhahn & \ 'Vl'igert. (J~!)() ) and Tussoul (JD7S). th att.ho global t.imoscal o for mi xin g by t.he meridional oircul at .ion . known aR the

Eddington-Sweet timescale, is of order TIm/X where TI<J] == GAr 2 / R], iRtheKelvin-Helmholtz timescale (L h(,ing Ow luminosity) and X ~ 0 '2 R:I/(.'f\J .

For tho Sun , Tl( ll iR about lO7 years, and X ~ 10- " ; :-;0 the E dd ing to ll­Sweet l.imescalo iHabou t JOl'2 years , Il11H:IJ lon ger than the Sun 's age . Local

circ ulation timescal os ca n 1Ir. much shor ter, however.For the Su n , the Eddingto n-Sweet timescale is much gren tc~r ( ~W~II th nll

the nuclea r timescale , hut this is not so for some more ma ssive s tars . Yett he observational ev idence does no t suppor t t he idea t hat th ese stars arc

mi xed , as th ese timoscalos would sug gest. The explanat ion (d . Kippcnhahn& Weigert J990 ) is t hat mi xin g is opposed a nd stopped by composit ion

gra dient s (and hence gradients ill the mean molecul ar weight) .It should he mentioned t hat , al though one CHn po stulate HOllie arhitra rv

rotation profile for the interior of a star , this will not necessarily lie sta hl«,and hence will not necessarily be realizable in a real st ar. An exam ple of a

stability consideratio n is the Rayleigh criterion (sec Section 4.3).

\

\

\

\

\

\

• il E i. , I

T hc01'Y of n ot,etting Bor/ics

1 ' 1 "-- '

0.'1 \ \\

C' \I-.. ' sI ..

,\i\COl I

II\\ \

I

Ii I, i ,! ' I

CH CUi O.B 1.0

T/R

1.0:-\ : ~o

{:'-)(1 \ f I

O.B :IHO ( ( ( I

\

\':\1\0) i

I

,, 0 (\I

o;0 .0-,

~ 1~~t)

. 1 t I (J98R) ' \R all!)]'llxi1ua tdvf .' Ij'lll' is ajv(:n bv Ulric 1 C , 0.. . , • ( , . ' "or ('X a ll, ' b ,-

n / 27f = (tiS]'!) - G5.:~ ('os2 (J - GG.7 ('OS·1 0) nll z . (:U :{)

. " , .' . . 1 from traekin g di fferent fea1.11res do not agreeThe rot ntlOll rates dc1Cll!lllleC . . ,', l ' Snuspots , for exa mp le ,

1 t ' with c'1('h other pt ecise }. ,wit,;]1 tho a love r a .e OJ . ," I "' " (3 4'3) ind icates a t low latitudes. Thcir

. t ' . t 10 15 nlIz Ias t.ei t iau . " I ' .rol.a .c c . • . ' I thc r t' t' rate in a some what c oep eimovement may ]) 0. more l!lcl!catlvc 0 1. ic 10 ,<1.ion "

1 tl ots m ay he rooted .suhsurface regioll ".rlCl.·C • re sp I " '. " " I . ' t (']I'ln O'('c! l .v 1Il111'l ' th an

1 '. tl e sllrhcc rat (' I,ll' lIO , , ,.., .fo r at. 1('ast n e(m ,m y , 1. , " . 1 f' 1(1/ ] 1'\ \ '( ' 11('( '11 d p\.pc! ,0,d as.' . f t he oj'( (' 1' () !O , , ' .

GpC'1' C(' l Jf.. However, vnna\.~ons 0 ' • ' 1 1.' j'· j' \ (I('~ t( > t he ('ll ua t,or with a" . t g from nn( - , I ,I ,I ,~, "

zona l hands of fi.ow nngl a , 1 ~1 _ . ' II d to rsiOll't1 osc.illa t ioJlS, thon gh' 1 t 11 " l'S 1 hese al e en e " .period of a ) 01 1 ' yc,), . . "

thi s is a 1M of a misnomer. . " l ' l ' t he past.The rot at ion rate of the intel·.ior its~lf hals l~eee" :~ ~.:;(el~~e~~ci::': of ~lo1Jal

1 I· . ... . , imagmg' llSlJIg 0 ) SCI V .. ' 'two dccad es by Ie 10SClsmlC " ' . '. l ' r in ti ll', . ' '.. (S r ]2 C) ) 'Wf\VC's pl opag,1 m gaeons!ic modes of t.he Snn see . cc ,10 n . ":, ' .. ,

. ' S I '16 in ferred bv helio1"ei611lo1o!,\~" '1'1 ", ('O ll tOlll,S

Fl'g, :1,1 R ota t.ion r a t e 1ll1"Id e the ....111 , ' f' ](J II Ti le h o r i7,Ol ll. :11 axis is i ll 1h e Sun s] I C :1 sp:1cm g 0 n 7" , . f t1

l:1hell<,d in n l lz aur iav " ' . . ' f 1 t.i ]ljs t 'lI1<'e1" a re in u ni ts 0 • ICare ( .. .. 1 is is the aXl f-' 0 rot .a .ion , . .r ~ , .

nnt.orial pl all e , the vcrt .icn aXI. , '. 'I t.1 \ ase of th o Suns COI\ VCc\.IOl l zone ,~\I:' s phot ospheric radiu s. T he dashed line m a r ( S ie )"

A st roph ysical Fluid Dimam ics46

(. . .) denotes an average over longitude. An equat ion for the conserva­tion ~f a.ngular momentum density J = p(r sin 8)2n can be obtained bymultiplying the ¢-component of eq. (3 ,44) describing the rate of change of

49

(3.49)

(3.48)

(3.47)

(:3.46)

T heory oj Rotating Bodies

r sin f.rEM = - - « pB, B,p) e,. + (pBoB,p) eo)

po

and viscous diffusion

the Reynolds st ress t erm that arises from non-zero correlations betweenturbulent fluctuations 11,' == 11, - (11,) in the velocity in the ¢-direet ion andthe other two directions . The remaining tenus·represent the transport due

to electromagnetic Maxwell stresses

.rv = - z;p(r sin (J )2\7D (3.50)

respect ively (e.g. Thompson et al. 2003). In deriving Eqs. (3.47 )-(3.50)

we have neglected longitudinal variations in p and 1/ .

Viscous forces ar e presumably negligible in the solar interior, andthe Max well stresses ar e also likely small in the bulk of the convectionzone (though possibly not in the radiati ve interior and tachocline, nor insunspots). Hence a steady-state rotation in the convection zone indi cates abalan ce betwen the divergences of the fluxes of angular momentum causedby meridional circulation and Re ynolds stresses due to turbulence. If theRossby number is small and the flow barotropic , then the Taylor-Proudmantheorem (Section 7.4 ) states that the rotation rate will be constant on cylin­drical sur faces aligned wit h the rotation axis . This is evidently not the casein the sola r convection zone (F ig. 3.1) except perhaps at low la titudes. Thisis at least partly du e to baroclinicity (\7px \7p =1= 0) driving a meridionalcirculat ion which red istributes angular momentum. Latitudinal variationsin heat tran sport du e to rotational modulation of the turbulence cause athermal wind (Section 7.4); but also the Rossby number is likely not sm all

the flux of an gul ar momentum due to the meridional circula.tion in the

(1', e)-directions , and

oJ = - \7 . (.rr,IC + .rRS + .rEM + .rv ) .at

Here the term on the right comprises (minus) the <livergeuce of various

angular-momentum fluxes. The first tw o are

mentum by the distance 7' sin 8 from the rotation ax is, and using the

InOcontinuity equat ion (1.3):

(3.45)

Astrophysi cal Flu id Dynamics

D(1' ) = no + (u</J) .rsin 8 '

48

same dir~ction a.s the rotation have a slightly higher frequency than thosepropagatmg .agamst the rotation, and the differ ence in frequency dep endson the rotation rate. The results of such imaging in the outer 60 per centor so of the .sola~· ill~erior are shown in Fig. 3.1. The outer 30 per centof ~he solar I.Iltenor IS the convcctively unstable convection zone. In thi sregion , the differential rotation with latitude is similar to tha t seen at thesur faee: so the contours of constant rotation are nearly radial. Only atlow latitudes do we see something like Taylor columns (see Section 7.4).By .contrast the radiative interior beneath the convection zone appears tobe III a st ate of nearly rigid-body rotation, to the exte nt that it ca n bemeasured at present using helioscismology, Between the two regions is alayer of strong she ar , called the tachocline, There is also st rong radial shearin the region just beneath the surface. The hclioseisinic findings and theirtheoretical interpretation are revi ewed by Thompson ei at. (20(J3).

.Young stars are observed to rotate much faster than the Sun, and it isbelieved ~hat l:itar~ lose angular momentum from their surface layers throughstellar winds . TIns loss is only communicated to the stellar interior if thereare ways to redistribute the angular momentum inside the star. As we sha llsee in. ~haPter 4, shear in a flow induces inst abilities (Sect ions 4.3, 4.4 ), soa ~ufflcJently stee p rotational gradient would become unstable . Turbulencemight then transport the angular momentum. Magnetic fields via t heac t ion of Alfveu waves , may also redistribute angular momentum: A weakmagnetic field may indeed be responsible for the nearly rigid rotation ofmuch of the radiative interior and may also stop the spread of the tacho clin egradient further down into the Sun.

Including magnetic Lorentz force j xB and a viscous term 'D, the mo­men.tum equat ion in a frame rotating with steady angular velocity no(which we take to he the mean solar rotation rate) is

au _ 1Pat - - p(u · \7 )11, - \7p + p\7iJ> - 2pnox u + - jx B + V. (3,44)

/10

As usual, 11, is the residual velocity in the rotating frame. The total angularrotation rate is

for some scales of motion in the t urbulent con vec t ion zone, breaking th econditions for t he Taylor-Proudrnan theorem to apply.

It is po ssible to model t he rotation in the convect ion zone using nH'an_field models (d . Section 5.3.2) but kn owin g how to prescribe the Reynoldsst ress es from the mean -field velo city is a d ifficulty with thi s approach . He­cent large-scal e numeri cal simulations (e.g. large-eddy simula t ions ) ca p­t uring some of the turbulen t nature of th e convection ZOlW , can producerotation profiles that are q ua lit at .ivoly simi lar to what is bein g found illhclioscisniology (see Thompson ci al. 200;l for a review).

3.7 Binary Stars

Theory of Ro/. n.t.ing Bodi es

M,II

II

L, poin t

L3 point

x

Many stars are found to h c in binary system», Tho orbit» of s1.;\.]"s in a

closo bin ary system tend to become circ ula r ovor ti rno , due to t idal foru 's.

Consider a binary system in which the two components an' in circular orbitsabout t heir common centre of l WI. SS 0 , and in which nw two star s corotat( ,so as to always show the same side to the other st ar. In this sys te m th en>is a rotating frame in which the st ars are completely stat ionary . If n is tlwan gular veloc ity of each sta r abou t 0 , in an inertial frame, then of coursen is also t he angular velocity of the rotating frame.

Suppose that t he separation distance between the two st ars is 0 , that.

t heir masses are IH 1 , A12 , and that their respective distances from 0 are110 awl (1 - 11)0. Since () is t he cent re of mass,

1' 1 I' t.wo s \'l rs wit II ll ,a SSI ' S 11 / , a ' \I' 1\ / '2 a lllll!!..J 1J ' H, d w 1',,11'1I U1 " , .., . . , I " ' \ ' I

I" . J 2 1\ <;111. 111 rOlll!.I " ) , , I ," 1 ' 1 . til", l)lIsi t.illll S illd lCa l ." '," , >.5 \I '• If!; . .) . . 'I'll ~ s1nrs 't n ' 0( , \ ,f .( ,l · , . . ' .

III<' li tH' join ill!!. t h e two ~1ars . . .' ..' " "1 " / L.) a lld /' :J, a l'l' La p;rHll)!;iali jloJll1S,'. , T h e s tat iOll a ry p oint.s , t""hL<1t" , '" ' 1 , -lilies.

') ,() 0) q, can h e writ t ()]l from Eq. (;P» as(-(1 - II (I " ,

- GAl (;A12 ---;-:-

q) = ( . )" -I 1 2 + z2) - ((,:1: + ( I _ /I)a )2 +)/2 + :0; 2 )( :1: - flO - - )/ .

1 2 2 2)- - Sl (x + y ,2

Also the gravit a t ional force on star .l towards star 2 (and hence towards

0 ) mu st be equa l to 1111(p.a.) rl2 , since 11.0 is the rad ius of it s circula r or bit;hence it is straightforward to show that

Now Eqs. (3.7) and (3.8) hold for this sys tem in the ro tating frame, wheret he gravitat ional poten ti al 1/; is given by t he sum of the potenti als due to thetwo stars . Choosing Cartesian coordinates (x , y , z) such that the ang ularvelocity of the frame is in the z-directi on , wit h th e stars at (p,a,0, 0) and

Ii

G( M 1 + M 2 )

a:)

(3.51)

(3.52)

I. r· 1 Here we have llHlde the same ap pr ox-whieh is c<,\lIc,cl the Ro che po e,n]',la 1.'1 " t' can usc Ow ulldistorted gravi-,

. . S t' 3 3 name v . I ,). , we J . ,

ima!.IOlI as III , ec .ion " , .... . , ] '1(' for cpntrally cOllclcnsed. I f ,I '1. ar: t Ius 1S reason a J ~ ,' •

tati ona.] potent1a 0 eac 1 s ·< . ." . trat ell nC'IT t he centre.. 1. f 11 e mass is conce n. c . • " "

stars ill w1nch most 0 , 1 , .' : ~ : , 1 " fun ction of :r :1!OUg the linoThe n oell(' potent.ia] (3,53) IS 11l1lSt.1 at.er as a . t ars . Fi e 'l 2

. ]. . ., t he c('nt n 's ol tw o s m 1', III h ""

() . '11ono the me jOllllllg . ~ , T Iy = "',' = , l. ( ~ . , . h " . £ I ' , \7q) = 0 are indicat fld. . , lest'. . t s £ £ 2 and :), w lPl fl ' , .

The LagrangwJI P 0111 ' . I , , .' Ir , whe re t he for ces of attraetJOllare e(lllilihrinlU points 111 the rO,t a.t mg l ame, r o in l , ]'II('C

t -if ., 1 force an ' III Jd a . . 'towarcls the two stars and the ccn n uga . ,

. ,1 Lar the surface of each st ar in the b inaryA . ' 1.1 case of a sing e s ar, ' ia] f

S III ie " . . 1 I th e surface p ot ent.1a 0' " _ .fac e of cons t,ant <I> . Now PIO\,]( ec . , ' . ,. ,'. , ", .

syst.elll is a 81U a ' . ' . . t1 ' £ l'1gTang,-ian poi nt, each st.atis l II I the 1)ot enha1 q> I at . 18 I .sc

each staris css .1a · . ~ " ~ " ' , ..Jl] , 1 '1 stars form a dctacJwd binary. , ]1' t-] n chc ])otcnt1a a uc \. 10 S ."OCCllpleS a.we III . ie .0 .

I . " (Fig 3 3c) It is1 bins . " r "contact nnary . . .• , ."common enve ope mary a . . . 3 4) '

we have a . t lot contours of constant (j) in the x - y plane (F Ig. . . .. ,t l'Uct lve 0 P . t l .tar canalso 1115 . . 11 l : Roche lobe is the maximum rcgJOn . ic s ..

shaded reg lOl1, ca e( a. . , . . . . 'the I"t it ar ts to lose mass to It.S compa.nlOn.

py belorc 1 s (> ", • •

aeeU

/cs iroptui sica! Flui d Dy nam ics

M2 Ml M2 M1I I I I

I I I I

~)'t~ ' ~lf"

T heory of Rotating Bodies53

".Fig. 3.3 Cuts throu gh the R oche potent ial of st ars along the line joini ng th e two starsillstrat ing three cases . (a ) The two stars (ind icate d by hat ching) form a det ached binar ;system . (b) One star has filled its Ro che lobe an d is now losing m ass to its companicJIlstar. (e) The two stars form a cont act. bina ry in whic h t he two stellar cores oceupy iu aCOIll IllO U enve lope.

Fig . 3 .4 A contour plot of t he R och e potential of two stars , in a pl an e co ntaini ng t hetw o stars. T he Ro che lob e of the star on the left is indicated by the ha tching.

system (Fig. 3.33) . Suppose though that 11,12 expands (perhaps at temptingto become a red giant) until it s sur face potential is equa l to (j) t. , (Fig. 3.3b) .Any further expansion will cause matter to fall from star 2 to st ar 1, sinceit will fall to the lower potential. Algol is an examp le of such a bin ary.Fina lly, if the sur face potenti als of both st ars are greater than (j) L l , t hen

T······ .·· '· ··· ·•• ••

Chapter 4

Fluid Dynamical Instabilities

It. is hardly possibl e to st udy astrophysical fluid dywlIlli cs wit.hout «onsirl­

ering fluid dynamical instahil iti cs. A fluid flow, either in reality or ill a!Hodel, may be un stable, in which case perturbati ons nmy grow qui ckly andchange the fluid configuration awl it s flow. Fluid dYlIami cs involve var iousill st .ahiJiti ( ~s , which may profoundly afl'I'Ct !11(' s l.I'1 I('I.I1 ],(· alld r-volut.iou orHst.]'ophysical obj ects ; A well known ('Xallll' !r' is t.hl: ('01lv('d i\,(' insl.ah ilit.ywhich lead s to convective cores and con voct.ivo onvolopes in runny stars , 1.1]('forlllnr aff(~cting their nu clear evolu ti on and tho laH( ~]'l eadiJJg to a vnri­cty of phenomena including magneti c activity cycles in th e Sun aw l other

lat e-ty pe stars .T he top ic of fluid dynamical insl.au ilit.ies is a large one and we shall only

(lisc:uss selected inst abilit ies here. vVe mention a few others elscwhoro ill thi shook: in particular t he J eans instability in a self-grav itnt ing flnid is t]'( ~ atcd

in det ail in Chapte r 10. Excellent further reading on fluid instabi liti es arcthe hook by Drazin & Reid (1981) an d , particularly for rotating st ars, the

review by Zahn (1993) .

4.1 Convective Instability

4.1.1 The S chwarzschild criterion

Convection plays an impor tan t ro le in ste llar interior s and planetary at­mospheres . Consid er a fluid at rest with densi ty st rati fica t ion (i(z ) andwith gra.vity g = - ge z acting "downwar ds" , so z increases upwards. Thepressure distr ibut ion in z is given by hyd rostatic equilibrium. Let us nowconside r what happens if a fluid parcel is displ aced slightly upward s fromheight z = Zo by an amount oz. We suppose t hat t.he displacement O ('C 1ll'S

55

57

(4.9)

(4.8)

(4.7)

(4.6)

(4.5)

dinT > 1 _ 1dlnp "(

Flu id D ynamical Instab il iti es

for a perfect gas (see Section 1.7)

becomes

• ", .r .. ,. f 'eque nc" one root leading( I · ,I ould imply oscillations WIth nnagmar y I J '

W II C 1 W , ilibri )t ial growth of the disp lacement from equui num .to exponen I'

If t he chemi cal composit ion is uni form , t he n

ln p = In p + InT + constant

and so the instability criter ion (4.4)

(1 dlnp _ dln p ) = pg

2(dln p _ ~) .

N2

= g '1 dz dz p d In p "(

d .f . . .'1, ad had b een defin ed to increase(Recall t hat z increases up war S : I illS ea z . , T .sed .)

1 ·1 " , of the two deri vatives in z would have b en re verse .downwan s, t re signs .' . 1 f ' ' N Thus< ' t ion (4.6) descr ibes simple harmonic mot ion WIt 1. l eque.n,cy,., "~qlla.. " 1 can oscillate in the vertical dir ection about Its equilibrium POSI­t .le pa~ cIL f. . N whi ch is known as the Brunt- Viiisiilii fr equenc y ortion WIt 1 l equency , . I '1, aves in a

f . This is the mechanism for int ern a gravi y wlJ'Uoyancy requency.stably stratified fluid (e .g. Sectio~ 12.4).

The instabi lity criterion (4.4) IS thus t hat

. E ' .t ly the same crit erion. ith de t h to be stable to convection. xactr 'rapIdly WI p iderir 1,1 e parcel moving downwards inst ead, t I.lOu.gh11 ' ult fro m consI errng 1 . . IWOll C res I ' t o reverse the inequality when dividing hy t iemust then remem)Cl .one

(negat ive) e5 z . " f ti. C' convec t ive ins tabili ty is thepotellti al energyThe energy 8 0 m ce or . I , . . . ,. .: " mi unstable strat ific aticlll. . .

of th e Ollgn . , . ' . ·t·' 1I ' t . t he above cr it.erion then t he accelerat ionIf the strat IficatIOn IS S a ) e .0 , .

of the parcel is given by

2 .1: ( UP)~. u Z = J + 8z~ 9 - (p + (j p) gP dt2 f UZ

l' '." ' . . wei,,·ht ) where we now conside r oz to be a function(blloyallcy 01 LL llllllllS o ,

of time: and hence

whcre

(4.1)

(4.2)

(4.:3)

(4.4)

1 d Inp<

"( dlnpi .e.,

Po +6p = (PO +6P) ' .Po Po

Hen ce , linea ri zing in perturbation quantities , th e density perturbation oft he par cel is

op = .f!.!!-.op = Po e5z dp ,,,(Po ' ''(Po dz

T he p ar cel finds it self heavier than its sur ro undings and hen ce sinks backtowards its original loca ti on if

i.e.,

sufficient ly slowly that the fluid parcel rem ains in pressure equilibrium withit s new surroundings . On t he ot he r hand we suppose that t he displacem entocc ures sufficient ly quickly that no heat is exchanged between the parceland it s surroundings, so t he properties of the parcel change adiabaticall y,The pressu re and density at the ori gin al posit ion z = zo are Po and Po , Bay.

At the new position z = zo + 6z , t he pressure and density of t he parcel ar e

Po + 0]) and Po + op, say. Now at z() + 6z the pressure of the surroundingsand hence also of the fluid parcel is Po + 6zdp /d z to first order in oz ; so

Jp = e5zdp /d z. T he density of t he surroundings is Po + ozdp/dz . But lJythe adiabati c assumption, the parcel's pressure and density perturbat ionsare re lated by

i.e., if

.• ~ ". Vl''' ysu;at Fttiui Vynam ics

dpPo + op > Po + oz dz '

1 dp 1 dp-- > - -,pdz p dz '

(since dp/d z < 0). Converse ly, if

1 d in p> --

"( d lnp

t hen the par cel finds itself lighter than the surrounding fluid a nd hen cecontinues rising. In the lat t er case, the original strat ificat ion is un stable

and fluid parcels will mo ve around, i.e. the st rat ification is convect ivelyun st abl e , Note that t he density can incre ase with depth (eq uivalen tly,

increase with pressure) and still be un stable: it has to increase suffic ient ly

~dS 1 d]J df>c]J 'Y ]J P (4 .16)

where cp is the specific heat a t cons tant pressure. Tl len from (4.7),

N 2 = go dS .c]J dz ' (4.17)

th eIlt~r is stable to convecti on if the sp ecific ent ro py increases upwards.

\7 \7 ~. ~ad the . tem perat nre gl:adien t is said to be 8upcradiabatic; if. < 11 ad :1, IS .'~'U, ba dZQ.bat~c. Convective motions transport hea t : comllJonlyIII ste n.T interior regions that are convectively unstable c t i .effici t i t l . . ' , onvec lOll IS very

cien III iat It requires only a very small superadiabat ic gradien t \7 - \7ad

\7 > \7a d

for the fluid to be un stable, where

\7 == dIII l' n _ (d In 1') 1and v ! 1d lnp a, = dln p s = - 'Y (4.]] )

Eq\l~ti~n (4.g) is Lh~ Schwarzschi ld cr it erion for conveetive instabi lity.

I1\101 e ~eneraI1'y, If t here is a vert ical gr adient of chemical COlIlp()~ i l ion

C mractenzed by , ,

;'9Fluid Dyn am ical Inst abilit ies

to t.ransp or t t he st ar' s entire heat flux. III that. casp 'V ~ \lad and N 2 r-: IJ

t.here ; also S is nearly constan t.]\,lost stellar-structure modd liug nfiCfi a s imp k- plll'uOUIl'lIo]ogic.<1 I d(,­

script ion called l ni:J:i n y- lr:ny lh tJU:017J t o ca lculate t.ho ]wa1 t.ran spor! 11yconvcction and hen ce the s trati fication re quire d in «onvoct.ivclv uns t.ah loregionfi in order to produ ce the necessary couvoc.l.iv« heat flux (l'.g . Kip­jleuhallll & vVeigart 1(90). T he idea is th at b lob s of convectol fluid travel adist.auce (1 from thei r posit ion of equilibrium and thon disrupt a nd d ispers e

into the IIl'W surroundings : f' ifi th e iuixing longt.h. The rnixi iu; !l'ngt h liasto he proscribed . In stollar «ouvectivo envelopes it is ron unoulv P]'('filll lll'dto he a fixed con st ant t.imo» t.h« local prossu ro seal<' I I( ~ i gh t: th o value o r t. h( ~

fixed r-onst.aut (th« niirinq-lcuqth. 1J(l'm:m.!dcl ·) ca n hi) adjusted fiOafi prodllcpe.g. a solar model or the CO!Ted. radius, or course, xinco there is JlO ]'('alth eory involved , we do not know t.hat 11)(' ini xiug-longf.h paramcte r should

he the same for different stars . Other proscriptions of 1J)( ~ mixing longth

are possible. In order to lise thi s mixing-length thcorv iu modelling stellar:-:: t r llct.urc, it is necessary t.o cal cu late tll(~ fi lW(~d aJ whi ch blobs 1I100'e andthe aIIIOll11t of heat they transport. 11. should !Jo)'J1l' in mind t.hill. t.his is a ll

fairly crude an d t hat the adjustment of t. ho rnixiug-lengt h parameter Ul k ('R

up the slack left by incxac t it.ude in the argume nt. T hus l.he spee d 71 of

th e blobs can be calculated from the equa tion of moti on (!J .G) (whore N 2

is negative for an unstable reg ion) assuming at its simplest t hat the blobstarts from rest and t ravels a distance I! at constant acceleration. The con­

vcct ivo heat transpo r t effect ed by the motion of the blobs is fJvc 1'6.T where(;1' is t he sp ecific heat capacity and 6.T is the te mperature excess of a blobover it s surroundings . Sin ce 6.1) = (J (tho blob is in pr essure equi libriumwith its surroundings}, the te mp era ture excess can be wri tten in tenus of

th e superndiaba t ic grad ient divided by the p ressure scale height , multipliedby th e dist ance f. travell ed by the hlob . Since it t urns out that v is p ro­

portional to the square ro ot of t he supcradiabatic grad ient , t he convectiveheat flux is proportional to the superad iahatic grad ien t raised to t he power

:~ /2 . As already stated , in deep convective envelopes of stars it turns outth at the sp ecific heat capacity is large en ough that a lilly snp('n l.dia,]la1.ic

temperature gradient is sufficien t to give 1.]10 nec essary convecti ve heat Jinx,so that N 2 ~ 0 there.

f!!r[[;

ti,

(1.12)

(4.1:3)

(4 .14)

Astrophysical F luid Dynamics<'>0

which is often wri t ten

N 2 = goI f (\7a d - \7 + \7/1)

l'

where H p is the pressure scale height (2.70) and

th en

d ln /l- d ]J)]J ~

(5 == (f) In f> )f) In1'p

(0 = 1 for a perfect gas). Then the criterion for instability is

\7 > \7ad + \7'1 ; (4. 15)

thi s ~s the Ledoux ~ri t~rion for conv ective inst ability (see Secti on 4. 1.2).1 he Schwarzsc:1nld mstability criterion for a region f if

sit ' , I 1 . ( " 0 Ulll on n comp o-, tl~n caIn a so Je expressed in t erms of the gradient of sp ecific pntro l)11 Sno mg' t rat " , , . ,J ,

4 .1.2 Effects of dissipation

Convect ive instability is a dynam.ical process: it do es not require a dis­sipat ive process , and can ther efor e be treated in the adiabatic, invi scidapproximatio n. As established by Raylei gh (1880) , dissipation modifies theinstability cr iterion only slight ly:

iIIIII

(4 ,19)

(4.20)

(4.21)

pi 2_ 9 + v\7 uo

(Jez

. u = radiative exchange t erm

1 ,- - \7p

p~IU

_u_ + u . \7u =oj\7 ·u =O

~ + u·\7T'at

H )( \7 - \7 ) is t he so-called superadi­(c.g . Gough 1977 ) where (J .=. (~{ IBoussine:~ approximat ion the pi / pinabati c lapse rat e. Moreover , m b l~TI I T where 0 = - (8 1n pI DlnT )p, i.e.the first equat ion gets rep laced , y ." 1 t o temperature fluctuations.

. t' 'e neglected compaJe tlpressure ftue:tua .ions 31 . . itifi ed j 1 labor atory convec-. . . tion can be JUs lIe n

T he Boussinesq approxlllla , d d :ltv are long compared1 . 11, f wessure an ens J

ti on , where t he scale lelg 1 s 0 ]d . ,t d and also in some geop hys ical1 f 1.1 fluid layer un er s u y, fi 11

to the dept 1 0 Ie . 11 . ct ion is not really jus ti a ) e," I . , rcat ion to ste ar conve . , .

applicatiOns. ts app 1 I,1 d mav vield some mSlght Illto. Id 'tractable pro ) em anJ J ,

but it does Yle a mOJ e . ) for a discussion of st ellar convect lOnthe full problem , See SpIegel (1971and the Boussinesq approximation .

3 l\!l odellin g convection: the B07l.ssinesq apprm.:imation

4. 1. . . . " '. veet ion we men tion a com111only adopted ap­Before leaving the tOpICof con ' . l ' , .t: hk j"(.,,'ions , T his i::; the Bons81,-

. ' . . o(kllillg convedlve y uns a ' -o " .]lrOXllllat lOll 1.,1.11.1l ' . i1 '1 ' I ' . of ])ressure ab ou t Its hori zon-. I ,t pi 1)(' the ur: .ua .ionnesq (L1I1Y1 '(),:I:i'ma~,uJ'li . J(' 'f' ,t'l " 't'll('l'oJJlodv nmn ie quantiti es. The ve.locity

1 . inilarlv or () , W I " ' .J • II 'ta1meall , anc Sl c • ' 1 ut 'I l'('f'pJ'{ 'ncp st.ate'. in whic 1 t ie ve-

' l 1 Ii duat l0n ano c c " , " .u is also COllSl(. ere( a . u " .., . t ' t11(' deusity fluctuations are

B ' " e::;q a pproxnlla ,lOn . , " .locit.y is zero. In the O UHS lll ., . , . . tl '-' eCluat ion of mot.ion . Specifica.lly,

1 . t 1 ' bur yancy t.erru III .ue " , '1retained on y III .ne ),' I. ', ' . . 1 ' II tile continuity equat iOn . 1. Ie

" fi t u: ti ns an' Ig1101CC I ' .t.hen , the rlcusity uc ua J( ~ , . , ' . t ' (Gough 19(9) and filter s out

1 " ,I' 'h e approxnlla .ion .rlat.ter is called t ie ane as , . , 1 " t hat may h e considered

. , , 1 >' ..ena such as sounr Wcl.ves ' " " . ,higlJ-freqn<'.!lcy p lenOlll , >. ' " f 1.1e flow. The set of equat iOns fOJ

. 't. t for t ransport pr op eJl.lcs 0 I ,unllupOl ,d ll . . . ~, , . 13 .:, 'q aj)jwoxnnatiOn ISHuetuations in the oussmes

Fl'uid Dynamical In stabili tiesGl

t I I ' 0 tilt'., ' . 1 e thermally IInS ,a ) e ;,; ,Tl't'll' r st rat IficatlOn may ) , ' . ' 1 '

last example. . le ~,c a ' . r insbbility is sat.isfied , hut stahle overal so

Schwarzschild cn ten on (4.10) fo '. c 1 :l' t, " not satisfied - T he result ing. ,' . , ,' (415)formsta Hl ,yls , " . ' .

th at the Led ou x (.l It o iou i -i . , . • : _ . " Ivec [iou in ::;t.(;l1ar as t.rophysll:S ,. t.ime ' referr ed tu as ,%UH <-(I I , 1

motion IS some ,U IlC;,; Z'1 (1993) 'mgnlar JJlOJllcntullI w ay abo play I. Je

As point ml out by a 111. ' . ' c ' 1' ff ' " more slowly than heat ., . le t o the sa lt, smce It too (I uses .

ana logouS 10 e '

\!

(4. 18)

'As trophysical Fluid D yn am ics60

Ct ytc!

for inst ability. Here c: is a positive constant of order unity, and t ; and td

are the dissipatiou tiiuescales associated with viscos ity and heat diffusion,respectively (see Zahn 1993). Const an t c: dep ends on the geomet ry of thefluid reg ion and on boundar y conditions . Insid e stars, t d is shorte r than t ;since th e therm al diffu sivi ty r: is gene rally mu ch larger than the viscosity 11 ;

t he Prandtl number PI' == 1/1'" is est imate d to be of order 10- 9 to lO -G in

the Sun: see Lignier es (1999) . In the convectively stab le regions , diffusiontends to damp oscillations. W hen t-: ("-' l2I ",) becom es compar able withN - 1 , where l is a characteris tic parcel size to b e damped , t he ent ro py st rut­ification is no longer effective in stabilizing the layer , t hough composit iongradients if present will st ill provide a restoring force,

Interesting competing effects (" double-diffus ive insta bility" ) occur in a

layer whi ch is dyn amically stable, i.e. N 2 > 0 in (4.13) but eit her thecomp osit ion stratifica t ion or thermal strat ificat ion on their own would beun st able. T he archetypal laboratory example is water heated eit he r from

ab ove or below a nd wit h a gradient in salinity. Heat diffuses much fas tert han the salt concentration, If it is t he salinity gradient that is destabilizing(sa lt water on to p of fresh wate r, but wit h t he top of the layer hott er thanthe bot tom), th en small-scale perturbati ons for which td '" N::' can grow ,producing so-called salt fingers . Eventuall y t he st ratification becomes alayered convec t ion with both temperature an d salin ity varying stepwise indepth _. see Zalm (1993). The opposite case is that of salty water beneath

fresh water, wit h a n othe rw ise unst abl e t emperature grad ient whi ch ca n beproduced by heating the water fro m below. One can envisage a displacedfluid par cel oscillating but coo ling down (due to the sho r te r timescale t c!

when it is above its equilibrium po sit ion a nd hea ting up when below theequilibr ium position. This causes the veloc ity at which t he parcel passest he equi librium level to increase and the amplit ude of t he oscillation to

grow. This is overstability: t his par ti cular example is call ed thermohalineconvection .

In stellar cores, helium may play t he analogous role to the salt in the

62A strophysical Pl uid Dynam ics

Fluid Dynamical lnst.oh ilit ies

4 .2 The Rayl ei gh-Tayl or Instability

F ig.4.J The se t-up for the R ayl eigh-Taylor in stability : one flui d of (un iform ] dCllsit.y PI

ovcl'ly illg another of densi ty PL' T h« gravitatio na l accelera tion is 9 :' Ild z is th o ver t ica lcoord ina te (he ight) . The dasl wd curve rcpl'esell1.s th e per tu rbed illt.erface between UJ('two fluids.

. . I 1" ) 2 < () ami so w is im agi nary: on eif > (lH" IVWr HUH Oll .op w 22) 1Thus 1 PI fi 2 , . . . , li ,tlly I-';J'Ow i lll-'; sollJt io1ls (-1 . , , :UH

. 1S ('o],J'l 'SjlOll<!S t o ('XjlOllUI . , ,, . . 'Z .of I.II(~ two 10,,0 " is uu sl.ab l '1'1 . ,., 11 11' HO:I/1I'il/h ,. 'lhyZ 01 ' 1.nsfab1.1.fy.Iie " j ion IS 11JJ s1.a1J ( ' , IlS IS ." , , 2 kSO th« COll 19m" " . . _ () th en W( ' ]'()cover w = .1/,

• • . . ' I)..'; that 1] W() put fi l .- • IWe note II I jJ.lSSIl " . ,/', ., ' " ,' 1\, waves that was deriver. ' ] r .rorsion relation ]01' sm dH )..'; 1.1\ I . . ,

whiol: IS t.1e I IS] . , ' ," , , '" j t - , . , J', 'S th a1. dispersion rel ation to' 2 (' F X])l'( 'ssJOIl (4.2,~) gUICl ,11Ze.ill Sed, lOll ,J . .J ., ,

ifor density upper layer. f ,illcllJde a JllU orm- , , '. . (C' t ' A 1) the ('llergy SOlJ]'('e 01

ti ins tahiliIy ,~p c ,lOll ' i . " . .. .

As with the convcc 'lve , . ' ". ]" '1' 1'.1 en ergy sto red ill Ow initialthe RayJp igb-Taylor instability IS 1.ic po .( Il .1,1, . .

cOIIfignratioll.

. . S t' 4 ,j so rather than. .cneral situat.ion III , PC .1011. " , , .We shall analyse a lJlOl c g , . . "j I , . , 1(' 1.1H' rr-snlt : {roll I

" ., I al 'sis w e Sl It l]') y l(]( . q110 . . .,oIlCat. the mathcmat.icn au ) Ii I t]I 'lI. f"r irrol at ionnl . 111 ..

I" " , ' I [I, ' (' I. 1,,0 zer o, W I' Illl . , ,Eq. (,1.:18) WIt h U

1.1 ]]( 2 s " .. I f .(,( 1H 'I W \' w 0 (' t hr: ]H'rl.llrl. :I!.lll1l' " S'I]I]r, p ort.nrhnt.ions t.he t l'llJjlOJ ,' J I " ."OJJl I)1 LS, ] h 1

" "I ' "C' Jlor i ~olll.al WaVPlJlJHI )( ~r ., l,i'is rein1.CI ,0 I ,." " ,

w2 = (fi2 - fi 1)gk,[J2 + /) 1

R ot ation al I nstability4.3

e of possible ins tabili ti es . Som e arc. . I I Ie new rang ." .Hotation intror uces a W 10 . , , . le . ' I ill the nex t section, For

1 .' I 1 r and shall he COnSl( E1en m rn e 1assoc iatec WI ,; 1 s lea , . L ' . 1. 'J ~ might. envisage in t. lC

idor lv a simple scenario , w 11L J W ,. .now we cousir cr au " ' . " "t, q varios on ly wl11 1 1.1](', . ., . , r ' n which th e rotation ra ,( L , " • • " ..

mtcrror of a s tar , sa) , I ' I " li: 1 coordinate JlJ a cylinch ltd ]f til ax is (so to IS 1 JP Jd( 1<1 ,, ' • •] Idistance w rom . e , ." , 1.1 . t Ifects of viscosity arc negligi ) c.li . 1· ' ' t ) \Ve assumc , M, e .··c " . , 'polar coon md ,c s~ , . " . . . 1,,1 ' oquilibr ium config urationI, " led grav ity so m , 1C\Ve also for sim p JCll;y neg , , '. ,pressure and cent rifuga.l forces balance:

1 ip (4.24).;-(-"" + wn2 = O .pdw

. . ) undergoing a sm all ra.diaJ d isplacelncn1. fr:)~nConsider now a pal ceJ of flul< . I' '1 1 (1 ])'Irc('l COllS('I,\,('S it s spcClhc

s: S· ", ,, , 'scosity IS ncg 191 ) C , . Ie , , ' . . 1w t o w + uW . , mee \ I." . ' 2' 0" "tl tl l' hand the pressure for ce t l,e

t I - w n n le 0 . 18. , ,an gu lar m0111en 11111 I = . . " . d ,t" . . led iJv the angu lar vcloc lty. ' " ". rrollndmgs IS I' ,C11J1 ]] "parc~1 feels Jl1 Its new Rll I . 1 I"11('e of for ce per unit mass. " E (4 24) Hence t Ie ml ) f\. « .,th ere , accordmg to q. . '

(4.22)

2 = 0

fJ ,

exp (?:kx - iwt) .

Several instahiliti es can OCcur a.t inte rfa.ces hetween fluids , Consider two flu­

ids of uniform (hut different) density with a pl ane interfa.ce between them,

with a uniform gravit at ional field perpendicular to th o interface (F ig . 4.1).If the dens er fiuid is on top, (so Pl > P2 in the notation defined in the fig­ur e) , the Rayleigh-Taylor instahility develops. This m ay see m an unlikely

configura t ion to occur in nature, but 9 can equally he an ef fcc tivc gravita­tional acce leration, e.g. at an accelerating shock front a nd this sit ua t ion can

be found in Rupem ovac , for exa.mple. The R.ayleigh-Taylor instability com­

mon ly occurs at the same t ime as the Kelvin-Helmholtz inst abili ty , whi charises from veloc ity shear between the two layers : the Kelvin-Helmholtzinstabili ty is discusRed in Sect ion 4.4.

To underRtand the develojlmen t of the R ayleigh- Taylor instabili ty. wecons ider wh at happen R if there is a. sm all perturbat ion t o the interface.If the p er turbation grows then the configm at ion is unst able. Since th e

background configuration is tran sla.tionally invariant hori zonta lly (F ig. 4,1),we may without loss of generality consider an individual Fourier compo­nen t , in the x-d irect ion say, so with x-dep endenc e e i h . Likewise the time­

indep endence of the background means that we seek a t empor al variation of

the form e-iw t

say. vVe suppose that any velocities, pressure varia.tions, etc.a.rise only from the p er t urbat ion from t he interface: hen ce all pert urbat ionvariables will be propor t ional to

\Vitho\1t loss of generality we take k to be positive.

64 A stroph ysical Fluid D yn am icsF lu id D ynam ical In stabilit ies

65

. .Rather. st rong ~ l j fi"erelltial rotations are requi red to trigger this iust a­blh~y aJ~d 111 practi ce other instabil it ies would usually set in sooner (Zahn1(93). Such a shear inst abilit y is discussed next.

z =()

pz

'vVe have already seen that in the case of no horizontal velocity such aconfigur ation is subjec t t o the R ayleigh-Taylor i ll::;tahility if t he fluid ill topis more dense than the fluid below (PI > (J2 ) . 'vVe shall now see that if th ereis shear hetween th e two layers (i.e. V J "I [h ) t hen t his fur ther de::;tabili ;,::esthe configurat ion, wh ich m ay then be unst able even if the lower layer is t hedenser one. Put anoth er way, a sufficiently stab le density st rat ifica t ion is

necessary to overcome t h e shear inst abili ty.As in Section 4.2, we consider a pertur bation z = (x) of the interface

of the form

Fig. 4.2 T he set -u p for t.h e K elvill-Hehn ho lt z illst abiJ ity : one fluid layer flowing over"Hother. T he upper fluid la y e r has densi ty PI an d hor izontal speed 'Ill , t he lower layerhas density P2 a wl horizo nt .u! sp eed '11 2 . For other det ails , see t he capti on t o F ig. 4.1.

ItIlI(

II

II

(4.25)

(4.26)

(4.27)1 d ( 4 '.- 3~d w O~) > O.co to

Shear and the Kelvin-Helmholtz Instability

( { W20 (W) }W + SW) (W + ow )2 - (W + OW){O (W + SW)}

= _ 1 d ( 4 2) - ,)- w :J dw w n Sw == -Nfl oW

to O(5rv). T hus the equation of moti on of the parcel is

d20rv 2dt2 + N0.(\w = 0

in the rad ial direction felt by t he parcel after it s displacement is

4.4

~V I J iC,h gives astable, oscillatory motion if Ng is posit ive but an instability

If .N~ is w:g.aLi ~(; . As in the convective am i Rayleigh- Ta.ylor illstahil i tie~ ,thi s .instah ility ~s a dyn amical instability . T hus for stab ility the rot at.ionpr ofile must satisfy

We shall consider incOln p ressible, irrot.ationa] per turbations in each layer ,so that the small perturbations u ' to the background velocity are express ible

in terms of a scalar p ot ential 4>:

~~10ar, ~ I~stabilities, whi~h Cal~ occur when a fluid 's velocity is not uniform,ale ve.l} eOl:unon. ThClr mam featur e is that they ext ract vorticity from~. la1llll1ar .(I.e. non-turbulent ) flow and t he vor tices can then grow andIIlt~ract WIth one another . An excellent reference is the bo ok by Drazin &Reid (19tH ); Zahn (1993) also provides a good reference. .

l~~re we rest rict our selves to an analysis of the Kelvin-Helmholt z in­stability, and some general comments about the onset of shear instabili ti esand about turbulence.

( = A exp (i kx - iw t) -

u ' = Vc/J ,

(4.28)

(4.29)

4.4.1 The Kelvin-Helmholtz instability

T he Kelvin-Helmholt z instability can occur when one fluid flows over an ­

otl~er.. Cons~der. a two-layer fiuid wit h a plan ar interface (z = 0) in au.l1Ifonn gr~vltatlOnal field g = - ge z . Each fluid has it s own uni form den­sity and u~llfo~' l1: , stead y, horizontal velocity (in the z-direction say). Theconfiguration IS Illustrated in Fig. 4.2 .

We shall use sub scr ip ts 1 and 2 to denot e quantiti es in the upper an d lower

layers resp ecti vely ; so , for example, the total velocity u is VI ex + \74>1 in

the upp er layer and U 2 € X + \74>2 in t he lower one.All perturbati ons are deriven by the interface d istorti on . Thus all per-

t ur bed quantities have t he same depend ence on x and t as in Eq. (4.28) ;and they all tend to zero far from the inter face. Without loss of generalitywe take k to be positi ve . T hen since the velocity p ot enti als ar e solutions of

A st rophysical Fluid Dynamics

Dc!)] D( o, ~([) z Dt +

d 'l:ur!J2 u( U

2?(

Dz Dt + (·1.:31)UT

07

(4 .:l8)

Fluid D yn am ical hist.olnliiir:s

Agai n it. is !lO W co nsis te nt to oval uato th is at :: 0-: . () : 11(' ))(' ( '

SlIbst.it llt ing [0]' (;1 and (.'2 fro m E qs , ( 4 . ;~ 2) give'S all c'(jlw1.i(ln that is ]J( )­

Illogmlcous in A : for non-t ri vial (11 =/: OJ solut.ions W( ~ ))IlISt haw' t hat

i.o ..

wher c tr = (PlUI + f!2Ih ) j (PI + f!2) is a density -weighted aW'l'age s!H'C'd .

TIlt' coufign ra t.ion is unst.nhlo if U I(' right-h.u «] side of Eq . (;1.:18) is

llcgati ve, s in ce then w w ill h: 1YC a 1IOll··;I, crO illJ:1gilla r ,v pa r l n11l 1 ou r- of t.l«­t.wo solu tions will corresp ond to exp onen ti al gl'owtl l wit.h timo, If 111 C O' 112we obtain the cri terion (11.2:1) Ior ti l(' Hay leigh -Taylo]' inst.ahility. ]l'lh =/:U2 , we see that the configura t.iou is s t.ablc only if th « first (st rat ificat ion)term on the right-hand side of (4.:38) is larger than the second , shea r , term

which always act s in t he sense 10 dcst.abilize the sys t em . Dist urbances

of suffi ciently sm all wavelength (high k) ar e always unstable if U I # [12 ,

though this is not t rue in a rea l layer of finite t h ickness, see Draz in Sr lh'id(1981) . ote that the 0 term 01 1 the left of Eq. (4.38) is nicrolv oquival r-nt.

to a Galilean transformatio n in t he a-direction wit h speecl U,If the shea r cau ses the configuration to be unstable by making j:]ll ' rigId

of (4.38) negative, this is called the Kel vin-Helmholtz insta.l! ilily . Not«that. we have only pr oved a sufficient condit ion for stability since wo havo

considere d only a subse t of possible dist ur bances, namely irr otational ones.

r··

....·

..

··

r ..,.J .__'~"-, <

II

(4.:~ ())

(4.33)

CJ exp (- iwt + ik» : - kz) ,(,'.2 oxp ]- iwt + i k:7: + k z )

Lap]ace 's equation (4. 29) , one ca ll im nH'diately say that

at t he interface Con·c'c·t l' f "l ' ] ., Ir ad firs t-orr j ' . ,. ' .0 II S '. orr or, sin ce terms in these equations aroarready llst-ordc'r sm all " 1'1'

. " ' , (JU d.I! :1 .ics , we m av evaluate Ec ~ (4. 31) ..t I' 1. ' j' I .' c , " . " 1,0. .• a , t.liounper U1 )ec. surface loca l.ion z - 0 1'1 ' E ( . 'dedu c 1·1 1 ',( '- ' . IUS using :J(Jfi. 4.28) and (4 .:30) we.e . la .

-k Cl = - iwA + ikUJA ., AC'~ 2 = - iwA + ikU2A . (4.3 2)

Allo~;)ler condition is that the norm al st ress across t he inter face musthe cont inuous , which here means lhat t.l ~ ' .. . . .' . ' , .N I ie pi essure p must. be continuons

ow 1. ie momentum equat ion ca ll be written ' .

( D ' )v--'E 1 2 1at + v (2U ) = - pv»

which can be int egl' ted tr " ( .a .ec ,0 grvo to [inear order )

for some constants C l and C« T he diffr>I'Cllt. , ' , .. - ' , . ,, - , SlgBS III t.he z-dependell c(~ens ures that ¢; -> 0 as z -> ±oo.

A kin ern.at.ic conditi on is that, Oll either side of the int o ·f·, . ' ar fI .- , ·t · ,] atf.I , " . . . In ,(J d c e , d ll Y IudI.MI

., Ie e , I. , t H: intcrfac « sllrfacc ~ must remain on tho "111'f"lC'C' whir..']1 ..tl I 11 . . - 'J , ,, ." . n)('a ust· . l~ . /c

1vc~·t.rcalcol1J jJolle ll t of t.he velocity I1J llSt match t he mater ial d()]' i va~

.ive a t io in te rface dlHjJlacemen t «( :/:, 1):

(dU j dz )2Ri

4.4 .2 Criti cal R ichardson and R eynolds numbers

\Ve have seen in Sec tion 4.4 .1 how a stable stratifica t ion hinders th e onse tof shear inst. abl ity. In a more general configurat ion, where st rati ficatiouand velocity vary wit h height z, t he st abilizing effect is measllJ'ed l,y tl w

Richard soll Illllnher

(4. 34 )a¢; + U a¢; =at D.T

p- gz + F (I) ,

P

whe re F (t) is " 1. " f '. a cons ant 0 IIltegration . Now since the . I , I 'dep end t t . .' .' " on } ,un c-

t-l .' Cl

fl ' quan ;lt les are the perturba t ions, an d t hese tend to zero fa r frOl ~ l

.ne int er ace we can deduce' tl ' 1. P(t ) ' . ] .• , ' . . ' . .~ , ],l . IS IC en t lcally zero . T hus cont inuitv

of pl essm e at the Illterface Implies , using E q. (4.3Ll) , t.ha t "

- P I (a~1 + Ul~~ l + g() = - P2 ( a¢;2+ u aeP2 + c)v .l: at 2 ax 9 . (4 .35 )

4.4 .3 Turbulence and the Kolmogorov spectrum

where L and U are characteristic leugthscale a nd speed of th e flow respcc­

t ively, and J/ is the kin euratic viscosity . The Reyuolds number measures tl w

re lat ive importance of inertial terms and the viscous t erm in the rnornentumequation . In th e ab"mH:e of other forc es (e.g. buoyancy), a laminar flowbecom es unstable when t he Reynolds nu mber exceeds some cr it ica l numberRl~e ' Generally Re., is of t he order of I000 , hut it dep ends ou boundarycondit.ious of the How and on the particular velocity profile. See Drazin &Reid (1981).

In the a bsence of dissip ation , a sufficient condit ion for instability for a

variet y of velocity and density profiles is that Hi is smaller than a critical

value which in 1/4. If heat can dissipate, however , this will weaken t.hebuoyancy force (see the dis cussion of double-diffusion, Section 4.1.2) a ndmake th e layer less st able; thus the critica l Richardson number is increa sed .See Zahn (1993 ) for a fuller d iscussion of t he iSSlJ().

Another important quautitity determining the on set of turbulen ce in

a viscous How is the crit ical Reynolds number. The Reyn old s munbcr isdefined 1Jy

69

(4.44)

(4.4:1)

Fluid J)ynamiwl In stabili t ies

wit.h H. eyno1ds number Re == 'Uoln/ v ,V>le also introduce wavenumbers k so

t:'1 I - I ko = () .I.'. = l - , 1.'." = ' v ,

. -umber space is called the i nertial ra'nge.. t ·1 k < k < k m waven ut

The IJl erva 0 '". E(k t ) is defined such that the aver ageThe kinetic ener'yy speet1"'lJ.m ..J ','

kineti c per unit ma ss is

11. . .. 1" 1 until event ually at some scale 1II it is

de to smaller and sma eI sca es . .'ca es . ' . t 1 , t he llnid's molecular VISCOSIty I I .

dissipated as heat )J . . t.lu t dissipat ive lcugthscal e l, does notIt seems reasonable to SllP.pose , 1,1· . 0 0 . • . f . . . , [ .] _ £ '2'1'. -:1

, • . 1 Now I he ChllWIlSIOllS () t ,11e t - . ,J 1 but only on E am IJ . ' I' . . . .

depellll 0 11 0 ' • . ' . . . . f lc '1'1 and time: and the C UJIenSIOnSL.» 1 T lonote dnnenslOllS 0 cug , I c 1

where aUC ,( ] . > . fore c (lJ'IIl('usional groulldS we deduce t !at[ ] _ I '21"- There 01 e on ' ,of v arc v - J •

l; rv (1)I/ff /4 . (4 .t1 1)

. . ' 1, ' S ' on scale In presmnahly dep ends only onTl ' kin etIC ene rgy pm UI1l rna s

It. 1 " 1Y dl'IIIC'l'ICj' onal arg ume nts, I . 1, lUS agalll ), . .,In ,1l11 c, '2/3 ('1.42)

'U6 rv (d o) .

. ' re ssjous yilJdsEl

' . l 'lt l'I'lg" E between these two ex]! .. , ' .; 1l111l , . . •

(H ,)--:1/4ll , rv .c 0

(4.4 0)He == L Ull}

A strophysi cal Fluid Dimomics68

(4.45)

(4.46)

In st abili ti es such as those discussed above can rapidly lead ill highReyn olds-number flow to turbulen ce, ill which neighbouring parcels of fiuidat son ic inst an t rapidly follow very differen t an d practicall y unpredi ct abletrajectories . (A How that is not turbulent is ca lled laminar .) The onsetof turbulen ce may be pictured as the flow developing smaller and smaller

scales of motion un til a t sufficient ly small scales molecular viscos ity se ts in.Turbulence is a very challeng ing pr obl em . We shall not consi der further thest ages by which turbulen ce develops, but focus instead on fully developedturbulence. Even there the challenges are formidable and we restrict ourattent ion to t he case where the statis tic al properties of the turbulen ce arehomogen eous, isotropic and ste ady. The basic ideas were conceived byKohnogorov (1941) a nd a classic text in which the t heo ry is developed isBatchelor (1953).

vVe envisage that t he t urbulent How is in a st at ist ically st ea dy stat e .

1 ' ; 1I( ~rgy ente rs the flow a t a rate E in motions on some lengthscale lo. T he

( 'lll ~r~'y per unit mass on this len gthscale is ~ u6 ' This energy then cas-

~ ('U2 ) = j'oo E dk .2 0

. . , .. . there is an upper cut-off to the integr al atBecause of VISCOUS dISSIpatIOn, . " e [E]= L3T-'2 , we deducek = k/l' On dimension al gro unds once more, sine

th at in the inertial rangeE = Cf.'2/ :1{,; - 5/ 3

. . tl at E is indel)endent 10 and L there . Thisc' ' '. stant assunnng 1 . .

where . IS a cons ' .' . " 1 inertial range the energy densi ty inis t he Ko lmogorov scaling, t hat m t~~/3homo gen eous turbulen ce scales as k .