Cosmic Gas Dynamics Lectures Notes(and Lectures Also )

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<' ru ul.iuu u m dl 'snip l i" l1 01' III(' flo w 01' ; 1 lal'/-',I' 1I11 l1l1 lf'r 01' Such a d ( ~ scripli o ll 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 o ok 11](' JI11 id undor co us i(kr a l ioJJ wil l l',e ll('I';ll ly 1H' a I',as, th ough t.J w equations of Iiuid clyu .uuics cau a lso IH ' ap p 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 . pro vicl.«! t.luu 11111' is iU{.I ·]'('s !.ed ill t.h« co llpdi vp ho huviour Oil s u flicie ld ly I:I I'/-',e sca les. T h e 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 lid s ill t.hat . so lid s have' ri gidi ty. llol.l! ,~ o l i d s alld llu ids d oIorru w he n a s tress is »pp liod t o them : hu t. unlil« - a so lid . a sim p le fluid has no lelJ
roiuovod . T he continniun descripti on is fuud mueut.al !.o th e Ji u id approach to r1pscribing 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 JJiJ H'd l,y cOlll p;n'illg 111<' "ollisioll al nu -au Jiu ' paUl I o f 1.1](' pa r1.if·]r,s w it h 1:he m a croscopic ]('Jlg !.!t SC; I I" l. 0 1' ij d (']'es t in t.hc pruhlcui , If I « L t.hon it is rensoJl ah lf' t o in t.rod ucx - 1,11(' cOJlcppl of a fluid volume element. wh oso lin ear siz e i,s much large 1' th an 11m! mu ch sm all er th an L . Th e number of parti cles inside a Ilu id clem en t is ]a1'g(·. a nd we call asso cia te wit.li the Iln id elouieut a hul k velocit y 11. Ind ividu a l p a r ti cl e veloc ities have a random co mpo ne nt in add ition (.o 11 but" 1)1'('a11SI ' th« moan frce pnt .h is smal l. t he r.uu k un motion d( H's ]JO( illllllPdiaf el y t.al«: th p p articl e 1'<11' from its ncig hbonri ng part icles l H'(',1l1,SC' th e p ;lrlil'1l ' 11 :nC'1s o n ly a clis tnuco of o rdr- r I ]JI'fo]'(' 11lI(k rg oiu/-', 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 insi d e 1:1](' olcmont. di v id ed 1Iy it s VOIlllJJC ). O ver a shor t t ime inte-rval lro in tim e I to time I +iSt

Basic Flui d Equations

A siroplujsicu! Fluid Dynll1Hics

we may define the fluid eleme nt to transfo rm by transla t ing ea ch point of t he element by a n amount u (r-, t )M, wh ere u tr, t ) is t he local mean velocity at the position r of t h e elem ent. By virtu e of th e above considerat ions, the fluid clement will still con t 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 same pa rt icles asI JCli)]'( ~ . Heucc Ow iuacro sco p !« properties of tJw fluid clem ent will evolve only slowly, am i by a diffu si ve process. For furt her discu ssion of the cont inu um descripti on a nd t he fluid a pproach , see e.g . Batchelor (19G7) aud Shu (1992) . If t he mean free path of t he particles is not. m uch smaller t.han the m acroscopic scale of intere st, t hen the approp riate description of t he collective properties of th e particles is kinetic: t heo ry. T he equat ions of fluid dynamics can indeed lw derived from t.he mic roscopi c basis of kinetic theory. .For a presentation of thi s approach , ::;ee Shu (1992). He re we shall ins t ead as su me t he conti nu um descr iption from the outset an d see how simp le considerations of the motion of the fluid , a nd the forces acting on it , lead to the fluid dyna mica l equations .

1.1

T h e Material D e riva t ive

T he fluid properties , such as its density p a nd veloc ity u., will in general he functions of p ositi on r and of t ime I . We sh all always use DI Dt to denote t he rate of change of some qu ant ity with resp ect to tim e at a fixed position in space. In describing fluids it is also very useful to define t he material derinaii oe, wh ich will b e denoted D I Dt: t his is th e rate of ch a nge of some qua ntity with respect to ti me but travelling a long wit h t he fluid . Let tir ,/.) be a ny quanti ty, for ex ample, temperature of t he flu id . It may h appen t hat the temperature of all indi vidual p arcels of fluid is not chang ing with time, so t he material derivat ive D f l D t is zero; but if some fluid is hotter t ha n other fluid t he n t he tem perat ure at a fixed p oint in space m ay st ill ch a nge wit h time as flui d of different temperature passes t he po int at wh ich the t emper ature is m easur ed . In fact , in that ca se , Df! fJt = - 11,' V f where u ir , t ) is t h e velocity of the flui d. More generally, the material deri vative is relat ed to the rate of change at a fixed p oin t in space as

Df = Dt

8.f

fJt

+

u . \7 f

.

Eq uation (1.1) ca n b e deri ved by cons ider ing t he ch ange in

(1.1)

f

when following

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

Df DI

1. 2

im (f(1' -+- 11, r5t ,t +<5I) -- .[ (1',1,) ) I1m St,- .o M

T he Cont.in u ity Equation

Consider a volume V , wh ich is fixed ill space, enclosed by a surface S' on which n is th o ou tward-poiut.iug )J()l'IlJ al vecto r (Fi g. ! . J). The to t.al m ass of fluid in \I is .f~! pdV , whe n : p(1', t) is th o de nsity of the fluid. The ti me derivat ive of the mass ill V is t he m as s flux int o V acro ss its su rface 8 , i.e.

= -

-Id / .' P (111 (t..

F

j'(pu ) . n
(1.2)

S

Since V is a volume fixed in sp ace, t he time deri vat ive on the left of Eq . (1.2) can b e t aken inside the integral and becom es a deri vati ve a t. fixed positi on in space. The surface ter m on t he right-h and side of t he equati on ca n b e IT-expressed as a volum e integr al usin g t he di vergen ce t heor em . Hen ce we obt ain

j'

1

8r P dV = \7 . (p11,) dV . v Dt .v

Sin ce this ho lds for any ar bit rary volum e \7 in the fluid , it follows t ha t

8p

Dt

+

V · (p11,)

=

o.

(1.3)

This is t he con tinui ty equation (or m ass conse rvation equat ion). Com bining Eqs. (1.1) a nd (1.3) t he cont inuity equat ion can also be ex pressed as

Dp Dt

1.3

+

pV . u =

O.

(1.4)

T h e Moment um Equation

One ca n simi la rly deri ve a m om entum equation, or eq uation of moti on , for t he fluid by co nsider ing t he rate of change of t he t otal momentum of the tluid in sid e a volu me V . It t urns out to b e easi est t o conside r a volume

A s l.TOl' hysi cal F lui d D yn nU/.u ;s

4

]\lasS of a fluid element a nd is invari ant followi n g tho ruoti on so it s 11l;11 t'ri nl d cr j vnt.i v( ~

is zero. 'Thus / . /h , [' -- - d I' ,\ l rt

d /. -fl U d lI ;co

d! , \ '

( I.( i )

nn rl 11( 'II Cf\ apP h'in g; t.hr: di \'C'l".l',c'IW( ' t.!W()]'('lll jo tl« : s1lr fac I' iIIL q l:r:d ill

v

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

f)7J

p - - dV =

,/ I i ] ) (

/ .

,

v

( -VII -I ['f )dV.

Since t his hold s for :I1lY a r bitary m ntcri nl volu me II , it. follows 1.1];11. (1. 7 ) F ig.1.J

A n a rbit rnrv vol ume of Ilnid \I , wit h s llrface,.') au d ou t.ward-point.in g n orui nl n .

'm oving wit h the [iuid , so th at no fluid is flowin g ac ro ss it s su rface into out of F . T he momentum of th e fluid ill F is p7J d F , a nd 1.Iw rate change of t his momentum if; eq ual 1.0 t.he net Iorc« ac ti ng on ti le flui d volnnw V. These .a re of t wo kinds. F irst th er e ar c h od y forces , s uch gra v ity, wh ich a ct o n the particles insi d e F : their net, effec t is a force

I, .

or of ill as

where f is the b ody force p er unit mass , (N ote that for ce p er uni t mass h as d imension s of ac ce ler ation .) For ex am p le, f could h e the gravit at iona l a cce ler ation g , The secon d kino of for ces a cting a rc s ur fa ce forces - forc es exerted on the su r fac e S o f F by the su rro u nd ing fluid . In a n ituns cul fluid , suc h as we shall mo stl y he cons ide r ing. th e s urface force a ct s norm all y t o t he s urfa ce an d it s net effec t is

j' - pndS ,

T his is i.lw m om entu m ('cj1 w t io ll for a ll i\lviscid lIuid. In a gClwra l viscous fluid (it docsn 't IlcC'd to I lf ' as c'x t.]'CllW all oxmnpl« as tr eaclel) t.he ith C'.()]llj HJl1 en t o f 1.Iw fo]'( '( ~ ex ( 'l' t.c ~d Oil ~·mr fa('" S' II.\' 111C' surrou nd ing fluid is no t. just f <; - 1) 'lJ i d8 ]mi. is .fs (Tij 'IJj d 8. whore (Tij is t.ho 8 1.1'1.:88 I.I.:n8 0 1'. [N o t.o her e thai. the s u m m nl.iou (,Oll VC ~ll t.io ll is w )('( L so tlmt, if a ll index is re]lC'aU~d it sho uld lw su m me d ove r. A lIo11-]'C']lc'ntC'd in d ex d eno tes a COlll]ion ent of a vec t.or or te-nso r. SC·t' A ppC'1 \C1ix A . A Iso 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)11t for it s ith com p on en t for m wr: al ways wri te :1:i . ) For g;asl's and sin rplo

liquids it is foun d th at (J .8)

whe r e p, is t he so-cal led dyna mica l viscos ity: sec ('. .g. E n tr-lrolor (l 9(j7) . Landau & Lifshitz (J95D). Also Il i i is the Kron eck er i1e11a: sel' Ap]l(:nd ix A, Now

j'

,S

(J i j H j

.

/. D

ciS =

'1

,,<;

>

1

(J ij

d \'

• F u :J.' j

p being t he pressure. Ther e is no flux of m omentum across the su r face

(d ive rg enc e t heorem ), a nd so if 1'. is a co ns tant it follows t haI. t ho eq u ation

ca r ried by flui d p ar cels m oving , since by d efin iti on none crosses the su rface of a m aterial volu me . E q uating force t o chan ge of momentum we obtain

of mo tion for a viscous flu id is

e!

rpudV = .Isr

CI t i F

- pn d S

+ / . pf dF .

'F

o« Pm

=

- -Vp

-+ pf +

II

( ,)

V - l1

+

1 ( \ ~ V V' 1/,)) .

( U I)

(1.5 )

Sinc e V is a llmt.el'ial vo lu me, whe n t he t ime d eri va t ive is t a ken insid e t he in t eg ral it becomes a m ateri al deri vative; but t he prod uct pdF is the

With viscosit y included , E q. (J .9) is cal led tl io Navicr-Stokos equation. Its inviscid form , Eq . (1.7) . is cal led tJIC Euler eq uation. T hroug ho ut most. of this b ook we sh all neglect visc osi t y : the justification of this app rox im at ion

(i

A s tro pliusi cal Fluid Dynamics

Basic Flui d Equations

ill as trophysica l contexts will be seen in Section 2.8. However , viscosity plays a key rol e ill some ap plica t ions , not ably in astrophys ica l accretion disks which we discus s in Chap ter 9.

7

(b bein g th e Dirac delt a fun ction in ;~ - D space), Eq. (] .1:-1 ) cau as a part ial differential equation, Poisson's cquaiiou:

I)(~

rew ritten

(1.15)

Newtonian Gravity

1A

A muss '11/ at. position 1" exerts OI l an y other mass In a t posit ion I ' au at.tructi vo force proportional to th e pr od uct of the two masses and inversely proportional to th e square of UIC dist an ce betwee n them , directed towards iu us s

111,':

F = '/ng (1')

Om rtf' (1' - r' ) [r- _.- '1"1 -

==

Grn 'In' (1' - '1" )

II' -- 1" 12

Note that (1' - - 1")/11' forc e , Now

-

11' _. 1" 13

(L/ O)

If one takos Newton's third law, F = uui = In (tl v /rl/. ) and multiplies hy velocity '0 , OJl( ~ ol.taius tltill. rat e of work of the forces, F u , is equal to tho rate of chau gc of kine ti c energy , tl ( ~ '11 I.'IJ'2) /d/.. Simil arl y, taking th e do t product of th e equation or i uot.iou for a fluid , (J .7), wit h t ho fluid velocity 'U yields

D

Dt.

IT- 1" 13

(1. 11)

(t he derivatives are wit h res pect to T': they treat r ' as a cons tant vector) , so the grav ita tional a cceleration g (I' ) can be written as th e grad ient of a po tential function 'I/{ r ):

=

- \7'(/"

where

'1/'

=

- Gm ' IT' - 1" 1

(1.12)

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

-

dj' (12

dz

where the int egrati on is over the whole volume of the fluid . T he gr avit ational acce leration is - \74'. Using the result

\7-

(

1

)

IT' - 1" 1

- 41Tb(1' - 1" )

(1.14)

(1'2 2) 'u

-

1

-- u · \7])

(i

-+-

'U ' f

.

(1.16)

Equation (1. 16) says that th e rate of change of th e kine tic energy of a uni t mas s of fluid is equal to the rate at which work is do ne on t he fluid by pressure aw l body forces. This is somet imes called th e mechan ical energy equation . An equ ation for t he total energy --- kinet ic a nd intern al t hermal energy _._- can be derived in the same man ner as was t he mom entum equ ation in Secti on 1.3. Let the internal energy p er un it mass of fluid be U. Then th e rate of change of kin etic plus internal ene rgy of a material volu me (i.e. one moving with the fluid ) must be equ al to the rate of work dono on the fluid by surfac e and body forces , plus the rate at which heat is adde d to the fluid . Heat can be added in two ways: one is by it s being generated at a 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 iati ve heat flux). Thus

(1.13 )

'J

The Mech a n ic a l a n d Thermal E nergy E q uat io ns

1" 1 is th e unit vect or alo ng the line of action of tho

- (1' - 1.1)

9

1.5

- '/L 2

-+- U ) pdV

(1.17)

V

=

isr 'U ' (-pn) dS

-+-

.r;7'U· f p d V

-+-

1;7fpd V

- .f~ F . nelS' .

In t he same way as for th e mom entum equa t ion, one rewrites all the surface int egrals in thi s equat ion as volume integr als, using the diverg ence th eorem. The re sulting equation holds for an arbitrary volum e V a nd so one dedu ces

B o.si» Fluid Equ ation s

A st rophy si cal F lu id J)'i/narni r."

8

that D P ( Dt

(12 ") + .J)U) Dt 11 ,-

= -

+

V . (p H)

pu. r]

+

pe -

V ·F .

(J .] 8)

wlWl"e 11 is a fixed volu m e onclos ing th o wh o le 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.nhli sh (1"(1111 I'q. ( 1. 1:l) (.hat (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]

Iv

by t.lic fluid . One ca n deri ve a ll equa tio n for the thermal ene rgy alone by di vidin g E q . (1. ]8) by the density a nd then s u btracti ng t h e kin eti c ener gy equation (1. lG) to obtain

DU DI

DfI -+ -]1 ._fI'2 Di

I

-

1 - V ·P . p

(L I D)

T he d iverg en ce of 11. h as h oon rep lac ed 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 reci procal of tho den sity, i.e . 11 = p--- l , we reco gnise t he therm a l ener gy eq ua t ion ( J .19) a :-.; a statemen t of th o first law of th ermodynamics : ellJ =

(- ]I)elV

+ 5Q ,

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

dt

[" i p« - V' . F )dl! ,

iF

VO] U Il H'

a(TOS:-'; th c surface.

1.6

A Little J\tIore Thermodynamics

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

(1.21)

( 1.22 )

M:2 = 'I'
(1.2())

t h at is, the ch ange 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 a re pro perties of t he 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'Operty as t he heat content a nd so we can not speak of t he ch a nge of he a t. cont ent . Instead , we ca n only spea k of t he heat ad ded, a nd we t herefor e use a different. no tation , i.e. 5Q . Equ a tions (1. 16)- (1. 19) can be gene ralise d t o include a visco us xtrcss t 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) a re repl aced by p- l 7/.iU(Ji:i ! D:rj and lI ;(Ji.i n .; resp ecti vely, where rTij is t h e st ress tensor as ill Eq. (1.8). T he conseque nc e for the t herm a l en er gy eq ua t ion (1.19) is that. kinetic en ergy is conv erted to he at by viscosi t y, so t h a t one obta ins an a dd it ional hea ting ter m sim ila r to f . This is discussed in more detail in Chapt er 9. W ith some effort, one ca n us e t he above eq uations to de rive an integral equation (som etimes a lso referred t o as the total ene rgy equation) for t.lio ra te of ch an ge of t he t otal en er gy (kinet ic plus intern al plus gr av it ationa l p oten tial energy) for t he whole fluid volume:

=

int.cgrnls of di v{']"!'/'lic e te n us ill Eq . (J .2 1) ('an 0 [" C(lm s(' surfar«- in t.q~r als . If t.l i o flu x in :-.; q ua n ' ] )]";\('kd .:-.; ill 1.1 ](' sccon d t erm on tho ]d'i of Eq. (J .2 1) vanish e-s at the sur[";)l'( ' (If F. which migh 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 ch all g(' tJl1'IllIgll iuu-rual he.u :-';(llIrcc:-.;jsiJl]':-'; (I ) or ]wnJ flu x ( F ) . T ho

1)(' j'( ~-( ~XPl"css( ~d 11.'';

whore S is a thormodvn aiui c s(.al,e va ri ab le, t.h« 8]iN4il: cn l'l'OJiY (i .1 '. 1.hc ent ropy p er u nit m as s). Comb in ing th is with Ow firs tla w, E q. ( 1.20), vio kls

dU

= Td S -

plW .

From this va rious relati on s h ot.woon 1hormod vn am ic dorivati vcs can h I' d (~­ .lucod. For example, it follows immodiatcly fro m E q , (1.2:{) 1lJ:11

T =

( ~~~) v

and - 11 =

C~~:

)

5' :

but a prop er ty of p ar t ial differenti ation is th at [j'2 .f! (hUll

( 1.24)

D'2.f ! DyD:):,

so we find th a t

(

DT )

all

s = -

(

a]l ) as ,, '

( 1.2:;)

Another usoful mauipula ti on th a t is a general property of part.i n 1.lorivat .ivr-» is 1.l1il1.

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

.

( i.2G)

This follows by rearran gin g d f = (Df ! ch:)y d x + (Df ! uy)", d y 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 vati ve (D:l:! ay).r . Variou s t he rn )(J(ly n Hm ie rol at ion s th at ar e useful

f. ~

10

..~

Astrop luisical Flu id D yn am ics

Basi c Fl uid E ou ai icnis

ill s tellar ph ysics a nd ast ro phys ica l fluid dyn amics ca n b e found in e.g. J\"ippcnhalm & 'vVeiger t (l9f)()) .

We defille th e udiabuiic expone n ts 1 1, 1 2, 7 J by

~Y1.

_ (DIIlP)

-

~('!. -

7-

.d lll (J

]

,

S.

/2

=

D hI T ) ( Dhl]J

8'

1. 7

P er fect Gases

A perfect gas is ow : for wh ich

1:1 -- ]

Uln T ) ( DIll P .'; .

Not« that all thoso partial derivatives ar c at constant specifi c (mj,m py : 'adialJiltic' he n : me a ns without exchan ge of he at, so MJ = () = d S , i.e, 5' is COll Seall/" 'J'h o quanti ty ((2 -- l ) /~f'2 == (lJl n T /D hlp)", is oft en referred to as '\7".1 ' We define (;1" the sp ecific he at at co ns tant pressure, to I)(~ L1w am ount of heat required to make' a unit in crease ill t mnper atllre, without th e p re ss urr: changing: tlm s, from E q . (1.22) , (; 1' = T (US/DT )p. Sim ila rly we define ell, t ho specific hea t at constant volume, to be t he amount of beat required t o in ak o a unit increase in temperature at cons t ant V. 'The f()]]owing three useful res n lts re late I.lw aruoun! of heat. added t o the clJanges in p airs of t lwfl llo(lyna m ic va ri ablns:

(R bein g

::;O l1l C

(1.28)

For ex ample, the first equa tion can be der ived by noting

Td S

T

OS) dp + ( -D ]J v

T

(DS) -::>.

oV

.

l'

dV

(1.2!J)

constant ), and

U( T) .

U

(J.:m)

It follows from Eq . (1.29) t h a t. dp Now for all adi ab atic chaugo

o=

dB =

dT

dV II

+

( 1.:31)

or S1l( ;]1 a gas ,

I

T (dU + pdV ) =

d U d'J' dT T

-+

1

+

n

(1.;)2)

(1.3:3)

dU/dT .

Elim inating dT between Eqs . (1.31) an d (1.32) gives that '/'] is given by t he same expression (1.3:3), a nd likewise for 12 (eliminat ing dV ). Thus for a perfect gaH, the three adiabatic: exponents arc equal. Hencefor ward in this b ook , since for a p erfect gas a ll t hree adiaba ti c exponent s are equal , we shall usc 1 t o denote all of them when no confusi on can ar ise . In fact , for a uionatomic gas (in whic h the m olecules a re simp ly point Inasses) on e ca n show th at 1 is equal to 5/3, as follows . For a monatomic gas, the internal ene rgy is just the t ranslational kinetic energy of all the molecules. Assuming the gas to be isot ro pic (all directions equivalent ) a nd all t he m olecul es identical , t he t otal internal energ y of the gas in volume V is

T ( op) - 1 rdP + T(DS/cW)l' d V] DS v (oS/op) Ii and using Eq. (1.25) on t he factor outs ide th e squa re brackets and Eq. (1.26) to m anipulate the last t erm, together with t he defini ti on s of 1 1 a nd 13. Note t hat In V = - In p . The other two above expressions for fJC) arc de r ived sim ila rly,

av

R 17- .

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

fJ C)

R'J'

Jill

(1.2 7)

fJ(J -

11

(1.34) where m is the m ass of a mol ecu le, N is the nu mb er of m olecules in V , and (for exa mple) is th e me an squa red velocity in t he direction . Supp ose t he volume V is enclosed by a rigid rect angul ar b ox of len gth I in t he xdirection (a nd of cross-sec t ional a rea A = V/ I ). The for ce on t he en d of th e b ox is pA. Cons ide r a sing le mo lecul e. It has some z- velocity V x ' In

v;

x

12

A s troph ysi cal F luid Dyn nmics

B asic Fluid E qn ntim :»

21 j v,,. it bOIIll(,('S off t hat end of tho box OJJce. III brJlln r~i!l g: its :r:- lllOl I W l l t lll Jl ch angr-s Ily an muount 2111 1':,. (ass llllJing all cl astic «ollision ). Thus , s ince lorco (= p ressmr: x area) is equal 1.0 t he r ate of challgr~ of m U]JWIl(,1II 11, s \Illll lling 0 \'('1' all molecules gi vr ~s

ti me t:lt

==

J)/I =

L -t:ll-

2 1/1.1':1

Nu n ;,'; and

U

L

=

11/, :2 - 11

1

I

1i0 .

from E (jli. (J .2!J) ,

:=.:

2R T

:r

(J .:lG)

'

( I.a:n and ( I .:H ),

:\

G

and

')'

If a gali iii l)J)(!ergo iJJg ioni zat.iou , dU j rlT if:

=:

:l

grC'a1r ~r

-"

/.

' \,

1"

'17 , /

v ( , (I(

'

/. j' , I' ,

T:

11'

,/ 1.' , 1"

'17

v

1' .

"'. (1' '-1C") ' / ',------------"

- "(/

'

111 .-(' " -' ,'

:2 ' ,

U

])1'

=

-

Dt

1' , ' "

-- -" 1 C; / ,'

T ile velocity 11, is th e rat.e of (:!lange of position [()llowill g f,llc flui d :

( 1.·11)

'\

1" I'~

(1(1

(/1 _.' (1'/) I ",..__'I) /1(1' ,) (II .; (11.1' '" 1' ,

., )el\; (1( 7 ,I )el lI '

- - -~ (/ / ', I'' , ( .~,-~ -c1(.- --..2.,I") 1

The Virial Theorem

iI:I' ; /( ):r ,

I ;{ / ' J!C 1I ' .

n el S

'!'s

,\7 , t :

\:\7C' supp ose tll at 1.1 )(: prossn r« van is h es at 1.11(' houud arv 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,

( U ri)

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 ) 1Jw arliabaii e I'Xj!ollent s .uo red uced ill value.

1.8

= - 1',711"

. r . V7)( W

./ v

JV 'III,--;_ _ '/) 2

C/:

U sing 1.1)(: r1 i v r ' I'),!, ( ~ n r: ( ' t.lu-orern and ill(' i.lou t.i t.v pressu]'c term ill Eq . ( I .:l! }) ('all 1)(' rr-wtit .t.c-u ;IS

I' 11' --

I"

_;_I '\"""1')" " ') (1(1') 111' ji(r ,

1

)(11

'1

1"

, - ----I --(! (1,) d, '\ ' (! ( l ' ' )I 1\'"

. v . 1"

2

(r'

1'"

11' - ri ll 1"

I

( 1.12) .

CU 7)

Hen ce Eq. (1.7), w it h f n~p]acrxl by gravi tati onal acceleratio n and llsing Eq, (1.12), can be rewr itten

(th o lat er steps exploit th e syn n ne try IH'twcCJI

I}J

==

~

l'

al II I 1") where

I' 1/' (I(T )elV

( I.

I:n

2, "

(1.:J8)

is the total grav it at.ion a l energy . P utting all thi s togd ,]](']' viold s

Taking the dot product. with r am] integrating over tho whole volum o of th e fluid gives ' , ./ F

])21'

T: >

- -peW 2 Dt

= -

/ . , 1

. r . vp dV

/ .

-

T' .

, "

v 1/' peW .

(1.39)

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

Dr

/ (Dr)

-d / r' -pdV -elt . F Dt ,F])t

2

2 dt -

+

•'3

I' 7 IV + IJI

.v

1(

•

(1.-1-1)

where I == '/;1/W2rlF . E q uation (1 . 4' 11). IS ' Je sea Ia]' t0 nil of th e uiriol ' 1J theorem. One can also deri ve a tI'JlSO], viri 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 on out of 1' :

2

T

()( 11·

=

d / . 11' 12 prlV --2T -21 ----:z dt ' \ '

,

(J .iJ [))

(lAO)

wh ere T == ~ P 11,2 elF is t he total kinetic cncrgy of th e flu id. (No te t hat here and in simi lar exp ressions we write 1)2 when wh at is meant is In12 , i.e. 1/, . U - - t ho quantity is a sc al ar. )

Iv

-1 -d 2 I') -_ ?T -

11 ca n t he n b e shown that 1 d 2 Ii j

:2

dt :!

-n: - 7:J + s.. · 7J I' ]lrlJ! + . l'

IIl i .l, .

( 1.,Hi)

14

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

Busic Fl uid Equation»

15

where

B ----.. · ~v, ( r

. p :c;:cj d If , ·/ 17

'r.,

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

(J.47)

L., l'

=

(:Cj - .J./ )

1 , / . / . (:1:; -

-' ~- C 2

.

v. \/'

-- -

err)

Ir' -

r

. . ; :j_L p(r} W p(1")<117'.

A

I

Note tl mt if E q . (l Au) is c()lIt.racted over i a nd j (i.e. multiplied by Ii; i a nd sun uuod over i a w l .i) th en t he :-;(;ala1' vir ial theor em ( ] .114) is recovered . Dori var ious of L!w virial thoorciu ill d iJf(:rmlt forms call be found ill C llilJldrasddwJ' ( l UW ) aw l Tas soul (l!J7I)).

.

~

u (r) F ig. 1.2 '1'1 1/, m ot.ir» o f J",iglt i>o llrillg p"i llt.~ A (at. p osition r ) and B (al. p ost ion r ·j aT'), wltid . lea d s 1.0 t.h« «vol ut.iou of ,.1 ", lJ1a1.e rial Jill" .,j"lJ1ell t (}.,. with t.iru «.

Vortici t y

1. 9

The flu id velocity at A is ·LI C,.) a nd al. 13 it is '11, ('1' + or); therefore af ter a short, tiuic 51. t ho separation of 13 from A has ch anged to

Au important. deri ved quant ity Ior a ilnid How is the vorticity (J,4i))

For a wl

il

Il uid ro ta t illg rigidly with ang ular veloc ity 0 , for ex am p le,

w =

20 ,

11.

=

O X 'I'

(1.49 )

us ing a stalld ard vect or identi ty (cf. Appendix A). Gen erally, in a fluid flow 11. the vorticit y at any location is eq ual to t wice the local ro t at ioll ra te of a fluid line elem ent at that location, as is proved below. This do es 110 1, mean however t.hat st rea m lilies have to be c ur ved for t he fl u id to p ossess llOIl-:6ero vort.ici ty, For examp le, cons ide r t he shea r How 1l = Cue; wher e C is a 1I011-:6Cro consta nt : t.liis is a ullidirecti on al shear How in the a- direction wit h m agnit.ude p ro por t.ion n] to y . Here as elsewhe re we use e e e to x, y , z den ote unit vectors in th e X -, y- a nd z-direc t ions, It. is a st raightforward exe 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 everywh ere par allel to t he flow) arc st raight lines.

To see t he rel ationsh ip bet ween vorticity a nd local rotation of t he fluid , we : :; lwll uow a nalyse the relati ve motion of a fluid in the vicinity of a point. Let A be a poi nt movin g with the fluid , whi ch a t a n ini tial time i is at p ositi on 1' ; a n d let. B b e ano th er p oin t whi ch a t t ime t is a t nearby positi on r + 0'1' (F ig. 1.2). At t im e t t he refore the p ositio n of B re lat ive t o A is Sr,

correct. to O ( lit.). T he las t. term call b e simp lified by ex pandi ng 11, (1' + 61') in a Taylor series ab out r and keeping on ly tenus up to li1'. Trea ting Sr ]l OW as a func ti on of t ime, a n equa t ion for t he rate of cha nge of lir' wit h time can b e ca n be ob t ai ned from her e by dividin g the difference b et ween the new sep aration and the old one by Of; and taking th e lim it as -> 0:

ot

DOr'

- - = DI

.

Or, · V 'U

(1.fiO)

where, since we a re following the sepa ra tion b et ween m at eri al p oin ts , we wri te t he deriva ti ve as a material derivat ive. This equat ion ther efore describ es the evoluti on of a m a terial elem ent 01'. The right-ha nd side ca n be exp ressed ill index not ation as (<5r') j ehtdoxj . T he te nsor \7u , like a ny other second-r a nk tensor , ca n b e split into a sy m iuetric par t and an ant isyu un otric part:

OUi =_ -1 ( -,OUi OXj 2 u:c;

) OUi + -Dai + -21 ( -ch , - - -,U0 U) -) o:r ; 1:;

(1.51)

j

The second t erm on t he right of E q . (1.51) is a nti-sym met ric: usin g the defiuit ion (1.48) of vor t icity, it ca n b e written as ~~Eij k Wk (d. Ap p en dix A) . Hence, subs ti t uting int o Eq . (1.50) it m akes a contribution t o t he right-

hand sid e which is oq ual t o - ~ Jr x w == ~ w x Sr. T h us, com paring this wi th the velocity due to a so lid-bo dy rot.at.ion , wo d educe t hat the antis viu metric part of \7tt. co nti b utos to tho motion of point 13 relative 1:0 point 1\ a motion wh ich is a rotation wit h angul ar veloc it y ~ w . Equival ently, 1lw vor t ici ty is give n by Eq . (1.49) where n is interpreted a s the local rotati on

rate. The symmetric p art of du, I D:l:j , i.o. th e first. term on th e ri ght of Eq . (1.51), is called the rate of strain te nsor Cij :

,

_

<',.1 --

1 2

;-

(DU j -I-, -

()'/J,:J', )

(L!j 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 r esents an ex pa ns ion or com p ression of the fluid in th e re gion of point. A. Th e remai nder of c i:i , n am ely e i,j - *e kk0 i.i' has ze ro trac e and ]'()prelsents a loc al shear of the fluid , An evo lution eq uati 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 rewr ite t he 11' \71), term in E q , (1.7) us ing a vector id en tity t o ob tain

-Datt.t

=

l/, XW -

\7

(1 2)

] ~ \7]J

-1), 2

p

-I-

f .

(1.53)

iI,!

,

at

= w·\7u - (\7 ·u)w

ata (w) p

1 -I- z \7px \7p -I- V xf ·

(1.55)

p

P

-I- 11·\7 (w) =

p

(w) ,Vu -I- p3 1 \7p xVp -I- 1 \7 xf·

p

(1.5G)

; ..

Is'

r

Finall y, using the con t inuity equation (1.3) to elim in a t e \7 . 'lJ" we obtai n

D (w) Dt P _=

CC'

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

since t he curl of a gr ad ient vector is zer o . Th e \7 x ('lJ, x w ) can be ex pande d using identit y (A A) from Appendix A . Noting that \7 . w van ishes by its d efinition (1.48 ), Eq. (1.54) becom es

aw -I- u·\7w

slIr faccs of co nst.aut density and of cons t.a nt ])J'f'SSl1r(' "oillCid(' . l1 11 d it is possible 1,0 write oithor variahlr- a s a fuu rt .ion sllkl.\, "f 1.111' " fl ll'r v.ui nl r]«. c ,g. p = p (y ). Cou vorsol y, 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 pres ent. is COll sI'l'vaj,ivI' (i.e, V xf (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. t wo terms ill Eq. (I .fiG) vanish and 11](' vOl'Licity oqu nt.iou (J .f-,ri) is of ti ll) sa il II , form a s 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,' vorte x lilies iuo vo wit h I.lw Ilu id . (/\ vo rtr -x lin « is n li nr: ov r-rvw h orr: pl1 r:dkl 1.0 1.11( ' vor1.icity. ) YVI ' C:III ddill e a '1101 1 1':1' 1'111)(' 1.(J 1)(" ]oosldy sjH':l,killg. a I1I1JJd]e of vorl.cx lines or IliOn ) pre'cisc'ly a tu l«: w1IOSf' sllr f:wr' is no wh cr« ('f'o ss(·d by vorI.ox lilies and whos(' Sl1I'fa('(' is it.sl,]f I:OlllpOSl'd of vor te x lines. III 1.1](' p]'( 'sellt, 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 gt h of a vortex 1,111)(, to Ill' w· ndS, wlwre oS' is a ny 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
C containe d wit hin t he fluid to b e

Taking the curl of this eq ua t ion gives

aw Dt =

17

I3lLsic Fluid R'I uo ti"" ,,

A stro plujsi col Fluid Dinuimi cs

16

E quat ion (1.5G) is ca lle d th e vort.icit y equat.ion. It d escribes h ow vort icity evolves in a flu id. A flu id for which V p x \lp = 0 ever ywher e is called barot.mp ic: sinc e the vector gr adients of d ensity a nd p ressure arc ever y wher e parallel , the

=

i

(1.:>7)

u ' rlr .

.C

If we cho os e C t o he a m aterial curve , moving wi t.h the Iluid. I h('11

-df = dl

f') -

])11,

.

c Dt

. dr

-I-

i'

.c

11, •

-D dr .

(U j8)

Di.

2)

The last term can b e re wri tten a s the integral a round C of \7 ( f;.U using Eq. (1.50) , a nd for an inviscid flui d 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 eqn at ion (1.7). \\Ie consid er only :~- D fiui d do mains for w hic h curve C can he sp
-df 1-' d

=

.

j's (z1 \7p x \7p

-I- \7 xf ) . ndS .

(1.59)

fJ

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

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's Lhooreiu, as a flow around a matciia] curve embedded in the walls of tIw tube awl encircling the tube's axis. Hellce the strength of a vortex tube in such 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 is constant. A fluid for which Vf/XVp cJ 0 is called baroclinic. This means that surfaces of constant density are inclined to the surfaces of constaut pressure.

C ha pter 2

Sirrrple Models of Astrophysical Fluids and Their Motions

In t.he first chapter we est.a]ilish ed the mo mon turu equat ion (1.7), th e (;011ti nuil,y cq na Lioll (J A) , POiSSOll'S eq uat.iou (1. J fJ) aIHI t.lio energy eq uation (1.19). Assuming t hat th e onl y hod y forces arc (h w t o self-gravity, so t hat f = ~ \74' ill E q, (1.7), th es« eq uati ons are:

Du

p-

=

tn Dp DL

+

- \7 p" p\7 I/' ,

p d ivu

\7 '2 'lf'

DU

- -

Dt

0,

E

-

(2.2) (2.::\)

47fG p,

p Dp = p2 Dt

(2. 1)

J - \7 · F .

(2.4)

p

Note that these cont ain seve n dependent variables, namely p, t he t hree components of u , p , Vi an d U . The three com po nents of Eq . (2.1) , toget her wit h E qs . (2.2) -(2.4) , pro vid e six eq uat ions, a nd a seve nth is th e equat ion of state (e.g. t hat for a perfect gas) which provides a re lation betwee n an y t h ree thermo dynamic state var ia bles , so t hat (for example) t he internal energy U and temperature T ca n b e wr itten in terms of p an d p. (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 t o solve t hese equations , given suitable boundary conditions . In practice this set of equat ions is intract able to exact solut ion , and one mu st res ort to nu merical solutions. Even these can b e extremely problem atic so t hat, for ex ample, understand ing t ur bulent flows is st ill a very challeng ing research area. Moreover , an analytic solution to It somewhat ideali zed pro blem may teach one mu ch more than a nu merical 19

A stroplnjsicu i Pl1Lid D yn am ics

20

Si m ple AIodels

so lution, O lJ(~ useful idoa lizati on is wh ere we assume t hat th o fluid velocity aw] all tiin c derivatives are zero. T hese are cal led equilibrium solut ions and des cri be a stea dy sta te . Alt ho ugh a t rue steady state iuay be rare in reali ty, tli« t ime-sc a le over whi ch all as trop hys ical sys tem evo lves may be very lon g, so that at an y particular t ime th o st ate of many ast rop hysical fluid bodies sucl: as st an; may he well re p resented by all equi libri um mod el. E V( )JJ when t.li« d ynamical b elravi our of t he body is im p or tant, it call oft.eu IH : descri bed iu t erms of sm a ll depart ur es Iro m a n eq uilibri u m st a te . Hence in thi s c:lJ ilptcr W(~ st.a rt Ly looking at some equ ilibri u m mod els aw l then dcri Ill' equ ation s describing small perturbations ab out all equilibriu m st at e .

2.1

Grn (T)

the cont inuit y equat ion beco mes trivial ; and Eq. (2.:3) is unchanged. A fiuid sa t isfying Eq. (2.5) i::; said to be in hydrostati c equilibrium . If it is self-gr av it ating (so t hat '¢' is determined by t he density distribution wit hi n the fluid ), the n Eq. (2.:3) must also be satisfied . Putting 'U = 0 and a/ot = 0 in E q. (2.4), we obtain that the heat sources given by f must he exactly balanced by the he at flux term p - 1 \7 . F. If t his IJOhl::;, t he n t.he fluid is a lso said to be in t hermal equilib rium . Since we have no t yet co nsi dered what. t he heat so urces mi ght be, nor th e de tails of t he heat fiux , we shall neglect cons ide ra t ions of t herm al eq uilibrium at this po int .

(2.9) (Note that rn( r) is t he mass inside a sp her e of rad ius '1', centred on the origin .) Now \7'1/) = (ch/)/d1') e ,. if 'II> is only a func ti on of 1', e r being a unit vector in t he rad ial direction . Hence E q. (2.8) im plies

== -

.

a( ')eN) 8 (. 1 + -1''2 sin -1 sm (} -8'1/!) + 8T () 8(} EJ(} '1' 2 sin 1'- -

(2.10)

Eq uat ion (2.10 ) st ate s that in the spherically symmetric case, the gr avitationa l accelerat ion at position r is du e only to the mass interior to r and indep endent of t he density dist rubt ion outs ide r : this is kn own as Newto n 's sphere t heorem . Also , hy Eq. (2.5), " _ vp -

Grnp -2 - e ,. . r

- -

(2.11)

The vector \7p po in ts towards t he or igin, so t he p ressur e decreases as increases.

T

One ca n only make further progress by ass um ing some rela t ion be tween pressur e and density. Supp ose then t hat the fluid is a per fect gas, so p =

R pT

-

/.L

-

==

2

a p ;

(2.12)

a is kn own as t he isothermal so und spe ed. Su ppo se fur t her t hat t he t em-

perature T and mean molecular weight /.L are both const ants throughout t he fluid , so a is also a constant . Then Eq. (2.11) becomes

Sph er-ically symmetr i c ca s e

I, = '1''2 8T

Gm. .,.'2

- - ,- e ,. .

\7'1jJ =

Gmp

In sp herical polar coo rd inates (1',0, ¢) (see Append ix B),

\7 '2 ,, '1/

(2.8)

where

9

(2.;» )

2.1.1

Int egrating on ce gives

H ydrostatic Equilibrium for a Self-gravit at ing Body

If we suppose th at u = 0 everywhere, and tha t all qu antiti es are indcpenden t of ti uio, th en E q, (2.1) becomes

21

- 72 2 ()

8'2 '1"/' EJ¢'2

(2 .6)

which im plies that

2

Let us seek a so lut ion where everything is independent of () and ¢ (and hen ce dependent only all t he radial variable 1' ). T hen Eq. (2.3) becom es

d~' ('r pa'2 ~~)

= -4nG1'2 p .

(2.13)

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

a2

(2.7) p =

')""f.!• • 2 '

(2.14)

22

Astropliusico l Fluid D yn am i cs

S im p le Mode ls

This is t he sing ular self-grav itati ng isoth erm al sp here solution . It is not physically realist ic at r = 0, where p an d (I are sing ula r , but non eth eless it is a useful an alytical mod el solution. Of course , in a real non degen erat e st a r , for ex ample, th e inte rior is not isot hermal : the t em perat ure in creases wit h depth , which in turn m eans t hat t he pressure inc reases and the star is pr evented fro m collapsing in uJlon itself wit hout. recourse to infinit e pressure an d de nsity at t he centre.

2.1.2

dp = - g p(z ) . dz

-

i Ir ,.

r

I

I . ,! :;. .i' :y ~

I

i

H

=

1-1

J( p I -l- l / n

1[1.1)('
=

1dp dz

- yz

- y

-I- constant. .

(n-l- l )K

(2. 1!I)

(2.20)

(J (wh ich cou ld 1)(' a rcn sonahlr - ap p ro x im nl.ion if z = 0 wore t ho snrlaco of il s t.ilr) 1.lH' /l I.h(' COJlst.anf. of inU'gra t. ioJl ill Eq . (2.20) is zero. Hence fOJ' a plau e- parall ol p nly t.rop « of fini te illcratun: increases linea rly wit h depth . C7

Equations of Stellar Structure

Alt hough it if) a n asi de, it. m ay be instruct ive t.o point out hr-ro the rc lati onship b etw een the fluid equations that we have der ived t h us fa r an d t he equa tions of stellar st r uct ur e describi ng a stat ic, sp her ically syn nnct.ri r: star. These ar e comm only used in studies of stellar str uct ure and evolution. T he equation of hy dr ostat ic equilibrium is j ust t he morn eutu m eq uation in the sta t.ir: case: wit h sphe ric al sy mmetry, so t h at. quantities are only functions of radial var iable 1' , t h is is giv en by Eq . (2.1 1 ): dp

(17· (2 .17)

Hen ce, in t his case, H = 0. 2 I 9 and is constan t . T hus p = Po exp( - z I H ). A useful fam ily of solution s is that of plan e-parallel ]!olytro pes, wh ere ]J =

__

and t his integrates to give

2.2

whe re t he constant Po is the de nsity a t z = o. The densit y scale height H is de fined by

-1 -dp p dz 1

1

p"

(2.15)

(2.1G)

!

, (n -l- J) 11

pI / 1I

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 h e integr a ted to give

I

/\

Plane-parallel layer under constant gravity

In m odellin g t he atmosphe re and ou ter layers of a star, Ute sp herica l g(~­ omctry ca n often h e ignor ed , so t hat such a regio n can h e approximated as a pl ane-pa rallel layer. Moreover , in the rarified oute r layers of a star the gr avi t ational acceleration 9 m ay he ap proxim ated as a con st ant vect or. T hus , in Cartesian coor d inates (:J:, y , z) we have a region in which everyt hing is a function of z alo ne and 9 = -ge z , whe re 9 is constant . Note that we t ake z t o b e height , so e z p oints upwa rd s. Hence Eq. (2.5) becomes

I

polyt rope of index n = J .5. Note t hat t he isotherm a l l a'y( ~r is ob t ai JH'd ill th e limit. n. --7 00 . Subst.it.uti us; Eq. (2 .1R) into E q. (2. 1G) giws

(2 .18)

(wit h bot h J( an d n constant); n is ca lled t he p olytropic index . For examp le, in the ad iabatically st rati fied convect ion zone of the Sun the pressur edensi t y rela ti on is well descri b ed by p = K p" where I = 5/ 3 (except in regions of partia l ioni zati on ) and hence , com pa ri ng with E q . (2.18) by a

G'm p

== -- - -

1'2

(2.21)

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

(2. 22)

In a spherica lly sy mme t ric star t he hea t flux F is p urely radial: F = F(T)e,.. The flux F(1') is rela te d to t he total lum inosity th rou gh a sphe re of radius T by L (r ) = 41r1' 2 F (T). An equa tion describin g t he rad ial va riation of lum inosity follows from Eq. (2.4) . Setting the deri va ti ves Oil th e left of t ha t equat ion to zero. a nd using the expression for d ivergen ce in sp her ical

24

Astrophysical Fluid Dy n am ics

Simple Mo dels

p olar coordinates (Appe ndix B), it follows t hat

-dL = dl"

2 4 7fT

pf. .

(2.23)

A fourth and final differenti al equat ion describes how heat is transported in t he star. In the bulk of a st ar like t he Sun this is by radiation. To combine radiative t ransfer wit h fluid dynamics in general is a substant ial t opic in it self, and is excellently expounded by Mihalas & Mihalas (1984). Here we only consider a st at ic case and moreover , in t he int erior of a st ar the rad iat ive transport is well described by a diffusion equat ion . The prototypical diffusive transport equat ion for the fiux j q of some qu antity q with density (I " per unit volume is (2.24)

where D is the coefficient of diffusion. Typically D can be related to t he mean free path l amd mean sp eed v of particl es carrying the fiux , by 1

D = "3v l .

25

More det ails of the derivation and use of these equat ions, and of st ellar structure awl evolution in gen eral, may be found in e.g. the book by Kippenha hn & Weiger t (1990). It. should be evident from the above discussion of the origins of the st an dard equa t ions of ste llar structure that, if the st ar is not stat ic or not spher ically symmetric, it is ap pro priate to ret urn to t he full fiuid dynami cal equations t o ob tain equations ap propriate for modelling t he star.

2.3

Small Perturbations about Equilibrium

In ma ny int er esti ng instance s, the motion of a fluid hod y may be considered t.o be a small disturbance about an equilibrium state. Suppose t hat in equilibr ium the pr essure, density and gravitat ional pot ent ial are given by P = Po , P = Po , 'l/J = 'l/Jo (all possibly functions of position , but independent of time) and u = O. Using Eqs. (2.5) and (2.3), t he equilibrium quant it ies sat isfy

(2.25) (2.28)

In the pr esent case of radiative diffusion of heat , t he particle velocity is the sp eed of light (which in this subs ectio n only we denot e by c), and ast rophysicists describe the mean free path in te rms of a mea n op acity t o radiatio n (1'), thus l = (I' p)- I . The dens ity of thermal energy in t he radiation is U = aT 4 , where a is t he radiation const ant . T hus finally we ob t ain t he four th differe nt ial equat ion aft er some rearrangement to make dT/ dr the subj ect of t he formula as dT

- 3 1'p L

dT

167facr 2 T 3

(2.26)

in regions of the star where heat is transported solely by radiation. In convectively unst able regions the heat transport is by convect ion (or some comb ination of convection and rad iation) . If convection is very efficient , this typ ically leads t o a strat ificat ion that is ver y close to marginal st ability (see Section 4.1.1) in which t he temperature gr ad ient is instead given to a very good approx imation by dT dr

(2.27)

Suppos e now t hat t he system und ergoes small mot ions about the equ ilibrium state, so p

= Po + p' ,

P

= Po + p' , 'l/J = 'l/Jo + 'l/J' ,

(2.29)

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

(Po + p' ) (~~

+u

. \lu) = - \l(po + p' ) - (Po + P')\I ('l/Jo

~(po at + p') =

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

+ 'l/J')

, (2.30)

\l2('l/Jo + 'l/J' ) = 4nG(po + P') . We suppose t hat the p ert ur ba tions (t he primed qua ntities and the velocity) are small; hence we neglect the products of two or 1110re small quant it ies, since t hese will be even smaller . This is known as lineari zation, because we only retain equ ilibrium terms a nd t erms t hat ar e linear in small quantities.

A strophusical Fl uid Dynam ics

26

Si m p le M od d s

This simplifies the above equat ions to

au

Po Df; = - \7 (Po

ap'

27

T he lineari zed form of this equat ion is

+ p')

(pO

+ p') \7'1/Jo

- Po\7'1}/ ,

-iJ,l,/ -I- u , "v 1Jo iJf

(2.:n)

- \7 . (Po1L ) ,

Dt , \7 (1jJo +'l/J ) = 41TG(PII -I- p') .

= Po - (() - (I' + 1/. ' \7(111 ) . Uf

PII

(2.:17)

2

Subtracting equilibri um Eqs. (2.28) leaves a set of eq uat ions all the terms of whic h are linear ill smal l qu antities:

Dn

(2.32)

p075t

Dp'

at \7 2 '1j/

= - \7 . (Po1/.) ,

(2.:1:3)

41TGp' .

2.3.2

Adiabatic [iuctuaiion»

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.im escalo of the p ort.url iat.ious. 'I' hcn we can Ray t hat. over a t.i moscak- T 1.h(: hea t gai lled or lost by a flu id eICIIH'llt is zero: (KJ = O. By E q . ( 1.2H) th is iUlplies t.hai

(2.:34)

dp

d ]J

1" --

]1

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 equation to clos e t he sys te m : t hat eq uation com es fro m energy considerations. In generality, we sho uld perturb t he ene rgy equation (2.4) in the same ma nner as Eqs. (2.1) --(2.3). But t here ar e two lim it ing cas es, isot hermal perturbat ions a nd adi ab atic p ert ur bations, which are sufficiently common t o be very useful ami are simpler than using the full p ertur bed equation (2 .4) because t hey don 't involve a det ailed descript ion of how E an d F are p erturbed .

Isoth ermal flu ctuations

Let t he typical time scale and length scale on whi ch the perturbations vary be T and A, resp ectively. Suppose that t he timescale on which heat can be exchanged over a di st ance A if> mu ch short er than T. Since heat tends to flow from hotter regions to cooler ones, efficient heat exchange will eliminate any t emper a t ure flu ctua tion s. Assuming a. per fect gas , pert ur bing equation (2.12) gives dp

dp

p

p

For iso t hermal fluctuation s , dT mater ial deriva tives,

=

dT -I-

(2.35 )

T

O. Hence dp jp

Dp

pD p

Dt

pDf

d p j p.

(2 .:11-\)

or in terms of materi a l de ri vati vos 1)]1

, ]1 Dp

Dt

fl

(2.:39)

i»

The lineari zed for m of t his equation is

-D,p,' +

of.

2.3.1

.

P

n

1l. . v

Po

= -, Po (D-.-f/ + 11 · \7 Po) Po

Dt

.

(2.40 )

(In the last equation, is al so an equilibrium qua nt ity because we have linearized , bu t for clarit y the zero su bscript has heen omittod.) We see t hat t his is of t he same for m as Eq. (2.37) hut wit h an additional facto r At. The adi abatic approximation will gener ally be a good on e when con _ sidcri ng dynamical motions of e.g. t he deep in teriors of sta rs, whor« the dyn am ical t imesca le is much shorter th an the t hermal timoscal« . In j.]ml case, differ en t iating Eq. (2.32) (t he linear ized equat ion of moti on) wit h respect to ti me, an d using E q .(2.40) t o elim inate p' and E q. (2.:33) (t he linearized continuity equat ion) /.0 eliminate p'; yields

In terms of

(2.36)

In the p en ul timate term, the eq uilibri um Eq s. (2.28) have been use d to eliminate \7'1/Jn in favour of \7vn.

2.4

Simple Models

Astrophysical Fluid Dynamics

28

Lagrangian Perturbations

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

lip

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

(2.42)

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

lip(ro) = p'(ro)

+

or· \7po

U

(2.44)

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

2.5

Sound Waves

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

'1//(1') =

j' -Gp'(r) dV , 11' -

shall do so in the remainder of this chapter. The ./// term is of course also absent in problems where self-gravitation is ignored altogether. However it is crucially important in the Jeans instability (see Chapter 10). Suppose now that we have a homogeneous medium, so that equilibrium quantities are independent of position (and hence in particular \7po = () = \71/!O)' Equations (2.32) and (2.33) can then he rewritten

8u

Po-.- = -\7p 8t

,

(2.46)

so taking the divergence of the first of these equations and substituting for \7 . u from the second gives ,)2 , (I

U

-a·~ =

t

2 ,

(2.47)

\7 p .

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

(2.43)

where the argument on the left is written 1'0 (rather than 1') and this is again correct in linear theory. Of course, Eq. (2.43) holds for any quantity, not just pressure. We note that, in linear theory, 81i II 8t = Do I I Dt (where I is any quantity). The material rate of change of the displacement or of a fluid element from its equilibrium position is equal to its velocity. Hence

Dol' 801' = Dt = 8t .

29

(2.45)

1'1

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

(2.48)

c6

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

8 2 p' 8t 2

2

=

2 ,

Co \7 p .

(2.49)

This is a wave equation (d. the I-D analogue 8 2p'18t2 = c6 8 2p' 18 x 2 ) and describes sound waves propagating with speed co. In fact, Co is called the adiabatic sound speed. If we had instead assumed isothermal fluctuations, we would have obtained a wave equation with Co replaced by a, the isothermal sound speed; d. Section 2.1.2. One can seek plane wave solutions of Eq. (2.49): p'

=

Aexp(ik· T'

-

iwt) ,

(2.50)

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 meaningful solution.) Substituting Eq. (2.50) into (2.49), one finds that a non-trivial solution (A i= 0) requires

A s tro phiisi cal Flui d D ynam ics

S im ple M"dds

This is known as th e dispersion relation for the waves. It. sp ecifies the relation t hat must. hold betw een the frcq oncy and wavenumber for t he wave to h e a solution of t he given wave eq uat ion . W ith a sui table choice of ph aso, one can deduce from (2.50) t h at

,

cos(k . r - wi ) ,

P

(l

]I'

n c~ eos(k . T:

-

wi.) ,

(2.fJ2)

L

r'ir

- (] ~ k sin (k . l ' f! IlW L

-

w I,) ,

(2 . fJ:~ )

for so me const a nt a mplit ude rr. Not.e t h at th e adia batic pre ssure and den sity fluctuations a re in p hase, where as th e di spl ace ment r if> 7r / 2 out of ph as e . A sound wa ve is call ed longitudinal, b ecause the fluid displacement is p arallel to the wave number I.~ .

2. 6

whe nce

f( z ) = Aexp(kz )

+-

l3ex p( :-k:: ) .

The fluid is infinit ely d eep , and t he solution sho uld not l)(~(:()]lJI' infinitr- ;IF: z ----> - 00; h onc« 13 = (). The bounda ry co nd it.iou a t. tlH' frr~e :-;urfal'f' is Ih at 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, 11 1(' sur face,

011 t he ot her h and, taking; Lllf' do t product of E q. (2.rd ) wit h c;, an d using; Eqs. (2.56 ) a nd (2.G8) with B = I), yi<'1 dF: (2.(jO)

S u r face G ravity W aves

As a se cond example of a sim ple wave solution of the lin earized p erturbed fluid equat ions, cons ider incom pressible motions (V . u. = 0) of a fluid of const ant density Po which oc cu p ies 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 gravit y 9 = -ge z is uni for m a nd points downwards , and that self-gravity is negli gible. Thi s is a reason a ble model for ocean waves on deep water, for example. Equation ( 2 . :~ 3) implies t h at p' = O. Hence Eq. (2.32) be comes (2 .54) a nd t a king the divergence of t h is gives (2 .55)

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

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

:{ I

(2.56 )

(wit h k > 0 for definiten ess), where f is an as yet. unknown fun ction; a nd wit ho ut loss of gener ality k > O. Sub stituting this into (2.55) gives (2.57)

P-l1CT7/wh cr c. Hen ce the bo u ndary con dition (2.59) can only 1)(' sarisfiod if w a nd k satisfy t he disper sion relat ion

(2.n] ) It is clear that these ar e surface waves: for t.hc p ortu rhorl quant.iti cs a ll decrease expon entially with dep th . In rea IiLy, of co urse, th e f uid canu: 1j, 1)(' infinitely deep , so B is not identica lly zero. In stead , A and 13 will hav e {,o he chosen such that sonic boundary cond it ion is sa tisfie d a t the bottom of the fluid layer. However , provid ed the dep t h of t he layer is much greater 1 than k - , it will gene rally h e t he cas e that B has to b e much less tha n A . If the layer has depth h , a nd t he conditi on at z = - h is that the ver ti ca l disp lacem ent is zero, B =I 0 a nd by t he first. pari of E q . (2.GO) th e lower bounda ry condition a mounts to requir ing that Dp' / G.: = 0 t he re. All the above equat ions hold , excep t the last pa rt of Eq. (2.60). It follows that 13/ A = exp( - k h) and t h e dispers ion relation is w

2

=

gkt.anh kh .

(2 .62)

In t he r egim e kh. » ] ( "deep layer" ) t hen t h is is approxim ated by Eq . (2.61) . In t he opposite limi t of kh «: 1 ( "shallow layer" ) the disp ers ion relation annroxirnatns t.n I,J 2 = (nhlk,2 i P I . ' = , r;;r; L·

32

Astrophysical Flu id Dynami cs

, 'W'

f ,

i

S imple M odels

,.~

33

-~.

2.7

. ,: ,

j

ii

I! !

f

i,

I I

II' j'

t

I

I ,

Phase Speed and Group Velo city

Before leaving the to pic of waves it is worth noting two differ ent concepts regarding the speed at which waves prop agate. Consider a wave wh ich is locally a plane wave , prop agating with wavenumber k a nd with frequ ency w. T hese two quanti t ies are related IJy a disp ersion relation , so w = w(k ). Such a wave is proportional t o 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 ph ase. One concept of the speed at which a wave propaga tes is t he pha se speed. Compare t he wave at some locat ion x and t ime t with the wave at a location x + n6.x and slightly different 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 first location and time if 6. :,{; = (w/ k . n )6.t. Henc e we refer to w/( k . n ) as the phase speed in dir ect ion n. In par ticular, the ph ase speed in t he xdir ect ion is Vp h x = w / k x , an d likewise for t he y- and z-direct ions, provided the waven umber has a non-zero com po nent in t hat direction . Sometimes the phase velocity is defined to be a quantity wit h the direction of k and magnitude equa l to the pha se speed in the direction of k ; but it should be noted th at the x-, y- and a-components of this 'vector' are not in general the same as the phase speeds in t he x- , y- and z-direct ions . Note also t hat in directions a lmost perpendicular to k (so k . n almost zero) the phase speed can beco me arbitrarily lar ge; but t his does not correspond to any physical transp ort at t hat spe ed . A second concept of t he speed of a wave is the group speed or group vel ocity. A packet of waves of different wave numbers but similar to k o say prop agates physically at a veloc ity v g given by Vg i = Dw/ o k i , or in shorthand v g = ow/D k , evaluated at k = k o. This is the group velocity . We can also speak of the magnit ude of t his vector as t he gro up speed. For a proof t hat t his is indeed the velocity at which a wave p acket would propagate, see for example t he book by Lighthill (1978) . This is t he velocity at which wave ene rgy propagates. In the case of pure sound waves, it is st raight forward to sho w fr om their dispersion re lat ion (2.51) that the gro up speed is Co and that the phase speed normal t o the wave fronts is also Co . Hence in t his case these tw o are equal. In t he cas e of sur face gravity waves considered in Section 2.6, the phase speed normal to the wave fronts is w/k, but differentiating t he dispersion re lation (2.61) gives that the gro up sp eed is on ly half of this, so in this case the two sp eeds are not equal. Although in t he case of pure sound waves t he gro up velocity is in the direction of the wave number, t his

{~ ~.

2.8

2.8.1

Order-of-magnitude Estimates for A s trophysical Fluids

Typical scales

A Elyst em call often be charact erized by a typic al lengt h scale L , time scale 'T and velocity U . T hese are usu ally related by U = £ /T. T he appropriate length scale E for a part icular motion may be different from the size of the who le syste m - e.g . for sound waves, L might be t he wavelength , 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, the time scale is T ~ (£/g)I / '2 . For mot ion in a star's gravitatio nal Held wit h g = G111/ R'2 , where R = £: is the radius of t he star an d III it s mass,

T ~

t dYlI

=

R:3 ) ( GIll

1/ 2

ex (mean density) -

, I/2 .

(2.63)

This is t he typical time sca le for oscillations of e.g. Cepheid variable stars. is called t he dynamical t imescale.

td Yll

2.8 .2

Importance of viscosity

Molecular viscosity, which provides tangent ial forces in fluids, comes about microscopically be cau se molecules from faster-flowin g fluid diffuse into slower-flowing fluid , and vice versa . As can be seen from e.g. Eq. (1.9) , the viscosity 11 has dimensions M L -1 T - 1 where ~M, L, T denot e mass, length and time respectively. Let the mo lecules have mean velocity v and mean free pat h l . Then on dimensio nal grounds , fi

~

pvl .

(2.64)

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 ial force at a plane int erface between two fluids moving at different speeds, assuming that such force comes about by molecu les diffusing a distance of orde r l across t he bou nda ry at a sp eed v and dep osit ing their mom entum in the new environment , noting the relation t hat force is equal to rate of change of momentum. To make fur ther progress, we need to relate v and l t o macroscopic properties of t he fluid . If t he collisional r.rnss-spl"t,inll for t hp mn! p('ll !p<: j c

34

871H1,1" M on d s

A st roph ys ical Fluid D yna.m ics

(not t o be confused wit h tho stress to nso rl}, and t he nu mber den sity of particl es is 11 (so that. (J = 11m , where In is the m ean molecul ar mass), t hen 0 11 average hot WC Ull collisions a particle sweeps out a cylind er of vol um« a] and thus such a cylinder mu st. contain 011 aver age on e particle: nlo ~ l . Hen ce I ~ tn.](J (Y . TI l<) mean kinetic energy of a molecule ~ k nT, when, k n is Boltzma nn 's cons t.ant., so (Y

'V

rv

(

kHT)

-

1/ 2

-

(2.(i[) )

In

Thus p.

~

(kJ3 m) 1/2

J/ 2

T.

(2.GG)

(Y

th e left-h a nd sidp. of S q. (1.9) to tho viscous term

(11.1"2 / 1: jill I £"2

I!m . Y u l

IIJV

2

ul

rtf:

0 11 t.lIl '

righ t- ha nd sidl':

HI' .

II

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 in side th e S1I1 1 ( ~ ](I " IUS-- I) . Hi ' CV HI I" , :\lJd ;;0 rv cu for subs t.ant ia lly subsonic Hj H'(,ds tl w H.eYllolds num b er is g('w' n illy very much greal.! ~r t.luu: unity. T his shows once ag ai n (.]mt n rolocu lnr V iS ( T O IlS effects are gencrally negligib le ill the std lm «on l.cx t. However, smnll-scal« tnr lmj(>nt. flows ca n h ave a n dfcd . on the mean lar gc-scale mot.ion sim ila r to t.ha! of viscosity: th is is known as t.nrhul cnt viscosity . Jt mnv woll 1)(' th e source of " v isr.osi tv" i ll JlII l.lIY a.c;1.rop liy si, ·;I! \'is l"l 'lI:< lWITl'i. io Jl d isks. IiI]· exa mp)('.

For a crude est im a te, o ~ (radius of a!.o lll? , so a bout 10 -- 2 1J1ll 2 . (This undercst.ima te s (Y in an ioni zed gas, where electrom agnetic inte rac ti on s arc im portant .] T he m ean mass t n. = ii.m,,, whe re ji here denotes t he moan molecular weig ht and m il is the at omic ma ss un it . Assu ming reas on abl y t hat a ll the constants im plied in t he ,~, relations above are of or der unity, this gives (2.G7) where T is in Kelvin and {J is in kg m -:i . (See Appendix A for the values of p hysical rous t.an ts. ] Now the left-hand side of t he Navier-St okcs equa t ion (U J) is pDul Dt . while a typi cal viscous te rm is ILy 2U . If viscosity were dominan t , then t he tim esca le of moti on wo uld be det ermined by its effect: (2.68) or, re a rranging, Tv ~ 1:2 / ,1 . For t ypi cal ste llar values (1' = lOGK , (I = lkgm- 3 , ji. = 1, I: = 10 8m ) we ded uce usin g (2 .67) tha t t he viscous timescal e is of ord er 10 2 1 s ~ 3 X 10 13 years. Even for a star t his is a very long time , so mo lecula r vis cosity is unlikely to be important on st ellar scales. T his will ge nerally be true for ast ro physical fluids, t hough some form of viscosity is imp or tant in e.g. accret ion disks (see Ch apter 9) . A com monly used mea sure of th e importan ce of viscou s effects is t he Reynolds number R.e, which is the ratio oft-he ad vect ion te rm (impli cit) on

2.8 .3

The adiabatic opproximaiion.

Suppose Tp is t he ti mescale Ior the tra nsfer of heat. (11.Y flux F ). if t his is much greater th an t he timescale of th e mot ion U W Il (1 )(' G I ll t.n'at. t.he motion as H.dial>at ic (oQ = 0). This is th e adinbati« nppro ximatiou . Feu Ll H' SIIIl , 7/.. ~ 10 7 years ill d IP. in l.erior, and a bou t ono dav noar t.li« snrfnr« . Th e fund amental 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-nt for describing oscillations of the SII11 . Til th e solar a t.mosphore , howeve r, T» c all he much shorte r , in fact so sho rt t hat there are Rome circ um st.au ro» in which one can treat t he mot ion as isotherm al (Section 2.3).

2.8.4

Th e approximation of incompreseilnliu]

Th e flow is incom pre ssibl e if Dpl D t = (J fur th en the density of a fluid element docs not change with t im e. By the conti nuity equa tio n (2.2) th is is equivalent t o Y . u. = O. (Some authors prefer to take Y . u = 0 as I.he definit ion of incompressibi lity, a nd lJ[II Dt. = 0 as t he conseq uence of th at. ] Ro ughl y speaking , the cond it ions for t his to hold a re that U is mu ch less th an the sound sp eed and that I: is much smaller than tho pr essure scale height. 1

H I'

= ]1

I

lip 1--dz .

(2.70)

Astrophysical Flui d D ynam ics

For exam p le , for the Earth's atmosphere Hp ~ 10 km . Of course, co m pressibilit.y cannot. b e ign ored for m odelling sound waves!

C hapter 3

Theory of Rotating Bodies

Most if not all obj ects in t he universe rotate , a nd t he effect s of rotation are impor tant to a n understanding of the s tru cture a nd dyn am ics of many astroph ysical syst ems. Rotation is indee d suliiciont.ly iiuportun t to the subject of astrophysk al tluid dynamics th at we ret ur n to it several times in ad dit ion to t he present chapter: in Chapt er 4 on fluid instabiliti es, in Chapt er 7 on th e dyn amics of planet ary atmospher es, in C ha pter D on accretion disks, and elsewhere . T he pr esent chapter est ablishes the equa tions of motion in a rota ting fram e of reference a nd considers t he equilibrium st r uct ure and shape of a slowly and unifor mly rot a t ing st ar (or gas eous plane t). Vo/ e shall also cons ider briefly t he int ern al dynamics of a rot at ing star, and some conseq uences of orbital rot ation of stars in a binary sys te m . A great deal of resea rch h as been made into eq uilibria of rotating bodies, particularly in t he case of bodies with unifo rm density, by such illustrious names as Lap lace, Jacob i, Liouville, Riemann , P oincare , Lord Kelvin and J ean s. Much interesting d et a il of the results and his tory can be found in e.g. Lyttleto n (1953) , Lebovitz (1967), Chand ras ekhar (1969) and Tassoul (1978). Just to give some brief hist oric al context, we mention that in the ease of bo dies of uni form rot a ti on t here are two families of equilibrium configurations. One consists of t he Maclaur in spheroids : these are axisy rnmetric configur ations . T he second family con sists of t he J acob i ellipsoids, which are non- axisymmet ric. When the rot ation is sufficient ly fast , as measur ed by t he quantity 2 / 27fGp where p is the density, then t he Maclaurin sequence terminates and for faster rotation the only equilibriu m configurations for homogeneous bodies ar e the t ri-ax ial J acob i ellipsoids.

n

;17

38

Astrophysical Fluid Dynamics Theory of Rotating Bodies

3.1

Equation of Motion in a Rotating Frame

When considering rotating systems, it is usually most convenient to work in 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 of change as measured in an inertial (nonrotating) fi'aI1W, and d/dt to denotp rate of change as measured in the rotating frame, then

mmetry. For slow rotation, the distorted body is axisymmetric about sy as one would expect. Although,. we shall the I'O t a t',Ion axis' , " . not. COIISHlpr " .. f, "or rotation can give rise to SOIlIn surprrscs, notahly th« Jacobi it here, dS",. , . ..., " .., • ," . . '1'S w.hich figures of oquilll niurn. For. a , h.Jllu (XjH)Slj]OJ! elhps()]C , are triaxial ' . of ,e", " . ~ the 1 S111·).1C . .ct.L, ,0. C'('(',I, 'J'J('.' classic texts hv.. CJmudras('khar (I%(J) and Lvtt.lr-t.ou

(195:3).

Dr Dt

dr

-dt +

Oxr

(3,] )

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

( ell.~

+

nx)2 r

d2 r dt 2

+

dr 2fhdt

nX(Oxr),

(:3,2)

n

Now in the inertial framA, Eq. (2.1) is the equation of motion; so substi2r/Dt 2 tuting for Du/Dt =:: D from Eq, (3,2), and identi(ying dr/ell. as the velocity as measured in the rotating frame, gives the following equation of motion in the rotating frame: 1 --\7p p

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

(3.3)

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

3.2

"

.'

' , ' '"

'I"

._

(:1 01)

+

where in the last step we have now assumed that the rotation rate does not vary with time. There are some subtleties to VActors in rotating frarnes and calculating their rates of change, and the reader who would like more details is referred to Chapter 3 of Jeffreys & .JefFreys (1956).

du dt

,

vVe work in a frame rotating WIth the body: III that h eUI]( 1, H. ,c(jm .' . , described lrv Since we arr: llJoddlmg a hbnuIll IS.,o .. .1 u = 0 and CJ/Dt = • 0, • mhorical coordinates (,., (),, 4)). ncar 1y sp . . configuration, ' . , we use spherical polar " . ordinates In t Irese co ) e, , " , WI'l' ting t. • " (where e • IS a unit vector alollg the 0 as., i le., z polar axis,

Tns . "velocit.v , , ',I of tli« fiuid as seen from the nonrotating Iraino, if 1 · IS ' J.usr'j the it IS , a'.j rc.. st in the rotating' frame. J'vlon~(Jvcr,

- Ox(Oxr)

(3.5)

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 the above would he simpler in cylindrical polar coordinates (tv, ¢, ;;) since coordinate tv (pronounced "pomoga") is the distance from the axis: Eq. (:3,5) would become

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

2 2) tv

.

(3.G)

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

\7p

Equilibrium Equations for a Slowly Rotating Body

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 in particular the case of a slowly rotating star; but the equations apply equally well 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

n2 7' sin 2 () e, + 0 2 , sin () cos () eo \7 (~n2r2 sin ' ()) ,

- p\71> ,

(3.7)

where 1> =

1 2 r 2 sin ' 28 1/) - 2'n

(3 .8)

is the total effective gravitational potential (gravitational plus centrifugal). vVe can argue qualitatively from Eq, (3.5) what the effect of rotation on 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

40

A st rophysical Fluid Dynami cs

Theory of R otuting B odi es

t~lC equa tor ((j = 1T/ 2). It thus re duces the effective gravitational a ccelera_ tion at th e e(~uator, i.e. t he centreward pull is not so st rong there as at t he pole~ and so .m stea d of be ing sp herical the bo dy "b ulges" at the equa tor. C ' . ~ ~le gra(~lCnt vector \7 f of a ny sca la r is p erpend icul a r to surfaces of _on,s t ant f ,. so a norm al n to the surface of consta n t .f satisfies n x \7 f = 0 It follows from Eq. (3. 7) t hat surfac es of constant p a re also surfaces of con stant !P, and vice ver sa, T hus we ca n write p = p(p ), an d so dp - \l p d1'

=

\lp

= -

(J

so

(J

is abo

11

Iuuct.ion of

q"

i.e, p

=

dp -d p ,

p( q,).

" I, 'J-' -

1 n2 .2 "

2 () .

sm IS constant. Let us approximate the gravitat IOnal potential t/J by what it wou ld be in the nonrotating cas e: "

'

-

2'

1

1/-'

= _ OM T

(3.11)

at t~~e,s~rface a nd o~tside . the star . This is equivalent t o approximating the gr ax itational potential as If all t he mass were at t he centre a nd is called t he R och e model. T l: is is ~ reason a ble approximation in the case of a centrally condens ed st ar , III which mos t of t he m ass is concentrated nea r t he centre . \ Ve suppose t hat t he surface of t he rotating st a r is described by

r = R(l

+

R(l

(:U:3)

+ f( B))

is constant (i.e. indep endent of B). The rotation is slow a nd t he distortion weak , so rfl and e ar e small and we neg lect pro du cts of smal l qua nt iti es . Then (:3.13) im plies tha t

OM - - ( 1 - f (O) ) R

f (B)) ,

f(fJ )

(:3.10)

The Roche Mod el

O n the surface P -

OM

q) S l lrfac(~

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

(:3 .14)

is indep ende nt of 0, i.e.

, , ~:Iencef~)rward , for definiteness, we sh all speak of t he b ody as being a star , but, It could equally be a gaseous pl a net , for example. T he outer surf~ce ~)f the star is a s urface of constant pressure (b ecause t he pressure out side IS const a nt, say zero) and so
3.3

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

(3.9)

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

41

(3 .12)

1

n2 j { 3

. 2

2" G !YT sin 0 + constant.

(3.15)

Note th at n2 R is t he equat orial acc eleration due to cent rifugal for ces ; an d GMI R'2 is the gravitational accelerat.ion. So, the di me nsio nless quantity 2 R 3 I (OM ) is the ratio of cent rifugal acceleration to gr avitation al ac cel-

n

era t.ion . T he radii at t he p ole and a t. t he equat or a re obtained from Eq. (3 .12 ) by pntting = 0 a nd 0 = 1T12 respectively. T hu s t he relative diffe rence between equat orial a nd p olar r adii is

e

R( l

+ f(1T12))

- R (l

+ f(O))

R

] n2 R 3 2" OM .

(3 .16)

Thus th e relative difference in radi i, which is a measure of t he shape distortion, is n 2 R'J I (0 fl.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 sho uld properly use the gravitational potential app ro priate to the distorted st ar . We proceed to do t his now.

3.4

Chandrasekhar-Milne Expansion

The missing ingredient in t he previous sect ion was a proper treatment of the gravita tional p otenti al of the dis to rted star. In t he Chandrasekhar-M ilne expansion, on e cons iders the O(n 2 ) p ert urbation not onl y to the shape of t he st ar b ut also t o it s gravitation al potenti al. T he procedure is des crib ed in more detail in Tassoul (1978) . We know that on the surface
42

A stmphysical Fluid Dy nam ics

Theoru of Rotating Bodi es

is straightforward to demonst.rate t hat

Bence Ell " ( .'{. ].8) b ecomes LJ 7rG1

(

I ) - :m:l "

Pu +
1l'1I

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

2

.1 7f 1~

Th e zero-orde r terms give \7 1/)"

Co u , ' whi le t he first-orrl er tori ns .i ].

yield The problem invol ves th e \72 oper at or , so it is mo re natural to write j,]1(. 2 B-d ep eJl(len ce not as F>i1l () but ill te rms of Legendre pol Yllomials of cos 0: P" (cos 0) , since

(:\.2(;)

Also,

v

l'''P', (cosO)

=

and

V = 1" - (n+ JJp,, (cosO)

=

1,

:J: ,

=

]

c

is constant

h. ' aft er expandin g is illdepml( lellrt,0 l' O', whic . Taylor series expan sion , gJvos

2

2" (.h: - 1) .

+

(j)1 (1',O) ,

p

=

p,,(7')

+

P'(1', B) ,

(3 .22)

wh er e 'lj)" and p" a re the gravit at ional p ot ential a nd deusity in t he sp he rically sym met ric, nonrot:ating star , and t he primed quantities are smal l perturbations, of order [2 2 R 3j (G.M ), induced by the rotation. As before, we ex press the ste llar sur face as 7' = R (l +E(B)), wh ere E is the same order as t he other small perturbations; and we neglect product s of small quantities . Recalling t hat p = p( (j)),

Pu

-+

P1 = P( 1/'"

=

dO)

Let us write

'1/;,, (1")

-+

') =

) P( 1/)u

+

dp q)1 dq)

I

_ ~R (~h/'" I) dr R

(,3 .2(J)

(3. 21)

(3.23)

. "lj,u

the surfac e, so (:\.27)

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

Thus (j) may be rewritten as

(j) =

Oil

(3.] 9 )

are solut ions of Lap lace 's eq ua t ion , \72V = O. The firs t three Legen dre polynomials are Po (.1:)

(j)

l j!, ."

,' jS,,

the firs t two j.('nus of a

I (l/ (R J i) ( -j- r:o lls !: 1l11) "

" for ij>' : on ce (1)' . (3 26) is an lnliomogencous differe nti al equation Equat.lOn " ,. f E ( '~ 28) I ' f the surface follows rom ' q. . . . I , I ti I1111F>t h e regul ar in t he interior , is found, the s, iape o . . " ' rt " q)' is t h at t i e so u ,IOn " . f 1 , I cond it ion , we mu st ensure th at On e cone I ,IOn on " . ti lar at l' - 0 To llll a seconu c ' . - . . I " tll v o n t o t he oxtcr ual gravi!<1and III par ,I ClI . ( "J' t'at,) ' 011' 1 1 po tential 1 /' m ate ics smo o ' ) .. . the gra" , ., ,. ' . I tional fiel d. Ou tsid e t he s tar,

GM '1/' - - -' -

+

T

00 "L..-J A n (~) n+1 P,, (cos(1)

11.=0

(3 .20)

T

. 0 f 0riel' since \72'1/' = 0 ther e by E q . (1.15), where A " IS ( . [22fl3 /GM . Insid e the st ar ,

1/' =

(j)

1 2r 2 {I + -[2

- 'P.2(eos(O) }

=

1/.!11 (1') -I- q/

+ ~n21'2 {l .3

- P2(COS(0)} )

3

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

(3.30

To allow complet e gene r ality, we would now wr it e 00

00

(3 .24 )

<]/ (1', B) =

L qJ n=()

n(1' )Pn (cosB) ,

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

(3.31)

where all fun ctions are eva luated at r = R . Now rec alling t hat

(see Tassoul 1978). However , to avoid needl ess algebra, we note that the problem for uniform rotation has only P'2 (cosB) and Po(cosB) angul ar dependence "-- see Eel"u . (o J ')6) an ' d (3 .30) -- an.d so we anticipate . . v.. the solut ion to be

qi' (r,O)

= Po(-r ) +

P '2 (r)P2(cos B) ,

E( r, B)

=

EO(r )

+

R (l

ef'l/Ju 2 d'ljJu __ ,_ + -- -_ dr '2

r

dP'2 __ dr

+

3 ;r R

- ~z

1 G P'/l ' L7r

tl7rR'2;r

-- p u ~ '2

M

5 n '2R . 3

- H

(:3.39)

= R. Equatio lJ (3.39) pro vide s the surface boundary condit ion t hat mu st b e applied to the d ifferential equat ion for P z· Taking just the P2 (cos 0) t erms

at

'I'

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

\

~..

,

- G Ad

dr

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

E2(r )P'2 (cos B) ,

+ EO + E2P2)) +

1/.'u obeys

poisson'S equation so

'. . . . (:3.:32) and similarly for th e external field (3.29) . The surfa ce b oundar y cond ition 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 ouo of 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 . ' '1/)'11 (

45

Theory of R ot atin g Bodies

A st roph ysical Fluid Dyn am i cs

'

'II.

1. "

~\ = d
1L

clp'/l / d'ljJu

dr

(3.'1 1)

dr

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

'ljJ ,, (R )

~i.

(3.34 )

dr

(rzd z) _ .Q. z = _ 47r.,.z dPu '2 dr ".2 rn(r ) dr '

(3.42)

with boundar y condi tion (3.39) at r = R and z regular at r = O. In t he general case it would be neces sary t o solve the ab ove equation numeri cally. However , in the spec ial case wh ere Pu is cons t ant, it P is straig ht forward to find t he solut ions of Eq. (3.42) in the form z = Ar for constant s 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) that 02 = - (5/6)0 2 R 3 / GM. Hen ce the difference b et ween the equat or ial and polar radii , divid ed by R, is (5 /4)0 2 R :3 / G M in the case of a homo gen eou s

Similarly, cont inuity of a ljJ / ar implies

To zero-order , these two equat ions give simply d'l/Ju (R) dr

rZ

GM R2 .

(3.36)

The first- order terms propo rtional to P2 (cos B) give two equations which after A 2 has been elim inate d between them , yields ' (3.3 7)

stellar model. 3.5

Dynamics of Rotating Stellar Models

We have not so far considered how energy is transp orted in the ro t at ing star. A well-know n result, whi ch is discus sed at leng th by Tassoul (1978), is that one cannot have a uniformly rotating star in strict radiative equ ilibrium .

T hc01'Y of n ot,ettin g Bor/i cs

A st roph ysical Fl uid Dimam i cs

46

·/l

Assuming t he co ntrary leads to what is kn own as von Zcipel '» po'/'aclox, The same is t rue if the ro t.atiou rate is a Iunc t.ion onl y of di stance Iron, the rotation axis. \Vc co nclude ther efore th at tl w ro t ation rate must hH V( ~ a mor e gene ral form, dep ending on cy lind rica l polar coordinate z aR well as di stance fWIII the a xis, or t hat stiict radiative oqui libri m u (IoCR 1101. hold . \<\10. cons ider now the la tter possibili ty. The VOIl Zeipol paradox in dl'c'et says th at t.ho radiati ve Hux ca n not h e b al anced everywhere by th e (~Jl( 'r gy ge nerat ion . Som e regions have n net influx of heat: these will Iwat Ill) aile! tend 1.0 rise under 1moya ncy, O th ers will cool a IHI sink , This t end s 10 set up motions in m cridiou nl pl anos: thi s is ca lled m orid ionnl cir oulat.iou . It ca ll he S b OW Il , e .g . Kippcnhahn & \ 'Vl'i gert. (J ~!)() ) a nd Tussoul (J D7S). th at t.ho global t.imoscal o for mi xin g by t.he m eridional oir cul at.ion . k now n aR the Eddin gton-S weet timescale, is of order TIm/ X where TI<J] == GAr 2 / R], iRthe Kelvin-Helmholtz timescal e (L h(,ing Ow lumin osity) and X ~ 0 '2 R:I/(.'f\J . For tho Sun , T l( ll iR about lO 7 years, and X ~ 10 - " ; :-;0 t he E dd ing to llSweet l.imescalo iHa b ou t J Ol'2 yea rs , Il11H:IJ lon ger than the Sun 's age . Local circ ulation timescal os ca n 1Ir. much shor t er, however. For the Su n , the Eddingto n-Sweet timescale is much gren tc~r ( ~W~II th nll th e n uclea r tim escal e, hut this is not so for som e m ore ma ssive s tars . Yet t he o bservationa l ev idence does no t sup p or t t he idea t hat th ese stars a rc mi xed , as th ese tim oscal os would sug gest. The exp lanat ion (d . Kippcnhahn & Weigert J 990 ) is t hat mi xin g is opposed a nd stop ped by com p os it ion gra dient s (and hence gradients ill the mean molecul ar weight) . It should he m entioned t hat , al thou gh one CHn po stulate HOllie a rhitra rv rotation profile for the interi or of a st ar , this will not necessarily lie sta hl«, a nd hence will not necessarily be realizable in a real st a r. An exam ple of a sta bility consideratio n is the Rayleigh criterion (sec Section 4.3).

3.6

Solar Rotation

Deta iled observat ion s have been ca rried out of t he Sun' s su rfa ce rot.atioll rate over m an y years, by t ra cking surface featllreH such as su nsp o ts a nd. more recentl y, using Doppler velocity measurellleu t s. The rotation ra l:c varies wit h latitude, t.he equ a to ria l region s h avin g a rota t ion peri od of a ho ut 25 d ays while high la ti tudes rotate 1110re slowly, wi th peri ods in ex cess of olle mon t h. T he surfa ce rotation rat e is comnHm ly expressed in a n ex p a nHion jn cos '2 wher e is co-lat it ude, e.g . Snod gr ass (J98:3). The Doppl er r nt e.

e,

e

1

·

.

' 1 " -- '

,

l

I

E

•

i

i

1.0 :-\ : ~ o

:'-)(1

O.B

:IHO

':\1\0

o;

-,

~

{

\

f

I

( (\

i

)

I

\

I

,, 0 (\

0 .0

I

(

\

1~~t)

0.'1 \

\

\

\

\ \

..

C'

'

\

Ii

CH

I

I

I,

COl

I I

\\ \ !i

CUi

'

s

\I-

\

,I

,\i

\

I

O.B

1.0

T/R . ' S I '16 in ferred bv helio1"ei611lo1o!,\~" '1'1 ", (' O ll tOlll,S d e t he . ..111 , ' f' ](J II Ti le h o r i 7,Ol ll. :11 axis is i ll 1h e S un s Fl'g, :1,1 R ota t.ion r a t e] 1ll1"I n 7" , . f t1 I C :1 s p:1cm g 0 :1hell<,d in n llz aur iav " ' . . ' f 1 t.i a re in u ni ts 0 • IC l are ( .. .. 1 is i s the a Xl f-' 0 rot.a .i on , ]ljs. t.r'lI1<'e1" ~ ,. nnt.o rial pl all e , th e vc rt .icn aXI. , '. 'I t.1 \ ase o f th o Suns COI\ VCc\.IOl l zo ne , ~\I:' s phot osph eric radiu s. T he dashed lin e m a r ( S ie )"

. 11 Ct, 0..I. (, J98R) ' \R all!)]'llxi1ua tdv Ulric • ( , . ' " 'lll' is aj v(:n bv ,.' a lIjl , ' b for ('X n / 27f = (tiS]'!) - G5.:~ ('os2 (J - GG.7 ('OS·1 0) nll z .

(:U :{)

. " , .' . . 1 from t r aekin g di fferent fea1.11res do not agree The rot ntlOll rates dc1Cll!lllleC . . ,', l ' Sn uspots , for exa mp le , 1 t ' with c'1('h oth er 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 w h at c oep ei movement may ]) 0. mor e l!lcl!catlvc 0 1. ic 10 ,<1 .ion " 1 tl ots m ay he rooted . '. " " I. ' t (']I'ln O '('c! l .v 1Il111'l ' th an suhsurface regioll ".rlCl.· C • re sp I " 1 '. tl e sllr hcc 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 , , ' . G pC'1' C(' l Jf.. However, vnna\.~ons 0 ' • ' 1 1.'j'·j' \(I('~ t( > t he ('ll u a t,or with a " l . t g from nn( - ,I ,I ,I ,~, " zona l hands of fi.ow nng a ,1~1 _ . ' II d to rsiOll't1 osc.illa t ioJlS, thon gh " l'S . period of' a 1) 01 1t' 11 yc,), . . 1 hese " al e en e " thi s is a 1M of a mi sn omer. . " l ' l ' t h e p ast. The rot at ion rate of the intel·.ior its~lf hals l~eee":~ ~.:;(el~~e~~ci::': of ~lo1Jal · . ... . , imagmg' llSlJI g 0 ) SCI V . . ' ' two dccad es by 1Ie I10SClsmlC " ' . '. l ' r in ti ll' , .' '.. (S r ] 2 C)) 'Wf\VC's p l opag,1 m g .. , aeons !ic modes of t.he Snn see . cc ,10 n . ":, '

49

T heory oj R otating Bodies Astrophysi cal Flu id Dynamics

48

same dir~ction a.s the rotation have a slightly higher frequency than those propagatmg .ag amst the rotation, and the differ ence in frequency dep ends on the rotation rate. The results of such imaging in the outer 60 per cent or so of the .sola~· ill~erior are shown in Fig. 3.1. The outer 30 per cent of ~he solar I.Iltenor IS the convcctively un stable convection zone. In thi s re gion , the d ifferential rotation with latitude is similar to t ha t seen at the sur faee: so the contours of constant rotation are nearly radial. Only at low latitudes do we see somet hing like Taylor columns (see Section 7.4). By .c ont rast the radiative interior beneath the convection zone appears to be III a st ate of nearly rigid-body rotation, to the exte nt that it ca n be measured at present using helioscismology, Between the two regions is a layer of strong she ar , called the tachocline, There is also st rong radial shear in the region just beneath the surface. The hclioseisinic findings and their theoretical interpretation are revi ewed by Thompson ei at. (20(J3). .Youn g stars are observed to rotate much faster than the Sun, and it is believed ~hat l:itar~ lose angular momentum from their surface layers through stellar winds . TIns loss is only communicated to the st ellar interior if there are ways to redistribute the angular momentum inside the star. As we sha ll see in. ~haPter 4, shea r in a flow induces inst abilities (Sect ions 4.3, 4.4 ), so a ~u fflc Jently stee p rotational gradient would become un stable . Turbulen ce might then transport the angular momentum. Magn etic fields via t he ac t ion of Alf veu wav es , may also redistribute an gular momentum: A weak magnetic field may indeed be responsible for the nearly rigid rotation of much of the radiative interior and may also stop the spread of the tacho clin e gradient further down into the Sun. Including magnetic Lorentz force j xB and a viscous t erm 'D, the mo men.tum equat ion in a frame rotating with steady angular velocity no (which we take to h e the mean solar rotation rate) is

au _

Pat - - p(u · \7)11, - \7p

+

p\7iJ> - 2pn o x u

+

1

- jx B /10

+

V. (3,44)

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

D(1' ) =

no +

(u
rsin 8 '

(3.45)

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

ntum by the distance InOme continuity equat ion (1.3):

oJ at

7'

sin 8 from the rotation ax is, and using the

= - \7 . (.rr,IC

+

.rRS

+

.rEM

+

.rv ) .

(:3.46)

Here th e term on the right com prises (m inus) the
(1', e)-directions , and (3.4 8) the Reyn old s st ress t erm that arises from non-zero correlations b etween turbulent fluctuations 11,' == 11, - (11,) in the velocity in the ¢-direet ion and the other t wo directions . The remaining tenus·represent the transport due to electromagnetic Maxwell stresses .rEM

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

(3.49)

po

and viscous diffusion .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 lon gitudinal variati ons in p and 1/ . Viscous forc es ar e presumably negligible in the solar interior, and t he Max well stresses ar e also likely small in the bulk of the convection zone (t hough possibl y not in the r adiati ve interior and tachocline, nor in sunspots). Hen ce a steady-state rotation in the convection zone indi cates a balan ce betwen the divergen ces of the flux es of angular momentum caused by meridional cir cul ation and Re ynolds stresses due to turbulence. If the Rossby number is small and the flow barotropic , then the Taylor-Proudman t heorem (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 case in the sola r conv ection zone (F ig. 3.1) except p erhaps at low la titudes. This is at least partly du e to baroclinicity (\7px \7p =1= 0) driving a meridional circulat ion which red istributes angular momentum. Latitudinal variations in heat tran sport du e to rotational modulation of the turbulence cause a t hermal wind (Section 7.4); but also the Rossby number is likely not sm all

Theory of R o/. n.t.in g Bodi es

for some scales of motion in th e t ur b ulent con vec t ion zone, breaking th e conditions for t he T ayl or-Proudrnan t heorem to app ly. It is po ssible to model t he rot ation in the convect ion zone usin g nH'an _ field m odels (d . Section 5.3.2) but kn owin g how to prescribe the Re ynolds st ress es from the m ean -field velo city is a d ifficulty with thi s approach . Hecent large-scal e numeri cal simulations (e.g. large-eddy sim ula tions ) ca pt u ring some of the turbulen t nature of th e con vection ZOlW , can produce rotation profiles t h at a re q ua lit at .ivoly simi lar to what is b ein g found ill hclioscisniology (see Thompson ci al. 200;l for a review).

3.7

M,

x

I I I I

L3 poi nt

L, poin t

Binary Stars

Many stars a re found to h c in b in ary system», Tho orbit» of s1.;\.]"s in a clos o bin ary system tend t o becom e circ ula r ovor ti rno , due to t idal foru 's. Conside r a binary system in which the two componen ts a n' in circular orbits a b out t heir common centre of l WI. SS 0 , a nd in which nw t wo star s cor otat( , so as to always show the same side to the other st a r. In this sys te m th en> is a rotating fram e in wh ich the st a rs a re comp letely stat iona ry . If is tlw an gular veloc ity of each sta r abo u t 0 , in an inertial frame , then of course is also t he angular veloc ity of the rot ating frame. Suppose that t he sep aration di stance b etween the t wo st a rs is 0 , that. t heir m asses a re IH 1 , A12 , and that their re spe ctive di stances from 0 are 110 awl (1 - 11)0. Since () is t he ce n t re of m ass,

' 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',,1 1'1I1U1 " , .., . . , I " '\ ' I 2 1\ <;111. 111 r Olll!.I . " ) , ~ s1 nrs , 't n ' I 0( ,", \ 1,f '.( 1 .,l til", l)lIsi t.illll S illd lCa l ." . '," ,>.5 \I ' 'I'll · , . . ' III<' li tH' joi n ill!!. t h e two ~1a rs . . .' ..' " "1 " / L.) a lld /' :J, a l'l' La p;rHll)!;iali jl oJll1S, '. , T h e s tat iOll a ry p oint.s , t""hL<1t" , '" ' 1 , lili es.

" I • If!; .

.J

.) .

n

n

(-( 1 - II

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

,

(; A12

- GAl

q)

=

( . )" -I 1 2 + z2 .) ( :1: - flO - - )/

-

((,:1: + ( I _ /I) a )2 +)/2

---;-:-

+ :0; 2 )

- -1 Sl2 (x 2 + y2) , 2

(3.52)

I. r· 1 Here we h ave llHlde the same ap pr oxcan usc Ow ulldistorted gravi-, whieh is c<,\lIc, cl the Ro che po e,n]',la 1.'1 " t' . . 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 on a.] potent1a 0 eac 1 s ·< . ." . trat ell nC'IT t he cent re. . 1. f 11 e mass is conce n. c . • " " stars ill w1nch most 0 , 1 , .' : ~ :, 1 " fun ction of :r :1!OUg the lino The n oell(' p otent.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 " " o the me jOllllllg . ~ , TI . '11on y = "',' = (), l. ( ~ . , . h " . £ I ' , \7q) = 0 are indicat fld. . , lest' . . t s £ £ 2 and :), w lPlfl ' , . The Lagr angwJI P 0111 ' . I , , .' Ir , whe re t he for ces of attraetJOll g are e(lllilihrinlU points 111 the rO,t a.t m l ame, r o in l , ]'II('C t -if ., 1 force an ' III Jd a . .' towarcls the two stars a nd the ccn n u ga .,

Now Eqs. (3.7) a nd (3.8) hold for this sys tem in the ro t ating frame, where t he gravitat ional p oten ti al 1/; is given by t he sum of t he potenti als due to t he tw o stars . Choos ing Cartesian coordinates (x , y , z) such th at the ang ula r veloc ity of t he frame is in the z-directi on , wit h th e stars at (p,a, 0 , 0) a nd

Lar the surface of eac h st a r in the b inary ' . . 1 I th e surface p ot ent.1a ia] f 0 ' " _ .fac e of cons t,ant . Now PIO\,]( ec . , ' . ,. ,' . , ", . syst.elll is a 81U a ' . ' . . t1 ' £ l'1 gTang,-ian p oi nt, eac h st.at is 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 ." OCCllp leS a. we III . ie .0 .

(3.51)

Ii

Also the grav it at ional for ce on star .l towards star 2 (and hence t owards

0 ) mu st be equa l t o 111 1(p.a.) rl 2 , since 11.0 is t he rad iu s of it s circu la r or bit; hen ce it is straightforward t o show that

G( M 1

+ M2)

a:)

.

,1

' 1.1ie case of a sing e s ar, AS. III "

T heory of Rotating Bodies

53

/cs iro ptui sica! Flui d Dy nam ics

M2 Ml

M2 M1

I

I

I

I

I

I

I

I

~) 't~ ' ~lf" F ig . 3.3 C uts throu gh t he R oche p ot ent ial of st ars along the lin e j oini ng th e t wo stars illstrat ing three cases . (a ) The two st ars (ind icate d by h at ching) form a d et ach ed binar ; system . (b) One star has filled its Ro ch e lobe an d is n ow losing m as s to its co m p anicJIl star. (e) The two stars form a cont act. bin a ry in whic h t he tw o st ellar cores oceupy iu a CO Ill IllO U enve lope.

Fi g . 3 .4 A contour pl ot of t he R och e potential of two stars , in a pl an e co ntaini ng t he tw o stars. T he Ro ch e lob e of the star on the left is indicated by the ha t ching.

system (F ig. 3.33). Suppose though t hat 11,12 expands (perhaps at tempting t o b ecome a red giant) until it s sur fac e pot ential is equa l to (j) t. , (F ig. 3.3b) . Any fur ther expansion will ca use matter t o fall from star 2 t o st ar 1, since it will fall to the lower potential. Algol is an examp le of such a bin ar y. F ina lly, if the sur face potenti als of b oth st ars are great er than (j) L l , t hen

. " (Fig. . 3 .•3c) bins . " r "contact Innary , . It is "common enve1ope mary a . . . 3 4) ' we have a . t lot contours of constant (j) in the x - y plane (F Ig. . . . . ,tl'Uct lve 0 P . t l .tar can also 1115 . . 11 l : Roche lob e is the maximum rcgJOn . i c s .. sha ded reg lOl1, ca e( a. . ,.. . .' the I" it ar ts to lose m ass t o It.S compa.nlOn. belor ct 1 s aeeUpy (>

".

",

•

•

•• ••

···· .· '· ··· T·

·

Chapter 4

Fluid Dynamical Instabilities

It. is hardly po ssibl e to st udy astrophysical fluid dywlIlli cs wit.hou t «onsirl-

ering fluid dynami cal instahil iti cs. A fluid flow, either in reality or ill a !Hodel, may be un stable, in which case perturbati on s nmy grow qui ckly and change the fluid configu ration awl it s flow. Fluid dYlIami cs involve var ious ill st.ahiJiti ( ~s , which may profoundly a fl'I'Ct ! 11(' sl.I'1 I('I.I1 ],(· alld r-volut.iou or Hst.]'o physical obj ects ; A well known ('Xallll' !r' is t.hl: ('01l v('d i\,(' insl.ah ilit.y which lead s to convective core s and con voct.ivo onvolopes in ru nny stars , 1.1](' forlllnr aff(~cting their nu clear evolu ti on and tho l aH( ~]'l e adiJJ g to a vnricty of ph enomen a includ ing ma gn eti c activity cycles in th e Sun aw l ot her 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 her e. vVe mention a few others elscwh oro ill thi s hook: in particular t he J ean s instability in a self-grav itnt ing flnid is t]'(~ atcd in det ail in Ch apte r 10. Excellent further re ading on fluid instabi liti es arc t he hook by Dr azin & Reid (198 1) an d , particularly for rotating st ars, t he review by Zahn (1993) .

4.1

4.1.1

Convective Instability

The S chwar zschild criterion

Convection plays an impor t an t ro le in ste llar int erior s and planet ary atmospheres . Con sid er a fluid at r est with densi ty st rati fica t ion (i( z ) and with gra.vity g = - ge z acting "d ownwar ds" , so z increases upwards. The pressure distr ibut ion in z is given by hyd rostatic equilibrium . Let us now conside r what happens if a fluid parcel is displ aced slightly up ward s from height z = Zo by an amo unt We suppose t hat t.h e displace ment O ('C 1ll'S

oz.

55

s ufficient ly slowly that the fluid parcel rem ains in pressure equilibrium with it s new surroun dings . On t he ot he r hand we suppose t hat t he displacem ent occ ures sufficient ly quickly t hat no heat is exchang ed between the parcel a nd 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 a nd Po , Bay. At the new position z = zo + 6z , t he pressure a nd den sity of t he parcel ar e Po + 0]) and Po + op, say. Now at z() + 6z the pressure of the surroun dings a nd hence also of the fluid parcel is Po + 6zdp /d z to first ord er in oz ; so Jp = e5zdp /d z. T he density of t he su rro u ndin gs is Po + ozdp/d z . But lJy the adiabati c assumption, the parcel's pressure a nd density perturbations a re re lated by

Po +6p Po

=

(PO +6 P) ' . Po

57

Flu id D ynamical In stab iliti es

.• ~ ". Vl''' ysu;at Fttiui Vyn am i cs

(4.1)

. E xactr ' .t ly the' same crit erion . ith de t h t o b e stab le to convection. rapIdly WI p iderir 1,1 e p arcel moving downw ards inst ead, t I.lOu.gh 11 ' ult fro m consI errng 1 . . I WOll C res I ' t o rever se the inequality whe n dividing hy t ie must then rem em)Cl . one

" f ti. C' convec t ive ins tability, is thepotellti al energy (negat ive) e5 z . The ene rgy 8 0 m ce or . I , . . . . .: " mi unstable strat ific aticlll. . . of thIfethe Ollgn . , . IS ' . S·t·'a 1)Ie' t.0. t,he above cr it.erion then t he accelerat ion st rat IficatIOn . of the parcel is given by

~ .2 u.1: Z =

P dt 2

( J

f

UP) + 8 z~ UZ

9 -

(p

+

(j p) g

' . . wei,,·ht ) , where we now cons ide r oz to be a function (blloyallcy l'01'." LL llllllllS o of time: and h en ce

(4.6)

Hen ce , linea ri zin g in p erturbation quantities , th e density perturbation of t he p ar cel is

op

=

Po e5z dp ,

.f!.!!-.op = ,,(Po

' ''(Po

dz

whcre

(4.2)

T he p ar cel finds it self heavier than its sur ro undings a nd hen ce sinks b ack towards it s original loca ti on if

dp Po + op > Po + oz dz '

(4.5)

(4.:3)

i.e., if

N

2

1 dlnp _ dln p ) = g( dz dz

'1

=

pg (dln p _ ~) . p d In p "( 2

(4.7)

.f . . .'1, ad h ad b een defin ed to increase . , T .sed .) (Recall t h a t z increas es up wa r d S : I illS ea z ·1 " , of the two deri vatives in z would have b en re verse . downwan 1s, t re sign s .' . 1 f' ' N Thus < ' t ion (4.6) descr ib es simple harmonic mot ion WIt 1. l eque.n,cy,., " ~qlla .. " 1 can oscillate in the vertical dir ection ab out Its equ ilibri um POSIt .le pa~ cIL f. . N whi ch is known as the Brunt- Viiisiilii fr equ enc y or tion WIt 1 l equen cy , . I '1, aves in a . This is the m ech anism for int ern a gravi y w lJ'Uoyan cy requency.

f

stably stratified fluid (e .g. Sectio~ 12.4). The instabi lity criterion (4.4) IS t hus t hat

i.e.,

1 dp

1 dp p dz '

-- > - -

,pdz

(since dp/d z

i .e.,

(4.8)

1 d Inp < "( dlnp

< 0). Converse ly, if 1

"(

d in p d lnp

> --

(4.4 )

t he n the par cel finds it self lighter than the surroun ding fluid a nd hen ce contin ues rising. In t he la t t er case, the ori ginal stratification is un stable and fluid parcels will mo ve a round, i.e. the st rat ification is convect ive ly un st abl e , Note that t he density ca n in cre ase with depth (eq uivalen tly, incr ease with pressure) a n d still be un st able: it has to in cr ease suffic iently

• " , .r .. , . oscillations WIth nnagm ar y fI'eque nc"J ' on e root leading (W III·C,I1 W oul d imply , ilibri ) to exp onen t ial I' growth of t he disp lacem ent from equui num . If t he chemi cal composit ion is uni form , t he n

ln p = In p

+

InT

+

cons tant

for a perfect gas (see Section 1.7) and so the instabilit y criter ion (4.4) b ecom es dinT > 1 _ 1 dlnp "(

(4.9)

<'>0

Astrophysical F lui d Dynamics Fluid D yn am i cal In st abilities

;' 9

which is often wri t ten \7 > \7 a d

for the fluid t o be un stable, where \7

Eq\l~ti~n

==

dd IIIlnpl'

n v

and

a,

!

_

=

(ddlnIn 1') p s

=

1

-

1

'Y

f

(4.]] )

Lh~ Schwarzschi ld cr it erion for conveetive insta bi lit y. 1\101 e ~eneraI1'y, If t here is a vertical gr adient of chemical COlIlp()~il ion (4. g) is

CImractenzed

by

, ,

!

!

r

[

[

;

t

d ln /l

i

(1.12)

d ]J)]J ~

-

th en N

2

go

=

If

(\7 a d

-

+

\7

l'

\7/1)

(4.1:3)

where H p is the pressure scale h eight (2.70) a nd

(5 ==

(f)f) In1' In f> )

(4 .14)

p

(0 = 1 for a p erfect gas). Then t he criterion for instabilit y is

> \7 a d

\7

+

(4. 15)

\7'1 ;

thi s ~s the Led oux ~rit~rion for conv ective inst ability (see Secti on 4. 1.2). 1 he Schwa rzsc:1nld mstability criterion for a region f if sit ' , I 1 .( " 0 Ulll on n comp o, tl~n caIn a so Je exp ressed in t erms of the gradient of sp ecific pnt ro l)11 S no mg' t rat " , , . ,J ,

~dS

1 d]J 'Y ]J

c]J

df> P

where cp is the specific heat a t cons tant pressure. Tl N

2

=

go dS

(4 .16)

len from (4.7),

.

c]J d z '

(4. 17)

th e Ilt~r is stable t o convecti on if the sp ecific ent ro py increases upwards.

~. ~ ad

the . tem p erat n re gl:adien t is said to be 8upcradiabatic; if Convective motions tr ansport hea t : comllJon ly III ste n.T interior regions that a re convectively un stable c ti . effici t i t l . . ' , onvec lO ll IS very cien III iat It requires onl y a very small superadiabatic gradien t \7 - \7 ad \7

\7

. < 11 ad

:1, IS .'~ 'U,badZQ.bat~c.

,

to t.ransp or t t he st ar' s entire heat flux. III t hat. casp 'V ~ \l ad a nd N 2 r-: IJ t.here ; also S is nearly consta n t. ]\,lost stellar-structure modd liug nfiCfi a s im p k- p lll'uOUIl'lIo]ogic.<1 I d(,scription called l ni:J:i n y- lr:ny l h tJU:017J t o ca lculate t.ho ]wa1 t.ran spor! 11y convccti on and hen ce the s trati fication re quire d in «on voct.ivclv uns t.ah lo regionfi in order t o produ ce the necessary cou voc.l.iv« he at flux (l'.g. Kipjleuhallll & vVeigart 1(90 ). T he idea is th at b lob s of convectol fluid t ravel a dist.auce (1 from thei r posit ion of equilibrium and thon di srupt a nd d ispers e into t he IIl'W surroundings : f' ifi th e iuixing longt.h. Th e rnixi iu; !l'ngt h lias to he proscribed . In sto llar «ouvecti vo envelo p es it is ron uno u lv P]'('filll lll'd to he a fixed con st ant t.imo» t.h« local prossu ro seal<' I I( ~i gh t: th o val ue o r t. h( ~ fixed r-onst.aut (th« niirinq-lcuqth. 1J(l'm:m.!dcl ·) ca n hi ) adjusted fiO afi prodllcp e.g. a solar mod el or the CO !Ted. radius, or course, xinco th ere is JlO ]'('al th eory involved , we do not k now t.hat 11)(' ini xiug-lon gf.h paramcte r sho uld he the same for d iffer ent stars . Other proscriptions of 1J)( ~ mixing longth are possible. In order to lise thi s mixing-len gth thcorv iu modelling stella r :-: t rllct.urc, it is necessary t.o cal cu late tll(~ fi l W(~d aJ whi ch blobs 1I100'e and th e aIIIOll11t of hea t they transp ort. 11. should !Jo)'J1l' in mind t.hill. t.his is a ll fairly cr ude an d t hat the adjus tment of t.ho rnixiug-lengt h parameter Ul k ('R up the slack left by in cxac tit.ude in the argume nt. T hus l.he spee d 71 of th e bl obs can be calc ulated from the equa tion of m oti on (!J .G) (whore N 2 is negative for a n unstable reg ion) assuming at it s sim p lest t hat the blob starts from r est and t ravels a distance I! a t cons tant acceleration. Th e convcct 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 ca p acit y and 6.T is the te mperature excess of a blob over it s surround ings . Sin ce 6.1) = (J (tho blob is in pr essure equi librium with its surround ings}, the te mp er a t u re excess can be wri tten in te nus of th e su p erndiaba t ic grad ient divided by the p ressure scale height , multiplied by th e dist a nce f. travell ed by the h lob . Since it t urns out t hat v is p roportion al to the square ro ot of t he supcradiabatic grad ient , t he convective heat flux is p roportional to t he sup er a d iahatic grad ien t raised t o t he p ower :~ /2 . As alre ady stated , in deep convective envelopes of stars it turns out th at the sp ecific heat capacity is large en ough that a lilly snp('n l.dia,] la1.ic temp erature gradient is sufficien t to give 1.]10 nec essary convecti ve hea t Jinx, so that N 2 ~ 0 there.

Gl Fl'uid Dynamical In st abilities

60

4 .1.2

'As trophysical Fluid D yn am i cs

., ' . 1 e thermally IInSt ,a I)Ie ;,;' 0 tilt', T l 't'll' r st rat IficatlO n may ) , ' . ' 1' last example. . le ~,c a ' . r insbbility is sat.isfied , hut stah le over al so n on (4.10) fo '. c 1 :l' t, " not satisfied - T he res ulting Schwarzschild cn te . ,' . , ,' (415)formsta Hl ,yls , " . ' . , . • : _ . " Iv ec [i ou in ::;t.(;l1ar as t.ro physll:S , th at the Led ou x (.l It o iou i -i . . t.im e ' referr ed tu as ,%UH <-(I I , 1 motion IS some ,U IlC;,; 1 (1993) 'mgnlar JJlOJllcntullI w ay abo pl ay I. Je As point ml out by a 111. ' . ' c ' 1' ff ' " mor e slowly t han heat . , . le t o the sa lt, sm ce It too (I uses . ana logouS 10 e '

Effects of dissipation

Convect ive in stability is a dynam.ical process: it do es not require a dissipat ive process , and can ther efor e b e treated in the adiabatic, invi scid approxim atio n. As established by R aylei gh (1880) , dissipation m odifies t he instability cr iterion only sligh t ly:

C t ytc!

Z'

(4. 18)

for inst a bility. Here c: is a positive cons tant of order unity, and t; and td are the dissipatiou tiiuescal es associated with viscos ity and heat diffu sion , re spectively (see Zahn 1993). Const an t c: dep ends on the geomet ry of the fluid reg ion and on boundar y conditions . In sid 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 b e of order 10- 9 to lO -G in the Sun: see Lignier es (1999) . In the con vectively stab le re gions , diffu sion t ends t o damp oscillation s. W hen t-: ("-' l2I ",) b ecom es compar able with N - 1 , where l is a characteris tic parc el size t o b e damped , t he ent ro py st rutification is no lon ger effective in stabilizing the layer , t houg h com pos it ion gradients if presen t will st ill provide a re storing for ce, Interesting competing effects (" doub le-diffus ive insta b ility" ) occur in a layer whi ch is dyn amicall y stab le, i.e. N 2 > 0 in (4 .13) but eit he r the comp osit ion stratifica tion or thermal strat ificat ion on their own would be un st able. T he archet ypal laboratory examp le is wa ter heated eit he r from ab ove or b elow a nd wit h a gradient in salinity. Heat diffu ses much fas ter t han the salt con centration, 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 t he layer hott er t han the bot tom), th en small-scale perturbati on s for which td '" N::' can grow , producing so-called salt fingers . Eventuall y t he st ratification b ecomes a layered convec t ion with b oth temper ature an d salin ity va rying stepwise in dep th _. see Zalm (1993). The op p osite case is that of salt y water b eneath fresh water, wit h a n othe rw ise unst abl e t emperature grad ient whi ch ca n b e produced by heating the water fro m below. One can envisage a di splaced fluid p ar cel oscillating but coo ling down (due to t he sho r te r timescale t c! when it is above its equilibrium po sit ion a nd hea ting up whe n b elow the equilibr iu m position. This ca uses the veloc ity at wh ich t he parcel passes t he equi librium level to increase and the amp lit ude of t he oscillation to grow. This is overstability: t his p ar ti cu lar example is call ed thermohaline convection . In stellar cores, helium may play t he a nalogous ro le to the salt in the

3

l\!l odelli n g convection: the B07l.ssines q apprm.:imation

4. 1. .. . " '. veet ion we m en ti on a com111only adopt ed a pBefore leaving the tOpICof con ' . l ' , .t: hk j"(.,,'ion s , T his i::; the Bons81,. ' . . o(kllillg convedlve y un s a ' -o " . ]lrOXllllat lOll 1.,1.11.1l. ' I ,t pi 1)(' the. i1 ur:'1.ua' I'.ion. of ])ressure ab ou t Its h ori zonnesq (L1I1Y1 '(),:I:i'ma~,uJ'li. J(' 'f' ,t'l " 't'll('l'oJJlod v n mn ie quantiti es. The ve.locit y 1 . inilarlv or () , W I " ' .J • II ' ta1meall , anc S l c • ' 1 ut 'I l'('f'pJ'{'ncp st.ate'. in whic 1 t ie ve'l 1 Ii duat l0n ano c " , " . u is also COllSl(. er e( a . u " .. , . t' t 11(' deusity fluctuations a re B ' " e::;q a pproxnll a ,lOn . , " . locit.y is zero. In the O UHS lll ., . , . . tl '-' eClua t ion of mot.ion . Specifica.lly, " , '1 . t 1 ' bu r ya ncy t.erru III .ue retained on1y III .ne ),' I. ', ' . . 1 ' II tile continu ity equat iOn . 1. Ie " fi t u: ti ns an' Ig1101CC I ' . t.hen , the rlcusity uc ua J( ~ , . , ' . t ' (Goug h 19(9) and filt er s out " ,I' 'h e a pproxnlla .ion .r lat.ter is called t 1ie ane as , . , 1 " t h at 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.,d ll t for t ransport pr op eJl.lcs .0 I , unllupOl . .~ , , . 13 . : , 'q a j)jwoxnnatiOn IS Huetuations in the ouss mes c

>. '

~I U

u _ _

oj

+

"

1

u . \7u = - - \7p p

,

pi _ 9

o

2

+ v\7 u

\7 ·u =O

~ + at

\

!

u·\7T'

(Je

z

.

u = radiative exchange t erm

(4 ,19) (4.20) (4. 21)

H )( \7 - \7 ) is t he so-called superadi(c.g . Go ugh 1977 ) wh er e (J .=. (~{ IBoussine:~ app roximat ion t he pi / pin a bati c lap se r at e. Mor eover , m b l~TI I T whe re 0 = - (8 1n pI Dln T )p, i.e. the first equat ion gets rep laced , y . " 1 t o temperature fluctuations. . t' 'e neglected com paJe tl pressure ftue:tua .ions 31 . . itifi ed j 1 labor ator y convec. .. tion can b e JUs lIe n d d :ltv are long com pare d T he Boussinesq approxlllla , 1 . 1 1, f wessure an ens J ti on , whe re t he scale lelg 1 s 0 ]d . ,t d a nd also in some geop hys ical 1.1 fluid layer un er s u y, fi 11 t o the dept11 0 f Ie . 11 . ct ion is n ot reall y jus ti a ) e, " I ., r cat ion to ste ar conve . , . I,1 d m av v ield some mSlght Illto applicatiOns. ts app 1 . Id ' tractab le pro ) em an J J , but it does Yle a mOJ e . ) for a discussion of st ellar convect lOn the full problem , See SpIegel (1971 a nd the Boussinesq approximation .

i I I I I I

62 A strophysical Pl uid Dynam i cs Flui d Dynamical lnst.oh ilit ies

fJ , 2=0

. III . ,SPC t'.1011.4",j , ,so rather than . .c ner al situat.ion . We sh all analyse a lJlOl c g , . . "j I , . , 1(' 1.1H' rr-snlt : {roll I " ., I al 'sis w e S l It l]') y l(]( . q 110 . . . ,oIlCat. th e mathcmat.icn au ' (')I. 1,,0 zer o, W I' IiIlll I t]I 'lI. f"r irrol a t ionnl . ,111 .. I" " , ' I [I, . , 1 Eq. (,1.:18) WIt h U .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 "OJJl I)1 LS, ] ,'h 1J I " . "is rein1.CI "I ',0 I ,." Jlori~olll.al WaVP lJlJHI )(~r ., l,i' " , " C'

w2 = F ig .4 .J The se t-up for th e R ayl eigh - Tayl or in st ability : o ne 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 l coord ina s. te (he ight) . The d asl wd curve rcpl'esell1.s th e p er tu rbed illt.erface between UJ(' two fluid

[J2

The Rayl ei gh-Tayl or Instability

>

.J

ex p (?:kx - iwt) . \Vitho\1t loss of generality we take k to be positive.

(4.22)

t-

, .,

., ,

ifor density upp er layer. f , illcllJde a JllU orm- , , '. . (C' t ' A 1) the ('llergy SOlJ]'('e 01 ti ins tahili Iy ,~ p c ,lOll ' i . " . .. . As with t he convcc 'lve , . ' ". ]" '1' 1'.1 en erg y sto red ill Ow initial . the RayJp igb -Taylor instability IS 1.ic p o .( Il .1,1, . cOIIfi gnratioll.

4.3

R ot atio n al I nstability .

.

I

I Ie n ew rang e of possible ." . ins tabiliti es. Som e arc

Hotation intror uces a W 10 . , , . le . ' I ill t he nex t section, For 1 .' I 1 r and shall he COnSl( E1en m rn e 1 1. 'J ~ mi ght. envisage in t. lC assoc iatec WI ,; 1 s lea , . idor lv a sim p le scenario , w 11L J W ,. . now we cousir cr a u " ' ." "t, q varios on ly wl11 1 1.1](' , . ., . , r ' n wh ich th e rotatio n ra ,( L

To under Rtand t he develojlmen t of the R aylei gh- Tayl or instability. we co ns ider wh at happen R if there is a. sm a ll perturbat ion t o t he interface. If the p er turbation grows t hen the configm a t ion is unst able. Since th e ba ckground confi guration is tran sla.tionally inva ria nt hori zonta lly (F ig. 4,1), we may without loss of ge nerality consid er an individual Fourier componen t , in the x-d irect ion say, so with x- dep en denc e e i h . Lik ewise the timeindep endence of the background m eans that we seek a t empor al variation of iwt the form esay. vVe suppose that any velocities, pressure varia.tions, etc. a.rise only from the p er t ur ba t ion from t he interfa ce: hen ce all pert urba t ion va riables will be propor t ion al t o

) gk,

fi 2

j

Sev eral in stahiliti es can OCcur a.t inte rfa.c es hetween fluids , Consider two fluids of uniform (hut different) density with a pl a ne interf a.ce between them, with a uniform gra vit at ional field p erpendicula r to th o int erface (F ig . 4.1). If the de ns er fiui d is on top, (so Pl > P2 in the notation defined in the figur e), the Rayleigh-Taylor in stahility develops. This m ay see m an unlikely config ura t ion to occur in nature, but 9 can equally he an ef fcc tivc gravit ational acce leration , e.g. at an accelerating shock front a nd this sit ua t ion can be found in Rupem ovac , for ex a.m p le. The R.ayl eigh-Taylor instability common ly occurs at the same t ime as the Kelvin-Helmholtz inst ability , whi ch a rises from veloc ity shear between the t wo layers : the K elvin-Helmholtz instabili ty is discusRed in Sect ion 4 .4.

+ /) 1

. HUH . I Oll 1".op ) w 2 < () am i so w is im agi nary: (lH" IVWr 22) on e1 ,. . . , li ,tlly I-';J'Ow i lll-'; sollJt io1ls (-1 . , , : UH . 1S ('o],J'l 'SjlOll
if Thus 1 PI

I.II(~

4 .2

(fi 2 - fi 1

'

.

L

,

"

•

•

"

..

mtcrror of a s t a r , sa) , I ' I " li: 1 coordinate JlJ a cylinch ltd ] f til ax is (so to IS 1 JP Jd( 1<1 , ' • •] I distance w rom . e , ." , 1.1 . t Ifects of viscosit y arc negligi ) c. li . 1· ' ' t ) \Ve assumc , M, e .··c " . , ' polar coon md ,c s~ " . . so m . 1,, 1 1C' oquilibrium config uration , ", . neg led \Ve also for sim p IJCll;y , grav , ity '. , pr essure and ce nt rifuga.l forces balance: 1 ip " + wn 2 .;-(-" pdw

=

O.

(4.24)

. . ) undergoing a sm all ra.diaJ d isplacelncn1. fr:)~n Consider now a pal ceJ of flul< . I' '1 1 (1 ])'Irc('l COllS('I,\,('S it s spcClhc s: S· ", , ,'s cosity IS nc g 191 ) C , . Ie , , ' . . 1 w t o w + uW . , mee \ I." . ' 2' 0" "tl tl l' ha nd the pressure for ce t l,e t I - w n n le 0 . 18. , , an gu la r m0111en . . " IS . d I',t",C11J1]] . . led iJv the angu lar vcloc lty . ' "11111 ". I = rrollndmgs " parc~1 feels Jl1 Its new Rll I . 1 I"11('e of for ce per unit mass . t"o E q . (4 .24) ' Hence t Ie ml ) f\. « . , th ere , accordmg

65

F lu i d D ynam i cal In stabilit ies A stroph ysical Flui d D yn am i cs

64

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

2 ( { W 0 (W) } W + SW) (W + ow )2 - (W + OW){O (W + SW)}

= _

-

1 d ( 4 2) w :J dw w n Sw

==

,) oW

-Nfl

(4.25)

to O (5rv). T hus t he equation of moti on of the par cel is 20

d rv dt 2

+

2

N0.(\w = 0

~V I J iC,h gives astable, oscillat ory motion if

(4.26)

is po sit ive but an instabilit y If .N~ is w:g.aLi ~(;. As in t he convective am i R ayleigh- Ta.ylor illstahilitie~, thi s .inst ah ility ~s a dyn amical instability . T hus for stab ility t he rot at.ion pr ofile must satisfy

1 ~ dd ( w 4 O~) '. > O. co to

. . Rather. stro ng

(4.27)

~ l j fi"erelltial

rotations are requi red t o trigger this iust aother instabil it ies would usu ally set in soone r (Zahn 1(93). Su ch a shear inst abilit y is discu ssed next.

blh~y aJ~d 111 practi ce

4.4

Shear and the Kelvin-Helmholtz Instability

~~10ar, ~I~stabilities, whi~h Cal~ occ ur when a fluid 's velocity is not uniform,

ale ve.l } eOl:unon. ThClr mam feat ur e is t hat t hey ext ract vorticity from ~. la1llll1ar .(I.e. non-turbulent ) flow and t he vor ti ces can then grow and IIlt~r act WIt h one another . An excellent reference is the bo ok by Drazin & Reid (19tH ); Zahn (1993) also provides a good referen ce. . l~~re we rest rict our selves to an analysis of t he Kelvin-Helmholt z inst a bility, and some general comments about the onset of shear instabili ti es and about t urbulen ce.

4.4.1

z =()

pz

I

Ng

- 3

I t

Th e Kelvin-Helmholtz instability

T he Kelvin-Helmholt z inst ability can occur when one fluid flows over an o tl~er.. Cons~der. a two-layer fiuid wit h a plan ar interfa ce (z = 0) in a u.l1Ifonn gr~vltatlOnal field g = - ge z . Each fluid has it s own uni form density and u~llfo~'l1: , stead y, horizont al velocity (in t he z-direction say). The configuration IS Illustrated in Fi g. 4.2 .

l

I (

II I I

Fig . 4.2 T he set -u p for t.h e K elvill-Hehn ho lt z illst abiJ it y : one fluid layer flowin g over "Hot her. T he u p per fluid la y e r has densi ty PI an d hor izontal speed 'Ill , t he lower layer has density P 2 a w l horizo nt .u! sp eed '11 2 . For other d et ails, see t he capti on t o F ig. 4.1.

'vVe have already seen that in t he case of no horizontal velocity such a config ur ation is subjec t t o the R ayleigh-Taylor ill::;tahility if t he fluid ill top is more dense t han the fluid below (PI > (J2 ) . 'vVe sh all now see that if th ere is shear hetween th e two layers (i.e. V J "I [h ) t hen t his fur t her de::;tabili ;,::es the configurat ion, wh ich m ay t hen b e unst able even if the lower layer is t he denser one. Put anot h er way, a sufficiently st ab le density st rat ifica t ion is necessary t o overcom e t h e shear inst abili ty. As in Section 4.2, we cons ide r a pertur bation z

= (x)

of the interface

of the form

( =

A exp (i kx - iw t) -

(4.28)

We shall consider incOln p ressi ble, irrot.ationa] per turbations in each layer , so that t he small perturbations u ' t o t he background veloc ity are express ible in terms of a scalar p o t ential 4>:

u'

= Vc/J ,

(4.29)

We shall use sub scr ip ts 1 and 2 to den ot e qu antiti es in t he upper an d lower layers resp ecti vely ; so , for exam ple, the t otal velo city u is V I e x + \74>1 in the upp er layer and U 2 € X + \74>2 in t he lower one. All perturbati ons are deri ven by the interface d istorti on . Thus all pert ur bed quantities have t he same dep end en ce on x a nd t as in E q. (4.28) ; and they all tend to zero far from the inter face . Withou t loss of gener ality we take k t o be positi ve . T hen since t he velocity p ot enti als ar e solutions of

r ..,.

J .__'~"-, <

A st rophysi cal Flui d D yn amics

··

Lap]a ce 's equation (4. 29) , one ca ll im nH'di ately say t hat

+ i k» : - k z) iwt + i k:7: + k z )

C J exp ( - iwt (,'.2 ox p ]-

r··

, (4.:~ ())

for some constants C l a nd C« T he diffr>I'Cllt. ,' , . . -' , . ,, - , SlgBS III t.he ens ures that ¢; -> 0 as z -> ±oo.

z- d ep en d ell c(~

A kin ern.at.ic con diti on is that, Oll either side of the int o ·f·, . ' ar fI . -, " . . . In ,(J d c e , d ll Y Iud MI·t · ,] atf.I , I.tl I., Ie11 e ,I. , t .H: intcrfac « sllrfacc~ m ust remain on tho . - "111'f"lC'C' whir..']1 ..n)('a us t·. l~ . vc~·t.rcalcol1JjJollellt of t.he velocity I1JllSt match t he m ater ial d()]'i va~ .ive a t 1io in te rfa ce dlHjJlacemen t «( :/:, 1):

/c

'J

Dc!)]

,

,,

."

-k C l = - iwA

,.

+

u( Dt

+

U2

AC'

i kUJA .,

?( UT

(·1.:31)

~

2 =

- iwA

+

ikU2A .

(4.3 2)

Allo~;)ler condition is that t h e norm al st re ss acros s t he inter face mus t he cont in uous , which h ere means lhat t.l ~ ' .. . . .' .' , . N I ie pi essure p must. be continuo ns '. ow 1. ie momentum equat ion ca ll be written

at + v

(2 1

2 U )

= -

p1 v»

(4.33)

which can be int egl'a ted " (to [inear . .ec tr,0 grvo order )

a¢; + U a¢; at D.T

=

p

P

-

gz

+

F (I) ,

(a~1

+

U

l ~~l v .l:

+

g()

I

I

SlIbst.it llting [0]' (;1 a nd (.'2 fro m E qs , ( 4 . ;~ 2) give'S all c'(jlw1.i(ln that is Illogmlcous in A : for non -t ri vial (11 =/: OJ solut.ions W( ~ ))IlISt haw' t hat

=

- P2

]J( )-

i.o ..

( a¢;2 at + u aeP2 ax 2

wher c tr = ( PlUI + f!2Ih ) j (PI + f!2) is a density -weighted aW'l'age s! H'C'd . TIlt' cou fign ra t.ion is unst. nhlo if U I(' right-h.u «] side of Eq . (;1.:18) is llcgati ve, s in ce th e n w w ill h: 1YC a 1I Oll··;I,crO ill J:1gilla r ,v pa r l n11l 1 ou r- of t.l«t.wo solu tions will corresp o nd to ex p o nen ti al gl'owtl l w it.h tim o, If 111 112 we obtai n the cri terion (11.2:1) Ior ti l(' Hay leigh -Taylo]' inst.ahility. ]l'lh =/: U2 , we see that the co nfigura t.io u is s t.ablc only if th « first (st rat ifica t ion) ter m on the right-ha nd side of (4 .:38) is la rger than the second , shea r , term which al ways 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 , t hough this is not t r ue in a rea l layer of finite t h ickness, see Draz in Sr lh'id (1981) . ote t hat the 0 ter m 01 1 the left of E q . (4.38) is nicrolv oquival r-nt. to a Galilean tran sforma tio n in t he a-dire ction wit h speecl U, If the sh ea r cau ses the configuration t o be unstable by ma king j:]ll ' rigId of (4.38) negative, this is called the Kel vin-Helmholtz insta.l!ilily . Not« t hat. we ha ve on ly pr oved a s ufficient condit ion for stab ility since wo havo considere d on ly a subse t of possibl e di st ur bances, namely irr otation al ones. C O'

(4. 34 )

whe re F (t) is. a " cons 1. ant " 0 f IIltegration ' . Now since t he . I , I ' t . .' .' " on } ,un cdep end t -l .' Cll ' quan ;ltles are the per turba tions, an d t hese tend to zero fa r frOl~l t.n e int er f ace we can deduce' tl ' 1. P(t ) ' . ] . • , ' . . ' . .~ , ],l . IS IC en tlcally zero . T hu s cont inuitv of p l essm e at the Illterface Impli es , using E q . (4.3Ll) , t.hat " - PI

() : 11(' ))(' ( '

(4 .:l8)

'

v--'E ( D ')

co nsis te nt to oval u ato th is at :: 0-: .

.

at t he interface Con· c'c·t l' f "l ' ] . , Ir ad firs t-orr .0 II S '. orr or, sin ce terms in these equations aro sm all " 1'1' arr eady llst-ordc'r " ' , (JU d.I! :1 .ics , we m av eval u ate Ec ~ (4. 31) ..t I '1. ' j' I. .' c, ". " 1,0. .• a , t.lio u np er U1 )ec. surface loca l.ion z - 0 1'1 ' E ( . ' dedu c 1·1 1 ' ,( ' - ' . IUS using :J(Jfi. 4.28) a nd (4 .:30) we .e . la . .

!lO W

. .. ·

D(

ur!J2 Dz '

Agai n it. is

07

o, ~( Dt + d 'l:

[) z

j

Flui d D yn am i cal hist.olnliiir:s

+

9

c)

.

(4 .35 )

4 .4 .2

Cr iti cal R ichardson and R eynolds numbers

\Ve have seen in Sec ti on 4.4 .1 how a stable stratifica t ion hinders th e onse t of shea r inst. abl ity. In a mo re general configuration, whe re st rati ficati ou and velocity vary wit h height z, t he st abilizing effect is measllJ'ed l,y tl w Richa rd soll Illllnh er Ri

(dU j dz )2

69

Fluid J)ynamiwl In st abilities 68

A strophysi cal Fluid Dimomics

In the a bsence of dissip ation , a sufficient con d it ion for instability for a variet y of velo city and density profiles is t h at Hi is smaller than a critical value which in 1/4. If heat can di ssipate, however , t his will weaken t.he buoyancy for ce (see the dis cus sion of double-diffusion, Section 4.1.2) a nd m ake th e layer less st a ble; thus the critica l Ri chardson number is increa sed . See Zahn (1993 ) for a fuller d iscussion of t he iSSlJ(). An other important quautitity determining the on set of turbulen ce in a viscous How is the crit ical Reynolds number. The R eyn old s munbcr is defined 1Jy

He == L Ull}

(4.4 0)

wher e L and U are characteristic leu gthscale a nd speed of th e flow respcct ively, and J/ is the kin euratic viscosity . The Reyuolds number m easures tl w re lat ive importance of in ertial terms a nd the viscous t erm in the rnornentum equation . In th e ab"mH:e of other forc es (e.g. buoyancy ), a laminar flow becom es unstable when t he Reynolds nu m ber ex ceeds some cr it ica l number Rl~e ' Generally Re., is of t he order of I 000 , hut it dep ends ou boundary condit.ious of the How and on the particular velocity profil e. See Drazin & Reid (1981).

4.4 .3

Turbulence and the Kolmogorov spectrum

In st a bili ti es such as those discussed ab ove ca n rapidly lead ill high Reyn old s-number flow to turbulen ce, ill which neighbouring pa rcels of fiuid at so n ic inst an t rapidly follow very differ en t an d practicall y unpredi ct able traj ectories . (A How that is not turbul ent is ca lled laminar .) The onset of turbulen ce ma y b e pi ctured as the flow developing smaller a nd smaller scales of motion un til at sufficient ly small scales molecul ar viscos it y se ts in. Turbule n ce is a very challeng ing pr obl em . We sh all not consi de r furth er the st ag es by wh ich turbulen ce develops, but focus instead on fully developed turbulen ce. E ven ther e the challenges are formid a ble an d we restrict our attent ion t o t he case wher e the statis tic al properties of the turbulen ce are homogen eous, isotropic and ste ady. The basic idea s wer e conceived by Kohnogorov (1941) a nd a classic text in which the t heo ry is developed is Batchelor (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 r ate E in motions on some lengthscale lo. T he ( 'lll ~r~'y p er unit mass on this len gthscale is ~ u6 ' This energy then cas-

. . .. 1" 1 until eve nt ually at some scale 1II it is de to sm aller and sma11eI sca es . .' ca es . ' . t 1 , t he llnid's mol ecular VISCOSIty II . dissip ated as heat )J . . t.lu t dissipa t ive lcu gthscal e l, do es n ot It seem s r easonable to SllP.pose , 1,1· . f . . . , [ .] _ £ '2'1'. -:1 ,• . 1 Now I he ChllWIlSIOllS () t ,11 e t . , J 1 but only on E am IJ . ' I' . . . . depellll 0 11 0 ' • . ' . . . . f lc '1'1 and time: and the C UJIenSIOnS L.» 1 T lonote dnnens lOllS 0 cu g , I c 1 where aUC ,( ]. fore c (lJ'IIl('usional gro u lldS we deduce t !at _ I ' 2 1"There 01 e on ' , of v arc [v ] 4 l; rv (1)I/ff / . (4 .t1 1) 0

>

J

0

.

•

.

.

•

. . ' 1, ' S ' on scale In presmnahly dep ends only on Tl ' kin etIC ene rgy pm UI1l rna s It. , I . 1,1lUS ag"alll 1Y ), dl'IIIC'l'ICj' . ., onal arg ume nts In ,1l11 c, '2/3 ('1.42) 'U 6 rv (d o) .

. ' re ssjous y ilJ ds ' . l ,'lt. .l'I'lg" E b etween these two ex ]! .. , ' . ; 1l111l • El H ,)--:1/4l l , rv ( .c 0

(4.4:1)

wit.h H. eyno1ds number Re == 'Uoln/ v , V>le also introduce wavenumbers k so I.'.

=

l- 1 ,

1.'."

=

I' v- I

,

ko =

t:' ()

.

(4.44)

. -umber sp ace is called the i ne rti al r a'nge. ·1 k < k < k m waven ut T he IJl erva 0 '" . E(k t ) is defined suc h that the aver age The kin etic ener'yy speet1"'lJ.m ..J ' , ' . t

kineti c per unit ma ss is

~ ('U 2 ) = 2

j'oo E dk .

(4.45)

0

. . , .. . ther e is an upper cut-off to t he integr al at Becau se of VISCOUS dISSIpatIOn, . " e [E] = L3T- '2 , we deduce k = k/l' On dimension al gro un ds on ce more, sine th at in the inertial range

E = Cf.'2/ :1{,;-

5/ 3

(4.46)

. . tl at E is indel)endent 10 and L there . This stant as su nn ng 1 . . where . IS a cons ' .' . " 1 inertial range the energy densi t y in is t he Ko lm ogorov scaling, t h at m t~~/3 homo gen eous turbulen ce scales as k .

c'' '.