Convective energy transport
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1 PH217: Aug-Dc Convctiv nrgy tranpt In tllar intri, onc th tmpratur gradint bcom larg, it may bcom m favourabl to tranpt nrgy via convction rathr than radiativ diffuion and conduction. Th critrion of convctiv intability can b wkd out a follow. r+dr,p+dp,t+dt, ρ+dρ ρ, T r,p,t, ρ Figur 1: Prturbation of mattr lmnt to tt f convctiv intability Lt u conidr a mattr lmnt at a radiu r in th tar, and diplac it upward to r + dr. Th lmnt would com to prur quilibrium with th nw urrounding, but it dnity and tmpratur would not ncarily b th am a tho of th urrounding matrial ( fig. 1). If it dnity i mallr than th urrounding matrial thn th lmnt would ri du to buoyancy. If th dnity i highr than th urrounding, it would ink back. Th ituation will b tabl againt convction if ( ) ( ) dρ dρ > 0 (1) dr dr whr th ubcript rfr to th lmnt and to th urrounding.
2 PH217: Aug-Dc Sinc P = ρkt/µm p, on ha dρ ρ = dp P dt T + dµ µ Igning compoition gradint f th tim bing, w can rwrit q. (1) a ( 1 P ) ( dp 1 dr T ) ( dt 1 dr P ) ( dp 1 + dr T ) dt > 0 dr Th trm containing prur gradint cancl du to th prur quilibrium tablihd btwn th lmnt and th urrounding, laving a tability condition in trm of th tmpratur gradint: ( ) ( ) d ln T d ln T > dr dr Writing in trm of drivativ w.r.t. prur P intad of r, ( ) d ln T < d ln P < ( ) d ln T d ln P inc d ln P/dr < 0. If th lmnt volv adiabatically thn = ad = γ 1 γ whr γ i th ratio of pcific hat. F monatomic ga with γ = 5/3 th valu of ad i 0.4, xcpt in rgion of partial ioniation whr addition of nrgy cau an incra in numbr of particl and hnc tmpratur incra lowr than nmal, dpring ad blow it tandard valu of 0.4. If indd all th tranpt do tak plac via radiation and conduction thn = rad
3 PH217: Aug-Dc W can thn writ th condition f tability againt convction a rad < ad (2) Thi i calld th Schwarzchild critrion f dynamical tability. If a compoition gradint i prnt, thn th tablility critrion i modifid to rad < ad + µ (3) whr µ = (d ln µ/d ln P). Thi i calld th Ldoux critrion f dynamical tability. If th condition ar violatd thn convction t in to tranpt nrgy and th tmpratur gradint i no longr givn by rad. rad now tand f th tmpratur gradint that would hav bn ncary to tranpt th whol flux by radiation(+conduction). Th radiativ(+conductiv) flux itlf i modifid now to F rad = 4acG 3 T 4 m κpr 2 whil th full flux, including convction, i F rad + F conv = 4acG 3 T 4 m κpr 2 rad by dfinition. A propr tratmnt of convctiv tranpt would b abl to dtrmin th actual tmpratur gradint. From th two rlation abov on may writ F conv = 4acG T 4 m 3 κpr ( 2 rad ) = 4ac ( T 4 1 r 2 dp ) ( 3 κρr 2 rad ) P dr whr w hav ud th hydrotatic quilibrium rlation m = (r 2 /Gρ)dP/dr. Introducing th Prur Scal Hight H P dr/d ln P, th abov can b writtn a F conv = 4ac T 4 1 ( rad ) (4) 3 κρ H P
4 PH217: Aug-Dc A convctiv lmnt riing with pd v and with an xc tmpratur DT ovr it urrounding contribut to a local convctiv flux of nrgy F conv = ρvc P DT (5) whr C P i th pcific hat pr unit ma at contant prur. 4πr 2 F conv (r) will giv th nt convctiv nrgy tranpt acro a phrical hll of radiu r if v and DT ar rplacd by appropriat avrag valu ovr all lmnt croing th hll. An ffctiv thy f timating th avrag i calld th mixing lngth thy. In thi, an lmnt i aumd to travl an avrag ditanc qual to th mixing lngth l m rtaining it idntity, and thn mrg into th urrounding. Ovr thi ditanc th xc tmpratur DT of th lmnt tart with zro and kp incraing until th idntity of th lmnt i lot. If w now conidr th urfac acro which w ar intrtd in valuating th convctiv flux, lmnt croing it would hav tartd at ditanc varying btwn 0 and l m. Th avrag ditanc travlld by a blob bf croing th urfac i thrf l m /2. Th avrag xc tmpratur DT carrid acro th urfac i thn givn by DT T = 1 d(dt) l m T dr 2 = l m 1 d 2 T dr (T T) = l m 2 (d ln T d ln P d ln T d ln P ) = ( ) l m 2 dr d ln P 1 (6) H P whr w hav aumd that DT/T 1 in writing th firt trm in th paranth. Thi xc tmpratur cau th lmnt to hav a dnity diffrnc from th urrounding Dρ/ρ = DT/T and a conqunt buoyancy fc on th lmnt (pr unit ma) k r = gdρ/ρ. Sinc Dρ tart from zro at th bginning of th lmnt motion, on an avrag k r /2 would hav actd upon th lmnt during th motion up to th croing.
5 PH217: Aug-Dc Th wk don by thi fc i pnt partly in incraing th kintic nrgy of th lmnt and partly in puhing away th pr-xiting mattr. Auming that half th wk go into ach, th vlocity v of th lmnt can b writtn a 1 2 v2 = 1 k r l m = 1 gl m DT 2 4 T v 2 = g( ) l m 2 (7) 8H P inrting v from hr and DT from q. (6) into q. (5) w find F conv = ρc P T g l m H 3/2 P ( ) 3/2. (8) To procd furthr, w will nd to valuat, th rat of chang of tmpratur of th lmnt a it ri. In abnc of any xchang of nrgy with th urrounding, th tmpratur chang will imply b that du to adiabatic xpanion. Howvr th xc tmpratur of th lmnt ovr th urrounding will cau radiativ lo of nrgy from th lmnt. Hnc ( dt dr ) = ( dt dr ) ad = ad λ ρvc P v λh P ρvc P vt whr λ i th nrgy lo pr unit tim from th lmnt of volum V to th urrounding. W now nd an xprion f λ. To do o, lt u conidr a blob of diamtr d which ha a urfac ara S and volum V. F radiativ (+conductiv) nrgy tranpt from th blob to th urrounding th ffctiv tmpratur gradint i approximatly 2DT/d. Th radiativ flux i thn and th nrgy lo rat f = 4acT3 3κρ 2DT d λ = S f = 8acT3 3κρ DTS d (9)
6 PH217: Aug-Dc λ = 8acT3 3κρ Subtituting thi in q. (9) on obtain ad = S d ( )T l m 2H P. (10) 4acT3 l m S 3κρ 2 C P v Vd ( ) (11) Th nrgy tranfr i hight by th largt blob. Th largt poibl blob undr th currnt circumtanc would b of diamtr d = l m. F th, l m S/Vd = 6/l m. An avrag valu f all blob that i oftn ud i 9/2l m. Uing thi, w find ad = = 6acT 3 κρ 2 C P vl m (12) 6acT 3 8H P 1 ( κρ 2 ) 1/2 C P l m g l m (13) whr v ha bn ubtitutd from q. (7). Introducing a dimnionl variabl U = 3acT3 8H P (14) κρ 2 2 C P l m g w can writ Th lft hand id of thi can b writtn a ad = 2U( ) 1/2 (15) ad = ( ad ) ( ), o q. (15) i a quadratic quation in ( ) 1/2, with a olution whr ( ) 1/2 = U + ξ (16) ξ = U 2 + ( ad ) (17) Th tru tmpratur gradint can b writtn in trm of ξ: = ξ 2 + ad U 2 (18)
7 PH217: Aug-Dc On lat tp now rmain in valuating ξ and hnc. Equating F conv from q. (4) and q. (8), 4ac 3 T 4 1 ( rad ) = ρc P T g l m κρ H P 4 2 H 3/2 ( P ) 3/2 2 ( ) 3/2 = 8 9 U( rad ). (19) Uing q. (16) and q. (18) in thi, w can writ ( U + ξ) 3 = 8 9 U[ (ξ2 + ad U 2 ) + rad ] (ξ U) 3 = 8 9 U[U2 ξ 2 + W] (20) whr W rad ad. Eq. (20) i a cubic quation in ξ that can b olvd f any givn t of paramtr U and W; and it turn out that it ha only on ral olution. From ξ, th tmpratur gradint can b obtaind uing q. (18). In limiting ca th natur of th olution can b aily n. W not that all th gradint,,, ad and rad ar finit inid th tar, and xcpt f rad thy ar all mallr than unity. In vry dn cntral part of th tar U fall to vry low valu. Stting U 0 in q. (15) giv ad, and hnc q. (19) yild ad. Hr th convction i a vry fficint mod of nrgy tranpt, a vry mall xc ovr ad i ufficint to tranpt th ntir luminoity. Nar th tllar urfac th dnity drop to low valu and U grow vry larg. In thi limit on can t U and q. (19) yild rad. Hr convction i vry infficint a a mod of nrgy tranpt. Evn maj convctiv motion carry too littl nrgy to mov ignificantly away from rad. In both th abov limiting ituation th valu of i known without having to olv th quation of th mixing lngth thy, and hnc uncrtainti of th thy do not affct th rult.
8 PH217: Aug-Dc In th ca in btwn th two limit th mixing lngth quation nd to b olvd, which yild a btwn ad and rad. In addition to th nrgy tranpt, convction play an imptant rol in vral othr apct of tllar tructur. F xampl, it homogni compoition ovr th ntir convction zon vn though nuclar raction ar confind to a fraction of thi volum. Thi i imptant in dciding th cour of volution of th tar. Convction nar th urfac i rponibl f gnration of trong magntic fild in unpot and tarpot, through which conal activity i powrd in th tar. On th Main Squnc, tar with ma largr than about 1 M (uppr main qunc) po a convctiv c, du to rapid gnration of nrgy via CNO cycl. In th lowr main qunc ( 1 M and lowr) tar hav a convction zon nar th urfac, rulting mainly from riing opacity at lowr tmpratur. Far to th right in th Hrtzprung-Rull diagram, cool tar (low-ma main qunc tar highr ma volvd tar) bcom fully convctiv: th convction zon tarting at th urfac pntrat all th way down to th c. Such tar can b dcribd, approximatly, a polytrop of indx 3/2 inc = ad = 0.4 vrywhr would imply P T 2.5 and hnc P ρ 5/3.
(1) Then we could wave our hands over this and it would become:
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