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Apracticalmodelofconvectivedynamicsfor

stellarevolutioncalculations

NeilMiller∗andPascaleGaraud†

∗DepartmentofAstronomyandAstrophysics,UniversityofCalifornia,SantaCruz

†DepartmentofAppliedMathematicsandStatistics,UniversityofCalifornia,SantaCruz

entmotionsintheinteriorofastarplayanimportantroleinitsevolution,since

theytransportchemicalspecies,rallgoalisto

constructapracticalturbulentclosuremodelforconvectivetransportthatcanbeusedinamulti-

dimensionalstellarevolutioncalculationincludingtheeffectsofrotation,shearandmagneticfields.

Here,wefocusonthefirststepofthistask:capturingthewell-knowntransitionfromradiativeheat

transporttoturbulentconvectionwithandwithoutrotation,aswellastheasymptoticrelationship

betwendthe

closuremodeldevelopedbyOgilvie(2003)andGaraudandOgilvie(2005)toincludeheattra王冕的诗代表作品 nsport

andcompareitwithexperimentalresultsofRayleigh-Benardconvection.

Keywords:Turbulentconvectionmodelling

PACS:;;;

INTRODUCTION

Tent

motionsareoftenthedominantmechanismofenergytransportinstellarconvective

zones,andcanincreasemomentumtransportandchemicalmixingbyseveralorders

rotatingastronomicalsystemswhereturbulenceisanisotropic,

Reynoldsstressesarethedominanttransportersofangularmomentumandtherefore

influencetheinternaldynamicsofthewholesystem.

Itiscurrentlynotpossibletoperforma3-Dnumericalsimul长安十二时辰张小敬结局 ationofconvectivemo-

r,whenconsideringstellarevolution

wearenotnecessarilyinterestedinthespecificdetailsoftheconvectivemotions,but

ratherinstatisticalpropertiessuchastheconvectivefl

long-termgoalistoconstructaclosedsetofevolutionequationsforthesestatistical

quantitiesintermsoflarge-scalesystemproperties(ity,rotation,shear),which

canbeusedinastellarevolutioncalculation.

InthispaperwefocusonmodellinghowtheconvectiveReynoldsstressesandheat

flspeciallyinterestedinapproximatelypredicting

theonsetofconvectionaswellastheasymptoticbehaviourofturbulenttransportfor

largeRayleighnumber;capturingtheonsetisrequiredforthestellarevolutionmodel

tte

predictionoftheasymptoticbehaviouroftheconvectiveturbulenceisimportantto

describetheamountofenergyandangularmomentumthatistransportedinthemajority

oftheconvectivezone.

Toquantifythequalityoftheproposedclosuremodelwetestitagainstlinearstability

ApracticalmodelofconvectivedynamicsforstellarevolutioncalculationsSeptember3,20071

analysis,numericalsimulationsandlaboratoryexperimentsoftherotatingRayleigh-

Benardproblem.

ROTATINGRAYLEIGHBENARDCONVECTION

ThetypicalRayleigh-Benardconvectionsetupisasfollows:tworigid,ideallyinfinite,

horizontalplatesseparatedbyadistanceDconfineaweaklycompressiblefluidbetween

raturedifference∆Tismaintainedbetweenahotbottomanda狡兔三窟比喻什么人 cooltop.

Thesystemisassumedtoberotatingwithaverageangularvelocity

Ω=Ωzwithgravity

g=−gerningequations

are

∂

i

u

i

=0,(1)

(∂

t

+u

k

∂

k

)u

i

+2

ijk

Ω

j

u

k

=−g

i−∂i

+∂

kk

u

i

,(2)

(∂

t

+u

k

∂

k

)=∂

kk

(3)

wherethedynamicalvariablesarethetemperatureoffset,thepressureperturbation

fromhydrostaticequilibriumandthefllowingparametersare

assumedtobeconstant:thecoefficientofexpansion,thekineticviscosityandthe

thermaldiffusivity.Sumsoverrepeatedindicesareimplied.

Thequalitativebehaviourofthesystemiscontrolledbythreedimensionlessquanti-

ties:theRayleighnumber,Ra≡g∆TD3/()(measuringtheratiobetweenbuoyancy

forcestoviscousstabilisingforces),theTaylornumber,Ta≡4D4Ω2/2(measuringthe

ratiobetweencentrifugal柳永雨霖铃背景故事 forcestoviscousforces),andthePrandtlnumber,Pr≡/

(measuringtheratiobetweentheviscousdiffusionrateandthethermaldiffusion经典语录 励志 rate)登鹳雀楼翻译 .

ForanygivenTaylornumberandPrandtlnumber,thereexistsacriticalRayleighnumber

(Ra

c

)abovewhichthesystemisconvectiveandbelowwhichthesystemisconductive.

WestrivetoconstructourmodeltomatchtheknownvariationofthecriticalRayleigh

verifythatatRayleighnum-

bersmuchlargerthancriticaltheNusseltnumberNu(ratiooftotalheatfluxtoconduc-

tiveheatflux)iscorrectlypredictedbythemodel.

CLOSUREMODEL

Ourgoalistocapturethebehaviourofturbulentconvectioninarotatingsystembothin

lopedasecond

orderclosuremodelusingthetechniquedescribedbyOgilvie(2003)andGaraudand

Ogilvie(2005).

Wewriteeachquantityasthesumofameanandfluctuatingpart(u=u+u′,=

+′,and=

+′)–intheRayleigh-Benardproblem,thesemeanquantitiesonly

varywithz.

Wedefinethefollowingcorrelationquantities:R

ij

=

′u′

i

,andQ=

Theexactequationsgoverningthemeanquantitiesare

∂

i

u

i

=0(4)

(∂

t

+u

k

∂

k

)u

i

+2

ijk

Ω

j

u

k

=−

g

i−∂i

+∂

kk

u

i−∂j

R

ij

(5)

(∂

t

+u

k

∂

k

)

=∂

ii

−∂

k

F

k

(6)

whilethosegoverningthesecondordercorrelationtermsaremodelledas

(∂

t

+u

k

∂

k

)R

ij

+R

ik

∂

k

u

j

+R

jk

∂

k

u

i

+2

ilm

Ω

l

R

jm

+2

jlm

Ω

l

R

im

+(F

i

g

j

+F

j

g

i

)

−∂kk

R

ij

=−C

1

−1

R

ij−C2

−1(R

ij−1

2

(+)∂

kk

F

i

=−C

6

−1

F

i−

1

d2

+

2

1NotethatC

1≃0.4,C2≃0.6,andC

≃12havealreadybeenfoundtogiveanadequatedescriptionof

theturbulentstressesinCouette-TaylorexperimentsbyGaraudandOgilvie(2005),butC

6

andC

7

remain

tobedetermined.

ApracticalmodelofconvectivedynamicsforstellarevolutioncalculationsSeptember3,20073

leigh-NusseltrelationadaptedfromChandrasekhar(1961).Thedashedlineisthe

modelpredictionforΩ=0,=10−3,=10−4generatedbyvaryingthetemperaturedifferencebetween

thetwoplates.

theboundaries,implyingR

ij

=F

i

=owerboundary=∆Tandattheupper

boundary=hetemperatureperturbationsarezeroatbothboundaries,Q=0.

InFigure1weshowtheNusseltnumber-Rayleighnumberrelationshipinthenon-

rotatingcaseforbothourmodelandtheselectionofexperimentaldatasummarisedby

Chandrasekhar(1961).Notethatourmodelshowsgoodagreementwiththeexperimen-

taldataforthecriticalRayleighnumberwherethetransitionbetweenconductive(Nu=

1)andconvective(Nu>1)elalsoreproducesthestan-

dardpowerlawrelationshipNu∝Ra1/3athighRayleighnumberwhichisanatural

consequenceofourselectionL=dwhenΩ=0(l’smixinglengththeory).

However,thegoodquantitativeagreementbetweentheexperimentaldataandthemodel

predictionforNuwasunexpectedsinceC

6

orC

7

,whichplayarolewhenthesystem

isconvective,sting

theseweshouldbeabletofurtherimprovethecorrespondenceofthemodeltoreality

forRa>Ra

c

.

Inrotatingsystems,convectionisknowntodelayoftheonsetofconvectionat

theory(asekhar(1961))predictsthatRa

c

∝

Ta2/3,withacoefficientofproportionalitywhichdependssomewhatontheboundary

re2,wecomparethisknown

relationshiptoourmodel(solidline).Thepredictedpowerlawmatcheslineartheory,

butthecoeffieless,

weconsidertheagreementsatisfactoryconsideringthesimplicityofthisclosuremodel.

Toconclude,wefindthattheclosuremodeladequatelydescribesboththeturbulent

convectiveheatfluxasafunctionofRayleighnumberintheabsenceofrotationaswell

rework,weintendto

comparethedegreetowhichthemodelmatchesnumericalsimulationsofdeveloped

convectionintherotatingRayleigh-Benardproblem(etal.(1996)).This

comparisonwillbeusefulindeterminingtheutilityofourmodelawayfromtheonset

ofconvectioninarotatingsystemandpermitthecalibrationofC

6

andC

7

,theremaining

freeparametersofthemodel.

ApracticalmodelofconvectivedynamicsforstellarevolutioncalculationsSeptember3,20074

idlineisthemodelpredictionforthecritical

RayleighnumberasafunctionofTaylornumberwith=10−3,=10−tedlinesarethe

asymptoticrelationsderivedfromlinearstabilityanalysisbyChandrasekhar(1961)forΩ=0andfor

Ta→∞forthedirectmodeofinstability(seeequation184onpage106).

ACKNOWLEDGMENTS

thankGordonOgilvieforhisguidancethroughoutthecompletionofthiswork.

REFERENCES

e,340,969–982(2003).

,e,JFM530,145–176(2005).

asekhar,HydrodynamicandHydromagneticStability,Oxford,1961.

,,iams,JFM322,243–273(1996).

ApracticalmodelofconvectivedynamicsforstellarevolutioncalculationsSeptember3,20075