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Two?andThree?DimensionalSimulations of

theThermon uclearRunaway in anAccreted

Atmosphere ofaC?OWhiteDwarf

A?Kercek

?

? W?Hillebrandt

?

andJ?W?Truran

?

?

Max?Planc k?Institutf?urAstrophysik?

Karl?Schwarzschild?Strasse??D??????Garching?Germany

email?ask?mpa?garching?mpg?de?wfh?mpa?garching?mp g?de

?

DepartmentofAstronomyandAstrophysics?EnricoFermi Institute?

University ofChicago?Chicago?IL??????USA

email?truran?nova?uchicago?edu

KeyWords?Convection?Turbulence?Thermonuclear Combustion

Abstract?We presentthe resultsoftwo? andthree?dimensionalcalculationsof

turbulentn uclearburningofhydrogen?richmaterial accretedontoawhitedwarf

of???M

?

? Themain aimof thepresentpap eristoinvestigatethequestionasto

whetherandhowthegeneralproperties oftheburningarea?ectedbydimension?

alit yandnumericalresolution e?ects?Inparticular?wewant toseewhetherornot

convectiveovershooting in tothe surfacelayers of theC?Owhite dwarfcanlead

toself?enrichmentoftheinitiallysolar compositionofthehydrogen?richenvelope

withcarb onandoxygenfromthe underlyingwhitedwarfcore?

?

eject

In

the

tro

en

duction

v

To day?weknow that novaearewhitedwarfsinclose binarysystemswith a

mainsequencestar astheircompanion?Thewhitedwarfcollectshydrogen?

rich materialfrom thecompanion which ?rstsettles into an accretion diskand

isthen accreted on to thewhite dwarf ?ssurface? Duetocompressional heating

this env elopecan reach temperaturessu?cien tly high to burnhydrogen into

helium? ?rstbythe proton?protonc hainandlaterby theCNO?cycles?where

C?N andOnucleiactascatalysts forthe burningofh ydrogen?Numeri?

calsimulations have shownthatthese reactionscan fuseallH intoHeand

elope? providedthe matteris electrondegenerateand theCNO?

abundances aresu?ciently high?severaltimes theirsolarv alues?Starr?eld

etal??????Starr?eld etal???????Iftheseconditionsare met?the outburst

resemblesthe observ edproperties ofclassicalnov aev erywell?

Itis animp ortantfeature thatthe presence of C? N?andOnucleienhances

theenergyproduction duringtheviolentstagesof thenovaoutburst dramat?

ically?Sphericallysymmetrichydro?dynamicalsim ulationsofthe accretion

processandthe follo wing outburst?Starr?eld etal??????Fujimoto????? have

shownthatitis crucialto havelargeoverabundanceofCNOnucleiinthe

H?richenvelopealreadyat theonset of the violent burning phaseinorderto

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getathermonuclearrunaway?TNR? atall?According tothesecalculations

amassfraction of CNOnucleiclose to ??? isrequired forastrongoutburst

anda fastnov a?Thecrucialquestion isthen? CanweexplainthehugeCNO

abundancesinthe accreted env elopeof thewhitedw arf? HereNis nota

problemsinceitis directlyproduced fromCandOduringH?burning? The

problem areC andO?

?Calculations

Herewe presen

calculations

t theresults

w

ofsim

with

ulations

PR

wehavecarriedout

explicit

fora

m

white

ulti?D

dwarfof?M

?

accretinghydrogen?richgas atarateof????

??

M

?

peryear?

The

a

calculations

higher

were

duno

carried

v

outin

with

acarthesian

an

system

Riemann?solv

ofco ordinates?

er?

Only

a fractionofthewhite dwarf?ssurfacewascov eredbythecomputationalgrid?

and

?????

equations

p erio

This

dic

h

de?ne

boundary

ydro

a

conditions

er

oundary

wereimpliedhorizon

generalized

coupling

tally?Tw

to

o?dimensional

more

computational

and

spacial

three?dimensional

dimensions

calculations

applying

werecarried

splitting?

out?

A

Thedimensions

equation

of the

calculated domainw as????x????km?radialxlateral? and ????x????x

????

under

a

km

consideration?

computational

?radialxlateral

an

grid?

xlateral?

arbitrarily

Therefore?

co

degenerate

eac

verdbyagrid

non

to

ofthe dimensions

electron

vided

???

gas

ad?

x

and

??? and

photon

???x???

The

x ???

co

for

also

the

con

t

tains

w o?dimensional

a

and three?dimensional

reaction

sim ulation?resp ectiv ely?

The ere doneOMETHEUS?an

Eulerian

The

h ydrocode? The

is

code

b

con

y

tains

dividing

a

the

hydro

computational

solverwhich

grid

is based

in

on

orderGo method exactnamelya

PPM?piecwise

part

parab

of

olicmetho d??sc

is

heme

the

according

ydro

to Colella

the

?W

dynamical

oo dward

solv

b

isone?dimensional

v

and than one

byop eratorrealistic ofstate

is

ditional

also part

zones

ofPR

to

OMETHEUS

uniquely

including

b

theBoltzmann

conditions

gases

ev

of

b

the

oundary

n uclei

relativistic

e

agas?denuclearnetw orkinclud?

ing

for

protons?

a

helium

blo

and??

the

CNO

in

isotopes?

of

solving

the

sti?

dynamical

ordinarydi?eren

equations

tial

on

equations tomodeln uclearreactionsbyaNewton?Raphsontechnique?

parallelizationdone tosmall

bloc ks eachcorresp ondingtoa processorelement ?PE??Themost c hallenging

in termsparallelizationhsolver sinceh ydro

alueproblemthewhole

grid? ingeneral?ThePPM?sceme is anappro ximation which needs only?ad?

de?neoundaryatery of

hblockhasbpro withfour

ditionalso?calledghost zonestouniquelyde?ne theboundaryv alueproblem

singlec k? Beforetegrationsh ydro

each blockfor a timestep aredone theghostzonesare?lledwithinformation

from theneighbouringblocks?Thismeansthattherehastobe communi?

cationb etweenneighb ouringPEs precedingeachh ydrodynamicaltimestep?

Thisis donebyexplicitmessagepassingusingtheMPP? and? ina more

recentversion ofPROMETHEUS?the MPI?library?

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Thehydrodynamicaltimestep isgivenbytheCFL?Courant?Friedric h?

Levy??condition ?Colella?Woo dw ard??????Thetimestep iscomputed in

ev eryzone andtheminimaltimestep accoun tsfor thefollowing globalh y?

drodynamicaltimestepapplied tothewholegrid?

Allcalculationsw ereperformedontheCRAYT?EoftheRechenzen trum

Garching?Thethree?dimensionalsim ulations consumed about?????? PE

hoursonthe full???PE system?Theparallele?ciencyisov er????

?NumericalResults

Firstsimulations weredone intwospatialdimensions?assuming thatall

ph ysicalquan titiesare independentofthe thirddimension?Inthissymmetry

conv ective eddiesarerepresentedbyin?nitelylong rolls?Some ofourresults

aredisplayedinFig??and??Fig??showssnapshotsoftheabsolutevalue of

thev elocities obtained inthis ?Dsimulation?Asmall temperatureincrease

w

ams

asimp

??????

osed on

is

one

for

zone

the

asan

time

initial

ho

p

er?

erturbation?

that

Already

deviations

??seconds

from

later

soundwav esemerging from the ignitionregionignited thewholeb ottom

layerof theenvelope ?Fig??a?? As timeprogressesquasi?stationaryaxially

symmetric

there

sim

?ow?elds

y

p

app

found

ear ?Fig??b?

in

one

whic

ysical

h dominate

applications

run

the?o

and

times

w

ha

patterns

v

higher

after

ortan

spatial

about ???s?Fig??c??Thesevorticesarevery stable andcarrya considerable

amount

a

of

direct

theturbulen

consequence

tkinetic

w

energy

cannot

? andthey

the

disapp

results

ear

of

only

the

when

sim

the

n

of

some

uclear

Glasner

detail

energygeneration

Livne

here?

reac

and

to

hes

Glasner

the

itsmaxim

et

umafter

?????

that?

sev

W

least

eral

?nd

in

h

considerably

?D?

undredseconds?

Theappearance ofaxi?symmetric quasi?stablestructuresin?D ?owsis

not

ergy

new

is

but

v

hasb

for

eenfound

the

earlier

?and

in

fastest?

?Dsim ulations

con

of

e

con

eddies?

v ection

Although

in plane?

???

parallel geometry? andhas alsob een observed in the Earth?soceans?McWilli?

It?rstwevsuchstandard

inertially

outburst?

driv

again

enturbulence?

in

where

to

theenergy

calculations?

isfedin totheturbulenteddies

onthe

In

largest

order

scales?

mak

cascades

sure

do

all

wn

relev

in to

an

the

t

small

scales

scales?

ha

and isdissipated

ed

by viscosit

w

?

erformed

areastroph

additional

eimpt

implications

resolution?

there?

but

Ase con?rmulations

????? al?eless

dredge?upofCandO? mainlybecauseonaverage lessturbulentkineticen?

a ailablelargestvectiv

seconds afterignitionaconsiderableamount of

??

Ccanbeseenintheen?

velope ?????themixing processhasbeentoo slow toyieldastrong nova

contrastearlier

to ethatveb eenresolv inour

ulationsewith ?ve

didnot?nd majordi?erences? asfar astheglobal properties

are concerned?Mixing w asslightlylesse?cien tthan inthe modeldescrib ed in

leadingconclusionatself?enrichment

ofanovaenvelopebyviolentconvectivemotionsisaratherunlikelyprocess?

Finally?wealsoperformed?rst?Dcalculations?againwiththesameinput

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physicsand similarresolution asb efore?butfound ev enless mixing?therefore

less energy generation anda completelydi?eren t ?owstructure ?Fig???than

in?D asmight haveb eensusp ected?Thestablevorticesdo notapp earin

the?Dcalculationssincethe?owis notrestrictedtotwo spacialdimensions

anymoreandthereforevorticityisnotconservedalonga ?uidparticlepath?

Werather observeconvectiveand turbulentpatternsasone mightexpectfor

a?Dconvective?ow?

Itisob vious?accordingtothedi?erencesbetween the?Dand ?Dcalcu?

lations? that?Dsimulationsof the TNRontopofa whitedwarf areindeed

necessarytomo delthe turbulent combustionprocess?

? Conclusions

Wehave demonstratedthatdirectn umerical simulations ofnovaoutbursts?

as faras thephysics isconcerned? require severe enric hmentofthe H?rich

env elope priortothe TNR?If forsomereasonthis does nothapp en the

whitedw arfwill enterintoa phaseof quieth ydrostaticburningandwill? if

the env elopeful?llscertain requiremen ts?resem bleasuper?soft X?raysource

more thananova?Wealso have sho wnthatdirectn umericalsimulations

of reactionhydro dynamicshavebecome feasible?evenin?D? andshouldbe

donewhen fuelandashesarew ell mixedandburning times arem uch longer

thanthoseof turbulent mixing?

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Gas?DynamicalSimulations? J?Comp?Physics??????? ??????

FujimotoM?Y??ATheory ofHydrogenShell Flasheson AccretingWhiteDwarfs?

I?Their ProgressandtheExpansionof the Env elope?Astrophysic al Journal? ????

??? ???????

Glasner S?A? andLivneE?? ConvectiveHydrogen BurningDuringa Nova Out?

burst?Astr ophysicalJournal? ???????L???????

GlasnerS?A?? LivneE?and Truran J?W??Reactive Flow inNovaOutbursts?

Astrophysic alJournal? ??????? ???????

McWilliamsJ?C?? TheEmergence of IsolatedCoherentVortices inTurbulent

Flo w? J? FluidMe ch????????? ???????

StarrfieldS?? TruranJ?W?? SparksW?M? andKutterG?G?? CNOAbun?

dancesandHydrodynamic Models oftheNovaOutburst? Astrophysic alJournal?

??????? ???????

StarrfieldS??Sp arksW?M?and TruranJ?W?CNOAbundancesand Hy?

drodynamic Modelsof theNova Outburst?II?????M

?

Models with Enhanced

CarbonandOxygen? Astr ophysicalJournalSupplement??????????????

StarrfieldS?? TruranJ?W? andSparksW?M??CNOAbundancesandHydro?

dynamic Models ofthe NovaOutburst?V?????M

?

Models withSmallMass

Envelopes?Astr ophysicalJournal???????????????

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a?

b?

c?

d?

Fig???V elocity?eld at di?erent stagesoftheevolution forthe ?Dmo delexplained

inthe text? Giv enincoloristhe absolutev alue ofthevelocity ateachpoin t?T?

denotesthetemperature of the hottestindividualzone inunits of??

?

K?