Session 3A:STARS Convection Comm. in Asteroseismology Vol. 157, 2008, WrocMl aw HELASWorkshop 2008 M. Breger,W.Dziembowski,&M. Thompson,eds.

Modelingstochasticexcitationofacousticmodes in stars: presentstatus andperspectives

R. Samadi1,K.Belkacem1,M.-J.Goupil1,H.-G.Ludwig2 ,and M.-A.Dupret1

1 Observatoire de Paris, LESIA, CNRS UMR8109, 92195 Meudon, France 2 Observatoire de Paris, GEPI,CNRSUMR 8111, 92195 Meudon, France

Abstract

Solar-likeoscillations have nowbeen detected formorethanten yearsand theirfrequencies measuredfor astill growingnumberofstars with variouscharacteristics (e.g. mass, chemical composition,evolutionarystage ...).Excitationofsuch oscillations is attributed to turbu- lent convection andtakes placeinthe uppermost part of the convective envelope. Since the pioneering work of Goldreich&Keely(1977),moresophisticated theoreticalmodelsof stochastic excitation were developed,which differfromeachotherbothbythe wayturbulent convection is modeled andbythe assumedsources of excitation. We briefly review herethe different underlyingapproximations andassumptionsofthosemodels. Asecond part shows thatcomputed mode excitation rates crucially depend on the waytime-correlations between eddies aredescribed but alsoonthe surfacemetalabundance of the star.

IndividualObjects: Sun, α Cen A, HD 49933

Introduction

Solar p-modes areknown to have finitelifetimes(afew days)and averylow amplitude(afew cm/s in velocity andafewppm in brightness).Inthe lastdecade, solar-likeoscillations have been detected in numerousstars, in different evolutionarystages andwithdifferent metallicity (see recent review by Bedding &Kjeldsen, 2007).Theirfinite lifetimesare aconsequence of severalcomplex dampingprocessesthatare notclearly identifiedsofar.Theirexcitationis attributed to turbulent convection andtakes placeinthe uppermost part of the convective envelope, whichisthe placeofvigorousturbulent motions. Measuringthe mode amplitudeand the mode lifetime enables to inferthe excitation rate, P (the energy whichissuppliedper unittimetothe mode). Deriving P put constraints on the theoreticalmodelsofmodeexcitationbyturbulent convection (Libbrecht, 1988). Goldreich&Keeley (1977, GK hereafter)haveproposedthe firsttheoreticalmodel of stochastic excitation of solaracousticmodes by the Reynolds stresses. Since thispioneering work,different improved modelshavebeen proposed (Dolginov&Muslimov,1984; Balmforth, 1992; Goldreichetal.,1994; Samadi&Goupil, 2001; Chaplinetal.,2005; Samadietal., 2003; Belkacem et al.,2006b,2008).Theseapproaches differfromeachothereither by the wayturbulent convection is described, or by the excitation process. In the present paper, we briefly review the mainassumptionsand approximations on whichthe different theoretical modelsare based. As shownbySamadietal. (2003),the energy suppliedper time unittothe modes by turbulent convection cruciallydepends on the wayeddies aretime-correlated. Arealistic R. Samadi, K. Belkacem,M.-J.Goupil, et al. 131 modelingofthe eddy time-correlationatvarious length scalesisthen an important issue, whichisdiscussedindetails below. Thestructureand the properties of the convective upper envelopealsohas an important impact on the mode driving. In particular,the surfacemetal abundance cansignificantlychangethe efficiency of the mode driving.

Theoreticalmodels

Most of the theoreticalmodelsofstochasticexcitationadopt GK’s approach.Thisapproach firstconsistsinsolving,withappropriateboundary conditions,the equation thatgoverns the adiabatic wave propagation(alsocalled the homogeneous wave equation). This provides the well-knownadiabatic displacement eigenvectors (ξ;(;r, t)). Then, one includes in the wave equation turbulent sources of drivingaswellasatermoflineardamping. Thecomplete equation (so-calledinhomogeneouswaveequation)isthen solved. Among the sources of driving, the contribution of Reynolds stresses, whichrepresents a mechanicalsourceofdriving,isconsidered. Goldreichetal. (1994, GMKhereafter)have proposed to include in addition the entropyfluctuations(alsoreferredtoasnon-adiabatic gaspressure fluctuations). It is generallyassumedthatthe entropyfluctuationsbehave as apassive scalar. In thatcase, Samadi&Goupil(2001, SG hereafter)haveshown thatthe contribution of the Lagrangian entropyfluctuationsvanishes.Onthe other hand, as shown by SG,the advection of the entropyfluctuationsbythe turbulent velocity field contributes efficientlytothe mode drivinginaddition to the Reynolds stresses. This advective term corresponds to the buoyancy forceassociated with the entropyfluctuations. Since it involves the entropyfluctuations, it canbeconsidered as athermal source of driving. Thesolution of the inhomogeneouswaveequation corresponds to the forced mode dis- placement, δ;r (orequivalentlythe mode velocity ;vosc = dδ;r /dt). Adetailedderivationofthe solution for radial acoustic modes canbefound in Samadi&Goupil(2001, SG herefater)or in Chaplinetal. (2005, CHEhereafter). It canbewritten as

˙ ¸ ; 2 2 ; 2 P || ξ || 2 2 || ;vosc || (ω )=|| ξ || = (C + C ), (1) 0 2 η I 16 η I 2 R S where ω0 is the mode frequency, P is the mode excitation rate (the rate at whichenergy is suppliedtothe mode), η the mode dampingrate(whichcan be derived from seismicdata), 2 2 I the mode inertia, andfinally CR and CS the contribution of the Reynolds stress andthe 2 2 entropyfluctuations, respectively.The expressionsfor CR and CS are

ZM „ «2 2 3 dξr CR =4πG dm ρ0 SR (r, ω0) (2) 0 dr Z 4 π3 H M α2 C 2 = dm s g S (r, ω ) (3) S 2 r S 0 ω0 0 ρ0 where αs =(∂Pg/∂s)ρ, Pg is the gaspressure, s the entropy, ρ0 the mean density, G and H aretwo anisotropicfactors (see theirexpression in SG), SR and SS arethe “sourceterms” associated with the Reynolds stressesand entropyfluctuationsrespectively, ξr the adiabatic mode radial eigen-displacement, andfinally gr (ξr , r)afunction thatinvolvesthe first and second derivatives of ξr (see itsexpression in SG). Thesourcefunctions, SR and SS ,involve the dynamicpropertiesofthe turbulent medium. Theexpression for SR is (see SG andSamadietal. (2005)): Z Z ∞ +∞ E 2(k) S (r, ω )= dk dω χ (ω + ω) χ (ω) (4) R 0 2 k 0 k 0 −∞ k 132 Modelingstochastic excitation of acousticmodes in stars: presentstatusand perspectives

where k is the wavenumber, E(k)the time averaged kineticenergy spectrum, χk (ω)isthe frequency component of E(k, ω)(the kineticenergy spectrum as afunction of k and ω). A similarexpression is derived for SS (see SG). Note thatthe function χk (ω)can be viewed as ameasure in the Fourierdomain of the time-correlationbetween eddies.Thisfunction is generallyreferredas’eddy time-correlation’ function. ThederivationofEqs.(1)-(4) is basedonseveral assumptionsand approximations.Among them,the mainones are: • quasi-Normal approximation(QNA):the fourth-order moments involvingthe turbulent velocity aredecomposedinterms of second-order ones assumingthe QNA, i.e. assuming thatturbulent quantities aredistributed accordingtoaNormaldistribution with zero mean.However,asshown recentlybyBelkacem et al. (2006a), thepresence of plumes causes aseveredeparturefromthisapproximation. Belkacem et al. (2006a)have proposed an improvedclosure model thattakes the asymmetrybetween plumes and granules as well as the turbulence inside the plumes into account. As shownbyBelkacem et al. (2006b), this improved closuremodel reduces the discrepancy between theoretical calculations andthe helioseismicconstraints.

• isotropic, homogeneous, incompressibleturbulence:the turbulent medium is assumed to be isotropic, homogeneous, andincompressible at the lengthscale associated with the contributing eddies.Thisassumption is justifiedfor lowturbulent Mach number, Mt (see SG). However, when the anisotropy is small, it is possibletoapplyacorrection that takesthe departurefromisotropyinto account. Such correctionshavebeen proposed by SG andCHE in twodifferent ways.However,inbothformalisms, the exactdomain over whichthesecorrections arevalidisunknown andremainstobespecified.

• radial formalism:radial modes areusuallyconsidered. However, generalizations to non- radial modes have been proposed by Dolginov &Muslimov (1984), GMKand Belkacem et al. (2008).

• passive scalarassumption:aspointed outabove the entropyfluctuationsare supposed to behave as apassive scalar(seethe discussion).

• length-scale separation:eddies contributing to the drivingare supposed to have achar- acteristiclength scale smallerthanthe mode wavelength. This assumption is justified forlow Mt (see SG andthe discussion).

Eddy time-correlation

Most of the theoreticalformulations explicitly or implicitly assume aGaussian function for χk (ω)(GK;Dolginov&Muslimov,1984; GMK; Balmforth, 1992; CHE).However,hydrody- namical3Dsimulations of the outer layers of the Sunshowthat, at the length associated with the energy bearingeddies, χk is ratherLorentzian (Samadietal.,2003).Aspointed outbyCHE,aLorentzian χk is alsoaresult predicted forthe largest, most-energetic eddies by the time-dependent mixing-length formulationofconvectionbyGough (1977).Therefore, thereare some numerical andtheoreticalevidences that χk is ratherLorentzian at the length scale of the energy bearingeddies. Theexcitationofthe low-frequency modes (ν ! 3mHz)ismainly due to the large scales. However, the higher the frequency the more importantthe contribution of the small scales. Solar 3D simulations show that, at small scales, χk is neither Lorentzian norGaussian (Georgobianietal.,2006).Then, accordingtoGeorgobianietal. (2006),itisimpossible to separate the spatial component E(k)fromthe temporalcomponent at all scaleswith the same simple analyticalfunctions. However, such resultsare obtained usingLarge Eddy R. Samadi,K.Belkacem,M.-J.Goupil, et al. 133

Figure 1: Solar p-modeexcitation rates, P,asafunction of ν = ω/2π.The dotscorrespond to seismic data from SOHO/Golf (filledcircles)and from theBiSON network(diamonds).The thicklines correspond 3D to semi-theoretical calculations basedondifferentchoices for χk :Lorentzian χk (solid line), χk i.e. χk directly derived from thesolar 3D simulation(dashed line),and aGaussian χk (dot-dashedline).

Simulation(LES). Theway the small scalesare treated in LES canaffect our description of turbulence. Indeed, He et al. (2002) have shownthatLES resultsinaχk(ω)thatdecreases at all resolved scalestoo rapidly with ω with respect to direct numerical simulations (DNS). Moreover,Jacoutotetal. (2008a)found thatcomputed mode excitation rates significantly depend on the adopted sub-grid model.Furthermore,atagivenlength scale,Samadietal. (2007) have shownthat χk tends toward aGaussian when thespatial resolution is decreased. As aconclusionthe numerical resolution or the sub-grid model cansubstantiallyaffect our description of the small scales. As shownbySamadietal. (2003),calculationofthe mode excitation rates basedon aGaussian χk resultsfor the Suninasignificantunder-estimationofthe maximumofP whereasabetter agreement with the observations is found when aLorentzian χk is used (see Fig. 1).Asimilarconclusionisreached by Samadietal. (2008a)inthe case of the star α CenA.

Up to now, only analyticalfunctionswereassumedfor χk (ω). We have hereimplemented, forthe calculationofP,the eddy time-correlationfunction derived directly from long time series of 3D simulationrealizations with an intermediatehorizontalresolution(≃50 km). 3D As showninFig.1,the mode excitation rates, P,obtained from χk ,are found comparable to thatobtained assuming aLorentzian one, except at high frequency.Thisisobviously the 3D direct consequence of thefactthataLorentzian χk reproducesrather well χk ,except at high 3D frequency where χk decreasesmorerapidly thanthe Lorentzian function.Athighfrequency, calculations basedonaLorentzian χk result in larger P andreproduce better the helioseismic 3D 3D constraints thanthose basedonχk .Thisindicates perhapsthat χk decreases toorapidly 134 Modelingstochastic excitation of acousticmodes in stars: presentstatusand perspectives with frequency thanitshoulddo. This is consistentwithHeetal. (2002)’s resultswho found thatLES predicts atoo rapiddecrease with ω compared to the DNS(seeabove). CHEalsofound thatthe useofaGaussian χk severely under-estimates the observedsolar mode excitation rates.However,incontrastwithSamadietal. (2003),they mention thata Lorentzian χk resultsinanover-estimationfor the low-frequency modes.They explain thisby the fact that, at agiven frequency,aLorentzian χk decreasestoo slowly with depth compared to aGaussian χk .Consequently, forthe low-frequency modes,asubstantial fraction of the integrand of Eq.(2) arises from largeeddies situated deep in the Sun. This mightsuggests that, in the deep layers, the eddies thatcontribute efficiently have ratheraGaussian χk (see thediscussion below).

Impactofthe surface metal abundance

We have computed two3Dhydrodynamicalsimulations representative –ineffective tem- peratureand gravity–of the surfacelayersofHD49933, astarwhich is rathermetalpoor comparedtothe Sun. One3Dsimulation(hereafter labeled as S0) hasasolarmetalabun- dance andone other (hereafter labeled as S1) hasasurfaceiron-to-hydrogen abundance, [Fe/H], ten timessmaller. For each 3D simulationwematch in the manner of Samadietal. (2008a)anassociated global1Dmodel andwecompute the associated acoustic modes. Therates P at whichenergy is suppliedinto the acoustic modes associated with S1 are found aboutthree timessmallerthanthose associated with S0.Thisdifference is related to the fact thatalowsurface metallicityimpliessurface layers with ahigher mean surface density. In turn, higher mean surfacedensityimpliessmallerconvective velocity andhence a less efficient drivingofthe acoustic modes (for details seeSamadietal.,2008b). This result illustrates the importance of taking the metallicityofthe star into account when computing P. This conclusionisqualitativelyconsistent with thatbyHoudek et al. (1999) who– on the basisofamixing-length approach –alsofound thatthe mode amplitudes decrease with decreasing metalabundance.

Discussionand perspectives

Theway mode excitation by turbulent convection is modeled is still very simplified. Some approximations must be improved,someassumptionsorhypothesis must be avoided:

• length-scale separation:Thisapproximationislessvalidinthe super-adiabatic region wherethe turbulent Mach numberisnolonger small (for the Sun Mt is up to 0.3). This spatial separation canhowever be avoided if the kineticenergy spectrum associated with the turbulent elements (E(k)) is properly coupled with the spatial dependence of the modes

• eddy time-correlationfunction, χk : Current modelsassume that χk varies with ω in the same wayatany length scalesand in anyparts of the convective zone (CZ).At the length scale of the energy bearingeddy andinthe uppermost part of the CZ,there aresomestrongevidences that χk is Lorentzian ratherthanGaussian.However,as discussedhere, it is notyet clearwhatisthe correct description for χk at the small scalesand alsodeep in the CZ.Use of more realistic3Dsimulations wouldbevery helpful to depict the correct dynamicalbehavior of the small scalesaswellasinthe deep CZ.

• passive scalarassumption:Thisisastrong hypothesis thatprobablyisnolonger valid in the super-adiabatic part of the convective zone wherethe drivingbythe entropyis important. Indeed, the super-adiabatic layeristhe seat of importantradiativelosses R. Samadi, K. Belkacem,M.-J.Goupil, et al. 135

by the eddies.Assuming thatthe entropybehavesasapassive scalarisnot correct. To avoidthisassumption,one needs to include eddy radiativelossesinthe model of stochastic excitation. Finally,westressthatsomesolar-likepulsators areyoung starsthatshowastrong activity (e.g. HD 49933) whichisoften linkedtothe presence of strong magneticfields.Astrong magneticfieldcan inhibitconvection (see e.g. Proctor&Weiss, 1982; V¨ogler et al.,2005). Furthermore,using 3D solarsimulations,Jacoutotetal. (2008b)havestudiedthe influence of magneticfields of variousstrengthonthe convective cells andonthe excitation mechanism. They found thatastrong magneticfieldresults in turbulent motionsofsmallerscalesand higher frequencies thaninthe absence of magneticfield, andconsequentlyinaless efficient mode driving. Further theoreticaldevelopments arerequiredtotakethe effects of amagnetic field into account in the theoreticalcalculationofthe mode excitation rates.

Acknowledgments. RS is grateful to the SOCfor the invitation to thisworkshopand ac- knowledges HELASfor the financial support. References

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DISCUSSION

Houdek: Thescalinglaw (L/M)1.5 in Houdeketal. (1999) refers to amplitudesatthe photosphere.At theheight of h =200 km,weobtained the(L/M)1.1 law. Bruntt: Youfind that lowmetallicityimplieslow pulsationamplitudes forsolar-likeoscillations.Haveyou triedtoinclude metallicityasaparameter in your scaling relationofamplitudes? Samadi: Thescalinglaw (L/M)0.7 proposed in Samadi et al. (2007) wasindeed derived usingstellar3D simulations with solarmetal abundances.Partofthe remainingdiscrepanciesbetween thescalinglaw (L/M)0.7 andthe observations maybeexplainedbythe fact that some starshaveametalabundance significantlydifferenttothatofthe Sun. This is particularly so forHD49933. Noels: Youtalked about “solar”abundances.Whatabundances andmetallicityare youreferingto? Samadi: We have considered both the“old” andthe “new” solarabundances.Atfixed [Fe/H],the “new” solarabundances result in alower totalmetal abundancethanthe “old”onesand henceinlower P.In thecase of HD 49933, mode excitation rates, P,computedbyassuming the“new” solarabundances are found ∼ 30 %smallerthanthosecomputedassuming the“old” solarabundances (see Samadi et al., 2008b). Noels: Couldwehavesomesolar-likeoscillations in starsasmassive as β Cepheid starsifaconvection zone appearsnearthe surfacedue to an accumulationofiron? Samadi: As faraswehaveasurfaceconvectiveenvelope, it is potentially possible to excite p modes. However,the excitation is efficientwhenthe characteristic eddy turn-over time is of thesameorder than theperiodofthe p modesconfinednearthe surface.

G¨unter Houdek enjoyingthe boat ride Comm. in Asteroseismology Vol. 157, 2008, WrocMl aw HELASWorkshop 2008 M. Breger,W.Dziembowski,&M. Thompson,eds.

Theeffect of convectiononpulsational stability

G. Houdek InstituteofAstronomy, UniversityofCambridge,Cambridge CB30HA, UK

Abstract

Areviewonthe current stateofmodephysics in classicalpulsators is presented. Two, currentlyinuse,time-dependent convection modelsare comparedand theirapplications on mode stabilityare discussedwithparticular emphasis on thelocationofthe DeltaScuti instabilitystrip.

Introduction Starswithrelativelylow surfacetemperatures show distinctive envelopeconvection zones whichaffect mode stability. Among the first problems of thisnaturewas the modellingofthe rededgeofthe classicalinstabilitystrip (IS) in the Hertzsprung-Russell (H-R) diagram.The first pulsation calculations of classicalpulsators without anypulsation-convection modelling predicted rededges whichweremuch toocooland whichwereatbestonlyneutrally stable. What follows were severalattempts to bringthe theoreticallypredictedlocationofthe rededge in better agreement with the observed location by usingtime-dependent convection models in the pulsation analyses(Dupree 1977;Baker &Gough 1979; Gonzi1982; Stellingwerf 1984).Morerecently several authors, e.g. Bono et al. (1995,1999),Houdek (1997, 2000), Xiong&Deng (2001, 2007),Dupretetal. (2005) were successful to model the rededgeofthe classicalIS. Theseauthors report,however,thatdifferent physical mechanisms areresponsible forthe returntostability. For example,Bonoetal. (1995) andDupretetal. (2005) report thatitismainly the convective heatflux,Xiong &Deng (2001) the turbulent viscosity, and Baker&Gough(1979) andHoudek (2000) predominantlythe momentum flux(turbulent pressure pt)thatstabilizes thepulsation modes at the rededge. Time-dependent convectionmodels Theauthors mentioned in the previous section used different implementationsfor modelling the interaction of the turbulent velocity field with the pulsation.Inthe past varioustime- dependent convection modelswereproposed, forexample,bySchatzman (1956),Gough

Figure 1: Sketch of an overturninghexagonal(dashed lines) convectivecell. Near thecentrethe gasraisesfromthe hot bottomtothe cooler top(surface) whereitmoves nearly horizontallytowards theedges, therebyloosingheat. The cooled gasthendescends along theedgestoclose thecircular flow. Arrowsindicate thedirectionofthe flowpattern. 138 Theeffect of convectiononpulsational stability

Table1:Summary of time-dependent convectionmodel differences. Balance between buoyancy &Kinetictheory of accelerating turbulent drag (Unno 1967, 1977) eddies (Gough 1965, 1977a) -accelerationterms of convective -accelerationterms included: w, fluctuations w, T ′ neglected T ′ evolve with growth rate σ -nonlinearterms approximated -nonlinearterms areneglected by spatial gradients ∝ 1/ℓ duringeddy growth -characteristiceddy lifetime:-τ=2/σ determined stochas- τ ≃ ℓ/2w tically from parametrized shear instability

-variation ℓ1 = δℓ/ℓ (Unno 1967):-variationofmixinglength ωτ < 1: ℓ1∼H1 accordingtorapid distortion ωτ > 1: ℓ1∼r1 theory(Townsend 1976),i.e. or (Unno 1977):variationalsoofeddy shape 2 2 −1 ℓ1 ∼ (1 +iω τ ) (H1−iωτρ1/3) (H is pressure scale height)

-turbulent pressure pt neglected - pt = ρ ww included in mean in hydrostaticsupportequation equ. forhydrostatic support

(1965, 1977a), Unno (1967, 1977),Xiong (1977, 1989),Stellingwerf (1982),Kuhfuß (1986), Canuto (1992),Gabriel (1996), Grigahc`e ne et al. (2005).HereIshall briefly review and compare the basicconcepts of two, currentlyinuse,convection models. Thefirst model is thatbyGough (1977a,b), whichhas been used, forexample,byBaker &Gough (1979), Balmforth(1992) andbyHoudek (2000).The second model is that by Unno(1967, 1977), upon whichthe generalizedmodelsbyGabriel (1996) andGrigahc`ene et al. (2005) arebased, with applications by Dupret et al. (2005). Nearly all of the time-dependent convection modelsassume the Boussinesqapproximation to the equationsofmotion. TheBoussinesqapproximationreliesonthe fact thatthe heightof the fluid layerissmallcomparedwiththe densityscale height. It is basedonacarefulscaling argument andanexpansion in small parameters(Spiegel &Veronis1960; Gough1969).The fluctuating convection equations foraninviscidBoussinesqfluidinastatic plane-parallel atmosphereare

−1 ′ ′ ∂t ui +(uj∂jui −uj∂jui)=−ρ∂ip+gαbTδi3, (1) ′ ′ ′ −1 ′ ∂t T +(uj∂jT −uj∂jT )=βw−(ρcp)∂iFi, (2) supplemented by the continuityequation foranincompressible gas, ∂j uj =0,where u=(u, v, w) is the turbulent velocity field, ρ is density, p is gaspressure, g is the accelerationdue to grav- ity, T is temperature, cp is the specific heatatconstantpressure, αb = −(∂ ln ρ/∂ ln T )p / T , Fi is the radiativeheatflux, β is the superadiabatic temperaturegradient and δij is the Kroneckerdelta.Primes(′)indicate Eulerian fluctuations andoverbarshorizontalaverages. Theseare the starting equationsfor the twophysicalpictures describing the motion of an overturning convective eddy,illustrated in Fig. 1. In the first physicalpicture, adopted by Unno(1967),the turbulent element, with a characteristicverticallength ℓ,evolves out of some chaoticstate andachieves steady motion very quickly.The fluid element maintainsexact balance between buoyancy forceand turbulent drag by continuous exchange of momentum with other elements andits surroundings. Thus ′ the accelerationterms ∂t ui and ∂t T areneglected andthe nonlinearadvection terms provide G. Houdek 139

Figure2:Modestabilityofan1.7 M⊙ DeltaScuti star computed with Gough’s(1977a,b)convection model. Left:Stabilitycoefficient η = ωi/ωr as afunction of surfacetemperature Teff across theIS. Resultsare shownfor thefundamental radial mode (n =1)and fortwo values of themixing-length parameter α.Positive η values indicate mode instability. Right: Integrated work integral W as afunction of thedepth co-ordinatelog(T )for amodel lyingjustoutsidethe cool edge of theIS(Teff =6813 K). Resultsare plottedinunits of η andfor α =2.0.Contributions to W (solid curve) arising from thegas pressure perturbation, Wg (dashed curve),and theturbulentpressure fluctuations, Wt (dot-dashedcurve), areindicated(W =Wg+Wt). Thedottedcurve is theratio of theconvectivetothe totalheatflux Fc/F. Ionization zonesofHandHe(5% to 95% ionization) areindicated(from Houdek2000). dissipation (ofkineticenergy)thatbalances the drivingterms. Thenonlinearadvection terms 2 ′ ′ ′ areapproximated by uj ∂j ui − uj ∂j ui ≃ 2w /ℓ and uj ∂j T − uj ∂j T ≃ 2wT /ℓ.Thisleads to twononlinearequationswhich need to be solved numerically together with the mean equations of the stellarstructure. Thesecond physicalpicture, whichwas generalizedbyGough (1965, 1977a,b)tothe time- dependent case,interprets the turbulent flowbyindirect analogy with kineticgas theory. The motion is notsteady andone imagines the convective element to acceleratefromrestfollowed by an instantaneousbreakup after the element’slifetime.Thus the nonlinearadvection terms areneglected in the convective fluctuation equations (1)-(2)but aretaken to be responsible forthe creation anddestruction of the convective eddies (Gough1977a,b). By retainingonly the accelerationterms the equationsbecomelinearwithanalyticalsolutions w ∝ exp(σt) and T ′ ∝ exp(σt)subject to proper periodic spatial boundary conditions,where t is time and ℜ(σ)isthe linearconvective growth rate. Themixinglength ℓ entersinthe calculationofthe eddy’s survival probability, whichisproportionaltothe eddy’s internalshear(rms vorticity), fordetermining the convective heatand momentum fluxes. Although the twophysicalpictures give the same result in astaticenvelope, the resultsfor the fluctuating turbulent fluxesina pulsating star areverydifferent (Gough1977a). Themain differences between Unno’sand Gough’sconvection model aresummarized in Table1.

ApplicationonmodestabilityinδScutistars

Fig. 2displaysthe mode stabilityofanevolving1.7 M⊙ DeltaScuti star crossing the IS.The resultswerecomputed with the time-dependent, nonlocal convection model by Gough(1977a,b). As demonstrated in the rightpanel of Fig. 2, the dominating damping termtothe work integral W forastar located nearthe rededgeisthe contribution from the turbulent pressure fluctuations Wt. Gabriel(1996) andmorerecentlyGrigahc`e ne et al. (2005) generalizedUnno’s time- dependent convection model forstabilitycomputationsofnonradial oscillationmodes.They included in their mean thermalenergy equation the viscousdissipation of turbulent ki- neticenergy, ǫ,asanadditional heatsource. Thedissipation of turbulent kineticenergy is introduced in the conservation equation forthe turbulentkineticenergy K := ui ui /2 140 Theeffect of convectiononpulsational stability

Figure3:Stabilitycomputations of DeltaScuti starswhich include theviscous dissipationrate ǫ of turbulent kineticenergyaccordingtoGrigahc`e ne et al. (2005).Left: Blue andred edgesofthe IS superposed on evolutionary tracks on thetheoristsH-R diagram. Thelocations of theedges, labelledpnBandpnR,are indicatedfor radial modeswithorders1≤n≤7. ResultsbyHoudek (2000, α =2.0,see Fig. 2) and Xiong &Deng(2001) forthe gravestpmodesare plottedasthe filledand open circlesrespectively. Right: Integrated work integral W as afunction of thedepth co-ordinatelog(T )for astable n =3radial mode of a1.8 M⊙ star (see ‘star’ symbol in theleftpanel). Contributions to W arising from theradiativeflux, WR,the convectiveflux, Wc,the turbulentpressure fluctuations, Wt (WR + Wc + Wt), andfromthe perturbation of theturbulentkinetic energy dissipation, Wǫ (WR + Wc + Wǫ), areindicated(adaptedfrom Dupret et al. 2005).

(e.g. Tennekes &Lumley1972, §3.4; Canuto 1992; Houdek &Gough 1999):

−1 ′ 2 ′ Dt K + ∂j (Kuj + ρ p uj ) − ν∂i K = −uiuj∂jUi + gαbujT − ǫ, (3) whereDtis the material derivative, Ui is the average(oscillation) velocity,i.e.the total velocity u˜i = Ui + ui ,and ν is the constant kinematic viscosity(in the limit of high Reynolds numbers the molecular transporttermcan be neglected).The first andsecond termonthe rightofEq. (3) arethe shearand buoyant productionsofturbulent kineticenergy,whereasthe 2 lastterm ǫ = ν(∂j ui + ∂i uj ) /2isthe viscousdissipation of turbulent kineticenergy into heat. This termisalsopresent in the mean thermalenergy equation, but with opposite sign.The linearizedperturbedmeanthermal energy equation forastar pulsating radiallywithcomplex angularfrequency ω = ωr +iωi canthen be written,inthe absence of nuclearreactions,as (‘δ’denotes aLagrangian fluctuation andIomit overbars in the mean quantities):

dδL/dm = −iωcpT (δT /T −∇adδp/p)+δǫ , (4) where m is the radial massco-ordinate, ∇ad =(∂ln T /∂ ln p)s and L is the total(radia- tive andconvective)luminosity. Grigahc`e ne et al. (2005) evaluated ǫ from aturbulent kineticenergy equation whichwas derived without the assumption of the Boussinesqap- proximation. Furthermore it is notobvious whether the dominantbuoyancy production ′ term, gαbuj T (see Eq. 3),was included in their turbulent kineticenergy equation andso in their expression for ǫ. Dupret et al. (2005) appliedthe convection model of Grigahc`e ne et al. (2005) to DeltaScuti and γ Doradus starsand reported well defined rededges.The resultsoftheirstabil- ityanalysisfor DeltaScuti starsare depicted in Fig. 3. Theleftpanel comparesthe location of the rededgewithresults reported by Houdek (2000, seealsoFig.2)and Xiong&Deng (2001). Theright panel of Fig. 3displaysthe individualcontributionstothe accumulated work in- tegral W forastar located nearthe rededgeofthe n =3mode (indicated by the ‘star’ G. Houdek 141

Figure4:Accumulated work integral W as afunction of thedepth co-ordinatelog(p). Resultsare shown forthe n =1radial mode of aDelta Scutistarlocated inside theIS(left panel) and outsidethe rededge of theIS(right panel).The stabilitycalculations include viscous dissipationbythe small-scale turbulence (Xiong 1989;see Eq.6). Contributions to W (solid curve) arising from thefluctuatinggas pressure, Wg (dashed curve),the turbulentpressure perturbations, Wt (long-dashed curve),and from theturbulent viscosity, Wν (dottedcurve),are indicated(W =Wg+Wt+Wν). Theionization zonesofHandHeare marked (adapted from Xiong &Deng2007). symbol in the left panel). It demonstrates the near cancellationeffect between the contribu- tionsofthe turbulent kineticenergy dissipation , Wǫ,and turbulent pressure, Wt,makingthe contribution from the fluctuating convective heatflux, Wc,the dominating dampingterm. Thenearcancellationeffect between Wǫ and Wt wasdemonstrated first by Ledoux &Wal- raven(1958, §65) (see alsoGabriel 1996)bywriting the sumofbothworkintegrals as:

Z M ∗ −2 Wǫ + Wt =3π/2 (5/3 − γ3)ℑ(δpt δρ)ρ dm , (5) mb where M is the stellarmass, mb is the enclosed mass at the bottomofthe envelopeand γ3 ≡ 1+(∂ln T /∂ ln ρ)s (s is specific entropy) is the third adiabatic exponent. Except in ionization zones γ3 ≃ 5/3and consequently Wǫ + Wt ≃ 0. Theconvection model by Xiong(1977, 1989) uses transportequations forthe second- order moments of the convective fluctuations.Inthe transportequation forthe turbulent kineticenergy Xiongadopts the approximationbyHinze (1975) forthe turbulent dissipation 2 2 3/2 rate, i.e. ǫ =2χk(uiuiρ /3ρ ) ,where χ =0.45isthe Heisenbergeddy couplingcoefficient and k ∝ ℓ−1 is the wavenumberofthe energy-containingeddies.However,Xiong does not provideawork integralfor ǫ (neither does Unno et al. 1989, §26,30) but includes the viscous dampingeffect of the small-scale turbulence in hismodel.The convection modelsconsid- ered heredescribeonlythe largest, most energy-containingeddies andignorethe dynamics of the small-scale eddies lyingfurther down the turbulent cascade. Small-scale turbulence does,however,contribute directlytothe turbulent fluxesand, under the assumption that theyevolveisotropically,they generateaneffective viscosity νt whichisfeltbyaparticular pulsation mode as an additional dampingeffect. Theturbulent viscositycan be estimated 1/2 as (e.g. Gough1977b;Unno et al. 1989, §20) νt ≃ λ(ww) ℓ,where λ is aparameter of order unity.The associated work integral Wν canbewritten in Cartesian co-ordinates as (Ledoux &Walraven 1958, §63) 142 Theeffect of convectiononpulsational stability

Z M » – 1 2 Wν = −2πωr νt e e − (∇·ξ) dm, (6) ij ij 3 mb where eij =(∂jξi+∂iξj)/2and ξ is the displacement eigenfunction. Xiong&Deng (2001, 2007) modelledsuccessfully the IS of DeltaScuti andred giantstars andfound the dominating dampingeffect to be the turbulent viscosity(Eq. 6).Thisisillustrated in Fig. 4for two DeltaScuti stars: oneislocated inside the IS (leftpanel), the other outsidethe cool edge of the IS (right panel). Thecontribution from the small-scale turbulence wasalsothe dominant dampingeffect in the stabilitycalculations by Xiongetal. (2000)ofradial pmodes in the Sun, althoughthe authors still found unstablemodes with ordersbetween 11 ≤ n ≤ 23.The importance of the turbulent dampingwas reported first by Goldreich&Keeley (1977) and later by Goldreich&Kumar(1991),who found all solarmodes to be stable only if turbulent dampingwas included in their stabilitycomputations. In contrast, Balmforth(1992),who adopted the convection model of Gough(1977a,b), found all solarpmodes to be stable due mainlytothe dampingofthe turbulent pressure perturbations, Wt,and reported thatviscous damping, Wν ,isabout oneorder of magnitude smallerthanthe contribution of Wt.Turbulent viscosity(Eq. 6) leadsalwaystomodedamping, whereasthe perturbation of the turbulent kineticenergy dissipation, δǫ (see Eq. 4),can contribute to both dampingand drivingofthe pulsations(Gabriel1996).The drivingeffect of δǫ wasshown by Dupret et al. (2005) fora γDoradus star.

Summary

We discussedthree different mode stabilitycalculations of DeltaScuti starswhich success- fully reproduced the rededgeofthe IS.Eachofthesecomputationsadopted adifferent time-dependent convection description.The resultswerediscussedbycomparing work inte- grals.All convection descriptionsinclude, althoughindifferent ways,the perturbationsof the turbulent fluxes. Gough(1977a), Xiong(1977, 1989),and Unno et al. (1989) didnot include the contribution Wǫ to the work integralbecauseinthe Boussinesqapproximation (Spiegel &Veronis1960) the viscousdissipation is neglected in the thermalenergy equation. In practise,however,thistermmay be important. Grigahc`e ne et al. (2005) included Wǫ but ignoredthe dampingcontribution of the small-scale turbulence Wν ,which wasfound by Xiong&Deng (2001, 2007) to be the dominating dampingterm. Thesmall-scale damping effect wasalsoignored in the calculations by Houdek (2000).Amore detailedcomparison of the convection descriptionshas notyet been madebut Houdek &Duprethavebegun to addressthisproblem.

Acknowledgments. Iamgrateful to Douglas Goughfor helpful discussionsand to the HELAS organizing committee fortravelsupport. Supportbythe UK Science andTechnologyFacilities Councilisacknowledged. References

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DISCUSSION

Christensen-Dalsgaard: Howdoesthe mixing lengthaffect thered edge of the γ Dorinstabilitystrip? Houdek: Thelocationofthe rededge is predominantly determinedbyradiativedamping whichgradually dominates over thedriving effect of theso-calledconvectiveflux blocking mechanism (Dupretetal. 2005). Achange in themixinglengthwill not only affect thedepth of theenvelopeconvectionzone butalso thecharacteristic time scale of theconvectionand consequently thestabilityofgmodes with different pulsationperiods.Acalibration of themixinglengthtomatch theobserved location of the γ Dor instabilitystrip will alsocalibratethe depthofthe convection zone at agiven surfacetemperature. Comm. in Asteroseismology Vol. 157, 2008, WrocMl aw HELASWorkshop 2008 M. Breger,W.Dziembowski,&M. Thompson,eds.

Stochasticwaveexcitationinrotatingstars

S. Mathis1,2,K.Belkacem3,and M. J. Goupil3

1 CEA/DSM/IRFU/SAp, Laboratoire AIM CEA/DSM, CNRS, Universit´e ParisDiderot,F-91191 Gif-sur-Yvette Cedex, France 2 Observatoire de Paris, LUTH,CNRS, UMR8102, F-92195 Meudon,France 3 Observatoire de Paris, LESIA, CNRS,UMR 8109, F-92195 Meudon,France

Abstract

Wave propagation, excitation andassociated transportare modifiedbythe Coriolis andthe centrifugal accelerations in rotating stars. In thiswork, we focus on the influence of the Coriolis accelerationonthe volumetric stochastic excitation in convection zones of rotating stars. First,wepresent the complete formalismwhich hasbeen derivedand discussthe different terms whichappeardue to the Coriolis acceleration. Then, we usethisformalismto compute the solarmodeexcitationrates andemphasize the peculiarbehavior due to rotation. Consequences on wave transportinrotatingstars areeventuallydiscussed.

Introduction

Themotivationofthisworkistoinvestigate the effectofuniform rotation on the stochas- tically excited mode amplitudes.Several issues canbeaddressed; is the amplitudeofa non-axisymmetric mode (m @=0)the same as foranaxisymmetric one(m=0)? Are pro- gradeand retrogrademodes excited in the same manner andwhatare the consequences? This canhavesomeimportant consequences from both an observationalpoint of view as well as atheoreticalone. As afirst step, we neglect the centrifugal accelerationthatinduces adeformationofthe star.Wethen focus our attention on the effectofthe Coriolis acceleration. We alsorestrict the study to uniform rotation.Inthe first section, we presentaformalismofstochastic excitation developed forarotating star,and we applyittothe solarcasebyperforming aperturbative development. Someconsequences on the angularmomentum transportby modes arebrieflydiscussedinathird section. Conclusionand perspectivesare provided in the lastsection.

Physical assumptionsand formalism

Following Samadi&Goupil(2001) andBelkacem et al. (2008),weestablish the inhomoge- neous wave equation “ ” ; ; ; 1 ∂t2 − LΩ ;υ osc + C (;υ osc, ;ut )=St(;ut). (1)

n 1Here thefollowing notation forpartial derivatives ∂ f = ∂ f is adopted. ∂xn xn S. Mathis,K.Belkacem,and M.J. Goupil 145

υ;osc is the velocity field of the wavesand ;ut is the turbulent oneassociated to the convective ; eddies.The C (;υ osc, ;ut )vectorfield, whichisnot detailedhere, is related to the wave- ; turbulence interaction thatcorresponds to the dynamicaldamping, η. LΩ is the operator that rulesthe wave dynamics in the case of the star free oscillations h i ; ; ; 2 ; ; LΩ (υ;osc)=∇ αs;υ osc · ∇s0 + cs ∇·(ρ0;υ osc) − ;geff ∇·(ρ0υ;osc) “ ” ; ; − ρ0Ω∂t,ϕυ;osc −2 ρ0Ω×∂t υ;osc −ρ0r sin θ ∂t υ;osc · ∇Ω beϕ . (2)

ρ, geff andΩarerespectively the fluid density, effective gravityacceleration (including the centrifugal one) andangular velocity. cs is the sound speed while αs =(∂p/∂s)ρ,where p is the pressure and s the macroscopic entropy. X0 and X1 (where X = {ρ, p, s} )are respectively the hydrostaticvalueofXandits wave-associated fluctuation. Finally,(r,θ,ϕ) arethe classicalspherical coordinates. Thesourceterms thatdrive the eigenmodes are h “ “ ” ”i ; ; ; ; St (;ut )=SSG − ∂t ρ1 Ω∂ϕ;ut +2Ω×;ut +rsin θ ;ut · ∇Ω beϕ . (3)

; Theterm SSG containsthe source terms as derived by Samadi&Goupil(2001)and Belkacem et al. (2008),inwhich the dominant ones arethe Reynolds andentropycontri- butions. Thethree lastterms arethose induced by rotation andcan be re-expressedsuch as “ ” h i ; ; ; ; ∂t 2 ρ1Ω × ;ut =2Ω×∂t(ρ1;ut)=−2Ω× ∇·(ρ0;ut);ut (4) h i ; ∂t (ρ1Ω∂ϕ;ut )=Ω ∇·(ρ0;ut)∂ϕ;ut (5) h “ ”i h i ; ; ; ∂t ρ1r sin θ ;ut · ∇Ω = −r sin θ ∇·(ρ0;ut);ut ·∇Ω, (6) where Ωi; ssupposed uniform andsteady on adynamical time scale. Theselastthree terms 3 scaleasMt (Mtis the turbulent Mach number),while Samadi&Goupil(2001) have shown 2 the Reynolds contribution scalesasMt.Thus the aboverotationalcontribution canbe ignoredinfront of the Reynolds one. Moreover,inthe case wherethe turbulent convective ; motionsare assumedtobeanelastic(∇·(ρ0;ut)=0),they canbeneglected. Therefore, the onlysourceterms we must retain arethe Reynolds andthe entropyones. Following the proceduredetailedbySamadi&Goupil(2001) andBelkacem et al. (2008), the powersuppliedinto the modes (P)isderived ` ´ 2 2 2 P = CR + CS + Cc / (8I ) (7)

2 2 2 where CR , CS ,and Cc arerespectively the contributionsofthe Reynolds stresses, of the entropyfluctuation advection, andthe crossedterms. Thecrossed terms areignored in front 2 2 of CR and CS (see Belkacem et al. 2008 fordetails). In addition,asshown abovethe source terms related to the rotation have been neglected. TheReynolds stressescontribution is givenby Z 2 3 CR=4π dmR(r)SR(ω0), (8) where Z Z dk S (ω )= E2(k) dωχ(ω+ω)χ(ω). (9) R 0 k2 k 0 k 146 Stochasticwaveexcitationinrotatingstars

E(k)and χk (ω)are respectively the kineticenergy spectrum andthe temporalcorrelation function whichare alsomodifiedthrough the action of rotation on turbulence. Thefrequencies ω0 and ω areassociated with pulsationand convection,respectively.Furthermore

R(r)=Rspheroidal + Rtoroidal (10) ˛ ˛ 11 ˛ dξ ξ ˛2 with R = L2 ˛ T − T ˛ toroidal 15 ˛ dr r ˛ ˛ ˛ „ « ˛ ξ ˛2 11 8 2 + ˛ T ˛ L2(L2 − 2) − F − L2 (11) ˛ r ˛ 5 5 ℓ,|m| 3 where R is givenbyEq. (23)ofBelkacem et al. (2008), spheroidal ˆ ˜ F = |m|(2ℓ+1) L2 − (m2 +1) , L2 = ℓ(ℓ +1), and(ξ,ξ,ξ)are the radial, ℓ,|m| 2 r H T horizontaland toroidal components of the eigenfunction corresponding to aspherical m harmonic(Yℓ). Rspheroidal corresponds to the non-rotating case.Itismodifiedbythe Coriolis accelerationthrough the modificationofξr andofξH it induces. Thesecond source term, the entropyfluctuation contribution is obtained as Z ˛ ˛ ! 4π3 H dr ˛ d (ln | α |) dD ˛2 C 2 = α2 ˛D s − ℓ ˛ + L2 |D |2 S (ω ) (12) S 2 2 s ˛ ℓ ˛ ℓ S 0 ω0 r d ln r d ln r

` ´ 2 with H an anisotropy factor(defined in Samadi&Goupil2001), D = 1 d r 2ξ − L ξ ℓ r2 dr r r H and Z Z dk S (ω )= E(k)Es(k) dωχ (ω +ω)χ (ω), (13) S 0 k4 k 0 k

Es beingthe spectrum associated to the entropyturbulent fluctuations.Asfor Rspheroidal no direct changes aredue to uniform rotation.

Applicationtothe excitationofsolar oscillationmodes

In thissection, we applythe formalismtospheroidalsolar oscillationmodes forwhich 2Ω/ω0 (where ω0 is (hereafter)the mode frequency in the non-rotating case)issuch thatthey are only slightlyperturbedbythe Coriolis acceleration. In this case,weget respectively foreach displacement eigenmodecomponent (cf.Unno et al. 1989) „ « „ « (0) 2Ω (1) 2Ω (1) ξα = ξα + m ξα and ξT = ξT (14) ω0 ω0

(0) (0) where α = {r, H}, ξα beingthe componentinthe non-rotating case forwhich ξT =0,and (1) (1) ξα and ξT aregiven by Unnoetal(1989). UsingtheseexpansionsinEq. (7), we get „ « (0) 2Ω (1) Pn,l,m = Pn,l,m + m Pn,l,m (15) ω0 so thatthe excitation rate is different forprograde(m<0) andretrograde (m > 0)2 modes since it depends explicitly on m.Tobetter quantify thisbias introduced by the Coriolis acceleration, we define δPm/P−m =(Pm−P−m)/P−m (16)

2 Themodephase is expandedasexp [i (mϕ + ω0t)]. S. Mathis,K.Belkacem,and M.J. Goupil 147

Figure1: Left panel: Bias between progradeand retrogrademodes (see Eq.(16))for ℓ =1modes, computed with astandard solarmodel. Rightpanel: Bias between progradeand retrogrademodes as afunction of themodeangulardegree(ℓ)and foraradial order n =5. whichisplotted in Fig. (1). In thisfirst application, the simplest turbulent spectrum of KolmogorovisassumedinevaluatingEqs (9) and(13).For low-frequency g modes that areexcited in the bottomofthe convection zone, where2Ω≈ωc (ωc beingthe convective frequency)the effect of rotation on the convective velocity field hastobetaken into account (workinprogress). First, δPm/P−m scalesas2Ω/ω0.Therefore, in the solarcase, we findthatacousticmode excitation rates areonlyweaklyaffected by the Coriolis acceleration while gravitymodes are affected up to 50 %for the most low-frequency modes.Onthe otherhand, foragiven m, δPm/P−m increasesfor decreasing ℓ (itbecomes maximum for l = |m|); in other wordsthe bias is stronger forlow-ℓ degrees.

Mode-induced transport

Let us nowexamine the mode-induced transportofangular momentum.The eulerian fluxof angularmomentum introduced by the Reynolds stresses, foreachazimuthalorder m is given by (see Lee&Saio 1993) Z ∗ e ; FAM;m = ρ0r sin θ ur;muϕ;mdΩwhere ;u = i ω0 ξ, (17) 4π dΩ=se in θdθdϕ beingthe solid angle. In the non-rotating adiabatic case,weget

FAM;m + FAM;−m =0; (18) therefore,modes do not transportany net fluxofangular momentum.Inthe rotating dissi- pative case,introducing Eq.(14)into Eq.(17),we get

FAM;m + FAM;−m = X nˆ ˜ ˆ ˜ o 2 2 (0) 2 (1) − 2D (η) ρ0r ω0 (2Ω/ω0) Im Al,m G1 + Al,m G2 @=0 (19) n,l where Im denotes the imaginarypart, and ˆ “ ” “ ”∗ ˜ 2 (1) (0)∗ (0) (1)∗ (0) m (1) m (1) G1 = m ξr;l ξH;l + ξr;l ξH;l + ξr;l αl−1ξT ;l−1 − βl+1ξT ;l+1 (20) 2 (0) (0)∗ G2 = −m ξr;l ξH;l (21) 148 Stochasticwaveexcitationinrotatingstars with s s (ℓ +1)2−m2 ℓ2−m2 α =ℓ and β =(ℓ+1) . (22) ℓ,m (2ℓ +1)(2ℓ+3) ℓ,m (2ℓ +1)(2ℓ−1)

D corresponds to the phase shiftbetween ur;m and uϕ;m due to dissipative processes (e.g.,the thermal diffusion or the viscousfriction) thatcausesanet transportofangular momentum (Goldreich &Nicholson 1989).Here, we assume thatthe dampingisquasi- independent of m since ω0 ≫ m Ωfor the considered modes.Then, the amplitudeisdevel- oped as forthe power, i.e. „ « ˆ ˜ ˆ ˜ 2 2 (0) 2Ω 2 (1) Aℓ,m = Aℓ,m + m Aℓ,m (23) ω0 „ « P(0) 2Ω P(1) = + m . (24) 2 2 2ηI ω0 ω0 2ηI ω0 Therefore, the Coriolis accelerationintroduces extrabiasesbetween progradeand retro- gradewaves throughthe modifications of the eigenfunctions(G1)and of the excitation rate 2 (1) ([Al,m] ).

Conclusion

In thiswork, we derivethe formalismthatallows to treatthe stochastic excitation of modes by convective regionsinpresence of rotation.Then, we appliedit, as afirst application, to the solarspheroidaloscillations.Weshowthatabias betweenpro-and retrogradewaves is introduced in the excitation by the Coriolis acceleration. It canberelativelyimportant for low-frequency g-modes while it is quite negligible foracoustic ones.Weshowedthatthe azimuthalasymmetriesbothineigenfunctions andtheirexcitationrates introduce an extra contribution.The associated mode-induced transportofangular momentum remainstobe quantifiedasinTalon&Charbonnel (2005). Futureworks must applythe formalismtothe case of rapidrotatorsfor both inertial and gravito-inertial modes (Dintrans&Rieutord 2000, Rieutord et al. 2001) andinclude the effect of differential rotation. References

Belkacem,K., Samadi,R., Goupil, M. J., &Dupret, M.-A.2008,A&A,478, 163 Dintrans,B., &Rieutord, M. 2000,A&A,354, 86 Goldreich, P.,&Nicholson, P. D. 1989, ApJ, 342, 1075 Lee, U.,&Saio,H.1993, MNRAS, 261, 415 Rieutord,M., Georgeot,B., &Valdettaro,L.2001, JFM, 435, 103 Samadi,R., &Goupil, M. J. 2001, A&A, 370, 136 Talon, S.,&Charbonnel, C. 2005, A&A, 440, 981 Unno, W.,Osaki, Y., Ando, H.,Saio,H., &Shibahashi, H. 1989, Nonradial oscillations of stars, UniversityofTokyo Press DISCUSSION

Guzik: Do yousee theasymmetryinexcitation introduced by theCoriolisaccelerationinany stellar g-mode data?Where shouldwelookfor it? Mathis: Ihavenot seen it yetinany data,but we have to look at this to eventually getanadditional observationalconstraintongravity mode behaviour. Comm. in Asteroseismology Vol. 157, 2008, WrocMl aw HELASWorkshop 2008 M. Breger,W.Dziembowski,&M. Thompson,eds.

Theroleofnegative buoyancyinconvectiveCepheid models. Double-mode pulsationsrevisited

R. Smolec

Copernicus Astronomical Center, Bartycka18, 00-716 Warszawa,Poland

Abstract

Thelongstandingproblem of modelingdouble-mode behaviourofclassicalpulsators was solved with the incorporationofturbulent convection into pulsationhydrocodes.However, the reasonsfor the computed double-mode behaviourwerenever clearlyidentified. In our recent papers (Smolec&Moskalik2008a,b)weshowedthatthe double-mode behaviour resultsfromthe neglect of negative buoyancy effects in some of the hydrocodes.Ifthese effects aretaken into account, no stable nonresonantdouble-mode behaviourcan be found. In these proceedings,wefocus ourattention on the role of negative buoyancy effects in classicalCepheidmodels.

Introduction

Since the earlydaysofnonlinearpulsation computations, modelingdouble-mode (DM) phe- nomenonwas oneofthe major objectives. However, the search fornonresonant double-mode behaviourwithradiativehydrocodes failed(seehowever ,Kov´a cs &Buchler 1993).The typicalmodal selection observedwas firstovertone(1O)pulsation at the hotsideofthe instabilitystrip (IS),fundamentalmode(F) pulsationatthe redside, andeither-ordomain (F/1O)inbetween. Theincorporationofturbulent convection into pulsationhydrocodes led to stable androbustdouble-mode pulsation (Koll´athetal. 1998, Feuchtinger 1998).Mostof the double-modemodelspublished so farwerecomputed with theuse of the Florida-Budapest hydrocode(e.g. Koll´a th et al. 2002).Thishydrocode adopts atime-dependent convection model basedonthe Kuhfuß (1986) work.Althoughthe Kuhfuß model wasalsoadopted in ourpulsation hydrocodes (Smolec &Moskalik 2008a), we couldnot findadouble-mode behaviour, despite ourextensivesearchfor it (Smolec&Moskalik 2008b). We linkedthe difference in the computed modalselection to adifferent treatment of negative buoyancy effects in both codes. In the Florida-Budapest code, negative buoyancy is neglected, while it is present in ourhydrocodes.The comparison of both treatments (performedwithour hydrocodes)allows us to understandthe reasonsfor double-mode behaviourcomputed with hydrocodes thatneglect negative buoyancy.

Turbulent convectionmodel

In the Kuhfuß model of turbulent convection,equationsofmotion andenergy conservation aresupplemented with an additional, single equation forgenerationofturbulent energy, et . Details of the model canbefound e.g. in Smolec &Moskalik (2008a). Here we focus our 150 TheroleofnegativebuoyancyinconvectiveCepheid models. Double-mode pulsations revisited attention on the turbulent energy equation andits crucial terms (turbulent pressure andflux of turbulent kineticenergy neglected): de t = S − D + E.V . (1) dt In the aboveequation, S is the turbulent source function (ordriving function), responsiblefor the drivingofturbulent eddies through the buoyantforces.The source function is proportional 1/2 to the superadiabatic gradient, Y = ∇−∇a,and to the speed of convective elements, ∼ et . 3/2 The D-termmodelsthe decayofturbulent eddies,through the turbulent cascade, D ∼ et . E.V .describes energy transfer througheddy-viscousforces.Itdescribes the interplay between turbulent motion andmeangas motion,being proportional to the speed of convective elements andtothe spatial derivativeofthe scaledmeanvelocityfield, U/R.Italwaysdampsthe pulsationsand contributes to the drivingofturbulent energy1. Thecrucial difference between our andthe Florida-Budapest formulationisthe treatment of the turbulent source function in convectively stable (Y < 0) regionsofthe model.Inour hydrocode, we allow fornegative values of the source function,justasinthe original Kuhfuß model (S ∼ Y ). In the Florida-Budapest approach,the source termisrestricted to positive values only(S∼Y+), whichisequivalent to the neglect of negative buoyancy.Following the convention introduced in Smolec &Moskalik (2008a), we will denote the convective recipe andmodelsignoringnegative buoyancy as PP models(Florida-Budapest approach), while convective recipe andmodelsincluding negativebuoyancy will be denoted by NN (our default formulation).

Consequences of neglectingnegativebuoyancy

Usingour pulsation hydrocodes,wehaveperformedadetailedcomparisonofmodels, differing only in the treatmentofthe source function (NNvs. PP models).Crucial differences are observed forsingle-mode limit cycle(full amplitude, monoperiodic oscillation) models. The amplitudeofthe modelsneglecting negative buoyancy (PP) is much lowerthanthe amplitude of the modelsincluding negativebuoyancy effects (NN).The lowering of amplitudeincase of the PP modelsisconnected with the eddy-viscousdampingofpulsationsinthe deep convectively stable regionsofthe model.Thisdampingisclearly visibleinthe nonlinearwork integrals presented in Fig. 1. For the PP model,asignificanteddy-viscousdampingbelow zone 70 (markedwitharrows in the Figure),not present in the NN model,isvisible.These internalzones areconvectively stable (Y < 0),however,inthe PP model significantturbulent energies arepresent in these zones,and hence eddy-viscousdampingispossible. Howare these turbulent energies built up? Thebottomboundary of theenvelopeconvection zone, connected with hydrogen-heliumionization, is located roughlyatzone70for both PP and NN models. Belowthisboundary,turbulent motionsare effectively braked in the NN model due to the negativevalueofturbulent source function (eq.1). Negative buoyancy slowsdown the turbulent motionsveryeffectively andeddy-viscousdamping is notpossibleinthe inner partsofthe model,due to negligible turbulent energies.The situation is different in case of PP models, in whichthe turbulent source function is setequal to zero in convectively stable layers. Therefore, in these layers, turbulent energies areset by the balance between the turbulent dissipation term(Dtermineq. 1),which dampsthe turbulent motions, and the eddy-viscousterm, whichdrivesthe turbulent energies. Duetothe neglect of negative buoyancy,turbulent motionscannotbebrakedeffectively.Onthe contrary, theyare driven at the cost of pulsation,through the eddy-viscousterm. Belowthe envelopeconvection zone turbulent energies areashighas109−1010 erg/g–only three ordersofmagnitude smaller

1Exactformofthistermdiffer in different pulsation hydrocodes,see Smolec &Moskalik 2008a, howeverresults presented here areinsensitive to thesedifferences. R. Smolec 151

Figure 1: Nonlinearworkintegrals plottedversusthe zone number. Surfaceatright. thaninthe center of the convection zone. Such high turbulentenergies extend to more than sixlocal pressure scale heights belowthe envelopeconvectionzone, leadingtosignificant eddy-viscousdamping, visibleinthe left panel of Fig. 1. Deeper in the envelope, turbulent energiesslowlydecay, reflecting thevanishingamplitude of the pulsations. To check theeffect of neglecting negative buoyancy on modalselection, we have computed several sequences of nonlinearCepheidmodelsusing both PP andNNconvective recipes. De- tails of mode selection analysiscan be found in Smolec &Moskalik (2008b). Each model was initialized (kicked) with severaldifferent initial conditions(mixtures of Fand 1O linearvelocity eigenvectors) andtimeevolution of the fundamentalmodeand first overtone amplitudes, A0 and A1,was followed. Exemplaryresults areshown in Fig. 2. Thecomputed trajectories are plotted with solid lines forthe PP model anddotted lines forthe NN model.Hydrodynamic computations coupled with amplitude equation analysisallow to findall stable pulsation states to whichtrajectories converge (attractors, solid squaresinFig.2), andall unstablesolutions, thatrepel thetrajectories (opensquaresinFig.2). For single-mode solutions, stabilitycoef- ficients arecomputed. Theseare γ1,0,which describes thestabilityofthe fundamentalmode limit cyclewithrespect to first overtone perturbation (switching ratetoward1O) and γ0,1, whichmeasuresthe stabilityofthe first overtone limit cycle. Thenegative valueofγmeans thatthe respective limit cycleisstable. If both coefficients, γ1,0 and γ0,1 aresimultaneously positive,bothlimit cycles areunstable, anddouble-mode pulsationisunavoidable.The run of stabilitycoefficientsacrossthe instabilitystrip,for sequenceofCepheidmodels, computed with both PP andNNconvective recipes, is presented in Fig. 3. ThearrowinthisFigure marksthe location of themodel,for whichhydrodynamicintegrationsare presented in Fig. 2. Theamplitude of the givenmodeisamainfactoraffecting itsstability. Thehigher the amplitude of the mode, the more able it is to saturatethe pulsation instabilityalone, and hence, the more stable itslimit cycleis. For NN models, the amplitude of the fundamental mode is much higher than the amplitude of the first overtone (see Fig. 2) across asignificant part of the instabilitystrip.Therefore, thestabilitycoefficient of thefundamentalmodelimit cycle, γ1,0,becomes negative,veryclose to the blue edgeofthe IS at temperature ≈ 6290K (Fig.3). Adouble-mode stateisnot possible, as first overtone becomes unstable (γ0,1 > 0) formuch lowertemperature(T≈6165K). In atemperaturerange in whichbothlimit cycles arestable, either-ordomain is observed, in whichpulsation in either limit cycleispossible. In case of PP models, amplitudes of both modes arereduced as comparedtoNNmodels, but nottothe same extent. As youcan seeinFig.2,the amplitude of the fundamentalmode is much more reducedthanthe amplitude of the first overtone. This effect is explained by ahigher amplitude of the fundamentalmode(in comparisonto1O) in convectively stable layers of the PP model,inwhich significantturbulent energiesare observed. As aresult, eddy-viscousdampingisstronger forthe fundamentalmode. Differential reduction of mode amplitudes is crucial in bringing up double-mode behaviourinthe PP model sequence. At the hotsideofthe IS,the amplitude of the fundamentalmodeislower,orcomparable to the amplitude of the first overtone. Hence,the fundamentalmodelimit cycleisunstable 152 TheroleofnegativebuoyancyinconvectiveCepheid models. Double-mode pulsations revisited

A1 PP 0.02 NN

0.015

0.01

0.005

0 0 0.01 0.02 0.03 0.04 0.05 A0

Figure2:Hydrodynamicintegrations foraparticular Cepheid model. Solid lines -model computed with PP convection, dottedlines modelcomputedwithNNconvection.

Figure3:The runofstabilitycoefficientsalong asequenceofCepheid models of constant mass/luminosity. Solid lines forthe PP models, dashed lines forthe NN models.

(γ1,0 > 0) in awidetemperaturerange. It becomes stable at much lowertemperature (T ≈ 6100K)incomparisontothe NN sequence. At thistemperature, the first overtone is already unstable. Consequently, in arelativelywidetemperaturerange (∼ 50K in Fig. 3), both limit cycles areunstableand double-mode stateemerges.

Summary

Neglectofnegative buoyancy hasserious consequences forthe computed Cepheidmodels. It leadstohighturbulent energies in convectively stable layers, andconsequentlytostrongeddy- viscousdamping. This dampingacts differentiallyonpulsation modes,which promotes the R. Smolec 153 double-mode behaviour. If buoyant forces aretaken into account, as theyshouldbe, no stable nonresonantdouble-mode behaviourcan be found (Smolec&Moskalik2008b). Therefore, the problemofmodelingF/1O double-mode behaviourinclassicalCepheids remainsopen.

Acknowledgments. Iamverygrateful formanyfruitfuldiscussionswithPawel Moskalik andWojciechDziembowski. PawelMoskalik is acknowledged forreading the manuscript. I am grateful to the referee, Geza Kov´acs, formanycomments on the presented results. The authorisgrateful to the EC forthe establishment of the EuropeanHelio-and Asteroseismology NetworkHELAS, as well as forthe financial support, whichmadethe participationofthe authortothisworkshoppossible. This work hasbeen supportedbythe PolishMNiSWGrant No.1P03D 011 30. References

Feuchtinger, M. 1998, A&A, 337, 29 Koll´ath, Z.,Beaulieu, J. P.,Buchler,J.R., et al. 1998, ApJ, 502, L55 Koll´a th,Z., Buchler, J. R.,Szab´o, R.,etal. 2002, A&A, 385, 932 Kov´a cs,G., &Buchler,J.R.1993, ApJ, 404, 765 Kuhfuß,R.1986, A&A, 160, 116 Smolec, R.,&Moskalik,P.2008a,AcA,58, 193 Smolec, R.,&Moskalik,P.2008b, AcA, 58, 233

DISCUSSION

Kov´acs: LetmenotethatpurelyradiativeRRLyrae models with aproperchoiceofthe artificial viscosity lead alsotosustained double-modemodels(seeKov´a cs &Buchler 1993).Furthermore,the convective models of Buchlerand coworkers(e.g. Koll´ath, Buchler, Szab´o et al. 2002) have been able to reproduce thebulkpropertiesofthe observed double-modeRRLyrae and δ Cepheistars.The formeroneswere successfully modeledalso by Feuchtinger(1998),who used similartypeofconvectivemodelingasKoll´ath, Buchler&Szab´o . Smolec: Radiativedouble-modemodelsofKov´a cs &Buchler were obtainedbydecreasingartificial viscosity. Thesemodelsare sensitivetonumerical details and do not reproduceall observational constraints.Detailed comparison of nonlinearconvectivedouble-modemodelsofBuchler andcoworkers with observations (Fourierdecompositionoflight curves)was not performed. UsingVienna pulsation hydrocodeFeuchtinger(1998) computed one double-mode RR Lyraemodel.Thisisthe only double-mode modelpublishedbythe Vienna group. As both Florida-Budapest andVienna codesare claimed to give essentially thesameresults in case of single-mode models (Feuchtinger, Buchler&Koll´a th 2000),we suspect that also in theVienna code negative buoyancywas neglected.However,aswehaven’t computed anyRRLyrae models, we cannot make definitestatementsabout double-modeRRLyrae models. Josefina Montalb´a nand Alosha Pamyatnykh have something interesting to discuss.