Astronomy&Astrophysicsmanuscriptno.sonoi17aph (cid:13)cESO2017 January26,2017 On the computation of eigenfrequencies for equilibrium models including turbulent pressure T.Sonoi1,K.Belkacem1,M.-A.Dupret2,R.Samadi1,H.-G.Ludwig3,4,E.Caffau4,andB.Mosser1 1 LESIA,ObservatoiredeParis,PSLResearchUniversity,CNRS,UniversitéPierreetMarieCurie,UniversitéDenisDiderot,92195 Meudon,France 2 Institutd’AstrophysiqueetdeGéophysique,UniversitédeLiège,Alléedu6Août17,4000,Liège,Belgium 3 ZentrumfürAstronomiederUniversitätHeidelberg,Landessternwarte,Königstuhl12,D-69117Heidelberg,Germany 7 4 GEPI,ObservatoiredeParis,PSLResearchUniversity,CNRS,UniversitéDenisDiderot,SorbonneParisCité,5PlaceJulesJanssen, 1 92195Meudon,France 0 2 January26,2017 n a ABSTRACT J 5 Context.The space-borne missions CoRoT and Kepler have provided a wealth of highly accurate data. However, our inabilityto 2 properly model the upper-most region of solar-like starsprevents us from making the best of these observations. This problem is called“surfaceeffect”andakeyingredienttosolveitisturbulentpressureforthecomputationofboththeequilibriummodelsand ] theoscillations.While3Dhydrodynamicsimulationshelptoincludeproperlytheturbulentpressureintheequilibriummodels,the R waythissurfaceeffectisincludedinthecomputationofstellaroscillationsisstillsubjecttouncertainties. S Aims.WeaimatdetermininghowtoproperlyincludetheeffectofturbulentpressureanditsLagrangianperturbationintheadiabatic computation oftheoscillations.Wealsodiscussthevalidityof thegas-gammamodel andreduced gammamodel approximations, . h whichhavebeenusedtocomputeadiabaticoscillationsofequilibriummodelsincludingturbulentpressure. p Methods.WeuseapatchedmodeloftheSunwithaninnerpartconstructedbya1Dstellarevolutioncode(CESTAM)andanouter - part by the 3D hydrodynamical code (CO5BOLD). Then, the adiabatic oscillations are computed using the ADIPLS code for the o gas-gamma and reduced gamma model approximations and with the MAD code imposing the adiabatic condition on an existing r t time-dependentconvectionformalism.Finally,allthoseresultsarecomparedtotheobservedsolarfrequencies. s Results.We show that the computation of the oscillations using the time-dependent convection formalism in the adiabatic limit a improvessignificantlytheagreementwiththeobservedfrequenciescomparedtothegas-gammaandreducedgammamodelapproxi- [ mations.Ofthecomponentsoftheperturbationoftheturbulentpressure,theperturbationofthedensityandadvectiontermisfound 1 tocontributemosttothefrequencyshift. v Conclusions.Theturbulentpressureiscertainlythedominantfactorresponsibleforthesurfaceeffects.Itsinclusionintotheequilib- 4 riummodelsisthusnecessarybutnotsufficient.Indeed,theperturbationoftheturbulentpressuremustbeproperlytakenintoaccount 4 forcomputingadiabaticoscillationfrequencies.Weproposeaformalismtoevaluatethefrequencyshiftduetotheinclusionofthe 2 termwiththeturbulent pressure perturbation inthe variational principle inorder toextrapolate our result toother starsat various 7 evolutionarystages.Althoughthisworkislimitedtoadiabaticoscillationsandtheinclusionoftheturbulentpressure,futureworks 0 willhavetoaccountforthenonadiabaticeffectandconvectivebackwarming. . 1 Keywords. Asteroseismology-Convection-Waves-Stars:oscillations-Stars:solar-type 0 7 1 1. Introduction thosesurfaceeffectsistheturbulentpressure.Instandardmod- : v els of stellar equilibrium structure and oscillations, it is gener- As shown by the space missions CoRoT (Baglinetal. i ally neglected because its modelling is difficult. Nevertheless, X2006b,a; Micheletal. 2008) and Kepler (Boruckietal. 2010; it is a key factor to obtain accurate frequenciesof stellar mod- rBeddingetal.2010;Chaplinetal.2011),solar-likeoscillations els and particularly for p modes that are very sensitive to the aareubiquitoustolow-massstarsfromthemain-sequencetothe surface layers. The crucial role of turbulent pressure in com- redgiantbranch.Theyhavebeenwidelyusedtoinfertheinter- puting stellar oscillations has been emphasized in many stud- nalstructureofthosestarsandhavepermittedustodramatically ies (Brown 1984; Zhugzhda&Stix 1994; Schlattletal. 1997; improveourknowledgeaswellastoputstringentconstraintson Petrovayetal. 2007; Houdek 2010). More recently, analyses stellarstructureandevolution(e.g.Chaplin&Miglio2013). of surface effects have been carried out using 3D hydrody- However, there are still some fundamental difficulties to namicalmodels(Stein&Nordlund1991;Rosenthaletal.1995, overcome so as to exploit the full potential of the asteroseis- 1999; Yang&Li 2007; Piauetal. 2014; Bhattacharyaetal. mic observations. Surface effects are likely to be the most im- 2015; Sonoietal. 2015; Magic&Weiss 2016; Balletal. 2016; portant.Thisgenerictermnamesthesystematicdifferencesbe- Houdeketal. 2017; Trampedachetal. 2017), because these tween the observedand modelfrequenciesdue to our deficient models providea realistic description of the equilibrium struc- physicaldescriptionof theupper-mostlayersofsolar-like stars tureincludingtheturbulentpressure. (e.g.Christensen-Dalsgaard2016).Oneofthekeyingredientsof However,whenturbulentpressureisincludedintheequilib- Sendoffprintrequeststo:[email protected] rium model, the computation of the related stellar oscillations Articlenumber,page1of11 A&Aproofs:manuscriptno.sonoi17aph Table1.Characteristicsofthepatchedmodel(PM) T [K] log g[gcm−2] T [K] p [gcm−1s−2] M[M ] Age[Gyr] α eff b b ⊙ 5775 4.44 1.53×104 3.66×106 1.01 4.61 1.65 becomes tricky and we have to care about the possible incon- ature at the bottom of the 3D model (T ). As for the last one, b sistency between the oscillation formalism and the equilibrium thetemperatureatthelevelhavingthesametotalpressurewith models.Toconsiderthisproblem,Rosenthaletal.(1995,1999) the bottom of the 3D model (p ) is matched with T . The free b b proposed two approximations, the gas-gamma model (GGM), parametersarethestellarage,totalmass(M),andmixinglength forwhichtheLagrangianperturbationoftheturbulentpressure parameter(α).TheresultingvaluesareprovidedinTable1.Our equals to the perturbation of the gas pressure, and the reduced matching point is deep enough since, at the bottom of our 3D gammamodel(RGM),forwhichtheLagrangianperturbationof model,thefractionoftheturbulentpressuretothetotalpressure theturbulentpressurevanishes.TheyhaveshownthattheGGM issmall enough(≃ 0.014)thatit doesnotaffectfrequenciesof frequenciesbetterreproducetheobservedfrequenciescompared acousticmodes,ofwhichamplitudeisconcentratedintheupper toonesobtainedwiththeRGMassumption.Theirresultimplies layers. thatitisimportanttotaketheperturbationoftheturbulentpres- PM is constructed by replacing the outer part of the opti- sureintoaccountinordertoobtainaccuratefrequencies.How- mized 1D model, which we call “unpatched” model (UPM), ever,theGGMassumptiondoesnotrelyonaconvincingprinci- withtheaveraged3Dmodel.Theadditionalsupportbyturbulent pleanddeservesmoreinvestigation. pressuremodifiesthehydrostaticequilibriumsothat,atthepho- In this work, we consider the computation of adiabatic os- tosphere,the radiusofPM islargerthanUPM byabout0.02% cillationsforanequilibriummodelincludingturbulentpressure. (seealsoTable2andFig.2inSonoietal.2015). To do so, we use a time-dependent convection (TDC) formal- ism thatenables us to accountfor the perturbationof turbulent 2.2.Computationofadiabaticoscillations:thegas-gamma pressure. We also discuss the validity of the GGM approxima- andreducedgammaapproximations tion.WeusetheTDCformalismdevelopedbyGrigahcèneetal. (2005),whichoriginatesfromtheworkofUnno(1967)andwas Following the work of Rosenthaletal. (1995, 1999), two ap- generalized for nonradial oscillations by Gabrieletal. (1975). proximationscan be adopted to accountfor the turbulent pres- This formalismhasbeen so far adoptedfor the computationof sure in the equilibrium model, namely the gas-gamma model the full nonadiabatic oscillations in order to explain the exci- (hereafter GGM) and the reduced gamma model (hereafter tation of the classical pulsators (e.g. Dupretetal. 2005, 2008), RGM). or to fit to the damping rates of the solar-like oscillations (e.g. TheGGMassumesthattherelativeLagrangianperturbation Dupretetal.2006a;Belkacemetal.2012;Grosjeanetal.2014). ofturbulentpressureequalstherelativeLagrangianperturbation Dupretetal.(2006b)developeditfortreatingthenonlocalcon- ofthermalpressure,whichisthesumofgasandradiationpres- vection. For our purpose, we will impose the adiabatic condi- sures,andhenceisequaltothatofthetotalpressure, tion on this formalism to see the validity of the GGM approx- δp δp δp δρ imation. Moreover,such an approachallows us to consider the turb ≃ tot ≃ th =Γ , (1) effectoftheturbulentpressureseparatelyfromthenonadiabatic pturb ptot pth 1 ρ effect,whichisalsoexpectedtoaffecteigenfrequencies(Houdek whereδdenotestheLagrangianperturbation, p istheturbu- turb 2010). lentpressure, p isthethermalpressure,and p (= p + p ) th tot th turb Thepaperisorganizedasfollows:Section2introduceshow isthetotalpressure. to computeeigenfrequencieswith turbulentpressure.Section 3 TheRGMapproximationintroducesthereducedΓ ,defined 1 discussesthe dominantcausesofthe frequencyshiftdueto the asΓr ≡ (p /p )Γ .Inthisapproximation,theLagrangianper- 1 th tot 1 perturbationof the turbulentpressure. Section 4 gives the con- turbationofturbulentpressureisneglected: clusion. δp turb =0. (2) p turb 2. Modellingeigenfrequencieswithturbulent Wehavethus pressure δp p δp δρ tot = th th =Γr . (3) 2.1.Equilibriummodel p p p 1 ρ tot tot th We use the solar “patched” model (PM) described in Figure1showstheadiabaticexponentinUPMandPM.We Samadietal. (2007) and Sonoietal. (2015). The inner part note that there is no difference between GGM and RGM for of this model was constructed using the 1D stellar evolution UPM,whichdoesnotincludeanyturbulentpressure.Inthe3D code CESTAM (Marquesetal. 2013) while the near-surface layerofPM,Γr haslowervaluesthanΓ asaconsequenceofthe 1 1 layers have been obtained using temporal and horizontal aver- presenceofturbulentpressure. ages of the 3D hydrodynamical simulation by the CO5BOLD Figure2showsthedifferencebetweentheobservedfrequen- (Freytagetal. 2012) code with the CIFIST grid (Ludwigetal. cies as given by Broomhalletal. (2009) and the computedfre- 2009). The turbulent pressure is thus included only in the 3D quencies obtained using the PM and UPM described in Sect. upper layers. The matching between the inner and outer layers 2.1. The larger radiusof PM makes frequencieslower than for have been computed through an optimization of the 1D model UPM.Then,thevalueof(ν −ν )ishigherforPM.Onthe obs model withaLenvenberg-Marquardtalgorithm.Theconstraintsforthe otherhand,theGGMtreatment,namelyincludingtheperturba- optimizationaretheeffectivetemperature(T )ofthe3Dmodel, tionofthe turbulentpressure,oppositelyincreasesthefrequen- eff the gravityaccelerationatthe photosphere(g),andthe temper- cies,andreducesthedeviationoftheRGMfrequenciesfromthe Articlenumber,page2of11 T.Sonoietal.:Onthecomputationofeigenfrequenciesforequilibriummodelsincludingturbulentpressure 1.7 15 UPM PM(GGM) 1.6 10 PM(RGM) 1.5 z] H 5 µ [ 1.4 el od 0 m ν 1.3 − BCZ TCZ obs −5 ν 1.2 Γ1 (UPM) −10 1.1 Γ1 (PM, GGM) Γr1 (PM, RGM) 1D 3D −15 1000 1500 2000 2500 3000 3500 4000 1.018 16 14 12 10 8 6 4 2 0 ν [µHz] log p (g cm 1 s 2) tot − − Fig. 2. Difference between the model and observed frequencies (Broomhalletal.2009)fortheradialmodes.Theerrorbarsstemfrom Fig. 1. Adiabatic exponent as a function of total pressure in UPM theobservation.ThemagentadashedlineisforUPM,andtheblueand (dashed magenta line) and PM (solid green and blue lines). For PM, greensolidlinesareforPMwiththeRGMandGGM,respectively.The ones for theGGM(Γ ,green) andfor theRGM(Γr,blue) areshown 1 1 modelfrequenciesarecomputedbyADIPLS. as functions of thetotal pressure. The vertical solid lineindicates the matchingpointbetween1Dand3Dmodels.Theverticaldashedlines indicatethebottomandtopoftheconvectionzonedeterminedbythe Schwarzschildcriterion,labelledasBCZandTCZ,respectively. tribution of turbulent pressure to the surface effects separately fromthenonadiabaticeffects. As mentioned at the end of Sect. 2.2, the phase lag oc- observed frequencies. As a result, the GGM frequenciesare in cursbetween the perturbationof the turbulentpressure and the betteragreementwiththeobservationthantheRGMones.This othervariableswhenweadoptaTDCformalism.Thephaselag resultconfirmstheresultofRosenthaletal.(1999). leadstoexcitationordampingofoscillationamplitude.Namely, ThedeviationoftheGGMfrequenciesfromtheobservation theeigenfrequenciesoftheoscillationbecomecomplex.Inthis isatmost∼ 6µHzinouranalysis.Thisisofthesameorderbut work, however, we only pay attention to the real part of the a little larger than those of the other analyses using the other eigenfrequencies, since we need nonadiabatic treatment to ex- 3D hydrodynamical models (∼ 4µHz in Rosenthaletal. 1999 actlyinvestigatethedampingrates.Weareawarethatthenona- andMagic&Weiss2016,and∼ 3µHzinBalletal.2016).The diabatic effects would be important not only for the damping deviationoftheRGMfrequenciesisatmost∼10µHz,similarly rates, but also for the oscillation frequencies. This is however toHoudeketal.(2017). outofthescopeofthepresentarticleandwillbeconsideredina Finally,wenotethattheGGMandRGMapproximationsare followingwork. easily implemented in an adiabatic oscillation code. However, Therefore,westartbyconsideringtheexpressionoftheper- the underlying assumptions are rather crude and deserve more turbationoftheturbulentpressure, attention.Particularly,theperturbationoftheturbulentpressure shouldbeoutofphasewiththatofthegaspressureanddensity δp δρ V δV (Houdek 2000; Houdeketal. 2017). Then, computations only turb,l = +2 r r, (4) withtherealpartoftheeigenfrequencysuchasthoseperformed pturb,l ρ Vr2 bytheADIPLScodearenotvalid.Therefore,onehastoprovide where p is the turbulentpressure as obtained in the frame- amodellingoftheperturbationoftheturbulentpressureandthis turb,l workofalocaltheoryofconvectionandV istheradialcompo- ispermittedbyusingatime-dependentmodellingofconvection r nentoftheconvectivevelocity.Theoverbarindicatesaveraging (TDC) as provided in the following section. Indeed, the phase in the coarse grain, which is much largerthan most convective lagbetweentheturbulentpressureandtheothervariablestakes eddiesbutmuchsmaller thanthe scale ofthe oscillationwave- placeincomputationwithTDC. length.Togofurther,weconsidertheperturbationoftheradial convectivevelocity(Eq.A.17orEq.21in Dupretetal.2006b) 2.3.Computationofadiabaticoscillations:theTDC intheadiabaticlimit,butforthesakeofsimplicity,welimitour- treatmentfornonlocalconvection selvestothecaseofradialoscillations(ℓ=0).Thisgives In the following, we adopt the TDC formalism developed by V δV 1 δc δQ δρ Grigahcèneetal. (2005) and Dupretetal. (2006b) to compute r r = · − p − − frequencies of PM. While it is usually used to compute non- Vr2 B+[(iΩ+β)στc+1]D ( cp Q ρ adiabatic oscillations, we consider the limit of adiabatic oscil- dδp dξ A iστ dξ 1 ξ lationsbysettingδs=0,wheresisthespecificentropy.Suchan + tot −(C+1) − c + dp dr A+1 ΩΛ dr Ar approachallowsustoproperlyconsiderboththeeffectsofturbu- tot ! lentpressureontheequilibriumstructureandofitsperturbation δω δl −ω τ D R +(D+1) , (5) intheadiabaticlimit.Moreover,thisclarifiestheindividualcon- R c ω l R ) Articlenumber,page3of11 A&Aproofs:manuscriptno.sonoi17aph 1.8 1.0 F /F c,l 1.6 F /F c,nl 1.4 0.8 1.2 1.0 0.6 A 0.8 0.6 0.4 0.4 0.2 0.2 0.0 6.0 5.5 5.0 4.5 4.0 3.5 log T (K) 0.0 4.1 4.0 3.9 3.8 3.7 3.6 Fig.3.AnisotropyparameterAasafunctionoftemperatureinlog-scale log T [K] intherangefromthebottom of theconvection zone tothetopof the atmosphere for PM.Theverticallineindicatestheupper boundary of Fig.4.Temporallyandhorizontallyaveraged3Dconvectiveflux,F , c,nl theconvectionzonedeterminedbytheSchwarzschildcriterion. anditslocalcounterpartobtainedbyEq.(13),F forPM.Thevalues c,l arenormalizedbythetotalflux,F. with localvaluesareobtainedbyperforminganaverage,thatis, iστ +ΩΛ B = c , (6) ΩΛ +∞ C = ωRτc+1 , (7) pturb,nl(ζ0)= pturb,le−b|ζ−ζ0|dζ, (10) (iΩ+β)στ +ω τ +1 Z−∞ c R c +∞ D = 1 , (8) Fc,nl(ζ0)= Fc,le−a|ζ−ζ0|dζ, (11) (iΩ+β)στc+ωRτc+1 Z−∞ wheredζ =dr/H ,andaandbarefreeparametersasintroduced whereσ (≡ 2πν)is the oscillationfrequencyin unitof rads−1, p byBalmforth(1992).Thetemporallyandhorizontallyaveraged τ is the convective timescale, c is the specific heat capac- c p valuesofturbulentpressureandconvectivefluxinthe3Dmodel ity at constant pressure, ρ is the density, ξ is the displacement, aresubstitutedinto p andF ,respectively.Thequantities turb,nl c,nl ω is the inverse of the radiative cooling timescale of convec- R p ,F standfortheirlocalcounterparts.Theseequationscan turb,l c,l tion eddies, l is the mixing length defined by Eq. (A.7), and berecastbytakingthesecondorderderivative Q[≡−(∂lnρ/∂lnT) ]isthevolumeexpandingrate. pth ThefreeparametersβandΩarerelatedtotheclosureofthe d2p /dζ2 =b2(p −p ), (12) turb,nl turb,nl turb,l TDC theory. The parameter β is a complex value and is intro- duced in Eq. (A.12). The parameter Ω is an adjusting function d2Fc,nl/dζ2 =a2(Fc,nl−Fc,l). (13) introduced in the closure terms of the momentum and energy equationsfortheconvectivefluctuations(Eqs.A.13andA.14). Equations (12) and (13) are then used to infer the values of a Forstationaryconvection,ithasthesamemeaningasinthefor- and b as well as the local values of the turbulent pressure and malism of Canuto&Mazzitelli (1991). This quantity is deter- convectiveflux fromthe 3D numericalsimulation. In the over- minedbymatchingwiththeresultsgivenbythe3Dsimulation shootingregion,thetwolocalquantities, pturb,l andFc,l,vanish, usingEqs.(A.11b),(A.15)and(A.16). so that a and b are obtained by fitting an exponential function TheparameterAstandsfortheanisotropyoftheturbulence totheturbulentpressureandconvectivefluxasgivenbythe3D andisdefinedas simulation.Fromour model,we geta = 6.975andb = 1.697. Subsequently, the local counterparts (p and F ) are easily turb,l c,l ρV2 obtainedbysolvingEqs.(12)and(13)intheconvectiveregion A= r , (9) (Fig.4andtoppanelof Fig.6).With theequationsforstation- ρ(V2+V2) aryconvection(Eqs.A.8,A.11bandA.16),wecanevaluateΩ, θ φ α and Γ as functionsof the depth, where Γ[= (ω τ )−1] is the R c whereV andV arethehorizontalcomponentsoftheconvective convectiveefficiency. θ φ velocity. In this work, this parameter is obtained directly from Theperturbationsofthenonlocalturbulentpressureandcon- the3Dsimulation.Forthelayersextractedfromthe1Dmodel, vective flux (δp and δF ) are obtained by solving the turb,nl c,nl wefixthevalueasgivenatthebottomofthe3Dsimulation.This eigenvalueproblemoftheperturbedhydrodynamicalequations quantityisdisplayedinFig.3. of mean flow combined with the perturbed equations of (12) For taking the non-locality into account, we adopt the ap- and(13).Ontheotherhand,theirlocalcounterparts(δp and turb,l proachofSpiegel(1963).Itconsistsinusingananalogywithra- δF )canbeevaluatedwiththelinearcombinationoftheeigen- c,l diativetransfer.Thelocalvalues,asgivenbythemixinglength functions,whichisgivenbyEqs.(4)and(5)forδp ,andEq. turb,l theory(MLT),areconsideredassourcetermsandthenthenon- (A.18)forδF . c,l Articlenumber,page4of11 T.Sonoietal.:Onthecomputationofeigenfrequenciesforequilibriummodelsincludingturbulentpressure 2.4.ComparisonoffrequenciesamongGGM,RGM,and 5 TDC 4 For the computationwith the TDC formalism,we need to give the value of the free parameterβ. The calibratedβ valueshave 3 been of the order of unity in the previousstudies (Dupretetal. ] z 2005,2006a,2008;Belkacemetal.2011;Grosjeanetal.2014). µH 2 Inthiswork,therealpartwererangedfrom0.2to2.0,whilethe [ imaginarypart from−2.0 to 2.0 at 0.2 intervals. The top panel odel 1 of Fig. 5 shows the results with the different values of β (the m ν blacklines,at0.4intervalsinboththerealandimaginaryparts −0 forvisibility).Evaluatingχ2 = (νn −νn )2 foreachβ,we obs n model obs ν foundthatthevalueofβ=0.2−1.2igivesthesmallestdeviation −1 fromtheobservedfrequencies(Ptheredline). Thebottompanelcomparesthecase ofβ = 0.2−1.2iwith −2 the GGM and RGM. First, compared to the GGM, the TDC TDC β=0.2−1.2i treatment improves the agreement with the observations, par- −3 1000 1500 2000 2500 3000 3500 4000 ticularlyfortheintermediateradialordermodes.Thedeviation ν [µHz] fromtheobservedfrequenciesisatmost∼4µHz.Althoughour analysisisadiabatic,itprovidesresultsofthesameorderasthe 12 RGM nonadiabaticanalysisofHoudeketal.(2017),whousedanother 10 GGM TDC formalism (Gough 1977b,a) and PM with a 3D model of TDC β=0.2 1.2i Trampedachetal. (2013) and reported ∼ 3µHz deviation from − 8 theobservedfrequencies. ] z Secondly,theGGMfrequenciesareclosertotheTDCones H µ 6 thantheRGMonesare.AlthoughRosenthaletal.(1995,1999) [ and our results in Sect. 2.2 (Fig. 2) show that the GGM repro- odel 4 ducedtheobservationsbetterthantheRGM,thisresultimplies m ν thattheGGM is superiorto theRGM alsofromthe theoretical − 2 pointofview.Namely,itwouldbeworthtakingtheperturbation obs ν oftheturbulentpressureintoaccountevenfortheadiabaticcom- 0 putationsfrombothobservationalandtheoreticalviewpoints. We note some difference between the TDC and GGM for −2 n&10,whiletheirfrequenciesarealmostidenticalforthelower radialorders.Thisdifferenceimpliesthattheturbulentpressure −4 1000 1500 2000 2500 3000 3500 4000 perturbationisnotassimpleasprovidedbyEq.(1),andthatthe ν [µHz] GGM cannot reproduce the influence of the turbulent pressure withenoughprecision.Wediscusssucheffectsinthefollowing Fig. 5. Same as Fig. 2, but for the model frequencies computed with section. TDCforPM.Top: theblacklinesarefordifferentvaluesoftheTDC freeparameter β.Therealpartof βarerangedfrom0.4to2.0, while theimaginarypartfrom−2.0to2.0at0.4intervals.Theredlineisfor β = 0.2−1.2i, which gives the smallest deviation fromthe observed frequencies. Bottom: comparison of the case of β = 0.2−1.2i with 3. Contributiontothefrequencyshiftintroducedby GGMandRGMforPM. turbulentpressureperturbation obtain In Sect. 2.2and Fig. 2, we have shown the frequencyshiftdue to theelevationoftheupperlayerduetotheturbulentpressure 1 M −1 M δρ∗(δp +δp ) in the equilibrium model, comparing the PM and UPM. Here, ν2 = |ξ|2dm Re th turb 4π2 ρ ρ wediscussthecontributiontothefrequencyshiftduetotheper- Z0 ! Z0 " # turbation of the turbulent pressure. First, we determine which −2g|ξ|2+ 2A−1 pturb,lRe ξ∗dξ dm. (14) region in the star contributes to the frequencyshift (Sect. 3.1). r A ρ r dr As shown in Eqs. (4) and (5), the perturbationof the turbulent " #! pressure consists in different perturbative processes. Secondly, Exceptforlow-ordermodes,thetermsinthesecondlineofEq. wedeterminewhichperturbativeprocessinconvectionisdomi- (14) hardly contribute since |ξ/r| ≪ |dξ/dr| and pturb ≪ pth. nant(Sect.3.2). Here,we discussthe termwith theturbulentpressureperturba- tion,δp .Weintroduce turb 3.1.Contributingregiontothefrequencyshift 1 M −1 m δρ∗δp N (m)= |ξ|2dm′ Re turb dm′, (15) turb 8π2ν ρ ρ To see the contribution of the turbulent pressure perturbation, Z0 ! Z0 " # we adoptthe variationalprinciple.Multiplyingξ∗ in bothsides so thatthe integraltothe surface, N (m = M), representsthe r turb oftheequationofmovement(B.3),usingEqs.(B.4) and(B.5), frequencyshifttowhichtheturbulentpressureperturbationcon- integratingoverthemassofthestarandtakingtherealpart,we tributes.Tobeexact,thistermincludessomepartoftheeffectof Articlenumber,page5of11 A&Aproofs:manuscriptno.sonoi17aph 0.25 n=28, ν=3987.2[µHz] p /p 104 0.20 turb,l tot 103 |ΠΞξδ/pRth/p| p /p | | 0.15 turb,nl tot 102 |δpturb,l/p|(=|Πδpth/p+Ξξ/R|) 101 |δpturb,nl/p| 0.10 100 0.05 10-1 10-2 0.00 12 n=6 10-3 a) z]10 n=15 10-4 H 8 µ [ 6 n=20 0.25 pRteur(bΠ,l/)pth b) b ur 4 n=28 Nt 0.20 2 0 0.15 5.0 4.8 4.6 4.4 4.2 4.0 3.8 3.6 log T (K) 0.10 0.05 Fig.6.Top:Ratiosofthelocal(red) andnonlocal turbulent pressures (green)tothetotalone.Bottom:cumulativecontributionoftheturbu- 0.00 lentpressureperturbationtotheeigenfrequency,Nturb(Eq.15),forfour 7 N0 radialmodeswithβ=0.2−1.2i.Thehorizontalaxisisthelogarithmof 6 N1 temperature. Theverticaldashed lineindicatestheupper boundary of N2 theconvectionzonedeterminedbytheSchwarzschildcriterion. ]5 N3 Hz4 N4 µ [3 theupperlayerelevation,whichappearsintheequilibriumvari- Ni2 c) ables,ρanddm.Nevertheless,itisusefultoseethecontribution 1 oftheturbulentpressureperturbation.Indeed,since p ≫ p th turb 0 and hence |δp | ≫ |δp |, most part of the elevation effect is th turb incluTdheedbiontttohmetpearnmelwoifthFiδgp.t6h.showstheprofilesofN forfour 0.20 RRee((GF/GΓΠ11)) RRee((FFGGΠΠ34)) d) radial modes. Here, the perturbation of the nonlocatulrbturbulent 0.15 Re(FGΠ2) pressuregivenbytheMADcodeissubstitutedintoEq.(15).The 0.10 integral N increases mainly at log T ≃ 4.0–4.4, just below turb the peak of the p /p ratio, shown in the top panel. By the 0.05 turb tot way,itslightlyincreasesevenintheovershootingregionabove 0.00 the boundarydetermined by the Schwarzschild criterion, since the nonlocal turbulent pressure contributes there. As the radial −0.05 ordernincreases,N increasesmoresubstantially.Theinertia turb 4.6 4.5 4.4 4.3 4.2 4.1 4.0 3.9 3.8 3.7 givesthemajorcontributiontothistendency.Withtheincreasing log T (K) radialorder,theamplitudebecomesconfinedinthenear-surface region.Becauseofthelowdensityofthisregion,thefrequency Fig.7. a) absolute values of eigenfunctions for then = 28 mode ob- becomeseasier to shift. We discussthe dominantcauses of the tainedwiththeMADcodewithβ = 0.2−1.2i;firstandsecondterms frequencyshiftinthefollowingsection. intherighthandsideofEq.(16)(blueandgreenrespectively),pertur- bation of the local turbulent pressure given by Eq. (16) (red), and of thenonlocalturbulentpressure(cyan)b)realpartofthecoefficientfor theperturbationofthethermalpressureinEq.(16),Re(Π),theratioof 3.2.Dominantperturbativeprocess thelocalturbulentpressuretothethermalone, p /p .c)cumulative turb,l th contributiontothefrequencyshiftofeachdecomposedcomponentofΠ In the previous section, we have confirmed that the zone just (Eq.17)definedbyEqs.(25)and(26).d)realpartsofthedecomposed below the peak of the p /p ratio dominantlycontributesto turb tot components of Π (Eq. 17). The vertical dashed line indicates the up- thefrequencyshift.Hereweidentifytherespectivecontribution perboundaryoftheconvectionzonedeterminedbytheSchwarzschild of the different processes to the total perturbation of turbulent criterion. pressure. For this purpose,we recast the expressionof the per- turbation of the local turbulent pressure (Eqs. 4 and 5). More with precisely,weexpressitasthelinearcombinationofthethermal pressureperturbationδpth andthedisplacementξ. Thedetailed 1 Π=G +F(Π +Π +Π +Π ) , (17) procedureisdescribedinAppendixB.Then,theperturbationof Γ 1 2 3 4 thelocalturbulentpressure(Eq.4)isexpressedas " 1 # Ξ=FH[Ξ +Ξ +Ξ +Ξ ], (18) 1 2 3 4 δp δp ξ 2 turb,l =Π th +Ξ (16) F = , (19) p p R B+[(iΩ+β)στ +1]D tot tot c Articlenumber,page6of11 T.Sonoietal.:Onthecomputationofeigenfrequenciesforequilibriummodelsincludingturbulentpressure p F p −1 for i = 1,2,3,4. Panel c) shows that the terms with G/Γ and G= turb,l 1− turb,l(2ω τ D+1)(D+1) , (20) 1 pth " 1+(στc)2 ptot R c # 7Π1shdoowmsitnhaenctlaysecoonftnrib=ut2e8t,othtehecofrnetqriubeunticoynsshoifftΠ. A,lΠthou,gahndFΠig. 2 3 4 and areevenmorenegligiblefortheotherlower-ordermodessince H = r pturb,l the mode amplitude is distributed in the inner region. Panel d) R p shows that all the terms except Π , related to the perturbation tot 2 F p −1 ofthemixinglength,certainlycontributeto the perturbationof × 1− turb,l(2ω τ D+1)(D+1) , (21) theturbulentpressureinthetoppartoftheconvectionzone.Par- " 1+(στc)2 ptot R c # ticularly,thelowconvectiveefficiency,namelythelowvalueof where the definitions of Π1,2,3,4 and Ξ1,2,3,4 are given by Eqs. Γ[= (ωRτc)−1],makesΠ3 contributiveneartotheupperbound- (B.10) to (B.17). The coefficientsΠ and Ξ correspondto the ary of the convectionzone. However,the physicalprocessesin 1 1 advection term in the equation of movement,Π and Ξ to the thetoppartoftheconvectionzonehardlycontributetothefre- 2 2 perturbationofthemixinglength,Π andΞ totheperturbation quencyshift.Then,Eq.(15)wouldbewrittenas 3 3 of the radiativecoolingtimescale of convectioneddies,and Π andΞ totheremainingparts. 4 1 M −1 4 ∆ν ≃ |ξ|2dm Panela)ofFig.7showstheabsolutevaluesoftheeigenfunc- turb 8π2ν tionsobtainedwiththeMADcode.Theperturbationofthelocal Z0 ! M p δp 2 1 turbulent pressure reduces to zero toward the boundary deter- × th th Re G +FΠ dm. (27) mined by the Schwarzschild criterion (red line). However, the Z0 Γ1ρ(cid:12) pth (cid:12) " Γ1 1!# perturbation of the nonlocal turbulent pressure has amplitude (cid:12)(cid:12) (cid:12)(cid:12) We note that we restrict(cid:12)ed ou(cid:12)r analysis to radial oscillations in even in the overshooting zone due to the nonlocal effects ex- (cid:12) (cid:12) thiswork.Fornonradialoscillations,weshouldadoptEq.(A.17) pressedasEq.(12)(cyanline). instead of Eq. (5). Besides, Eqs. (B.4) and (B.5) are no longer As shown in panel a), the second term of Eq. (16) is neg- validinthederivation.Forhighℓmodes,thequantityℓ(ℓ+1)ξ ligible (green line). Then, Eq. (16) simplifies to δp /p ≃ h turb,l tot maybecomeimportant. Πδp /p .Usingtheadiabaticrelationδρ/ρ=δp /p /Γ ,Eq. th tot th th 1 (15)becomes 1 M −1 4. Conclusion N (m) ≃ |ξ|2dm′ turb 8π2ν Previous studies (Rosenthaletal. 1995, 1999) have found that Z0 ! m p δp 2 thefrequenciesobtainedwiththegas-gammamodel(GGM)ap- × th th Re(Π)dm′. (22) proximation better agree with the observations than those ob- Γ ρ p Z0 1 (cid:12) th (cid:12) tained with the reduced gamma model (RGM) approximation. (cid:12) (cid:12) Although Π is a complex(cid:12)num(cid:12)ber, we should pay attention to Thistreatmentis easy to adoptforcomputingthe adiabaticos- (cid:12) (cid:12) onlyits realpartto discus(cid:12)s the(cid:12)frequencyshift. Panelb) shows cillations of models including the turbulent pressure. However thattherealpartofΠhasapeak(log T ≃ 3.96)locateddeeper this crude approximationhas no clear physical background.In than p /p , which correspondsto the GGM approximation. thisstudy,wecomputedthefrequencieswithaTDCformalism turb,l th However,their valuesare of the same order. It implies that the imposingtheadiabaticcondition.We foundthattheGGMpro- GGMtreatmentgivesagoodpredictiontosomeextent. videscloserfrequenciestotheTDConescomparedtotheRGM. Indeed,wecananalyticallyunderstandthattheGGMisvalid Itimplies that the GGM is superiorto the RGM from notonly in the bottom part of the convection zone. Since στ ≫ 1 ≫ observationalbutalsotheoreticalviewpoints.Besides,theTDC c ω τ insuchapart,wehave computationsreproducedthefrequenciescloserto theobserva- R c tion thandid the GGM, regardlessof the valuesof the free pa- ΩΛ p F → , G→ turb,l, (23) rameterβ.AlthoughourworkislimitedtotheSun,itisworthex- iστ p c th trapolatingourresultsobtainedbytheTDCtootherstars.Using andG/Γ ,whichcorrespondstothedensityperturbationinEq. the variational principle, we found that the perturbation of the 1 (4),andGFΠ aremuchlargerthantheothertermsinEq.(17). density and advection term mainly contribute to the frequency 1 Therefore,wecanderive shiftduetotheperturbationoftheturbulentpressure.Equation (27)canbethenusedtoevaluatethefrequencyshiftforadiabatic δp p 1 2A δp p δp turb,l → turb,l 1+ th ∼ turb,l th, (24) radialoscillations. ptot pth Γ1 A+1! ptot pth ptot As discussed in previous studies (e.g. Brown 1984; Rosenthaletal. 1999; Sonoietal. 2015), the turbulentpressure which impliesthatthe situationis close to the GGM (Eq.1) in in the equilibriummodelaffectsthe frequenciesbecauseof the thebottompartoftheconvectionzone. To see the contribution of each componentof Π to the fre- elevationoftheouterlayers.Howeveritsperturbationisalsoim- portantforthefrequencies,asdiscussedinthispaper.Although quencyshift,weintroducethevariationalprinciplelikeEq.(15): thissubjecthasbeenalreadyshownbyHoudek(2010)usingthe 1 M −1 m p δp 2 G equilibrium convection models and the TDC formalism based N (m)= |ξ|2dm′ th th Re dm′ (25) 0 8π2ν Γ ρ p Γ onthetheoryofGough(1977a,b),ourstudyusedtheconvection Z0 ! Z0 1 (cid:12) th (cid:12) 1! profilesobtainedwiththe3Dsimulations.Asafirststep,welim- (cid:12) (cid:12) and (cid:12)(cid:12) (cid:12)(cid:12) itedourselvestoadiabaticoscillationsandtheeffectofturbulent (cid:12) (cid:12) 1 M −1 pressure. However, future works should consider nonadiabatic N(m) = |ξ|2dm′ effects as well as the effect of convectivebackwarming.As for i 8π2ν Z0 ! thelatter,Trampedachetal.(2013,2017)reportedthatthehigh m p δp 2 temperaturesensitivity of the opacity in the top of the convec- × th th Re(GFΠ)dm′ (26) Γ ρ p i tionzone causeswarmingby upflowsof convectionsurpassing Z0 1 (cid:12) th (cid:12) (cid:12) (cid:12) (cid:12)(cid:12) (cid:12)(cid:12) Articlenumber,page7of11 (cid:12) (cid:12) A&Aproofs:manuscriptno.sonoi17aph coolingbythedownflowscoupledwiththenon-linearnatureof (A.3): radiativetransfer.Theresultantnetwarmingleadstotheeleva- →− tionof the outerlayersaswellas turbulentpressure.Theyalso ρV ∆ρ Λ = ∇·(∆β +∆β +∆β) reportedthatthecontributionofthebackwarminghasasimilar τ ρ g R t c magnitudeasthatoftheturbulentpressure. −∇·(∆β +∆β +∆β), (A.4) g R t For the asteroseismology of solar-like stars, we need cor- ∆s →− rect model frequencies. Since stellar ages are substantially af- ρT = −ρTV ·∇s−ρǫ +ρǫ fected by the surface effect, many studies have adopted the τc 2 2 empirical relation based on the solar frequencies proposed by →− →− +(ρT∇s)·V −(ρT∇s)·V, (A.5) Kjeldsenetal. (2008). However Sonoietal. (2015) found that →− this solar-calibratedrelationhas difficultyin correctingthe fre- ∇·∆F = −ω ∆sρT, (A.6) R R quencies in different stellar models and at different evolution- →− ary stages. Then, it may be preferable to find a method of the l = αHp =α|dr/dlnptot|=|V|τc. (A.7) correctionbased on a strong physicalapproach.Especially, the We adoptthe Boussinesq approximation,in whichthe pressure convectiveeffectsbothintheequilibriumstateandperturbation fluctuations are neglected except in the equation of movement andalsothenonadiabaticeffectmaybeimportantforthisprob- (Eq.A.2)andthedensityfluctuationsareneglectedintheequa- lem. Therefore, we will extend the work of Sonoietal. (2015) tionofcontinuity(Eq.A.1).Besides,thespatialvariationinthe includingtheseeffects. densityis assumedto be muchsmaller thanthatin the convec- Acknowledgements. T.S.hasbeensupportedbytheANR(AgenceNationalede tive velocity in Eq. (A.1). The closure approximations (A.4), laRecherche) programIDEE(Interaction DesÉtoilesetdesExoplanètes)and (A.5), and (A.6) follow the assumption that turbulentviscosity CNES(CentreNationald’ÉtudesSpatiales).H.G.L.acknowledgesfinancialsup- and thermal conductivity due to smaller eddies are expressed portbytheSonderforschungsbereich SFB881“TheMilkyWaySystem”(sub- with the typical scale given by a representative convectiveele- projectA4)oftheGermanResearchFoundation(DFG). mentincludingthem.Equation(A.7) isthe usualclosureequa- tion of the MLT. Assuming constantcoefficientsand Λ = 8/3, theaboveequationsgivethestationarysolutionconsistentwith AppendixA: Time-dependentconvectionformalism theMLT: fornonlocalconvection Γ(Γ+1)=A(∇−∇ ), (A.8) ad FollowingDupretetal.(2006b),weintroducethewaytoadopt 9 Γ3+Γ2+Γ=A(∇ −∇ ), (A.9) the results given by the 3D simulations of nonlocal convec- 4 rad ad tion to the time-dependent convection (TDC) formalism of α2c ρT P p Γ(∇−∇ ) 3/2 Grigahcèneetal.(2005). F = p T tot ad , (A.10) c 4 2P ρ Γ+1 s ρ " # AppendixA.1:Hydrodynamicalequationsforlocal p = α2 PTptot Γ (∇−∇ ), (A.11) convection turb 8 2P Γ+1 ad ρ TheTDCformalismofGrigahcèneetal.(2005)originatesfrom where A = P p /(2P ρ)[κc ρ2gl2/(12acT3p )]2 and Γ = T tot ρ p tot theoneproposedbyUnno(1967).Later,Unno’sformalismwas (ω τ )−1. As mentioned above, the Boussinesq approximation developedfornonradialoscillationsbyGabrieletal.(1975).The R c includes the neglect of the density fluctuations in Eq. (A.1). classical mixing length theory (MLT) of Böhm-Vitense (1958) However,this assumption is not valid in near-surface layers of is the description for convection in the hydrostatic equilibrium solar-likestarssinceconvectivevelocitycanbecomparablewith state.Ontheotherhand,theTDCformalismincludesvariation soundspeed of surroundingmaterials. Besides, the assumption of convection on the dynamical timescale. However, if we im- that the spatial variation in the density is much smaller than pose the stationary condition on the formalism, we can obtain that in the convective velocity is invalid in the deep part of a consistentresultswiththeMLT. convection zone, where the surrounding structure is no longer First,wederivetheequationofconvectionintheequilibrium homogeneous in the representative scale of convective eddies. state.Wethusdecomposethephysicalvariablesinthehydrody- However, this is a standard hypothesis made in most TDC ap- namicalequationsintothemeanflowandconvectivefluctuations →− →− →− proaches.Withoutsuchassumption,itisdifficulttobuildaTDC as y = y+∆y forthe scalarsand v = u + V for thevelocity. formalism. Besides it is a consequence of the adoption of the InUnno’sformalism,theconvectivefluctuationpartsofthehy- MLT. drodynamicalequationsofthecontinuity,movement,andenergy conservationaregivenby AppendixA.2:Perturbativetheoryforlocalconvection →− ∇·V =0, (A.1) To consider the behaviour of convection with the oscillations, →− →− weperturbtheaboveformalism,Eqs.(A.1)–(A.3),whichallows dV ∆ρ →− →− ρV ρ = ∇p −∇∆p −ρV ·∇u −Λ , (A.2) us to evaluate the perturbation of correlated quantities of the dt ρ tot tot τ c convectivefluctuations.Howevertheclosuredescribedaboveis ∆(ρT)ds +→−V ·∇s=−ωRτc+1∆s, (A.3) crude, and many complex physical processes are neglected in- ρT dt τc cluding the whole cascade of energy. Then, uncertainty cannot beavoidedwhenperturbingtheclosureterms.Becauseofsuch wherethenotationsfollowthedefinitionsintroducedinSect.2.3 uncertainties, the unphysical, short wavelength oscillations ap- of this paper. To obtain the above equations, the following ap- pear in the eigenfunctions of the differential equations for the proximationshavebeenmadefortheclosureofEqs.(A.2)and oscillations.Todealwiththisproblem,Grigahcèneetal.(2005) Articlenumber,page8of11 T.Sonoietal.:Onthecomputationofeigenfrequenciesforequilibriummodelsincludingturbulentpressure proposedtointroduceafreecomplexparameterβinthepertur- By adjusting Ω and α, we can fit these equationsto the results bationofthethermalclosureequations: givenby3DsimulationsincombinationwithEqs.(12)and(13). ThequantitiesF , p ,(∇−∇ )andotherthermodynamic ∆s ∆s δ∆s δτ c,nl turb,nl ad δ = (1+βστ ) − c . (A.12) quantitiesarededucedfromthe3Dsimulations,andwetakeap- τc ! τc " c ∆s τc # propriatehorizontalandtimeaverages.UsingEqs.(12)and(13), thelocalcounterpartsofturbulentpressureandconvectiveflux, Introducingthisparameterleadstophaselagsbetweentheoscil- p andF ,areobtainedbasedonp andF .Usingthese lationsandthewaytheturbulencecascadeadaptstothem. turb,l c,l turb,nl c,nl local counterparts, we obtain appropriate values of Ω and α at Then, we search for the solutions of the perturbed convec- →−→− eachlocationwithEqs.(A.16)and(A.11b),respectively. tivefluctuationequationsoftheformδ(∆X) = δ(∆X)→−eik·reiσt, TogeneralizetheperturbativetheorypresentedinSect.A.2, k assumingconstantcoefficientswithinthecoarsegrain,whichis wereplacetheequationsofmovementandenergyconservation much larger than most of convective eddies but much smaller forthelocalcase,(A.2)and(A.3),withtheonesforthe3Dcase, than the scale of the perturbation wavelength. Next, we inte- (A.13)and(A.14).WefollowthesameprocedureasinSect.A.2. grate these particular solutions over all values of kθ and kφ so Assumingagainconstantcoefficientandsearchingforsolutions thatk2+k2 = Ak2,keepingAconstantandthateverydirection in the form of plane waves, we obtain the new expressions for θ φ r →− theperturbedlocalconvectivequantitiessuchastheconvective ofthehorizontalcomponentof k hasthesameprobability.The fluxandturbulentpressure. value of A is the free parameter,given by Eq. (9) based on the Themainuncertaintiesinthisapproachappearinthewayto 3D simulationin thisstudy.We haveto introducethisdistribu- →− perturbΩandα.ThefreeparameterβintroducedinEq.(A.12) tionof k valuestoobtainanexpressionfortheperturbationof is also somehow related to these uncertainties. At present, we the Reynolds tensor which allows the proper separation of the have no theoretical prescription how to perturb Ω and α, and variablesintermsofsphericalharmonicsintheequationofmo- thenweneglecttheirperturbations.Howeverthereisnoreason tion.Finally,the obtainedvaluesoftheperturbationofthecor- toexpectthemtobesmall,andweshouldnotbetoooptimistic relatedvaluesareimplementedintothedifferentialequationsof whenusingthisnewperturbativetreatment. theoscillations. Here, we do not discuss the derivation which is very simi- lar to those of Grigahcèneetal. (2005). The finalresults of the perturbationoftheradialcomponentsofthelocalconvectiveve- AppendixA.3:Procedurefortakingequilibriumvaluesgiven locitiesandconvectivefluxaregivenasEqs.(A.17)and(A.18). by3Dsimulationintoaccount Theyare not so differentfromthe former expressions(Eqs. 12 3D hydrodynamic simulations (e.g. Stein&Nordlund 1991, and18inGrigahcèneetal.2005): 1998; Rosenthaletal. 1999; Yang&Li 2007; Piauetal. 2014) providemuchmorerealisticprofilesoftheconvectionzonesthan VrδVr = 1 withtheMLT.Herewediscusshowtoextendtheaboveformal- V2 B+[(iΩ+β)στc+1]D ismtothenonlocalcasefollowingDupretetal.(2006b). r δc δQ δρ dδp dξ As discussed in Sect. A.1, most of the uncertaintiesare in- · − p − − + tot − r cluded in the closure terms (Eqs. A.4–A.7). Then, we mod- ( cp Q ρ dptot dr ify these terms introducing a free function varying with depth, Q+1δs dδs dξ Ω, which has the same meaning as in the formalism of −iΩστcD +C − r Q c ds dr Canuto&Mazzitelli(1991).Itcanbeassumedtobeafunction p " # of the convective efficiency Γ following Canuto&Mazzitelli. − A iστc dξr + 1 ξr − ℓ(ℓ+1)ξh Wealsosettheusualmixinglengthαasanadditionalfreefunc- A+1 ΩΛ dr A r 2A r " # tionvaryingwiththedepthorΓ.Moreprecisely,wemultiplythe δT δc δκ δρ p lefthandsideofEq.(A.4)byΩ(Γ)andthelefthandsideofEqs. −ω τ D 3 − − −2 R c T c κ ρ (A.5)and(A.6)by1/Ω(Γ).Then,Eqs.(A.2)and(A.3)become p ! →− →− δl ρdV = ∆ρ∇p −∇∆p −ρ→−V ·∇→−u −Ω(Γ)ΛρV, (A.13) +[(iΩ+β)στc+3ωRτc+2]D l , (A.17) dt ρ tot tot τ ) c ∆(ρT)ds + d∆s +→−V ·∇s=−ωRτc+1∆s. (A.14) δFc,l = δρ + δT −iΩστ DQ+1δs +C dδs − dξr ρT dt dt Ω(Γ)τc Fc,l ρ T c Q cp " ds dr # In the stationarycase, these new equationshave a formsimilar δT δcp δκ δρ −ω τ D − − −2 totheoldones(Eqs.A.2andA.3).Equation(A.8)remainsun- R c T c κ ρ changed,givingthesamemeaningtoΓasinthepreviouscase. p ! VδV Equation(A.11)isstillverified(withvaryingα),butEqs.(A.9) +[(iΩ+β)στ +2ω τ +1]D r c R c and(A.10)areslightlymodified: V2 r 9 δl 4Ω(Γ)Γ3+Γ2+Γ=A(∇rad−∇ad), (A.15) +(2ωRτc+1)D l . (A.18) We note that Eq. (A.17) becomes Eq. (5) for adiabatic radial F = Ω(Γ)α2cpρT PTptot Γ(∇−∇ad) 3/2, (A.16) oscillations. c,l 4 2P ρ Γ+1 Ontheotherhand,theperturbationofthenonlocalturbulent s ρ " # pressure and convective flux is obtained by solving the eigen- α2 P p Γ valueproblemof thedifferentialequationsof oscillationscom- p = T tot (∇−∇ ). (A.11b) turb,l 8 2P Γ+1 ad biningtheperturbedequationsof(12)and(13). ρ Articlenumber,page9of11 A&Aproofs:manuscriptno.sonoi17aph AppendixB: Recastingtheexpressionofturbulent showninEq.(17).TheexpressionofΠ’sis i pressureperturbation A iστ 1 Π = c , (B.10) Here,wedescribetheprocedureforrecastingtheexpressionof 1 A+1 ΩΛ Γ 1 turbulentpressureperturbation,whichisrequiredforthediscus- D+1 p 2A−1 p 1 sion in Sect. 3.2. We begin with Eqs. (4) and (5) and aim to Π2 = 1+(στ )2 pth + A ρtugrbr,l − Γ , (B.11) expressthem as the linear combinationof the thermalpressure c " tot 1# perturbation,δp ,andthedisplacement,ξ. 2 th Π = −ω τ D 3∇ −c −κ − For the perturbationof the mixinglength,we adoptthe ex- 3 R c ad p,ad ad Γ 1 pression, 2 p 2A−1 p 1 − th + turb,l − , (B.12) δl 1 δHp 1+(στc)2 "ptot A ρgr Γ1#! = , (B.1) C 2A−1 p l 1+(στc)2 Hp Π4 = −cp,ad−Qad+ Γ + A ρtugrbr,l. (B.13) 1 wheretheperturbationofthepressurescale heightisdescribed Ontheotherhand,thecoefficientsofξ/R,Ξ,are by A iστ 2A−1 δHp = δptot − dδptot + dξ. (B.2) Ξ1 = A+1 ΩΛc A , (B.14) Hp ptot dptot dr D+1 σ2r 2A−1 p Ξ = +2 turb,l , (B.15) Tocancelthetermdδptot/dptot,weadopttheperturbedequation 2 1+(στc)2 " g A ρgr # ofmovement(Eq.D.3in Grigahcèneetal.2005),neglectingthe 2ω τ D σ2r 2A−1 p Ξ = R c +2 turb,l , (B.16) perturbationofthedivergenceoftheReynoldstensor: 3 1+(στ )2 g A ρgr c " # ddδpptot =−σg2rξr + g1ddδrΦ + δρρ + 2AA−1 pρtugrbr,lddξr, (B.3) Ξ4 = −σg2r −22AA−1 pρtugrbr,l +2C. (B.17) tot and the perturbed equation of continuity (Eq. D.1 in References Grigahcèneetal.2005), Baglin,A.,Auvergne,M.,Barge,P.,etal.2006a,inESASpecialPublication, δρ 1 d Vol.1306,ESASpecialPublication,ed.M.Fridlund,A.Baglin,J.Lochard, + r2ξ =0. (B.4) &L.Conroy,33 ρ r2dr Baglin, A., Auvergne, M., Boisnard, L., et al. 2006b, in COSPAR Meeting, (cid:16) (cid:17) Vol.36,36thCOSPARScientificAssembly,3749 Forradialoscillation,thePoissonequationbecomes Ball,W.H.,Beeck,B.,Cameron,R.H.,&Gizon,L.2016,A&A,592,A159 Balmforth,N.J.1992,MNRAS,255,603 1dδΦ dξ Bedding,T.R.,Kjeldsen,H.,Campante,T.L.,etal.2010,ApJ,713,935 = . (B.5) Belkacem,K.,Dupret,M.A.,Baudin,F.,etal.2012,A&A,540,L7 g dr dr Belkacem,K.,Goupil,M.J.,Dupret,M.A.,etal.2011,A&A,530,A142 Bhattacharya,J.,Hanasoge,S.,&Antia,H.M.2015,ApJ,806,246 Withtheadiabaticconditionδρ/ρ=δp /p /Γ ,wecanexpress Böhm-Vitense,E.1958,ZAp,46,108 th th 1 Eq.(5)asalinearcombinationofδp ,ξandδp .Wecatego- Borucki,W.J.,Koch,D.,Basri,G.,etal.2010,Science,327,977 th turb,l Broomhall,A.-M.,Chaplin,W.J.,Davies,G.R.,etal.2009,MNRAS,396,L100 rizethetermsinEq.(5)intofourpartsasfollows.Thefirstpart Brown,T.M.1984,Science,226,687 isthetermwhichstemsfromtheadvectiontermintheequation Canuto,V.M.&Mazzitelli,I.1991,ApJ,370,295 ofmovement: Chaplin,W.J.,Kjeldsen,H.,Christensen-Dalsgaard,J.,etal.2011,Science,332, 213 A iστ dξ 1 ξ δp ξ Chaplin,W.J.&Miglio,A.2013,ARA&A,51,353 − A+1 ΩΛc dr + Ar!=Π1 pthth +Ξ1r. (B.6) CDhurpirsetet,nMsen.-AD.,alBsgarabaardn,,JC..2,0G1o6u,pAilr,XMiv.-eJ-.,peritnatsl.,2as0t0ro6-ap,hin:1E6S0A2.0S6p8e3c8ialPublica- tion,Vol.624,ProceedingsofSOHO18/GONG2006/HELASI,Beyondthe Thesecondpartistheperturbationofthemixinglength: sphericalSun,97.1 Dupret,M.-A.,Goupil,M.-J.,Samadi,R.,Grigahcène,A.,&Gabriel,M.2006b, δl δp δp ξ in ESA Special Publication, Vol. 624, Proceedings of SOHO 18/GONG (D+1) =τ turb,l +Π th +Ξ . (B.7) 2006/HELASI,BeyondthesphericalSun,78.1 2 2 2 l pturb,l pth r Dupret,M.-A.,Grigahcène,A.,Garrido,R.,Gabriel,M.,&Scuflaire,R.2005, A&A,435,927 The third partis the perturbationof the inverse of the radiative Dupret,M.A.,Quirion,P.O.,Fontaine,G.,Brassard,P.,&Grigahcène,A.2008, coolingtimescaleofconvectioneddies: JournalofPhysicsConferenceSeries,118,012051 Freytag,B.,Steffen,M.,Ludwig,H.-G.,etal.2012,JournalofComputational Physics,231,919 δω δp δp ξ −ω τ D R =τ turb,l +Π th +Ξ . (B.8) Gabriel,M.,Scuflaire,R.,Noels,A.,&Boury,A.1975,A&A,40,33 R c ω 3 p 3 p 3r Gough,D.1977a,inLectureNotesinPhysics,BerlinSpringerVerlag,Vol.71, R turb,l th ProblemsofStellarConvection,ed.E.A.Spiegel&J.-P.Zahn,15–56 Gough,D.O.1977b,ApJ,214,196 Thelastpartcorrespondstotheremainingterms: Grigahcène,A.,Dupret,M.-A.,Gabriel,M.,Garrido,R.,&Scuflaire,R.2005, A&A,434,1055 − δcp − δQ − δρ + dδptot −(C+1)dξ =Π δpth +Ξ ξ.(B.9) Grosjean,M.,Dupret,M.-A.,Belkacem,K.,etal.2014,A&A,572,A11 c Q ρ dp dr 4 p 4r Houdek,G.2000,inAstronomicalSocietyofthePacificConferenceSeries,Vol. p tot th 210,DeltaScutiandRelatedStars,ed.M.Breger&M.Montgomery,454 Houdek,G.2010,AstronomischeNachrichten,331,998 Finally,weobtaintheexpressionofδp /p asEq.(16).The turb,l tot Houdek,G.,Trampedach,R.,Aarslev,M.J.,&Christensen-Dalsgaard,J.2017, coefficient of δpth/ptot (Π) consists in Πi’s (i = 1,2,3,4) as MNRAS,464,L124 Articlenumber,page10of11