Calibration of the Mixing-Length Theory for Convective White Dwarf Envelopes
P.-E. Tremblay, H.-G. Ludwig, B. Freytag, G. Fontaine, M. Steffen, P. Brassard
aa r X i v : . [ a s t r o - ph . S R ] D ec Accepted for Publication in ApJ, 2014 November 27
Preprint typeset using L A TEX style emulateapj v. 05/12/14
CALIBRATION OF THE MIXING-LENGTH THEORY FOR CONVECTIVE WHITE DWARF ENVELOPES
P.-E. Tremblay Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD, 21218, USA
H.-G. Ludwig
Zentrum f¨ur Astronomie der Universit¨at Heidelberg, Landessternwarte, K¨onigstuhl 12, 69117 Heidelberg, Germany
B. Freytag
Department of Physics and Astronomy at Uppsala University, Regementsv¨agen 1, Box 516, SE-75120 Uppsala, Sweden andCentre de Recherche Astrophysique de Lyon, UMR 5574: CNRS, Universit´e de Lyon, ´Ecole Normale Sup´erieure de Lyon, 46 all´eed’Italie, F-69364 Lyon Cedex 07, France
G. Fontaine
D´epartement de Physique, Universit´e de Montr´eal, C. P. 6128, Succursale Centre-Ville, Montr´eal, QC H3C 3J7, Canada
M. Steffen
Leibniz-Institut f¨ur Astrophysik Potsdam, An der Sternwarte 16, D-14482 Potsdam, Germany andP. Brassard
D´epartement de Physique, Universit´e de Montr´eal, C. P. 6128, Succursale Centre-Ville, Montr´eal, QC H3C 3J7, Canada
Accepted for Publication in ApJ, 2014 November 27
ABSTRACTA calibration of the mixing-length parameter in the local mixing-length theory (MLT) is presented forthe lower part of the convection zone in pure-hydrogen atmosphere white dwarfs. The parameterizationis performed from a comparison of 3D CO5BOLD simulations with a grid of 1D envelopes witha varying mixing-length parameter. In many instances, the 3D simulations are restricted to theupper part of the convection zone. The hydrodynamical calculations suggest, in those cases, thatthe entropy of the upflows does not change significantly from the bottom of the convection zone toregions immediately below the photosphere. We rely on this asymptotic entropy value, characteristicof the deep and adiabatically stratified layers, to calibrate 1D envelopes. The calibration encompassesthe convective hydrogen-line (DA) white dwarfs in the effective temperature range 6000 ≤ T eff (K) ≤ ,
000 and the surface gravity range 7 . ≤ log g ≤ .
0. It is established that the local MLT isunable to reproduce simultaneously the thermodynamical, flux, and dynamical properties of the 3Dsimulations. We therefore propose three different parameterizations for these quantities. The resultingcalibration can be applied to structure and envelope calculations, in particular for pulsation, chemicaldiffusion, and convective mixing studies. On the other hand, convection has no effect on the whitedwarf cooling rates until there is a convective coupling with the degenerate core below T eff ∼ Keywords: convection – hydrodynamics – stars: evolution – stars: fundamental parameters – stars:interiors – white dwarfs INTRODUCTION
In late-type stars, giants, and cool white dwarfs, theconvective outer envelope has a significant impact onthe observed properties. The physical principles explain-ing convective energy transport in stars are well under-stood, although the non-local and turbulent nature ofconvection has delayed the development of precise mod-els for convective stellar layers. The mixing-length theory(B¨ohm-Vitense 1958, hereafter MLT) has proven rathersuccessful despite presenting a very simple description [email protected] Hubble Fellow of convection. In this picture, the condition that dis-tinguishes between convective and stable layers is theSchwarzschild criterion, and the convective efficiency, theratio of convective and radiative fluxes, is computed fromlocal quantities. In the super-adiabatic convective layersthat define the atmosphere of most stars, the predictedconvective efficiency is very sensitive to the underlyingmodel describing the radiative energy losses, the lifetime,and the geometrical shape of individual convective struc-tures. These quantities are not well constrained by theMLT and must be calibrated from observations.In recent years, three dimensional (3D) radiation hy-drodynamical (RHD) simulations have provided predic-
Tremblay et al. tions for the surface convection that are in very goodagreement with the observed solar granulation (see, e.g.,Wedemeyer-B¨ohm & Rouppe van der Voort 2009). Fur-thermore, various studies relied on 3D RHD simula-tions to improve the predicted photospheric structuresand spectroscopic abundance determinations for theSun and other stars (Asplund et al. 2009; Caffau et al.2011; Scott et al. 2014a,b). In addition to a bet-ter representation of the surface inhomogeneities, 3Dmodel atmospheres feature non-local effects, such asthe so-called top overshoot layers, which are com-pletely missing in local 1D MLT models (Unno 1957;Ludwig et al. 2002; Nordlund et al. 2009; Freytag et al.2010; Tremblay et al. 2013c).The deep convection zone, where the stratification be-comes essentially adiabatic, is not sensitive to the convec-tion model. It is however the entropy jump in the super-adiabatic layers that completely defines the asymptoticentropy value of the deep, adiabatically stratified struc-ture, hence also the depth of the convection zone. Onepossibility to model these layers is to rely on RHD simu-lations to determine the asymptotic entropy value forthe deep convection zone (Steffen 1993; Ludwig et al.1999). This arises from the prediction that upflowsformed at the base of the convection zone follow an adi-abat almost up to the visible surface (Stein & Nordlund1989). The 1D MLT envelopes are then calibrated fromthe multi-dimensional asymptotic entropy, a techniquethat has been employed for late-type stars and giants(Ludwig et al. 1999, 2008). The calibrated 1D struc-tures nevertheless neglect the overshoot layers predictedat the base of non-local convection zones (B¨ohm 1963;Chan & Sofia 1989; Skaley & Stix 1991; Freytag et al.1996), which for deep convective envelopes, impacts theconvective mixing into the nuclear burning core. In thiswork, we are interested in the calibration of 1D envelopesof DA white dwarfs with a pure-hydrogen atmosphere.All currently available white dwarf structures rely onthe local MLT with a fixed parameterization (see, e.g.,Tassoul et al. 1990; Fontaine et al. 2001; Renedo et al.2010; Salaris et al. 2010).Surface granulation in DA white dwarfs is qualita-tively very similar to that seen in the Sun and stars(Tremblay et al. 2013b), albeit with shorter lifetimesand smaller characteristic sizes, which are roughly in-versely proportional to gravity. Convective instabili-ties due to hydrogen recombination develop in the at-mosphere of these pure-hydrogen stellar remnants at T eff ∼ ,
000 K, although convective energy fluxesonly become significant at T eff ∼ ,
000 K for log g =8. The convection zone eventually grows to sub-photospheric, and essentially adiabatic layers, at slightlylower effective temperatures. White dwarfs in the range14 , & T eff (K) & g values previouslyderived from spectroscopic observations of cool convec-tive white dwarfs (Bergeron et al. 1990). In particular, the top overshoot region was found to have a crucial im-pact on the spectroscopic predictions.The convection zone in DA white dwarfs remains lim-ited to the thin hydrogen envelope until it reaches thedegenerate core at T eff ∼ − M H /M tot (Tassoul et al. 1990). Before one of these events takesplace, the cooling process is regulated by the radiativeinterface layer just above the largely isothermal degen-erate core, which is in some sense the bottle-neck forthe energy transport. The evolutionary calculations con-verge to the so-called radiative zero solution, hence theyare insensitive to the details of the convection model(Fontaine & van Horn 1976), which is unlike earlier evo-lutionary stages (see, e.g., Freytag & Salaris 1999). Thesituation is different below T eff ∼ indirect effect on observedages, since they are often derived from spectroscopicallydetermined atmospheric parameters that are modified by3D effects.There are a number of cases where 3D effects on struc-tures are expected to have a direct impact. Non-adiabaticpulsation calculations depend critically on the structureof the convective layers, especially for the determinationof the edges of the ZZ Ceti instability strip of pulsatingDA white dwarfs (Fontaine et al. 1994; Gautschy et al.1996; van Grootel et al. 2012). Chemical diffusion ap-plications (Paquette et al. 1986; Pelletier et al. 1986;Dupuis et al. 1993) and convective mixing studies (see,e.g., Chen & Hansen 2011) also depend critically on thesize and especially the dynamical properties of the con-vection zone, e.g. the root-mean-square (RMS) verticalvelocity in the convective overshoot layers at the baseof the convection zone (Freytag et al. 1996). In orderto characterize white dwarfs accreting disrupted planets,it is likely important to account for the currently ne-glected convective overshoot (Koester 2009). The totalmass of the chemical elements mixed in the convectionzone (hereafter mixed mass), and to a lesser degree theirrelative abundances, depend on how rapidly these ele-ments diffuse in the deep overshoot region. Through theremaining of this work, overshoot refers only to the re-gion at the base of the convection zone, since the topovershoot layers have no direct relevance for white dwarfenvelope and structure models.This study proposes a calibration of the MLT free pa-rameters for the size of the convection zone in 1D en-velopes of DA white dwarfs from a comparison withCO BOLD 3D simulations previously computed for spec-troscopic applications (Tremblay et al. 2013c). We em-phasize that our proposed calibration has little in com-mon with the spectroscopic parameterization of theMLT. In both cases, the free parameters of the MLT areemployed to mimic specific properties of the mean 3D alibration of the Mixing-Length Theory for Convective White Dwarf Envelopes WHITE DWARF MODELS
3D Model Atmosphere Simulations
We rely on CO BOLD 3D simulations that were pre-sented in earlier works (Tremblay et al. 2013a,b,c, here-after TL13a, TL13b, and TL13c, respectively). The 70simulations cover the range 6000 ≤ T eff (K) ≤ , ≤ log g ≤ g = 8 presented in TL13a. We have demon-strated that sequences at different surface gravities pos-sess rather similar properties (TL13b, TL13c), largelybecause 3D effects depend mostly on the local density,and the same range of densities is found at all surfacegravities, albeit with a shift in T eff .The numerical setup of the 3D model atmospheresis described in detail in TL13a, and more broadly inFreytag et al. (2012) in terms of the general propertiesof the code. We provide a brief overview in this sec-tion. The 3D simulations rely on an equation-of-state(EOS) and opacity tables that are computed with thesame microphysics as that of standard 1D model atmo-spheres (Tremblay et al. 2011). We employed a grid of150 × ×
150 points in the x, y, and z directions, where z is used for the vertical direction and points towardsthe exterior of the star. The grid spacing in the z direc-tion is non-equidistant and the total horizontal extentis chosen in order to have about 10 granules at the sur-face. The structure of the deep convection zone is largelydetermined by the radiative energy losses in the photo-sphere, which also fix the T eff of a simulation. As sug-gested by Brassard & Fontaine (1997), Hansen (1999),and Fontaine et al. (2001), non-gray atmospheres are anessential boundary condition for precise envelope calcu-lations. The 3D simulations solve the non-gray radiativetransfer using 8 to 13 opacity bins, which has provenadequate for spectroscopic applications (TL13c). Thissetup is likely more than sufficient for a comparison with1D structure calculations which are less sensitive to theoptically thin layers.The implementation of boundary conditions is de-scribed in detail in Freytag et al. (2012, see Sect. 3.2).In brief, the lateral boundaries are periodic, and thetop boundary is open to material flows and radiation.We rely on bottom conditions that are either open orclosed to convective flows. The lower boundary is closed(hereafter closed simulations ) when the vertical extent ofthe convection zone can be fully included in the simula-tion. This is the situation for the 3D simulations with T eff & , , , , , , , g = 7.0, 7.5, 8.0, 8.5, and 9.0, respectively. In thosecases, we impose zero vertical velocities at the bottom,and a radiative flux is injected from below.For cooler simulations, the bottom layer is open toconvective flows and radiation (hereafter open simula-tions ), and a zero total mass flux is enforced. We specifythe entropy of the ascending material to obtain approxi-mately the desired T eff value (an indirect quantity com-puted from the resulting emergent flux of the simulation).Figure 1 shows that the entropy from 1D envelopes (seeSect. 2.3) at the lower boundary of the convection zoneincreases monotonically with T eff . Convection is essen-tially adiabatic in deep convection zones, and the entropyvalue in the lower part of the convection zone is assumedto be the same as that of the upflows at the bottom ofthe simulations (see Sect. 2.2). Figure 1.
Entropy at the bottom of the convection zone as afunction of T eff for DA white dwarf envelopes at log g = 8. The 3Dresults are shown in black, with the h i entropy extracted directlyat the Schwarzschild boundary for closed simulations (open circles,see Sect. 4.1), and asymptotic s env values for open simulations(filled squares, see Sect. 4.2). We also display 1D sequences (solidlines, see Sect. 2.3) with the MLT parameterization varying fromML2/ α = 0.4 (red) to 2.0 (blue) in steps of 0.2 dex. Additionally,we present sequences where gas degeneracy effects are neglected(dotted lines), which largely follow the former sequences. In all models, the top boundary reaches a space- andtime-averaged value of no more than a Rosseland opticaldepth of τ R ∼ − . The bottom layer was generallyfixed at τ R = 10 , well below the photosphere, i.e. theline-forming regions. A few models were extended todeeper layers when the bottom of the convection zonewas too close to the simulation boundary. We cover atleast ∼ H P ) below the unstableregions when the bottom of the simulation is closed tomass flows. Properties of the Deep Convection Zone
The physical conditions at the bottom of convectionzones can be extracted from 3D simulations even if wedo not simulate the full zones. We rely on the techniquepresented in Ludwig et al. (1999), for which a demon-stration is shown in Figure 2 for a DA simulation at T eff = 10 ,
025 K and log g = 8. We present the local3D values of the entropy in convective structures (blackdots) as a function of geometrical depth with the stellar Tremblay et al. surface on the right-hand side. We also display the aver-age entropy profile over constant geometrical depth (solidred line). We observe significant entropy fluctuationsat all depths, although there is a constant asymptotic upper limit, hereafter s env . According to the scenariodeveloped in Stein & Nordlund (1989) and Ludwig et al.(1999), the gas in central regions of broad ascending flowsis still thermally isolated from its surroundings until itreaches layers immediately below the photosphere. Inother words, convective upflows keep an imprint of thephysical conditions at the bottom of the convection zone.The averaged 3D entropy, on the other hand, is not a con-served quantity due to radiative losses and the presenceof downdrafts created in the photosphere. Figure 2.
Local 3D entropy values (black dots) as a function ofgeometrical depth for a subset of a simulation at T eff = 10025 Kand log g = 8. The h i entropy profile, averaged over constantgeometrical depth, is shown with a red solid line. We also displaythe 1D entropy (dashed red line) with the MLT parameterizationcalibrated from the 3D simulation (ML2/ α = 0.69, see Table 2).We highlight τ R values at 100, 1.0, and 0.1 (cyan points, valuesidentified in the legend) as a guide. The asymptotic 3D entropyvalue s env is 2.082 × erg g − K − . The above technique only applies if the center of up-flows remains adiabatic, hence a minimum requirementis that the conditions at the bottom of the convectionzone are adiabatic. We have observed that the adiabatictransition takes place when the bottom of the convec-tion zone reaches layers deeper than τ R ∼ . For allof our simulations with an open bottom, we can recoveran asymptotic value. For closed simulations, there isno significant entropy plateau since conditions are neveradiabatic, although in those cases we can directly extractthe properties at the bottom of the convection zones.We also overlay in Figure 2 the 1D model atmospherewith the MLT calibrated from a comparison of s env witha grid of 1D envelopes (see Section 4.2). The 1D modelatmospheres and envelopes calibrated is this way are onlymeant to recover the conditions at the bottom of the con-vection zone, although by construction they also providean accurate mean structure for the essentially adiabaticparts of the convection zone. On the other hand, there isno guarantee that the calibrated 1D models will providea good match to the mean 3D stratification in super-adiabatic layers. Fortunately, in the case of white dwarfsin contrast to main-sequence stars, the super-adiabatic Figure 3.
Mass of hydrogen integrated from the surface ( M H )with respect to the total stellar mass (logarithmic value) as a func-tion of T eff for DA envelopes at log g = 8. The 3D results are shownwith black symbols using different definitions for the bottom of theconvection zone (see Sect. 3 and 4). For closed simulations, we con-sider the Schwarzschild boundary (open circles), the flux boundary(filled circles), and a v z , rms decay of 1 dex (open triangles) belowthe value at the flux boundary. For open 3D simulations, the filledsquares represent the values calibrated by matching s env with the1D entropy at the bottom of the convection zone. We also display1D sequences (solid lines) with the MLT parameterization varyingfrom ML2/ α = 0.4 (red) to 2.0 (blue) in steps of 0.2 dex. Thebottom of the stellar photosphere ( τ R = 1, 1D ML2/ α = 0.8),which roughly coincides with the top of the convection zone, isrepresented by a dotted black line. layers have little direct impact on applications that re-quire the use of 1D envelopes. As it was custom untilnow, it is generally sufficient to employ 1D envelopeswhere the MLT parameterization is based on the deeplayers, and rely on a different set of models, e.g. 3Dsimulations, for atmospheric parameter determinations.An inspection of Figure 2 demonstrates that if needed,a connection of the 1D and mean 3D structures at largedepth could also be a fairly good approximation.
1D Envelope Models
For the purpose of this work, we computed 1D en-velopes relying on the MLT for the treatment of convec-tion, similar to those presented in Fontaine et al. (2001)and van Grootel et al. (2012). The models employ theML2 treatment of MLT convection (Bohm & Cassinelli1971; Tassoul et al. 1990) and an EOS for a non-ideal pure-hydrogen gas (Saumon et al. 1995). Realis-tic non-gray temperature gradients are extracted fromdetailed atmospheric computations and employed asupper boundary conditions (Brassard & Fontaine 1997;van Grootel et al. 2012). The non-gray effects on thesize of the convection zone are shown in Figure 5 ofvan Grootel et al. (2012). In order to compare the en-velopes to 3D simulations, we have varied the mixinglength to pressure scale height ratio ML2/ α = l/H P from values of 0.4 to 2.0 in steps of 0.2. ML2/ α is selectedas a proxy for all MLT free parameters since changes inthe other parameters have similar effects on the struc-tures. We use the same range of surface gravities and ML2/ α has the same functional form as the more commonlyused α MLT for stars but it also specifies the choice of auxiliaryparameters of the MLT formulation (Ludwig et al. 1999). alibration of the Mixing-Length Theory for Convective White Dwarf Envelopes T eff (steps of 0.5 dex and 100 K, respectively) as our setof 3D calculations.From the 1D envelopes we have extracted the physicalconditions at the bottom of the convection zone. Figure 3depicts the hydrogen mass integrated from the surface( M H ), with respect to the total white dwarf mass, forthe log g = 8 case. Clearly, the MLT parameterizationhas a strong effect on the size and mass included in theconvection zone at intermediate temperatures, where theatmospheric layers are super-adiabatic.To ensure that we share a common entropy zero pointin all calculations, we computed all entropy values us-ing the same EOS as the 3D simulations, based onthe Hummer & Mihalas (1988) non-ideal EOS, where wehave also accounted for partial degeneracy. The en-tropy values at the bottom of the convection zone areshown in Figure 1 (solid lines), along with additionalsequences where we have neglected partial degeneracy(dotted lines). The degeneracy effects are very small inthe convection zone ( η <
0, where ηkT is the chemicalpotential of the free electrons). This is largely due tothe fact that the degeneracy level is constant for an adi-abatic process. For the essentially adiabatic structure ofcool white dwarf convection zones, degeneracy is chang-ing very slowly as a function of depth (see Eq. (13) ofB¨ohm 1968). Furthermore, degeneracy effects are stillnegligible at the lower T eff limit where the calibration ofML2/ α is performed in this work (see Section 5.1).Our proposed calibration of the MLT is performed bycomparing 3D simulations to 1D envelopes. We also relyon 1D MLT model atmospheres (Tremblay et al. 2011)for illustrative purposes in cases where we display a de-tailed comparison of 1D and mean 3D stratifications asa function of depth. The 1D model atmospheres and en-velopes provide very similar results, within a few percent,below the photosphere. DEFINITION OF CONVECTIVE LAYERS
In the following, we rely on mean 3D values, hereafter h i , for all quantities except for the asymptotic entropy s env . h i values are the temporal and spatial average of3D simulations over constant geometrical depth. We use250 snapshots in the last 25% of a simulation to make thetemporal average. While our earlier studies have reliedon averages over constant optical depth, the geometricaldepth is better suited to extract convective fluxes andovershoot velocities.Before comparing 3D simulations and 1D envelopes, itis crucial to define what we refer to as the convectionzone. In the local MLT picture, the convective regionsare clearly characterized as the layers where the radiativegradient ∇ rad = (cid:18) ∂ ln T∂ ln P (cid:19) rad , (1)is larger than the adiabatic gradient ∇ ad = (cid:18) ∂ ln T∂ ln P (cid:19) ad , (2)with T the temperature and P the pressure. All otherparts of the structure are fully static. This is a rathercrude approximation of the dynamical nature of convec-tion, where material flows do not vanish abruptly when the thermal structure becomes stable. In this section, wereview the different regions that are found in non-localmodels of the lower part of convection zones (see alsoSkaley & Stix 1991; Chan & Gigas 1992; Freytag et al.1996). Table 1 formally defines the regions discussed inthis section, and we give an example of the geometricalextent and mass included in these layers based on the12,100 K and log g = 8 simulation.To further illustrate the profile of 3D convection zones,Figure 4 displays the RMS vertical velocities for closed-box simulations at log g = 8. We start from the verticalvelocity v z = u z − h ρu z ih ρ i , (3)where the mass flux weighted mean velocity (second termon right-hand side) is removed from the directly simu-lated velocity u z to account for the residual numericalmass flux. The latter results from the presence of plane-parallel oscillations and an imperfect temporal averagingdue to the finite number of snapshots. The correspond-ing RMS vertical velocity is v , rms = h v i = h u i + h ρu z i h ρ i − h ρu z ih u z ih ρ i , (4)where all averages are performed over constant geometri-cal depth . Furthermore, Figures 5 and 6 show the h i and 1D convective flux profiles. The h i convective fluxis the sum of the enthalpy and kinetic energy fluxes, F conv = h ( e int + Pρ ) ρu z i + h u ρu z i − e tot h ρu z i , (5)where e int is the internal energy per gram, ρ the density,and u the 3D velocity. The mass flux weighted energyflux (third term on right-hand side of Eq. (5)) is sub-tracted to correct for any residual non-zero mass flux inthe numerical simulations as in Eq. (4). This correctionis a small fraction of the convective flux for all simula-tions. The total energy is defined from e tot = h ρe int + P + ρ u ih ρ i . (6)We use the logarithm of the temperature as an indepen-dent variable since it is a local quantity, while opticaldepth and mass are integrated from the top of the con-vection zone, and are more sensitive to differences in thephotosphere.The proper convection zone in 3D (open circles inFigs. 4-6) is defined in the same way as in 1D fromthe Schwarzschild (stability) criterion. In this region,the entropy gradient is negative with respect to geomet-rical depth (increasing towards the exterior). In thefollowing, we define the bottom of this region as the Schwarzschild boundary . In the 3D simulations, con-vective flows are largely created, horizontally advected,and merged into narrow downdrafts in the photosphere This differs from the RMS velocity fluctuation h v i − h v z i where h v z i is expected to be non-zero due to a correlation betweenvelocity and density fluctuations in the convection zone. Tremblay et al.
Table 1
Regions in the Lower Part of Convection ZonesRegion dsdz a dsdz a F conv /F total F conv /F total v z v z ∆ z b ∆ log M H /M tot b (3D) (1D) (3D) (1D) (3D) (1D)Zone 1 < < > > = 0 = 0 − − Zone 2 > > > = 0 0 0 . H P > > < = 0 0 ∼ . H P ∼ . > > ∼ = 0 0 > H P > . a The coordinate z points towards the exterior of the star. b Ranges are taken from the simulation at T eff = 12 ,
100 K and log g ∼ . | F conv /F total | < . Figure 4.
Vertical RMS velocity as a function of the logarithm ofthe temperature for 3D simulations at log g = 8 (solid red lines).The T eff values are identified on the top right of the panels. Weshow the position of the Schwarzschild boundary (open circles), theflux boundary (filled circles), and the v z , rms decay of 1 dex (opentriangles) below the value at the flux boundary. We also display1D model atmospheres with the calibration of the MLT param-eters (see Table 2) for the Schwarzschild (dotted black) and fluxboundaries (dashed blue). For the models warmer than 13,000 K,we rely on an asymptotic parameterization of ML2/ α Schwa = 1.2and ML2/ α flux = 1.4, respectively (see Sect. 5.1). (Freytag et al. 1996). Large entropy fluctuations are pro-duced by the radiative cooling in these layers, whichdrives the convective motions. For cool convective whitedwarfs ( T eff . ,
000 K, log g = 8) with deep convec-tion zones, entropy fluctuations are smaller in the photo-sphere and the dominant role of the downflows is dimin-ished. The descending fluid form a hierarchical structureof merging downdrafts due to the increase of the pressurescale height with depth (Asplund et al. 2009).In the 3D simulations, downdrafts at the base of theconvection zone (according to the Schwarzschild crite-rion) still have large momenta. They are also denserthan the ambient medium, albeit with a decreasing dif-ference. As a consequence, the convective cells are stillaccelerated in the region just below the unstable lay-ers. Mass conservation guarantees that there is warm Figure 5.
Ratio of the convective energy flux to total flux as afunction of the logarithm of the temperature at log g = 8. The h i fluxes are represented by solid red lines and T eff values forthe simulations are identified on the panel. The ratio is exact forthe 12,100 K model, but other structures are shifted by one fluxunits for clarity. The symbols are the same as in Figure 4. Wealso display 1D model atmospheres matching the Schwarzschildboundary (black, dotted) and the flux boundary (blue, dashed).Parameters for the Schwarzschild boundary are ML2/ α Schwa =0.88, 1.07, and 1.32, for the 12,100, 12,500, and 13,000 K models,respectively. The values are ML2/ α flux = 1.00, 1.25, and 1.50 forthe flux boundary at the same temperatures. material transported upwards, hence there is a positiveconvective flux is this region. These layers are equiv-alent to a convection zone in thermodynamical terms.We define the bottom of this region as the layer where F conv /F tot = 0 and refer to it as the flux boundary (filledcircles in Figs. 4-6). The typical size of the region be-tween the Schwarzschild and flux boundaries is a bit lessthan one pressure scale height, or ∼ alibration of the Mixing-Length Theory for Convective White Dwarf Envelopes Figure 6.
Similar to Figure 5 but for the 3D simulations (solidred) at 13,500 and 14,000 K. The convective to total flux ratiois exact for the model at 13,500 K and shifted by 0.1 flux unitsfor the 14,000 K case. The symbols are the same as in Figure 4.In this regime, the MLT is unable to replicate both the 3D sizeof the convection zone and the maximum F conv /F tot ratio. Wedisplay instead 1D ML2/ α = 0.7 model atmospheres (black, dot-dashed), which correspond to the average MLT parameterizationto reproduce the maximum F conv /F tot ratio for shallow convectionzones (see Sect. 4.1). ambient medium and they carry a net downwards (nega-tive) convective flux, or in other words, the temperaturegradient in these layers is larger than the radiative gradi-ent. That follows from the change of sign of the velocity-enthalpy correlation. However, Figure 5 demonstratesthat this negative overshoot flux is always a small frac-tion ( . H P . While the convective fluxbecomes rapidly energetically negligible, the convectivevelocities still have mixing capabilities in much deeperlayers. This situation is due to the extreme ratio betweenconvective and diffusive time scales (see Section 5.2). Intypical cases for DA white dwarfs, convective velocitiesare of the order of v z , rms ∼ − at base of the con-vection zone, while overshoot velocities of the order of 1m s − still dominate over the slower diffusive speeds, andcan efficiently mix elements (Freytag et al. 1996). Thisimplies that microscopic diffusion timescales are likelyto dominate only in the deep overshoot layers, i.e. a few H P below the flux boundary. The exact layer where thishappens depends on the diffusing trace chemical elementand the atmospheric parameters of the model, althoughit is clear that the mixed region can be much larger thanin the 1D approximation. In Figures 4-6, we identifythe position of a 1 dex velocity decay with respect tothe velocity at the flux boundary (filled triangles), which is generally close to the bottom of the simulation. Oursimulations evidently provide a truncated picture of theovershoot layers and we review this issue in Section 5.2. COMPARISON OF 1D AND 3D CONVECTION ZONES
Closed 3D Simulations
We first proceed with a direct comparison of h i and 1D stratifications in the case of shallow convectionzones, completely enclosed within the simulation domain.Figures 7 and 8 present the h i logarithmic values ofthe temperature and pressure, respectively, characteriz-ing the bottom of the convection zone for simulationsat log g = 8. We rely on three different definitions forthe size of the convection zone as discussed in Section 3,with the same symbols as in Figures 4-6. These regionscorrespond to the Schwarzschild boundary (open circles),the flux boundary (filled circles), and a v z , rms decay of1 dex (open triangles) below the reference value at theflux boundary. Figures 7 and 8 also display 1D sequences,with values ranging from ML2 /α = 0.4 to 2.0 in steps of0.2, using the Schwarzschild boundary to define the sizeof the convection zone (solid lines). Figure 7.
Logarithm value of the temperature at the bottomof the convection zone as a function of T eff , for DA white dwarfenvelopes at log g = 8. The h i results are shown with blacksymbols using different definitions for the bottom of the convectionzone (see Sect. 3). We consider the Schwarzschild boundary (opencircles), the flux boundary (filled circles), and a v z , rms decay of1 dex (open triangles) below the value at the flux boundary. Wealso display 1D sequences with the MLT parameterization varyingfrom ML2/ α = 0.4 (red) to 2.0 (blue) in steps of 0.2 dex. The solidlines represent the bottom of the convection zone defined by theSchwarzschild boundary while the dotted lines stand for the layersbelow which the convective flux becomes energetically negligible( F conv /F tot < . Figures 7 and 8 demonstrate that the 1D envelope thatbest matches the bottom of a 3D simulation is generallyindependent of whether the matching is performed ontemperature or pressure. The pressure is proportional tothe 1D mass column, where only thermodynamic pres-sure contributes to hydrostatic equilibrium, while in 3Dsimulations one must also account for the turbulent pres-sure. Figure 3 depicts the h i and 1D comparison interms of the hydrogen mass, and the results are similar tothose presented for the temperature and pressure at thebase of the convection zone. It implies that even though Tremblay et al.
Figure 8.
Similar to Figure 7 but for the thermal pressure (loga-rithm value) at the bottom of the convection zone as a function of T eff at log g = 8. h i and 1D models have different profiles in the photo-sphere, due to the top convective overshoot and turbulentpressure, differences in the integrated mass column aresmall in the lower part of the convection zone. In thefollowing, the calibrated ML2 /α is the average value ofthe two 1D models that best match the h i pressureand temperature at the bottom of the convection zone,respectively, within a prescribed boundary. The masscolumn can be directly extracted from the envelopes cal-ibrated in this way.In terms of the Schwarzschild boundary, Figures 7and 8 demonstrate that the mixing-length parameter in-creases rapidly with T eff , with values of 0.88, 1.07, and1.32 at 12,100, 12,500, and 13,000 K, respectively. In this T eff range partially covering the ZZ Ceti instability strip,the MLT variation is significant compared to the usuallyassumed constant value of ML2/ α = 1.0 for envelopes(Fontaine & Brassard 2008). Our calibration of ML2/ α is meant to represent the h i temperature and pressureat the Schwarzschild boundary, and by construction, itprovides an estimation of the average temperature gra-dient for the full convection zone. However, the photo-spheric temperature gradient of a calibrated 1D envelopeis not expected to correspond to that of the 3D simula-tion.For the convection zone defined in terms of the h i flux boundary, ML2/ α values have to be increased to1.00, 1.25, and 1.50 for the same T eff values as above.The derived efficiency is significantly higher than thatfound for the Schwarzschild boundary. One should becautious since an inspection of Figure 5 for 1D model at-mospheres calibrated for the flux boundary (blue dashedlines) reveals that while the zero point of convective fluxis by definition in agreement with the 3D simulations, theoverall shape of the h i convective flux is not very wellreproduced for shallow convection zones. Our calibrationof ML2/ α is mostly useful to characterize the depth atwhich convection becomes energetically insignificant andthe velocities start to decay exponentially with geometri-cal depth. Finally, Figure 9 demonstrates that the h i versus 1D results (temperature only) at other gravitiesare fairly similar, albeit with a shift in T eff . As a con-sequence, the previous discussion applies most generally to white dwarfs with shallow convection zones. Figure 9.
Similar to Figure 7 but for log g = 7.0, 7.5, 8.5, and9.0, with values identified on the panels. For the very warm simulations, e.g. 13,500 and14,000 K at log g = 8.0, the Schwarzschild andflux boundaries are essentially in the photosphere( τ R , bottom < α value forthese layers becomes coupled with the MLT parameteri-zation used in spectroscopic applications (TL13c). Both3D simulations and 1D models show new patterns in this T eff regime. The h i convective flux becomes negligi-ble outside of the unstable layers, and there is a rever-sal of the flux and Schwarzschild boundaries, with theSchwarzschild boundary moving below the flux bound-ary with increasing T eff . In this regime, efficient radi-ation transport is able to smooth temperature fluctua-tions. This diminishes the flux of internal energy (firstterm in Eq. (5)) over a shorter distance from the topof the convection zone than the velocity field becomessymmetric in up- and downflows. The significant mo-mentum of the narrow downdrafts produces a negativekinetic energy flux (second term in Eq. (5)). This flux re-mains large near the mean Schwarzschild boundary sincecool downdrafts get convectively stable at larger geomet-rical depths than the upflows. As a consequence, themean total flux becomes negative slightly above the meanSchwarzschild boundary. We have verified that thereis no reversal of the flux and Schwarzschild boundarieswhen the kinetic energy flux is neglected. The MLT doesnot account for the kinetic energy flux, hence we do notexpect a similar reversal in 1D.For convective 1D models at large T eff , the size of theunstable regions becomes insensitive to the MLT param-eterization according to Figure 7, hence it is not possibleto calibrate the MLT based on the Schwarzschild bound-ary. This picture is somewhat misleading since the MLTconvective fluxes, and associated velocities, remain verysensitive to the value of the MLT parameters. Figure 7shows that the 1D convective flux drops to very smallvalues ( F conv /F tot < .
01, dotted lines) much higher inthe photosphere than the 1D Schwarzschild boundary.Our results would naively suggest that convective effi-ciency increases with T eff but the 3D simulations presenta more complex picture. At high T eff , non-local effectsfrom strong and deep reaching downdrafts create h i alibration of the Mixing-Length Theory for Convective White Dwarf Envelopes α in order to reproduce the maximumvalue of the h i convective flux, which peaks in the pho-tosphere, for shallow convection zones. Clearly, a muchsmaller mixing-length parameter is necessary to matchthe h i convective flux in the photosphere in compari-son to the Schwarzschild or flux boundaries. The valuesof ML2/ α = 0.6-0.8 are consistent with the commonlyused spectroscopic parameterizations (TL13c). Never-theless, the parameterizations for the Schwarzschild andflux boundaries offer a better representation of the con-ditions at the bottom of the convection zones. Figure 10.
Calibration of ML2/ α for the maximum F conv /F tot ratio as a function of T eff and log g (represented by different colorswith the legend at the bottom). The calibration is based on the1D model that best replicates the maximum h i convective fluxof closed simulations. This calibration can not be performed fordeep convection zones since all 1D models have F conv , max /F tot ∼ We have already discussed the fact that the convec-tion zone is drastically deeper when defined in terms ofthe h i convective velocities. This is also seen in Fig-ures 7 and 8 where we show the position of the one orderof magnitude decrease for v z , rms below the flux bound-ary (open triangles). It is inappropriate to parameterizethe 1D MLT for the highly non-local overshoot veloci-ties, and it would produce spurious stratifications in theunstable regions. Instead, we propose an overshoot pa-rameterization that does not directly involve the MLT inSection 5.2. Open 3D Simulations
For open 3D simulations, we have extracted the asymp-totic entropy values s env characterizing the deep adia-batic layers using the technique described in Section 2.2. s env is directly derived from the specified entropy of theascending material at the bottom boundary of the simu-lations. We have verified that this matches the observedasymptotic value below the photosphere (see, e.g., Fig-ure 2). We then assume that s env also corresponds tothe entropy value at the bottom of the unstable layersin 1D envelopes. The s env and 1D entropy values arecompared in Figure 1 for the log g = 8 case. The cali-bration of ML2/ α is directly performed from a match of s env with entropy values interpolated from the grid of 1Denvelopes. In Figure 3, we show the resulting hydrogenmass integrated from the surface. At low temperatures ( T eff . g = 8), DAwhite dwarfs have extremely small super-adiabatic at-mospheric layers, and the structure remains essentiallyadiabatic from the bottom to the top of the convectionzone. Since the top of the convection zone is higher thanthe photosphere ( τ R ∼ . DISCUSSION
1D MLT Calibration
Figure 11 (top panel) presents the MLT parameteri-zation for the lower part of the convection zone in or-der to recover the Schwarzschild boundary (hereafterML2/ α Schwa ) of the 70 3D simulations in our grid.We illustrate with different symbols the calibration de-rived directly from closed 3D simulations (open cir-cles) and inferred from a match of s env (filled squares).We also present in Figure 11 (bottom panel) the cal-ibration matching the h i flux boundary (hereafterML2/ α flux ). The latter calibration is directly performedfor closed simulations, and in those cases, α flux is 16%larger than α Schwa with a relatively small dispersion of3%. Therefore, we simply assume that ML2/ α flux =1 .
16 ML2 /α Schwa for open 3D simulations. This is likelya good approximation in the transition region betweenclosed and open 3D simulations, and at lower T eff , the1D envelopes depend less critically on the MLT parame-terization.The calibration is not performed when the 1D massincluded within the convection zone varies by an amountsmaller than 0.2 dex for the range of ML2 /α between 0.4and 2.0. This defines the upper and lower T eff boundariesin Figure 11, which depend on log g . At the cool end, wepropose to keep ML2/ α constant, since the value is irrel-evant for structure calculations. Similarly, at T eff valuesabove those in the calibration range, it is likely accept-able to keep the value constant for most applications.The choice of the asymptotic ML2/ α value is not ob-vious, however, because of its rapid variation with T eff .As a compromise, we adopt values of 1.2 and 1.4, forML2/ α Schwa and ML2/ α flux , respectively, at T eff valueslarger than our calibration range. If one is interested inthe detailed properties of shallow convection zones above T eff ∼ ,
000 K at log g = 8.0, it may be preferable tocombine the h i and 1D structures at some depth belowthe convection zone where the convective flux is negligi-ble. The MLT does not reproduce very well the extendedbut inefficient 3D convection zones in this regime. For T eff values above our calibration range, most of the 3Deffects will be from the overshoot at the base of the con-vection zone since contrary to the small convective fluxes,velocities remain significant well below the photosphere.Table 2 provides the tabulated MLT parameteriza-tions, which are valid for 1D envelopes with an EOS,opacities, and boundary conditions similar to those em-ployed for our grid. Physical conditions at the bottom ofour calibrated envelopes (mass, temperature, and pres-sure) are also given as a reference point. Moreover, wepropose fitting functions for ML2/ α Schwa and ML2/ α flux ,0 Tremblay et al.
Figure 11.
Top:
Calibration of ML2/ α Schwa for the lower part ofthe convection zone as a function of T eff and log g (represented bydifferent colors with the legend at the top). The calibration is basedon the 1D model that best replicates the Schwarzschild boundaryof a 3D simulation, either from a direct comparison (open circles)or by using the s env calibration (filled squares). The dotted linescorrespond to the proposed fitting function (Eq. (9)). Bottom:
Cal-ibration of ML2/ α flux based on the 1D model that best representsthe flux boundary of a 3D simulation (filled circles). For open 3Dsimulations, we use ML2/ α flux = 1.16 ML2/ α Schwa . The dottedlines correspond to the proposed fitting function (Eq. (10)). respectively, where the independent variables are definedas g = log g [cgs] − . , (7) T = ( T eff [K] − / − . g , (8)and the functions are as follow with numerical coefficientsfound in Table 3ML2 /α Schwa = (cid:16) a + ( a + a exp[ a T + a g ])exp h ( a + a exp[ a T ]) T + a g i(cid:17) + a exp (cid:16) − a ([ T − a ] + [ g − a ] ) (cid:17) , (9)ML2 /α flux = (cid:16) a + a exp h ( a + { a + a exp[ a T + a g ] } exp[ a T ]) T + a g i(cid:17) + a exp (cid:16) − a ([ T − a ] + [ g − a ] ) (cid:17) . (10)The proposed functions are presented in Figure 11 alongwith the data points. Similarly to our 3D atmosphericparameter corrections in TL13c, we have adopted func-tions that do not retain the fine details of the 3D and 1Ddifferences. Small scale fluctuations may be due to inac-curacies in the grid of 3D simulations. Ultimately, the calibrated 1D structures do not provide the detailed h i convective flux profile and neglect the turbulent natureof convection. It is not well constrained how much these3D effects impact chemical diffusion and pulsation calcu-lations. Finally, we remind the reader that 1D structurecodes typically make approximations for the non-gray ra-diative transfer in the atmospheres, which may introducea slight offset in the size of convection zones. The ML2/ α offset is at most a few percent for our setup (see Section2.3). As a consequence, we believe that a calibrationwithin 5% is sufficient. Parameterization of Overshoot Velocities
We have so far neglected the convective overshoot be-low the flux boundary. In most cases, the quantity ofinterest is the overshoot velocity, which does not exist inthe local MLT. In the following, we aim at providing aparameterization for overshoot in regions below the 1Dconvection zone.The spatial scales and timescales involved in convec-tion and microscopic diffusion differ by many ordersof magnitude in typical white dwarfs. It is thereforenot possible for multi-dimensional simulations to modelboth effects simultaneously. Instead, we depict the farovershoot regime as a random walk process character-ized by a macroscopic diffusion coefficient, which sim-ply counter-balances the microscopic diffusion coefficientin 1D calculations. The mixed regions are those wheremacroscopic diffusion dominates over microscopic diffu-sion. Freytag et al. (1996) studied this random walk pro-cess with tracer particles in 2D RHD simulations. Theyfound that the particles are immediately mixed withinthe convection zone, but that the RMS vertical spread δz overshoot in the overshoot layers could be described from δz = 2 D overshoot ( z ) t , (11)where D overshoot is the macroscopic diffusion coefficient D overshoot ( z ) = v , rms ( z ) t char ( z ) , (12)with t char a characteristic timescale. Just based on MLTmodels or even with detailed RHD simulations, v z , rms isnot directly available for the deep overshoot regions of in-terest. As a consequence, Freytag et al. (1996) propose,from a match to 2D simulations and physical consider-ations, that v z , rms has an exponential decay below theconvection zone. The resulting diffusion coefficient thentakes the form D overshoot ( z ) = v t char exp(2( z − z base ) /H v ) , (13)where v base is the velocity at the base of the convectionzone and H v the velocity scale height. In the following,we assume that the base of the convection zone is theflux boundary as determined by 3D simulations and the1D ML2 /α flux parameterization. Closed 3D Simulations
For closed 3D simulations, it is possible to verify theproposed exponential decay of overshoot velocities, aswell as calibrate Eq. (13) by extracting v base , t char , and H v . Figure 12 demonstrates that over the three pres-sure scale heights typically included in our simulations alibration of the Mixing-Length Theory for Convective White Dwarf Envelopes Table 2
ML2/ α Calibration for DA Envelopes T eff log g ML2/ α Schwa a log M H /M tot a log T a log P a ML2/ α flux b log M H /M tot b log T b log P b (K) [K] [dyn cm − ] [K] [dyn cm − ]6112 7.00 0.53 − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − Note . — T eff is the spatial and temporal average of the emergent flux. The RMS T eff variations are found in Table 1 of TL13b. log M H /M tot ,log T , and log P are extracted at the bottom of the convection zone from calibrated 1D envelopes. a Corresponds to the position of the h i Schwarzschild boundary for closed simulations (see Section 4.1). For open simulations, the calibration isperformed by matching the 3D s env value with the 1D entropy at the bottom of the convection zone (see Section 4.2). b Corresponds to the position of the h i flux boundary for closed simulations. For open simulations, we simply assume that ML2/ α flux =1 .
16 ML2 /α Schwa (see Section 5.1). Tremblay et al.
Table 3
Coefficients for Fitting FunctionsCoefficient ML2/ α Schwa
ML2/ α flux a − − − − −
02 1.0083984E+00a − − − − −
02 5.4884977E+00a − − − − − − − − − − −
01 2.9166886E − −
01 3.6544167E − − − − − below the flux boundary, the velocity decay is nearly ex-ponential. The velocity scale height is very close to onepressure scale height (dotted black line), although it is ac-tually changing with depth. It is larger than one pressurescale height immediately below the flux boundary, andbecomes subsequently smaller. As a consequence, taking H v = H P is very likely to overestimate macroscopic dif-fusion in the deep overshoot layers, and gives an upperlimit to the mixed mass. Finally, Freytag et al. (1996)demonstrate that the timescale of overshoot for shallowconvection zones is the same as the characteristic convec-tive timescale in the photosphere, since this is where thedowndrafts are formed. As a consequence, it is possibleto use directly the characteristic granulation timescalescomputed in TL13b and TL13c. In Table 4, we present v z , rms at the flux boundary ( v base ) and the characteris-tic granulation timescales ( t char ) for closed simulations,which can be used in Eq. (13) for shallow convectionzones. The velocity scale height can be directly evalu-ated from the 1D pressure scale height in the envelopessince this quantity is not significantly impacted by 3Deffects, although we also include the local h i values atthe base of the convection zone in Table 4.The overshoot coefficients in Table 4 are limited bythe T eff range of our 3D simulations. Figure 13 com-pares the maximum velocities, which peak slightly belowthe photosphere, for h i and 1D ML2/ α = 0.7 mod-els at log g = 8. We applied the MLT parameteriza-tion that best represents the maximum convective flux ofthe warmest 3D simulations (see Figure 10 and TB13c).The MLT suggests that velocities in the photosphere for14 , < T eff (K) . ,
000 are still of the same order ofmagnitude as in cooler models, although the upper T eff limit depends critically on the MLT parameterization.The large photospheric velocities are likely to supportstrong overshoot layers in DA white dwarfs above ourwarmest 3D simulations, even though convection has anegligible effect on the thermal structure. Open 3D Simulations
For open 3D simulations, we can not directly extractquantities to calibrate Eq. (13). Furthermore, the as-sumption that the overshoot timescale is the same asthe surface granulation timescale is unlikely to be valid,since the downdrafts have time for merging into the hier-archical structure observed in simulations of deep, con-vective envelopes (Nordlund et al. 2009). We propose in-
Figure 12.
Vertical RMS velocity decay as a function of pres-sure (natural logarithm values) for 3D simulations at log g = 8.The reference point is the flux boundary for which we define∆ ln v z , rms = 0 and ∆ ln P = 0. The simulations are color-codedfrom T eff = 12,100 (red), 12,500, 13,000, 13,500, to 14,000 K (blue).The − H P . The velocity decay at ∆ ln P >
Figure 13.
Maximum v z , rms velocity within the convection zonefor 3D simulations (filled points, red) and 1D ML2/ α = 0.7 modelatmospheres (open points, black) at log g = 8. The points areconnected for clarity. stead that t char = H v /v base , with the velocity scale heightequal to the pressure scale height as above. Therefore, v base is the only quantity that remains to be evaluated.For deep and essentially adiabatic convection zones,the MLT and 3D simulations agree on the temperaturegradient. An examination of the MLT equations demon-strates that for very efficient convection ( F conv ∼ F tot ),velocity is proportional to ρ − / , along with a depen-dence on heat capacity and molecular weight in the pres-ence of partial ionization. While the 1D velocity model isonly an idealization of the complex 3D dynamics, we sug-gest that the v /v ratio remains very similar acrossthe deep convection zone. This is seen in Figure 4 forthe cooler 10,025 K model where convection is reason-ably adiabatic below the photosphere. Figure 14 showsthe h i versus 1D ML2 /α flux velocity ratio for opensimulations and a reference layer identified by the crite- alibration of the Mixing-Length Theory for Convective White Dwarf Envelopes Table 4
Overshoot Parameters for Closed 3D Simulations T eff log g v base a log t char b log H P c (K) (10 cm s − ) [s] [cm]10540 7.00 3.36 − − − − − − − − − − − − − − − − − − − − − − a Corresponds to h i v z , rms at the flux boundary. b Same as the decay time in Table A.1 of TL13c. c Corresponds to h i P/ ( ρg ) at the flux boundary. rion log τ R = 2 .
5. This region is deep enough for con-vection to be largely adiabatic, and far away from thebottom boundary to prevent numerical effects. We ob-serve small variations around a mean value of v /v =1.5 for the DA white dwarfs with a deep adiabatic con-vection zone. We suggest that this calibration remainsvalid down to the bottom of the convection zone, as longas F conv ∼ F tot . We still face the problem, however,that by definition v MLT , base = 0. We recommend insteadto take a characteristic velocity v MLT , base ∗ one pressurescale height above the bottom of the convection zone. Insummary, for the T eff range below the one covered byTable 4, we propose the following overshoot parameteri-zation D overshoot ( z ) =1 . v MLT , base ∗ H P exp(2( z − z base ) /H P ) , (14)where all quantities are extracted from 1D ML2 /α flux structures as described above.We confirm the results of Freytag et al. (1996)that overshoot is significant and present for all DAwhite dwarfs with convectively unstable layers ( T eff . ,
000 K). The total mass of hydrogen included in theovershoot region may be a few orders of magnitudegreater than the mass included in the proper convectionzone. This effect is totally neglected in local MLT mod-els, and our proposed parameterization provides an or-der of magnitude estimate (upper limit) of the overshootvelocities and macroscopic diffusion coefficients. The re-sulting effects on the chemical abundances of mixed ele-ments, for instance in accreting white dwarfs in a steadystate, depend on the outcomes of chemical diffusion cal-culations.
Improvements to the Local MLT
Figure 14.
Ratio of the 3D v z , rms and 1D ML2/ α flux velocities atlog τ R = 2.5 as a function of T eff and log g (represented by differentcolors with the legend at the top). The points are connected forclarity. The ML2/ α flux calibration is presented in Table 2. The previous sections have revealed that the local MLTonly depicts a rough portrait of the underlying dynami-cal nature of convection, which is illustrated by the needof having different parameterizations for different appli-cations. We note that non-local 1D MLT models couldprovide a better match to the 3D results. In particu-lar, the models discussed in Spiegel (1963), Skaley & Stix(1991), Dupret et al. (2006), and St¨okl (2008) naturallydeliver the Schwarzschild and flux boundaries, as wellas (partial) overshoot layers. In these non-local MLTmodels, the more realistic physics is recovered at the ex-pense of adding more free parameters. In some sense,this is a more elegant and accurate way of obtainingthe Schwarzschild and flux boundaries than we have pro-posed in this work. While it does require some modifi-cations of existing 1D model atmosphere and structurecodes, this should be investigated in the future.Montgomery & Kupka (2004) have also presented anon-local convection model for white dwarfs, although inthis case it is not an extension of the MLT theory. How-ever, one issue for all non-local 1D models discussed hereis that they have not been very successful at modelingovershoot velocities reproducing the exponential decayobserved in RHD simulations, which is the main part ofthe 1D models that we would like to improve.
ZZ Ceti Instability Strip
The spectroscopically determined atmospheric param-eters of pulsating ZZ Ceti white dwarfs have been dis-cussed in TL13c, as seen in the light of our grid of h i spectra. We found that the dominant 3D effect is onthe spectroscopically determined surface gravity, with anaverage shift of ∆ log g = − T eff corrections depend critically on the calibrationof the MLT parameters in the reference 1D model at-mospheres. Based on the 1D ML2/ α = 0.8 calibration,we observed a 3D shift of ∆ T eff = −
225 K on average,although this is in the same range as the uncertaintiesin the 3D corrections. The spectroscopic blue edge atlog g = 8, below which white dwarfs are pulsating, is lo-cated at T eff ∼ ,
500 K when relying on h i spectra,while it is slightly warmer by 100 K based on 1D ML2/ α Tremblay et al. = 0.8 model atmospheres. On the other hand, the h i red edge is located at T eff ∼ ,
000 K for log g = 8. Over-all, the observed position of the instability strip is notchanged significantly compared to earlier investigations(Gianninas et al. 2006, 2011). We remind the reader thatthe observed edges are defined from only a few pulsatingand constant objects, and that the individual errors onthe spectroscopic atmospheric parameters must also beconsidered.Non-adiabatic asteroseismic models provide predic-tions for the position of the blue edge of the ZZ Cetiinstability strip, although the results are highly sensitiveto the parameterization of convection (Fontaine et al.1994; Gautschy et al. 1996). Recently, van Grootel et al.(2012) relied on a non-adiabatic code including time-dependent convection to study the driving mechanism.Compared to earlier studies (Fontaine & Brassard 2008,and references therein), their approach neither assumesfrozen convection nor an instantaneous convection re-sponse during a pulsation cycle. Using 1D ML2/ α = 1.0white dwarf structures similar to those discussed in thiswork, they find a seismic blue edge at T eff = 11 ,
970 Kfor log g = 8. Since the convective flux contribution iscritical in the non-adiabatic perturbation equations, wecan compare their results with our ML2/ α flux calibra-tion in Figure 11. We find that ML2/ α flux ∼ g = 8, in very close agreement withthe value generally used to predict the blue edge of theinstability strip, based on seismic models. There seemsto be a slight discrepancy between the observed and pre-dicted blue edges, the latter being cooler by about 500 K.We note, however, that the current agreement is stillfairly good considering the uncertainties in the 3D sim-ulations and spectroscopically determined atmosphericparameters. It would be interesting to review the non-adiabatic pulsation calculations with the new calibrated1D envelopes or a direct use of the h i convective fluxprofiles (Gautschy et al. 1996). Finally, dynamical con-vection effects that are missing from both current andnewly calibrated 1D envelopes could also have an impacton pulsations (van Grootel et al. 2012).At the red edge of the instability strip,van Grootel et al. (2013) recently revived an ideaof Hansen et al. (1985) originally applied to the blueedge. They suggest that the red edge of the g-modeinstabilities is reached when the thermal timescale in thedriving region (bottom of the convection zone) becomesof the order of the pulsation period. Beyond this limit,outgoing g-waves are no longer reflected back by theatmospheric layers, and will lose their energy in theupper atmosphere. Using this argument for g-modesof spherical-harmonic degree l = 1, the red edge liesat ∼ g = 8 with ML2/ α = 1.0 1Denvelopes. In this range of T eff , we predict a slightlyshallower 3D convection zone, although it is unlikelyto impact in a qualitative way the results presented invan Grootel et al. (2013). CONCLUSION
We have presented a comparison of our grid of 3DRHD simulations for 70 DA white dwarfs, in the range7 . ≤ log g ≤ .
0, with 1D envelope models based onthe mixing-length theory for convection. While MLTonly provides a bottom boundary of the convection zone based on the Schwarzschild criterion, the 3D stratifica-tions are more complex. In 3D simulations, convectivestructures are still accelerated just before reaching theSchwarzschild boundary and the convective flux remainssignificant in layers below the classical definition of theconvection zone. In addition, we confirm that DAs havestrong lower overshoot layers, where vertical velocitiesdecay exponentially with a velocity scale height of theorder of the pressure scale height.We proposed two functions to calibrate ML2/ α val-ues in 1D envelopes that best reproduce the 3DSchwarzschild and flux boundaries, respectively, as afunction of T eff and log g . The calibration was performedfrom a direct comparison for closed simulations with shal-low convection zones. For cool white dwarfs with deepconvection zones, the 3D simulations use an open bot-tom boundary condition, and therefore do not includethe lower part of the convection zone. We rely on thefact that below the atmosphere, upflows still evolve underadiabatic conditions. We have extracted the 3D asymp-totic entropy values that correspond to the conditions inthe lower part of the convection zones, which were thenemployed to calibrate ML2/ α of 1D MLT envelopes.We have found that for shallow and inefficient con-vection zones ( T eff & ,
000 K at log g = 8), theMLT parameters for the bottom of the convection zonepoorly reproduce the overall h i convective flux profilethrough the convection zones. Mean 3D stratificationsshould be used for studies that require detailed convec-tive flux profiles. For applications such as chemical diffu-sion and convective mixing, the dominant convective ef-fect is likely to come from the overshoot velocities, whichare completely missing from local MLT envelopes. Theextreme ratio between convective and microscopic diffu-sion timescales prohibits the usage of 3D simulations toprecisely calibrate the deep overshoot layers. Instead,we reintroduce in the context of white dwarfs the ana-lytical overshoot parameterization initially proposed byFreytag et al. (1996), with new constraints based on the3D simulations. The next step will be to apply our cali-brations to non-adiabatic pulsation models as well as spe-cific cases of white dwarfs with convection zones contam-inated by metals accreted from former disrupted planets.Support for this work was provided by NASA throughHubble Fellowship grant alibration of the Mixing-Length Theory for Convective White Dwarf Envelopes15REFERENCES