arXiv:astro-ph/0208584v1 30 Aug 2002 · arXiv:astro-ph/0208584v1 30 Aug 2002...

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Astronomy & Astrophysics manuscript no.(will be inserted by hand later)

Numerical simulations of surface convection in a late M-dwarf

Hans-Gunter Ludwig1,2, France Allard2, and Peter H. Hauschildt3,2

1 Lund Observatory, Box 43, 22100 Lund, Swedene-mail: [email protected]

2 C.R.A.L., Ecole Normale Superieure, 69365 Lyon Cedex 7, France3 Dept. of Physics and Astronomy & Center for Simulational Physics, University of Georgia,Athens, GA 30602-2451

Received date; accepted date

Abstract. Based on detailed 2D and 3D numerical radiation-hydrodynamics (RHD) simulations of time-dependentcompressible convection, we have studied the dynamics and thermal structure of the convective surface layers ofa prototypical late-type M-dwarf (Teff ≈ 2800K, log g = 5.0, solar chemical composition). The RHD modelspredict stellar granulation qualitatively similar to the familiar solar pattern. Quantitatively, the granular cellsshow a convective turn-over time scale of ≈ 100 s, and a horizontal scale of 80 km; the relative intensity contrastof the granular pattern amounts to 1.1%, and root-mean-square vertical velocities reach 240m/s at maximum.Deviations from radiative equilibrium in the higher, formally convectively stable atmospheric layers are found tobe insignificant allowing a reliable modeling of the atmosphere with 1D standard model atmospheres. A mixing-length parameter of αMLT=2.1 provides the best representation of the average thermal structure of the RHDmodel atmosphere while alternative values are found when fitting the asymptotic entropy encountered in deeperlayers of the stellar envelope (αMLT=1.5), or when matching the vertical velocity field (αMLT=3.5). The close cor-respondence between RHD and standard model atmospheres implies that presently existing discrepancies betweenobserved and predicted stellar colors in the M-dwarf regime cannot be traced back to an inadequate treatment ofconvection in the 1D standard models. The RHD models predict a modest extension of the convectively mixedregion beyond the formal Schwarzschild stability boundary which provides hints for the distribution of dust grainsin cooler (brown dwarf) atmospheres.

Key words. convection – hydrodynamics – radiative transfer – stars: atmospheres – stars: late-type

1. Introduction

Late-type M-dwarfs are fully convective stars where theconvective flows penetrate far into the atmospheres reach-ing optical depths as low as 10−3 (Allard & Hauschildt1995). Allard et al. (1997) have reviewed the physical,spectroscopic, and photometric properties of these ob-jects. In the past, model atmospheres have typically failedto reproduce their spectroscopic and photometric proper-ties in two respects: i) the near-IR spectral distribution(JHK colors) where, independent of the source of watervapor line data used, models all agree to predict an under-luminous K-band (relative to J), and ii) the optical MV

vs V − I color-magnitude relation, where all models sys-tematically predict bluer colors (i.e. being overluminousin V ) than observed.

Brett (1995) raised the possibility that this near-IRproblem was due to models being “too cool in the upperphotospheric layers”, and suggested two possible causes:

Send offprint requests to: Hans-Gunter Ludwig

chromospheric heating and/or the treatment of convectionbased on mixing-length theory (MLT).

Hydrodynamical simulations of solar and stellar gran-ulation including a realistic description of radiative trans-fer have become an increasingly powerful and handy in-strument for studying the influence of convective flowson the the structure of late-type stellar atmospheres aswell as on the formation of their spectra (e.g. Nordlund &Dravins 1990; Steffen & Freytag 1991; Ludwig et al. 1994;Freytag et al. 1996; Stein & Nordlund 1998; Asplund et al.2000). Hitherto, model calculations have been exclusivelyperformed for atmospheres where atomic lines are dom-inating the line blanketing. A possible next step in thedevelopment of hydrodynamical models is towards cooleratmospheres where molecular absorption dominates theatmospheric energy balance. Constructing hydrodynami-cal model atmospheres for cooler stars can shed light onthe presently pressing shortcomings of the classical mod-els mentioned above. Regarding the considerable improve-ments in the quality of the molecular opacities and relatedatmospheric models, it becomes more and more important

http://arxiv.org/abs/astro-ph/0208584v1

2 Ludwig et al.: Numerical simulations of surface convection in a late M-dwarf

to determine whether the treatment of convection by MLTis at the origin of the observed discrepancies.

The basic questions we want to answer in this theoret-ical investigation are: Is mixing-length theory adequateto handle convection in the atmospheres of M-dwarfs?And if so, which mixing-length parameter αMLT is nec-essary to reproduce the various thermal and dynamicalproperties of an atmosphere (temperature profile in theline forming region, surface boundary condition connect-ing to stellar evolution models, convective velocities)? Westart by describing some methodological aspects and theapplied computer codes, in particular discuss the criticalquestion of how accurately we can describe the complexradiative transfer within the hydrodynamical simulations.We continue by presenting our results which give some in-sight in what granulation looks like on the surface of anM-dwarf. We proceed with quantitative estimates of themixing-length parameter, and discuss the consequences forconventional atmosphere modeling. Finally, we extrapo-late slightly beyond the existing hydrodynamical modelsproper, and suggest a scenario for the transport of dustgrains in brown dwarf atmospheres due to convective over-shoot which is motivated from our present simulations athotter temperatures. Often we refer to the Sun as ourbenchmark for comparison and assume some familiaritywith its atmospheric properties.

2. Methodological aspects, computer codes

The aim is to model the atmospheric structure of a pro-totypical late M-dwarfs as realistically as possible, witha focus on the interplay of convective flows and radiativetransfer. Being well aware of the limitations in our models,we took, whenever possible, a differential approach in try-ing to reduce the influence of systematic uncertainties onthe outcome of our investigation. This concerned mostlythe dimensionality of the problem: the multi-dimensional,time dependent approach adopted in the hydrodynamicalsimulations versus the one-dimensional, time independentapproach adopted in classical stellar atmospheres. We en-sured that the numerical treatment (i.e. implemented mi-crophysics, representation of radiative energy transport)in the two “worlds” was as similar as possible. We em-ployed various computer codes whose names and maincharacteristics we introduce below. We elaborate on spe-cific aspects critical for the investigation in more detaillater.

RHD: A radiation-hydrodynamics code developed byA. Nordlund and R.F. Stein (see Stein & Nordlund 1998,and references therein) for modeling stellar atmospheresin two or three spatial dimensions. It implements a con-sistent treatment of compressible gas flows together withnon-local radiative energy exchange. The radiative trans-fer is treated in LTE approximation, the wavelength de-pendence of the radiation field is represented by a smallnumber of wavelength bins (see below). Open lower andupper boundaries, as well as periodic lateral boundariesare assumed. The effective temperature of a model (i.e. the

average emergent radiative flux) is controlled indirectly byprescribing the entropy of inflowing material at the lowerboundary. Magnetic fields are neglected.

LHD: A 1D Lagrangian hydrodynamics code devel-oped by one of the authors (HGL) used to calculate stan-dard stellar atmospheres which can be compared withresults obtained with RHD. Besides the reduced dimen-sionality, the adopted physics (equation of state, radiativetransfer scheme) is the same as in RHD. The convectiveenergy transport is based on MLT. In this paper we adoptthe MLT formulation of Mihalas (1978). Excluding oneexception, all values of the mixing-length parameter aregiven with reference to Mihalas’ formulation.

PHOENIX: A 1D model atmosphere code developedby two of the authors (PHH & FA, for a detailed descrip-tion see Hauschildt & Baron 1999). It implements a treat-ment of the wavelength dependence of the radiation fieldwith high resolution based on direct opacity sampling. Inthis investigation PHOENIX served as opacity data base,and was used for assessing the quality of the simplified ra-diative transfer employed in the hydrodynamics codes.

ATLAS6: A version of the 1D model atmosphere codedeveloped by R. Kurucz (Kurucz 1979). It served as addi-tional opacity data base.

Table 1 summarizes the properties of the hydrodynam-ical models discussed in the paper. Model A-3D is ourM-dwarf reference model. The twice as large model B-3Dwas primarily calculated for checking effects of the do-main size, the solar model S-3D was added for assessingscaling properties with effective temperature and gravita-tional acceleration. We note that the solar model is notstrictly differentially comparable to the M-dwarf modelssince it is employing different opacity sources and equationof state which stem from the Uppsala stellar atmospherepackage (Gustafsson et al. 1975). We do not consider thisas particularly critical since the physical conditions in theatmospheres of M-dwarfs and the Sun are so different thatone looses the advantages of a differential approach any-way. The 2D models C-2D and D-2D are models whichwere considered in the forefield to investigate effects of thenumerical resolution. They are based on ATLAS6 opac-ities without contributions of molecular lines and employgrey radiative transfer. Due to the different input physicstheir behavior is qualitatively different from the more re-alistic 3D models. Despite their shortcomings, they areof interest for qualitatively understanding the interplayof convection and radiative transfer in optically thin re-gions, and therefore will be discussed in more detail. Asis apparent from the table, all hydrodynamical M-dwarfmodels show very small fluctuations around their aver-age effective temperature. This reflects the fact that thehorizontal and temporal fluctuations of all quantities aresmall compared to the Sun — a central feature of the at-mospheres of late-type M-dwarfs.

Ludwig et al.: Numerical simulations of surface convection in a late M-dwarf 3

Table 1. The hydrodynamical models discussed in the paper: Name is used to identify a model, Dim. the dimen-sionality of the model, Mesh the number of grid points (XxYxZ, where Z denotes the vertical, X and Y the horizontaldirections), Size the geometrical size of the computational domain, Opacities the employed opacity source, NOBM

the number of OBM bins (“Opacity Binning Method”, for a description see Sect. 2.2) for calculating the radiativetransfer, Teff the resulting effective temperature of the model including an estimate of its RMS fluctuations, log g thegravitational acceleration, and Modelcode an internal model identifier.

Name Dim. Mesh Size [km] Opacities NOBM Teff [K] log g Modelcode

A-3D 3 125x125x82 250x250x87 PHOENIX 4 2789.1 ± 0.7 5.0 d3gt30g50n18B-3D 3 250x250x82 500x500x116 PHOENIX 4 2799.8 ± 0.2 5.0 d3gt30g50n19S-3D 3 125x125x82 6.0x6.0x3.2 (Mm) Uppsala 4 5640 ± 14 4.44 sun3dC-2D 2 125x82 250x80.3 ATLAS6 1 2774 ± 1.6 5.0 d2gt30g50n8D-2D 2 251x162 250x80.3 ATLAS6 1 2770 ± 1.6 5.0 d2gt30g50n9

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Fig. 1. Comparison of the specific heat at constant pres-sure from various equations of state in a representativeM-dwarf: RHD (solid), and PHOENIX (dash-dotted).If one artificially removes the contribution of H2 molecu-lar formation cp is largely reduced (dotted). For furthercomparison the run of cp in a solar model atmosphereis depicted (dashed). For the M-dwarf model a logarith-mic Rosseland optical depth scale is indicated by the tickmarks near the abscissa.

2.1. The equation of state

Figure 1 shows that the equation of state (EOS) whichis employed in the hydrodynamical simulations and LHD

models is very similar to the PHOENIX EOS. We do notexpect significant systematic differences by applying thesetwo different equations of state in our various model cal-culations. The equations of state of RHD and PHOENIX

treat ionization and molecular formation assuming Saha-Boltzmann statistics. Since non-ideal effects are not pro-nounced in atmospheres of M-dwarfs the inclusion of H2

molecular formation is the main ingredient required to ob-tain a realistic description of the thermodynamics of thestellar plasma.

Indeed, Fig. 1 demonstrates that in the M-dwarf at-mosphere H2 molecular formation is the most importantcontributor for increasing the specific heat above the value

associated with the purely translatorial degrees of free-dom. In the Sun the dominant contributor is the hydrogenionization which has an even more dramatic effect on theheat capacity of the stellar plasma. I.e. from the perspec-tive of the content of latent heat in the gas flows M-dwarfatmospheres are not particularly extreme objects.

2.2. Radiative transfer

An important problem when modeling M-dwarf envelopesis the treatment of the large number of absorption linesin their atmospheres which are mostly of molecular origin.The complex wavelength dependence of the radiation fieldis illustrated in Fig. 2. While it is already a formidable taskto treat the frequency dependence of the radiation fieldin 1D model atmospheres, this is even more the case inhydrodynamical models where one has to account for the3D geometry of the flow field and its temporal evolution.Present computer capacity allows only for a very modestnumber of frequency points to be included the modelingof the radiation field within a hydrodynamical simulation.However, the situation is somewhat alleviated. For theinteraction of hydrodynamics and radiative transfer onlythe frequency integrated net amount of radiative heating(or cooling if negative)

Qrad = 4πρ

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(ρ denotes the density, χν the monochromatic absorp-tion coefficient, Jν the angular average of the intensity,Sν the source function) is of relevance. I.e. we do notneed to know the detailed frequency dependence of theradiative heating. Perhaps surprisingly, one can obtaina reasonably accurate description of the radiative heat-ing with very few frequency points by a judicious choiceof the frequency discretization: frequencies with a simi-lar run of monochromatic optical depth are grouped to-gether into opacity “bins”. In the context of hydrody-namical stellar atmospheres this Opacity Binning Method(OBM) was originally introduced by Nordlund (1982), andsome variants and refinements were subsequently devel-oped (Ludwig et al. 1994). The basic idea is to group fre-


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Fig. 2. Scatter plot of standard optical depth where themonochromatic optical depth reaches unity as a func-tion of wavelength in a PHOENIX model atmospherewith Teff=2900K, log g=5.3, and solar chemical compo-sition. Standard optical depth is the optical depth scalein the continuum plus opacity contributions of H2 andH2O molecular lines at λ = 1.2µm. The horizontal linesmark the thresholds which were used for defining the OBMopacity bins (see text). The curve in the lower part of thepanel is the emergent flux (arbitrarily scaled) of the model.

quencies which reach monochromatic optical depth unitywithin a certain depth range of the atmosphere into onefrequency bin. Effectively, this combines frequencies intothe same bin which belong to the continuum, weaker, andstronger spectral lines. The opacity in each bin is an aver-age (Rosseland or Planck) over the frequency range whichis represented by that bin. With these average opacities,supplemented by the frequency integrated source function,the equation of radiative transfer has to be solved onlyonce per bin to obtain the formal solution of the radiativetransfer for a given flow field. The binning procedure is il-lustrated in Fig. 2 where the horizontal lines separate thedepth ranges — as measured by a frequency independentstandard optical depth — which are sorted into the vari-ous OBM bins. In the present investigation we used 4 binsin our non-grey hydrodynamical models, representing thecontinuum and line wings, as well as the cores of weak,medium strong, and strong spectral lines.

It is clear that the sorting procedure described before isspecific to the stellar atmosphere under consideration andhas to be repeated as soon as the atmospheric parametersdiffer widely from the one where the sorting was done.We used a model atmosphere calculated with PHOENIX

at Teff = 2900K and log g = 5.3 as reference atmospherefor the grouping. It is sufficiently close to the atmosphericparameters of the hydrodynamical models. This has beenchecked by studying the performance of the sorting whenapplied to differing atmospheric parameters (log g downto 3.0, and Teff up to 3300K).

The sorting criterion that τλ = 1 should fall withina certain depth range in the atmosphere does not guar-

antee that the overall depth dependence of τλ is similarfor all frequencies grouped together — a condition forallowing the interchange of the solution of the transferequation with the frequency integration when evaluatingQrad. In particular, the simultaneous presence of atomicand molecular lines can lead to significantly different func-tional forms of the monochromatic optical depth at differ-ent frequencies: deeper regions of the atmosphere too hotto allow for molecule formation might be dominated byatomic lines while higher and cooler regions which allowfor molecule formation might be dominated by molecularlines. If the atomic and molecular lines emerge from differ-ent elements there is no physical connection between themleading to an uncorrelated behavior in optical depth ofdeeper and higher layers with frequency. Such a situationwould be unfavorable for the OBM. However, the OBMis rather successful in reproducing the heat exchange be-tween radiation and matter — as evident from Fig. 3. Thisis linked to the statistical dominance of molecular absorp-tion in all radiative layers of the rather cool atmosphereunder consideration.

Another point concerning the present implementationof the OBM is our usage of global Rosseland means — i.e.Rosseland averages over the whole frequency range — forrepresenting the average opacity in each bin. For the thecontinuum we took the Rosseland means themselves whilewe scaled them by factors of 101, 102 , and 103 for thebins representing the successively stronger lines. The scal-ing factors correspond to the factors of 10 in optical depthselected as thresholds for the sorting procedure which arelog10 τstandard = −0.5,−1.5, and − 2.5 (see Fig. 2). Thebasic assumption behind this approach is that the lineopacity scales with temperature and pressure like the con-tinuous opacity. The source function has been integratedover the frequency ranges of the individual bins. While itis certainly not the optimal representation of the opaci-ties it was dictated by the lack of a detailed tabulationof the monochromatic opacities over the full temperature-pressure range of interest. Work is presently under way togenerate such tabulation which is a non-trivial task due tothe enormous amount of line data which need to be pro-cessed. Considering the various approximations describedbefore one might ask whether the OBM is a real improve-ment beyond a grey description. Eventually, this can betested quantitatively by comparing 1D atmospheres com-puted with high frequency resolution or the OBM. Sucha comparison serves as ultimate indicator of the perfor-mance of the OBM.

Figure 3 shows a comparison of 1D model atmospherescalculated with PHOENIX employing opacity samplingwith a wavelength resolution of ≈ 2 A, and LHD employ-ing the OBM as described before. Comparing flux con-stant models in radiative and radiative-convective equilib-rium shows a good correspondence of the resulting equi-librium temperature profiles. While the radiative equilib-rium models are less important for the investigation of M-dwarf atmospheres, they were added to the comparisonto show the similarity in the radiative transport proper-


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Fig. 3. Comparison of 1D standard model atmospheres(Teff=2800K, log g=5.0) in purely radiative (dashedlines) and radiative-convective (αMLT = 1.0, solid anddotted lines) equilibrium. The thick lines are based onLHD atmospheres using the approximate OBM radiativetransfer, the thin lines are based on PHOENIX modelsusing opacity sampling with high wavelength resolution.The dotted line depicts an LHD model employing greyradiative transfer. A logarithmic Rosseland optical depthscale is indicated by the tick marks near the abscissa.

ties independent of influences of the convective transport.More important are the models in radiative-convectiveequilibrium since they are closer to the actual physicalsituation. For judging the correspondence between opac-ity sampling and OBM models one should take the greymodel as benchmark which represents a strongly simplified(equivalent to one frequency bin) description of radiativetransfer. Clearly, the OBMmatches much more closely thePHOENIX opacity sampling model. The cooling of thehigher atmosphere by lines is reasonably represented aswell as the backwarming of the deeper layers.

The most important deviation between opacity sam-pling and OBM model occurs in the layers where the tran-sition from convectively to radiatively dominated energytransport takes place (around logP = 5.3). The OBMstratification becomes noticeably cooler. This was tracedback to an insufficient heating of the gas in the OBM con-tinuum bin. This in turn is probably related to the useof Rosseland averages for the continuum opacity in theseoptically thin regions. Planck averages would be more suit-able. But again, at the time this work was performed onlyglobally averaged opacities were available. Rosseland andPlanck averages differ by at least a factor of 100 in theseregions, making an ad-hoc switching from one to the otherproblematic. However, we consider the remaining devia-tions not as vital, in particular with respect to the cali-bration of the mixing-length parameter which we describelater in this paper. The calibration is a result of a differ-ential comparison of models which all base on the OBM.

For completeness, we finally remark that only theexchange of energy was considered within the radiative

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Fig. 4. Various characteristic time scales as a functionof pressure in an M-dwarf model: Brunt-Vaisala period(solid), Kelvin-Helmholtz time scale (dashed), radia-tive relaxation time based on ATLAS6 Rosseland opaci-ties (dotted), PHOENIX Rosseland (dashed-dotted),and PHOENIX Planck (triple-dot-dashed) opacities.A logarithmic Rosseland optical depth scale is indicatedby the tick marks near the abscissa. The kink in the runof the Brunt-Vaisala period is located at the boundarybetween the convectively stable and unstable part of themodel.

transfer. The exchange of momentum was neglected; theprevailing relatively high mass densities combined withlow radiative fluxes render radiation pressure unimportantfor the structure of M-dwarf atmospheres.

3. Characteristic time scales

To get an overview of the problem Fig. 4 shows var-ious characteristic time scales in a representative M-dwarf model (LHD model with Teff=2790K, log g=5.0,αMLT=1.0). The expressions which we applied in the cal-culation of the time scales are summarized in the ap-pendix. The time scales have been evaluated under simpli-fying assumptions, and should therefore be taken as orderof magnitude estimates only.

The radiative time scales based on ATLAS6 andPHOENIXRosseland mean opacities are rather similar inthe deeper layers while being very different in the opticallythin regime. This emphasizes the great influence of molec-ular absorption which becomes important at cooler tem-peratures and which is not included in theATLAS6 opaci-ties. The radiative time scales from PHOENIX Rosselandand Planck opacities differ also significantly. Comparingthe radiative time scales to a dynamical or convective timescale as given by the Brunt-Vaisala period leads us to ex-pect that one gets a qualitatively different behavior de-pending on the treatment of the radiative transfer. Wewill see that models based on a frequency-independentRosseland opacities show significant deviations from ra-diative equilibrium conditions. A more realistic treatment— in the optically thin regions more closely represented


by the Planck mean opacities — results in an atmosphericstructure closer to radiative equilibrium. In all cases, oneexpects an almost adiabatic structure in the deeper layerssince the radiative relaxation time is much longer than theconvective time scale.

3.1. A remark on thermal relaxation

RHD uses an explicit numerical scheme for advancingthe solution in time. It is well known that in an explicitscheme the time step is limited to a fraction of the dy-namical time scale, more precisely to the limit given bythe Courant-Friedrichs-Levy stability condition. Togetherwith the amount of available computer — and wallclock —time, this limits the time interval which can be covered bya 3D RHD model to about 104 s of stellar time. Figure 4seemingly implies that it is impossible to obtain a ther-mally relaxed hydrodynamical model within this time in-terval since the Kelvin-Helmholtz time scale of the deeperlayers is at least an order of magnitude larger. Contraryto the impression given in Fig. 4, in the multi-dimensionalhydrodynamical simulations this does not pose a problemsince the thermal relaxation of the model is not governedby the time scale for the exchange of energy as expressedby the Kelvin-Helmholtz time. In fact, the thermal evolu-tion of the deeper layers of the RHD models is governedby the time scale for the exchange of mass in these layerswhich is much shorter (as quantified below, see Fig. 16).Due to the exponential run in density of the atmospherethe mass exchange consists primarily of the replacementof mass by fresh material stemming from deeper layers.In the hydrodynamical model it is ultimately fed into thecomputational domain at the lower boundary. The con-vective energy flux is the net effect of the energy trans-ported by counteracting, opposing mass currents in andout of a test volume. The advective — as opposed to dif-fusive — nature of the mass transport leads to a situationwhere there can be a large imbalance in the energy con-tent of the mass currents: the in-coming mass elementscarry the energy (strictly speaking the specific entropy)of much deeper layers (in the hydrodynamical model theentropy at the lower boundary), while the out-going el-ements just carry the local energy density. This impliesan energy flux much larger than the one close to equi-librium conditions. The system is driven much faster to-wards equilibrium than implied by the classical Kelvin-Helmholtz time scale, which assumes an energy flux asencountered close to equilibrium conditions.

We have argued from the perspective of our hydrody-namical models. But the fast thermal relaxation is drivenby a physical mechanism implying that a real stellar con-vection zone does not operate any differently. Hence, wewant to stress that to our understanding the thermal re-laxation of a convective layer is generally not governedby the classical Kelvin-Helmholtz time scale but by theusually shorter time scale of mass exchange.

In the one-dimensional LHDmodels no mass exchangetakes place and the convective energy flux is computedfrom MLT, in which convection is modeled as diffusionprocess. Under such circumstances the time scale of thethermal evolution of the convective layers is indeed com-parable to the Kelvin-Helmholtz time scale as shown inFig. 4. Of course, covering the time interval necessary forthe slower thermal relaxation in a 1D model run poses noproblems due to the largely reduced computational costs.

4. 2D precursor models

Models C-2D and D-2D are 2D models (see Tab. 1) whichwere constructed to gain experience regarding the sizeof the computational domain and grid resolution. Theyare based on grey radiative transfer utilizing ATLAS6

opacities which do not include contributions of molecularlines. The choice of the ATLAS6 opacities was dictatedby the lack of more realistic opacities during the initialphase of the project. From Fig. 4 one might readily con-clude that their atmospheric structure will be dominatedby convection since the radiative relaxation times in theatmosphere are long as compared to the dynamical timescale. Indeed, when starting from a temperature structuretaken from a LHD model based on MLT, we find a rapidcooling of the originally radiatively stratified atmosphericlayers by convective overshooting (see Fig. 5). Convectiontends to transform the stratification into a purely adia-batic one since the radiative heating is too weak to keepthe temperature close to the radiative equilibrium value.The transformation into an almost adiabatic stratificationtakes too long to be covered within the multi-dimensionalhydrodynamical simulations. However, we conducted nu-merical experiments with LHD where an ad-hoc velocityfield mimicking the convective overshoot was put into theatmospheric regions which are formally stable accordingto the Schwarzschild criterion. The LHD models indicatethat the asymptotic temperature profile tends to a fullyadiabatic stratification.

Due to the unrealistic opacities, the models cannotgive a good representation of a real M-dwarf atmosphere.They are nevertheless interesting from a numerical pointof view. The resolution of the less resolved model C-2Dis already sufficient to represent the convective transportproperties, in particular in the important transition re-gion from convectively to radiatively dominated energytransport: after an initial relaxation phase we find in bothmodels that the minimum temperature drops linearly witha rate of dT

dt = −0.011K/s. Looking at further diagnosticsat comparable instances during the secular evolution ofthe model runs we observe that RMS velocities are simi-lar within a 20% level. The higher resolved model showssomewhat more small-scale features, and its downflowsappear more concentrated. However, we are primarily in-terested in the balance of convective to radiative energytransport. The temperature drop rate is a convenient mea-sure of the relative efficiency of both processes. Hence, weconclude from the similarity of the drop rates that the


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Fig. 5. Temporal evolution of the average temperatureprofile of model C-2D in steps of 5 ks showing a suc-cessive cooling of the layers around logP = 5.5 (solidlines). For comparison a LHD model with αMLT = 1.0is plotted (dash-dotted). All models are based on greyradiative transfer employing ATLAS6 opacities devoid ofmolecular contributions. All models have the same atmo-spheric parameters (Teff=2770K, log g=5.0). A logarith-mic Rosseland optical depth scale is indicated by the tickmarks near the abscissa.

resolution of the less resolved model applied in 3D sim-ulations is sufficient to model the transport properties ofthe convective flows.

Another conclusion which can be drawn from thesemodels is that standard MLT models can give quite mis-leading predictions of the atmospheric temperature struc-ture if the radiative relaxation time is long in comparisonto convective time scales. In other words, one should becautious when the coupling of the temperature structureto the radiative equilibrium temperature is so weak thatovershoot of low amplitude happens essentially adiabat-ically. In the following we shall see that M-dwarf atmo-spheres are unlikely to exhibit such conditions.

5. 3D models: results and discussion

5.1. General flow structure

Figure 6 shows a typical snapshot of the emergent inten-sity during the temporal evolution of model B-3D. Forcomparison, Fig. 7 shows a similar snapshot from the so-lar run S-3D. Note, that the spatial and intensity scalingis very different in the reproductions, hence, only rela-tive geometrical properties should be directly compared.The average relative RMS intensity contrast of the gran-ular pattern amounts to 1.1% in the M-dwarf as opposedto 16% in the solar case. The first thing to note is thatsurface convection in an M-dwarf produces a granular pat-tern qualitatively resembling solar granulation: bright ex-tended regions of upwelling material which are surroundedby dark concentrated lanes of downflowing material. Thedark lanes form an interconnected network. Looking more

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Fig. 6. Typical snapshot of emergent intensity during theevolution of model B-3D. The intensity contrast amountsto δIrms = 1.2% at this instant in time. Note, that the con-trast of the intensity pattern in the graphics is enhancedwith respect to its real appearance.

closely, granules are less regularly delineated in M-dwarfs,the inter-granular lanes show a higher degree of variabilityin terms of their strength. A feature which is uncommonin the solar granulation pattern are the dark “knots” (e.g.at x = 100km and y = 230km in Fig. 6) found in orattached to the inter-granular lanes. The knots are associ-ated with strong downdrafts which carry a significant ver-tical component of angular momentum. The width of theinter-granular lanes to the typical granular size is smallerin M-dwarfs. Inspecting the velocity field (not shown) invicinity of the continuum forming layers shows less pro-nounced size differences. This indicates that the relativelybroader lanes in the solar case are the result of a strongersmoothing of the temperature field due to a more intenseradiative energy exchange, i.e. the effective Peclet num-ber of the flow is larger around optical depth unity inM-dwarfs.

The different magnitude of the intensity contrast al-ready indicates that horizontal fluctuations of the ther-modynamic quantities are small in M-dwarf atmospheres.Figure 8 shows the run of the relative temperature andpressure fluctuations in model A-3D. Plotted are long termtemporal and horizontal 1 RMS averages. As we will arguelater, the fluctuations in the higher atmosphere are likelyto be overestimated in the model. But even taken at facevalue they are quite modest. The low level of fluctuations

1 The horizontal averaging is performed on the geometricaldepth scale. Due to the small horizontal fluctuations, alterna-tive choices (e.g. on the optical depth scale) give similar results.


0 1 2 3 4 5x [Mm]

0

1

2

3

4

5

y [M

m]

Fig. 7. Like Fig. 6, but for the solar model S-3D. The in-tensity contrast amounts to δIrms = 16.7% at this instantin time. Note, that for reproduction purposes spatial aswell as intensity scale are altered in comparison to Fig. 6.

in the thermodynamic quantities is accompanied by smallflow velocities, the maximum Mach number amounts to6.5% in model A-3D. MLT models show that Mach num-bers drop to even lower values as one goes to lower effec-tive temperatures which are encountered in the regime ofbrown dwarfs. Our findings may have a direct bearing onthe dust formation conditions in such objects. Our hydro-dynamical simulations support the view that dust form-ing layers in cool main sequence objects experience onlysmall variations around their mean thermodynamic state.This view is clearly at odds with the scenario discussedby Helling et al. (2001) who study the dust formation inturbulent brown dwarf atmospheres. Helling et al. assumethermodynamic fluctuations of order unity.

Another important feature, which distinguishes M-dwarf atmospheric conditions from solar ones, is the ex-tent of the convective layers. As evident from Figures 11and 12, the convective motions reach much lower opticaldepth (log τRosseland ≈ −1.5) in the M-dwarf than in theSun (log τRosseland ≈ 0). Two factors contribute to thisbehavior. The temperature gradient in non-grey radiativeequilibrium is significantly steeper in the M-dwarf thanin the Sun, presumably due to efficient cooling of the at-mospheric layers by molecular lines. Furthermore, the H2

molecule formation reduces the adiabatic gradient in theM-dwarf atmosphere, while in the Sun the hydrogen re-combination is essentially completed in subphotosphericlayers. The different radiative and thermodynamic condi-tions favor the presence of convection in M-dwarf atmo-spheres.

3 4 5 6 7 8log10 Pgas [dyn/cm2]

0.00

0.01

0.02

0.03

0.04

δTrm

s , δ

Prm

s

3 2 1 0-1-2-3

Fig. 8. Relative horizontal temperature (solid) and pres-sure (dashed) fluctuations of model A-3D. The fluctua-tions in the region logP < 5 are likely to be overestimatedin the model, i.e. should be considered as upper limits.The dotted line indicates the approximate location of theSchwarzschild boundary. A logarithmic Rosseland opticaldepth scale is indicated by the tick marks near the ab-scissa.

In the deeper layers we observe the tendency — alsoknown from solar simulations (see Stein & Nordlund 1998)— that the granular network of downflows decays into iso-lated downdrafts. Our models are rather shallow reachingonly 2.3 pressure scale heights below optical depth unity.Thus we cannot follow the change of flow topology as faras has been done for solar models, but within the limiteddepth range comprised by our models we do not see in-dications of a qualitatively different behavior as found inthe Sun.

5.2. Horizontal scales

In the following we shall discuss spatial power spectra ofintensity and vertical velocity. Note, that in the Figures 9and 10 we display power per logarithmic wavenumber in-terval and not per unit wavenumber — the more commonchoice. This allows visually for a more direct identificationof the power carrying scales. However (in humble respectof Kolmogorov’s achievements), we labeled the power lawsdrawn for comparison with the familiar spectral index ofpower per unit wavenumber. The spectra are temporal av-erages over one to a few convective turn-over times andmany convective cells, so that they are statistically repre-sentative.

Figure 9 shows a comparison of power spectra of theemergent intensity (more precisely: the intensity in theOBM continuum bin in vertical direction at the upperboundary of the computational volume) for the models A-3D, and B-3D as well as the solar model. Figure 10 showsa corresponding comparison of the vertical velocity com-ponent measured at the layer where its RMS value reachesthe maximum. To facilitate a comparison between the M-


0.01 0.10 1.00 10.00Horizontal wavenumber [rad/km]

10-13

10-12

10-11

10-10

10-9

10-8

d(P

)/dl

n(k)

[(in

tens

ity)2 ]

-5/3

-17/3

Fig. 9. Power spectrum of the emergent intensity patternfor model A-3D (solid), and B-3D (dashed), as well asthe solar model S-3D (dotted). The spectrum of the so-lar model was scaled in power as well as wavenumber tomatch the peak of power in the M-dwarf models. Lineswith slopes dP

dk ∝ k−5/3 and dPdk ∝ k−17/3 are shown for

comparison with expectations from simplistic turbulencetheory. The total relative intensity contrasts amount toδIrms = 1.12% (A-3D), 1.12% (B-3D), and 16.2% (S-3D).

0.01 0.10 1.00 10.00Horizontal wavenumber [rad/km]

101

102

103

104

105

d(P

)/dl

n(k)

[(m

/s)2 ] -5/3

-17/3

Fig. 10. Power spectrum of the vertical velocity pattern,line styles as in Fig. 9. The solar power spectrum wasscaled to match the peak of power of the M-dwarf mod-els. The velocity pattern was taken from the layer wherethe RMS vertical velocity reaches its maximum, with abso-lute values of vrms,max = 236m/s (A-3D), 241m/s (B-3D),2600m/s (S-3D).

dwarf models and the solar model, the solar model wasarbitrarily scaled in power and wavenumber so that themaxima and position of the power distributions matched.For the M-dwarf models the hump in power at the highestwavenumbers is likely an artifact of an imperfect choiceof parameters controlling the small scale dissipation andshould be ignored.

Fluctuations in the Sun are an order of magnitudelarger and had to be reduced accordingly in order to match

the height of the maxima in the M-dwarf models, while thehorizontal scale of the solar model had to be shrunk by afactor of 15. This factor closely corresponds to variation ofthe pressure scale height at the surface (factor 12.5) andfollows the scaling found for hotter models by Freytaget al. (1997). Different from Freytag et al., we accountedfor the change of the mean molecular weight of the stellargas since the H2 molecular formation leads to a significantincrease in M-dwarf atmospheres. The peak power of theM-dwarf models is located at an absolute scale of 80 kmfor intensity and 63 km for velocity structures.

Comparing the width of the power distribution at 1 dexbelow peak level, the M-dwarf model B-3D displays arange of scales of 1.4 dex, the solar model of 1.1 dex. Asis evident from the figures, the solar scale distribution isslightly but noticeably narrower. This might be tracedback to the lower Peclet number of the plasma encoun-tered at the surface of an M-dwarfs as opposed to solarconditions. The formation of small scale structures is morestrongly suppressed by the radiation field under solar con-ditions, leading to an overall narrower power distribution.Model A-3D apparently lacks some larger scale power indi-cating that the computational box is somewhat too smallin this case. However, since the total fluctuations are verysimilar in the models we conclude that the relevant struc-tures are captured in model A-3D which serves as ourstandard model.

Towards high wavenumbers the M-dwarf spectra donot show a clear power law behavior. In any case, theycannot be fitted with a −5/3 slope, and are at best roughlycompatible with a −17/3 slope. We would like to empha-size that despite the very different physical parameters ofM-dwarf atmospheres this is qualitatively what was al-ready found in solar models, and is at odds with expec-tations for homogeneous and isotropic turbulence. Whilesmaller scales in the hydrodynamical models are definitelyinfluenced by the artificial viscosity, it is still somewhatsurprising that the asymptotic behavior in the Sun as wellas in the M-dwarf is so similar despite their quite differentcharacteristic flow speeds (i.e. Mach numbers). While ourpresent models cannot make definite statements about theproperties of small scale turbulence, a “non-Kolmogorov”behavior cannot be ruled out neither. See Nordlund et al.(1997) for an in-depth discussion of this issue in a solarcontext.

5.3. Mean temperature stratification and

corresponding effective mixing-length parameter

Figures 11, 12, and 13 show a comparison of the ther-mal structure of our “best” hydrodynamical model A-3D with standard 1D atmosphere models computed withLHD. Again, we point out that the treatment of the mi-crophysics (equation of state and opacities) is equivalentbetween RHD and LHD models. Differences lie mainlyin the dimensionality of the models — 3D versus 1D —and the related MLT treatment of the convective energy


3 4 5 6 7 8log10 Pgas [dyn/cm2]

1500

2000

2500

3000

3500

4000

4500

Tem

pera

ture

[K]

3 2 1 0-1-2-3

Fig. 11. Overview of the temperature-pressure relation ofhydrodynamical model A-3D (solid) and three standardmixing-length model atmospheres (dash-dotted) withαMLT = 1.0, 1.5, 2.0. The αMLT = 1.0 model is the hottestat high pressure while it is coolest around logP = 5. Alogarithmic Rosseland optical depth scale is indicated bythe tick marks near the abscissa. All models have identicalatmospheric parameters (Teff = 2790K, log g = 5.0, solarchemical composition). The dotted line indicates the lo-cation of the Schwarzschild boundary in the αMLT = 1.0model. See Fig. 12 for an enlargement of the transitionregion from convective to radiative transport.

4.4 4.6 4.8 5.0 5.2 5.4 5.6log10 Pgas [dyn/cm2]

2000

2200

2400

2600

2800

Tem

pera

ture

[K]

-1-2

Fig. 12. Temperature-pressure relations like in Fig. 11 butfocusing on the transition region from convective to ra-diative energy transport. In this region the αMLT = 1.0model is the coolest, αMLT = 2.0 the hottest mixing-lengthmodel.

transport in LHD. In LHD effects of the turbulent pres-sure due to convective velocities computed from MLT areneglected. This approximation is well justified since theturbulent pressure — as found in the RHD simulations —nowhere exceeds 0.5% of the gas pressure.

For the hydrodynamical model we have plotted in factsix different mean stratifications in the afore mentionedfigures, each one a horizontal and temporal average over a250 s time interval — comparable to the turn-over time of

4 5 6 7 8log10 Pgas [dyn/cm2]

8.80

8.90

9.00

9.10

9.20

9.30

9.40

Spe

cific

Ent

ropy

[108 e

rg/g

/K]

Fig. 13. Entropy-pressure relation of hydrodynami-cal model A-3D (solid) and three standard mixing-length model atmospheres (dash-dotted) with αMLT =1.0, 1.5, 2.0. The αMLT = 1.0 model has the largest entropyat high pressure while it has the lowest around logP = 5.The entropy level of material entering the computationalbox of hydrodynamical model A-3D at the lower boundaryis indicated by the dashed line. All models have identicalatmospheric parameters (Teff = 2790K, log g = 5.0, solarchemical composition).

the convective cells. The statistical variations are so smallthat the profiles are virtually indistinguishable on the scaleof the plots demonstrating again that we are dealing withatmospheres showing little temporal and horizontal fluc-tuations. Fig. 11 shows that the mean thermal structure ofthe hydrodynamical model corresponds closely to those ofmixing-length models. The mixing-length models them-selves do not show a strong dependence on the mixing-length. In the deeper, convective layers small horizontalentropy fluctuations are sufficient to carry the convectiveflux, and the radiative energy transport is unimportant.Here the stratification has to follow essentially an adiabatin the pressure-temperature plane. The same reasoningholds for the hydrodynamical as well as MLT models. Thesituation is somewhat different in the radiative layers. Byconstruction, the mixing-length models have to approach atemperature profile corresponding to radiative equilibriumconditions. Since the differences among the mixing-lengthmodels in the convective region are small the radiativeequilibrium profiles are almost identical as well. In thehydrodynamical model the temperature of the higher at-mospheric layers (logP < 5.3) is controlled by a competi-tion of radiative heating and adiabatic cooling of materialwhich overshoots into regions with formally stable entropygradient ds

dP < 0 (see Fig. 13), and, analogously, the ra-diative cooling and adiabatic heating of downflowing ma-terial. The relative time scales involved in adiabatic andradiative temperature changes determine how the temper-ature will adjust between the upper extreme — the radia-tive equilibrium temperature — and the lower extreme —


the temperature achieved during adiabatic expansion of arising mass element in the higher atmosphere.

Obviously, in the present case, radiative processesdominate and the hydrodynamical model closely followsthe radiative equilibrium temperature. As we have seen inthe context of the hydrodynamical models adopting greyATLAS6 opacities this is not necessarily so. In fact, wetrace back the short radiative time scales (see Fig. 4) tothe presence of molecular absorption which provides theclose coupling of the stratification to the radiative equi-librium temperature. We speculate here that the situationmight change significantly when one goes to metal poorobjects. Convective velocities are likely not to be reducedas dramatically as the atmospheric opacities, shifting therelative importance towards adiabatic cooling. This mightlead to a significant deviation from radiative equilibriumin the atmospheric layers of such objects. Indeed, for metalpoor dwarfs at about solar effective temperatures such ef-fects are found in hydrodynamical models (e.g. Asplund& Garcia Perez 2001).

Quantitatively, there are small differences in the ther-mal structure between the hydrodynamical model and themixing-length models. As one can expect the differencesare most pronounced in the transition region betweenconvectively and radiatively dominated energy transportwhere the detailed balance between both processes decidesabout the resulting temperature profile. Figures 12 and 13show that the hydrodynamical model has a mean ther-mal structure which is close to the profiles of the mixing-length models but cannot be matched exactly. This cer-tainly comes not as a big surprise when one considers thesimplistic nature of mixing-length theory. However, cer-tain aspects of the hydrodynamical model can be matchedby choosing a mixing-length model from the set which areparameterized by αMLT. This is most easily done consid-ering the entropy since it emphasizes model differences(see Fig. 13). The entropy of inflowing material of the hy-drodynamical model which we interpret as the entropy ofmaterial located in the deep almost perfectly adiabaticallystratified part of the stellar envelope is matched best bya mixing-length model of αMLT(evo) = 1.5. In global stel-lar structure models the entropy of the envelope controlsto some extent the radius of a star, making αMLT(evo)most relevant for evolutionary models — hence, our nam-ing “evo”. For further justification and discussion of ourinterpretation see Steffen (1993) and Ludwig et al. (1999).

If one wants to match the entropy jump of the hy-drodynamical model αMLT(∆s) = 2.1 provides the closestfit. The entropy jump is the entropy difference betweenthe atmospheric entropy minimum and the asymptoticentropy of the deeper layers. The entropy jump providesa qualitative measure of the overall available convectivedriving, and in the present case also gives a match tothe temperature gradient of the deeper photosphere withlog10 τross > −1. The internal uncertainties in the deter-mination of αMLT(evo) and αMLT(∆s) = 2.1 are small(≈ ±0.05) as estimated from the statistical fluctuationsobserved in the models. However, in the present M-dwarf

model small deviations from equilibrium conditions deter-mine the mixing-length parameters. This is a challengingsituation for any numerical scheme, and we consider itpossible that the actual calibration errors are dominatedby related systematic uncertainties.

5.3.1. Importance of the MLT formulation

Throughout this paper mixing-length parameters aregiven with respect to the formulation of MLT accordingMihalas (1978) (see Ludwig et al. (1999) for details of theimplementation). The formulation of Mihalas is widelyused, in particular in the context of stellar atmospherework. But it is by far not the only MLT “dialect” around.Another widely used formulation is the original one byBohm-Vitense (1958) which is often encountered in thecontext of stellar evolution work. We would like to em-phasize that generally the details in the MLT implemen-tation have a direct and significant influence on the cal-ibration of the mixing-length parameter. This is particu-larly true in the present case where convection penetratesfar into optically thin regions. Typically it is the treat-ment of the radiative energy exchange of the convectiveelements in the optically thin limit that distinguishes thevarious MLT formulations. For enabling inter-comparisonin other context, we like to mention that the MLT formu-lation in PHOENIX is similar but slightly different fromthe Mihalas formulation. However, test calculations haveshown that numerical differences are negligible in the M-dwarf regime. Furthermore, in the ATLAS6 stellar atmo-sphere code the Mihalas formulation is implemented. MLTformulations as given by Kippenhahn & Weigert (1991)and Stix (1989) are identical to the Bohm-Vitense formu-lation.

Figure 14 shows a comparison of the entropy profilesof standard mixing-length models of the same αMLT = 1.5based on the the Mihalas and Bohm-Vitense formulation.We concentrate on the entropy jump since it is sensitiveto the convective transport in the optically thick and thinlayers. The asymptotic entropy level of the deeper layers islargely controlled by the entropy change over the opticallythick layers only. Mihalas and Bohm-Vitense formulationdiffer little here, implying the rather similar outcome. Theformulations differ significantly in the optically thin layers.Clearly, the entropy jump comes out to be almost a factorof 2 larger in the Bohm-Vitense formulation! To reducethe entropy jump to a level comparable as obtained fromthe Mihalas formulation, one had to increase the mixing-length parameter noticeably. Again, this emphasizes thatαMLT is only well defined with reference to a specific MLTformulation.

The Bohm-Vitense formulation assumes optically thickconvective elements throughout. Their radiative energyexchange is described in diffusion approximation. It mightbe surprising that the entropy jump obtained from theBohm-Vitense formulation is larger than from the Mihalasformulation which distinguishes between optically thick


4 5 6 7 8log10 Pgas [dyn/cm2]

8.70

8.80

8.90

9.00

9.10

9.20

9.30

Spe

cific

Ent

ropy

[108 e

rg/g

/K]

Fig. 14. Entropy-pressure relation of hydrodynamicalmodel A-3D (solid) and standard mixing-length modelatmospheres with αMLT = 1.5 based on two differentformulations of MLT: Mihalas (1978) (dot-dashed) andBohm-Vitense (1958) (dotted). Note, the significantlydifferent entropy jumps in the MLT models employing thesame mixing-length parameter. The entropy level of ma-terial entering the computational box of hydrodynamicalmodel A-3D at the lower boundary is indicated by thedashed line. All models have identical atmospheric pa-rameters (Teff = 2790K, log g = 5.0, solar chemical com-position).

and thin elements. The optically thin elements loose (orgain) energy in a non-diffusive fashion. A larger entropyjump implies larger radiative losses of the convective ele-ments. Intuitively, one would expect that a diffusive trans-port is rather inefficient, implying a smaller energy ex-change and smaller entropy jump. However, Eq. (A.3)shows that the situation is reversed: the thermal adjust-ment time of convective elements becomes independent oftheir optical thickness in the optically thin limit, while thediffusion approximation predicts (inaccurately) a decreaseof the adjustment time with decreasing optical thickness.Since the optical thickness becomes very small in the op-tically thin limit, the diffusion approximation underesti-mates the thermal relaxation time, and overestimates theradiative losses which leads to a larger entropy jump.

While it appears at first glance arbitrary which MLTformulation one chooses, Fig. 14 also shows that in thepresent case the formulation of Mihalas is perhaps supe-rior to the formulation of Bohm-Vitense. The shape of theentropy profile of the Mihalas model resembles the resultof the hydrodynamical model more closely. This comesat no surprise since the formulation of Mihalas considersoptically thick and thin convective elements — i.e. bet-ter represents the actual situation. Of course, our choiceof relating our αMLT to the formulation of Mihalas wasmotivated by this property.

5.3.2. Comparison to the Sun

It is interesting to compare equivalent mixing-length pa-rameters derived for the thermal structure of the hydro-dynamical models with other ones found in hotter objects,especially the Sun. Using the MLT formulation of Bohm-Vitense, Ludwig et al. (1999) found in a study of theconvective efficiency based on 2D hydrodynamical modelsαMLT = 1.6 for the Sun which has likely to be correctedto αMLT = 1.8 if one considers 3D models. In the Sun nodistinction needed to be made between fitting the entropyjump or the asymptotic entropy. The solar mixing-lengthmodels match the photospheric entropy minimum foundin the hydrodynamical models quite well. This is a con-sequence of the fact that convection does not reach veryfar up into optically thin regions. Together with the rel-atively large entropy jump which makes the exact valueof the entropy minimum less important this leads to the“one αMLT fits both” situation. Contrary to the situationin M-dwarfs differences between the Bohm-Vitense’s andMihalas’ MLT formulation are small in the Sun, again, aconsequence of the confinement of convection to opticallythick regions. This allows us to inter-compare the mixing-length parameter obtained by Ludwig et al. (1999) for theSun with the values derived here.

The mixing-length parameter of 1.8 for the Sun and1.5 (matching the asymptotic entropy) or 2.1 (matchingthe entropy jump) for the M-dwarf are not vastly differ-ent, the solar value even falls within the range spanned bythe M-dwarf values. In MLT the convective flux scales asthe square of αMLT, hence we are considering a flux vari-ation within a factor of 2 only. Convection is efficient inM-dwarfs in the sense that the whole envelope is almostadiabatically stratified. The convective energy transportprocess itself does not work much differently than in thesolar case — taking the similar mixing-length parametersas a measure. The adiabatic behavior follows mostly fromthe reduction of the transported flux which can be ac-commodated with smaller horizontal entropy fluctuations.This ultimately leads to the smaller entropy jump whichamounts to ∆s = 7×106 erg/g/K, or only ≈ 4% of the so-lar entropy jump. This is remarkable since as evident fromFig. 13 about half of the entropy jump happens to takeplace in the optically thin region which is qualitativelydifferent from the Sun. Moreover, the MLT captures sur-prisingly well the convective transport properties undersuch conditions.

We note that an extrapolation of the entropy jump asfound by Ludwig et al. (1999) (their Fig. 4) for F-, G-,and K-dwarfs leads to a significant overestimation of theentropy jump for M-dwarfs. This means that the trend inthe entropy jump found in hotter objects changes at somepoint when moving towards lower Teff . This, again, is plau-sible because the dominant atmospheric opacity source aswell as contributors to the specific heat change in the M-dwarf regime.


Table 2. Synthetic broad band colors V-I, R-I in theCousins system, as well as J-K, and J-H in the CIT systemfor the models displayed in Fig. 11. The effective temper-ature T ′

eff corresponds to the total flux as computed bythe detailed spectral synthesis.

Model T ′

eff V-I R-I J-K J-H[K] [mag] [mag] [mag] [mag]

RHD, A-3D 2754 3.651 1.974 0.749 0.455LHD, αMLT = 1.0 2743 3.733 2.010 0.739 0.449LHD, αMLT = 1.5 2744 3.701 1.996 0.746 0.452LHD, αMLT = 2.0 2745 3.686 1.989 0.749 0.454

5.4. Synthetic broad band colors

Table 2 provides synthetic optical and near-infrared col-ors for the models discussed before and shown in Fig. 11.The colors of the hydrodynamical model have been cal-culated from the average pressure-temperature structure.In the calculation of the colors the radiation field hasbeen sampled with a resolution of 2 A, i.e. the treat-ment of frequency dependence is different and more de-tailed than the OBM used in the models proper. Herewe are concerned with the relative behavior among themodels. For the V-I and J-K colors appreciable discrepan-cies exist between observations and model predictions byPHOENIX (“NextGen” model grid) in the Teff-range3600K > Teff > 2300K (Chabrier et al. 2000; Allardet al. 2001). In V-I the offset amounts to ≈ 0.2− 0.3magin the sense that the NextGen models are too blue. InJ-K the offset amounts to ≈ 0.2 − 0.5mag depending onthe source of water line opacity data used in the models.The treatment of the convective energy transport in theMLT framework — employing a standard αMLT of 1.0 —was suspected to be one possible reason for the discrep-ancy. Table 2 shows that a detailed treatment of convec-tion does not remedy the situation. The difference betweenthe V-I color of the hydrodynamical model to the LHD

αMLT = 1.0 amounts to 82mmag only. Moreover, the hy-drodynamical model is even bluer than the mixing-lengthmodel, i.e. when assuming the same differential behav-ior for models employing detailed radiative transfer, thediscrepancy between theory and observations gets evenslightly worse. As expected from the thermal structureαMLT = 2.0 gives the closest match in color among thethree LHD models.

Our results strengthen those of Allard et al. (2000)and Chabrier et al. (2000) who explain most of the visualdiscrepancy in terms of missing opacities. We would alsolike to emphasize that an optical color mismatch of thismagnitude does not indicate a major shortcoming of thePHOENIX models. The V-band, which is likely to beresponsible for the color mismatch, is formed high up inthe atmosphere. The radiative energy flux in the V-band isa small fraction of the total radiative output of the star.

Hence, the problem of the optical colors is related to apart of the atmosphere that is of minor importance forthe overall energetics.

Similarly, the discrepancies between observed and pre-dicted near-infrared colors cannot be traced back to aninadequate treatment of convection with MLT. Possibledifferences between MLT and hydrodynamical modelsamount to 10mmag only.

Table 2 allows us to address a side point, namely thequestion of how the hydrodynamical model A-3D and themixing-length models shown in Fig. 11 can have the sameeffective temperature (2790K) despite the fact that thehydrodynamical model is almost always noticebly hotterthan the mixing-length models. Indeed, when calculatingthe total flux within the detailed spectral synthesis thecorresponding effective temperature T ′

eff of the hydrody-namical model appears to be slightly (≈ 10K) hotter thanthe one of the mixing-length models — with an offset tothe nominal Teff due to the more detailed treatment ofthe radiative transfer. We interpret this as an effect of thesmall remaining non-linearities when interchanging the av-eraging of the temperature-pressure structure with the so-lution of the radiative transfer. Strictly speaking, in orderto determine the colors of the hydrodynamical model onemust perform the detailed calculation of the 3D radia-tion field for many instances in time and had to averagesubsequently horizontally and temporally. However, in thepresent context, the smallness of the effects did not appearworth the dramatically increased computational effort.

5.5. Mean velocity and corresponding effective

mixing-length parameter

In the previous sections we discussed mixing-length pa-rameters associated with the thermal properties of ourhydrodynamical models. We now want to turn to a dy-namical feature — the velocity profile. Figure 15 showsa comparison of velocities of hydrodynamical model A-3D with convective velocities from mixing-length models.In the case of the hydrodynamical model RMS fluctua-tions of the vertical velocity, for the mixing-length modelsthe MLT convective velocity are displayed. Six differenttemporal windows (each 250 s in length) have been usedfor averaging the hydrodynamical data, showing noticablevariations of the mean velocity particularly in the higheratmospheric layers. The velocity plateau in the uppermostlayers is an artifact of the upper boundary condition whichstipulates the same velocity at the boundary and the gridpoints next to the boundary. In order to match the convec-tive velocities in the convectively unstable layers we findthat αMLT(v) = 3.5 gives a reasonable match around thevelocity maximum. The hydrodynamical model has thetendency to retain higher velocities towards smaller at-mospheric pressure as compared to the MLT models. Thebehavior of the velocities towards high pressure shows afaster decline than the MLT models. The deeper hydrody-namical model B-3D (not shown) exhibits a similar veloc-


3 4 5 6 7 8log10 Pgas [dyn/cm2]

0

50

100

150

200

250

RM

S V

ertic

al V

eloc

ity [m

/s]

3 2 1 0-1-2-3

Fig. 15. Velocity-pressure relation of hydrodynamicalmodel A-3D (solid) and five standard mixing-lengthmodel atmospheres (dash-dotted) with αMLT = 1.0 to3.5 in steps of 0.5. The velocities of the mixing-lengthmodels increase monotonically with αMLT. For the hydro-dynamical model RMS vertical velocities from six differenttemporal windows are plotted. A logarithmic Rosselandoptical depth scale is indicated by the tick marks near theabscissa. All models have identical atmospheric parame-ters (Teff = 2790K, log g = 5.0, solar chemical composi-tion).

ity maximum as model A-3D but a decline which is notas fast. We conclude that the decline in model A-3D is tosome extend influenced by the lower boundary conditionand an asymptotic behavior towards high pressure closerto MLT predictions is likely.

The hydrodynamical model shows an appreciableamount of overshoot into the formally stable atmosphericregions. Of course, by construction, standard MLT is un-able to make any predictions about velocity amplitudesthere. Two different components contribute to the veloci-ties in the higher atmosphere: one contribution are wavesexcited by the stochastical fluid motions in the deeperlayers and travelling upwards, the other contributions areadvective motions overshooting into the stably stratifiedlayers (see also Ludwig & Nordlund 2000).

Qualitatively, we expect that velocity fluctuations v′

associated with advective motions decay with increas-ing distance from the Schwarzschild boundary, roughlyv′ ∝ P f where f is a positive parameter of order unity.For undamped sound waves we would expect a height de-pendence of roughly v′ ∝ P−

1

2 . This means that the rel-ative importance of both contributions shifts from advec-tion dominated motions to wave dominated motions withincreasing height. If the wave motions are not stronglydamped we would ultimately expect an increase of thevelocity fluctuations with height. Indeed, according toFig. 15 this is what we observe in our model. The wavemotions mostly arise from standing waves — i.e. excitedacoustic eigenmodes of the computational box. The de-tailed run of the velocity in the overshoot layers reflects

the location of nodes and anti-nodes of the eigenmodes invertical direction which explains the non-monotonic be-havior of the atmospheric velocity amplitude.

The spectrum and the structure of the eigenmodes areto some extent determined by the geometry of the com-putational domain. This means that the velocities in thehigher atmosphere where the velocity amplitude is wavedominated are model dependent and should not taken forreal without skepticism. In the next sections we will evenargue that the velocity amplitudes of the acoustic modesobserved in our particular models are probably stronglyinfluenced by the numerics, and in this sense are mostlyartificial. In order to study the transport properties of theatmospheric velocity field we will try to filter out as muchof the acoustic contribution to the velocity field as possi-ble.

5.6. Atmospheric overshoot and transport time scales

Stellar atmospheres around Teff ≈ 2800K are too hot toallow for a significant formation of dust grains. However,already at slightly cooler effective temperatures, grain for-mation sets in and dust grains become major opacity con-tributors, i.e. an important factor in determining the ther-mal structure of the atmosphere. In fact, the spectral en-ergy distribution in the range of effective temperatures2500K ≤ T ≤ 1500K is crucially linked to the distributionof dust grains in the atmosphere (Allard et al. 2000). Theamount of dust which is present is determined by chemi-cal condensation and evaporation processes as sources andsinks, as well as macroscopic transport processes whichcarry dust grains away from their sites of formation. InM-dwarf atmospheres the transport is dominated by twoopposing processes: gravitational settling of dust grainsand their mixing due to the presence of velocity fields, ei-ther related to convection or global circulations inducedby rotation (T. Guillot 2000, private communication). Inour models, no dust formation takes place but we nev-ertheless find it worthwhile to give a characterization ofthe atmospheric mixing due to convection and convectiveovershoot. This can give at least a first order approxima-tion of how convective mixing might operate when dust isactually present. The basic question which we want to ad-dress is whether convective overshoot can provide enoughmixing to prevent dust grains from being completely re-moved from the atmosphere by gravitational settling.

Formally, we are interested in a statistical representa-tion of the mean transport properties of the convectivevelocity field in vertical direction. The horizontal advec-tion of dust grains by the convective velocity field probablyproduces horizontal inhomogeneities in the dust distribu-tion. We neglect these here since i) we are targeting atthe application of our results in standard 1D stellar at-mosphere models, ii) the horizontal inhomogeneities aresmall scale (the size of a convective cell), i.e. hardly ob-servable and iii) the uncertainties in our understanding ofthe dust formation process itself perhaps limits the achiev-


3 4 5 6 7 8log10 Pgas [dyn/cm2]

-4.0

-3.5

-3.0

-2.5

-2.0

log 1

0 M

ass

exch

ange

freq

uenc

y [s

-1]

3 2 1 0-1-2-3

Fig. 16. Mass exchange frequency as a function of pres-sure for various degrees of subsonic filtering. The solid

curves from top to bottom correspond to no filtering,vphase ≤ 1.0, 0.5, 0.25km/s, respectively. The dashed lineis an extrapolation towards the asymptotic behaviour cor-responding to a complete removal of acoustic contribu-tions. The dashed-dotted line depicts the dependence ifone assumes a velocity field calculated from linear theory.A logarithmic Rosseland optical depth scale is indicatedby the tick marks near the abscissa. The dotted line in-dicates the location of the Schwarzschild boundary.

able accuracy anyway. In view of the last point we presenta proxy of the convective mixing only, without trying toderive a detailed statistical description of the transportproperties of the convective velocity field, i.e. extractingits effective transport coefficients (c.f. Miesch et al. 2000,as an example of a more stringent treatment).

Figure 16 shows the mass exchange frequency fexchangewhich provides an approximate measure of the convectivemixing. We define this quantity as

fexchange(z) ≡〈F up

mass〉(z)

〈mcol〉(z)(2)

where F upmass is the upward directed mass flux, mcol the

mass column density, and z the geometrical height. 〈.〉denotes temporal and horizontal averages. Note that theaverage total mass flux 〈Fmass〉 vanishes

〈Fmass〉 = 〈ρuz〉 = 〈F upmass + F down

mass 〉 = 0. (3)

ρ denotes the density, uz the vertical velocity component.fexchange is the frequency with which the mass above acertain height z is replaced by the flux of fresh materialfrom below. Mass conservation and quasi-steadiness de-mand that the upward directed mass flux from below iscompensated by a downward directed flux from above. Aslong as the flow field is sufficiently disordered this massexchange should also go hand in hand with mixing of ma-terial; in this way, fexchange can be interpreted as mixingfrequency of atmospheric layers above a certain referenceheight z. The assumption of a disordered velocity fieldwas checked qualitatively by studying the behaviour of

passive tracer particles being advected with the flow. Thedefinition of fexchange was motivated from our expectationthat the amplitude of the velocity field would strongly de-cline with distance from the convectively unstable zone.Freytag et al. (1996) find in hydrodynamical models thatthe velocity amplitude due to convective overshoot de-clines exponentially with distance from the Schwarzschildboundary, and promote this result as a generic feature ofovershoot. In such a situation, the mass flux in or out ofa control volume is dominated by the flux through thesurface closest to the convectively unstable region. This isthe reason why we consider just one surface (or height)through which mass is transported.

As mentioned previously, the atmospheric velocity fieldhas strong contributions of acoustic waves generated in thedeeper layers of the models. From the comparison (notshown) of the hydrodynamical models A-3D and B-3D —which has a slightly greater extent in depth — we con-cluded that instabilities associated with the lower bound-ary condition contribute to the excitation of waves, i.e.the waves are mostly a numerical artifact. We could nottrace down the responsable process, but the behavior wasapparently related to the high degree of adiabaticity thestratification exhibits close to the lower boundary. In theSun the boundary condition operated stably. Besides thenumerical findings, from a purely theoretical point of viewone would expect that the generation of acoustic energyis rather inefficient in a flow with as low a Mach num-ber (few percent) as in the present case. Finally, it turnsout that the Richardson number associated with the hor-izontal shear introduced by the acoustic modes in the at-mosphere is not low enough to render the shear flow un-stable. No small scale turbulence is induced by the wavesand their oscillatory velocity fields do not contribute sig-nificantly to the mixing of material. Hence, we concludedthat the wave contribution to the velocity field should bediscarded if one wants to measure its mixing efficiency.

We therefore removed the contribution of waves beforeevaluating the average mass flux by subsonic filtering — atechnique invented in the context of solar observations forcleaning images from “noise” stemming from the 5minuteoscillations (Title et al. 1989). In short, one considers atime sequence of images and removes features with hori-zontal phase speeds greater than a prescribed threshold.This is achieved by Fourier filtering of the spatial-temporaldata in the k-ω domain. In our case we worked with 150snapshots of the flow field sampled at 10 s time intervals.The total length of the sequence was long in comparisonto the periods of the excited acoustic modes. For everydepth layer we performed a 3D Fourier analysis (one tem-poral, two spatial dimensions) of the vertical mass flux.We selected velocity thresholds

vphase =|ω|

|k|= 1.0, 0.5, 0.25km/s. (4)

With these specific choices of vphase we tried to removeacoustic features which had phase speeds in the order ofthe sound speed without affecting the slow convective mo-


tions which have maximum velocities of about 0.25 km/s(see Fig. 15). Note that the technique accounts only forthe horizontal component of the phase speed. This meansthat e.g. purely radial oscillations could not be removedby this procedure. An improvement could, in principle, ex-tend the filtering technique to four dimensions (one tem-poral, three spatial dimensions). But this would requirea much higher computational effort, in particular sinceone would first have to determine a suitable set of basisfunctions for the representation of features in the verticaldirection, where simple harmonic functions do not forma proper set of eigenfunctions. Luckily, the standard sub-sonic filtering worked well enough in our case so that wedid not have to pursue this issue further.

Figure 16 shows the resulting mass exchange fre-quency fexchange as a function of height (expressed interms of the mean pressure) for our hydrodynamicalmodel A-3D. Lowering the threshold vphase leads to a sig-nificant decrease of the mass exchange frequency in the at-mospheric layers while the deeper layers remain essentiallyunaffected by the subsonic filtering. This again shows thatthe velocity field of the higher atmospheric layers stemsprimarily from wave motions. Only the lowest threshold of0.25 km/s starts to affect the deeper layers since also con-vective flow features reach such velocities. The dashed linein Fig. 16 indicates our extrapolation of what to expect incase of a perfect removal of the acoustic contribution. Itsslope is given by the envelope of the sequence of curves forincreasing degree of subsonic filtering. We want to empha-size that by the very nature of the problem the filteringprocedure does not apply a small correction to the sig-nal but removes even most of the original signal in somelayers. Wave motions almost inevitably dominate the flowvelocity above a certain atmospheric level. Large correc-tions as exhibited in Fig. 16 are to be expected, but do notpose particular problems so that we are confident aboutthe filtering procedure.

As an additional check of our approach we calculatedfor an LHD mixing-length model with αMLT = 2.0 lineareigenmodes describing the growth of a perturbation un-der the action of the convective instability. αMLT = 2 waschosen since the model gave among the available mixing-length models the closest match to the thermal struc-ture of the hydrodynamical model. As horizontal wave-length of the pertubation we chose the horizontal extentof model A-3D of 250km, i.e. the largest horizontal wave-length which could be accomodated in this hydrodynam-ical model. Freytag et al. (1996) showed that the velocityfield of convective eigenmodes gives a reasonable repre-sentation of the velocity field in the overshoot region aslong as the horizontal wavelength is chosen in the vicinityof the actually present convective scales. The dash-dottedline in Fig. 16 shows the shape of fexchange associated withthe linear eigenmode. The velocity amplitude was scaledto match the maximum of fexchange in the deeper layers ofthe hydrodynamical model. Besides a systematic shift thefunctional behavior predicted by the mode is quite similarto the one of the hydrodynamical simulation. Moreover,

the time scale for the growth of the perturbation comesout to 120 s which is of the same order as the convec-tive turn-over time scale. The similar rate of decline offexchange in the atmospheric layers strengthens our con-fidence in the robustness of our extrapolation procedure.However, changing the horizontal wavelength of the per-turbation within reasonable limits as well as experiment-ing with different background models did not improve thecorrespondence between linear modes and non-linear hy-drodynamical model beyond the quality shown in Fig. 16.

Strictly speaking, the mixing frequency displayed inFig. 16 is a lower limit since we removed all contributionsassociated with waves. However, the arguments given be-fore make it very likely that the contributions of wavemotions to the mixing are negligible: the Mach number ofthe flow is low, and the weak waves motions which may beexcited are associated with little turbulence. Therefore webelieve that the derived mixing efficiency is close to whatis encountered in a real M-dwarf atmosphere.

What are the possible consequences of the exponentialdecline of the mixing time scale due to convective over-shoot? Following the work of Rossow (1978), one arrivesat an estimate of about 104 s for the typical time scale ofdust sedimentation in M-dwarf atmospheres. According toour hydrodynamical models the convective mixing couldcounteract the sedimentation up to a layer of about twopressure scale heights above the formally convectively un-stable layers. The convective turn-over time scale is of theorder of 102 s. This is slow enough to allow for dust nucle-ation in the upper regions of the convective envelopes ofM-dwarfs — provided that the thermodynamic conditionsallow for nucleation in the first place. From these consider-ations we would expect that dust clouds in M-dwarf (andbrown dwarf) atmospheres are confined to layers in thevicinity of the upper Schwarzschild boundary of the con-vective envelope.

6. Concluding remarks and outlook

We used elaborate 2D and 3D radiation-hydrodynamicssimulations to study properties of convection on the sur-face of a prototypical late M-dwarf (Teff ≈ 2800K, log g =5.0, solar chemical composition). Despite the significantdifferences in the physical conditions encountered in thesolar and an M-dwarf atmosphere we obtained the strik-ing result that M-dwarf granulation does not look qual-itatively different from what is familiar from the Sun(see Fig. 6). Quantitative differences (intensity contrast1.1%, horizontal scales ≈ 80 km, maximum RMS veloci-ties≈ 240m/s, convective turn-over time scale≈ 100 s) re-main within the expectations derived from mixing-lengththeory.

Connected to this basic finding is the — for practicalpurposes — perhaps most important result that the tem-perature structure of the higher atmospheric layers is de-termined by the condition of radiative equilibrium, and isnot very much affected by processes usually not accountedfor in standard stellar atmosphere models. Convective


overshoot as well as energy transport by waves do not sig-nificantly affect the temperature structure outside of theregion of convective instability. We expect that this find-ing also holds for main-sequence objects of higher effectivetemperature where radiation becomes relatively more im-portant.

Answering the questions posed in the introduction weconfirm that in late M-dwarfs mixing-length theory can beapplied to obtain a realistic description of the convectiveenergy transport in a 1D stellar atmosphere code, pro-vided one can remove uncertainties related to the choiceof the mixing-length parameter. If an independent cali-bration is available we expect that 1D atmosphere modelsallow reasonable accurate predictions (on a level as dis-played in Fig. 13) of the atmospheric temperature struc-ture and ultimately the stellar spectrum to be made.Effects on the stellar spectrum related to horizontal tem-perature inhomogeneities are expected to be very smallin the M-dwarf studied here, primarily due to the smallhorizontal fluctuations of the thermodynamic variables.Even in the Sun — with the much higher temperaturecontrast present at its surface — effects related to hori-zontal temperature inhomogeneities are rather subtle (cf.Steffen & Ludwig 1999). We expect that 1D stellar atmo-sphere models provide an acceptable overall approxima-tion to the spectrum of main-sequence objects betweenthe Sun and late M-dwarfs. Only if one demands for aprecision exceeding commonly adopted levels or desires tostudy effects in principle not included in standard modelatmospheres (e.g. spectral line shifts and asymmetries),one has to go to more sophisticated modeling. We em-phasize that this refers to objects at solar metallicity. Formetal poor objects the situation is clearly different (seeAsplund et al. 1999).

A downside of our findings is that we cannot traceback the remaining differences between theoretical and ob-served colors for M-dwarfs to shortcomings in the treat-ment of the convective energy transport. The resolutionof the problem has to be found elsewhere, as pointed outearlier, deficiencies in the molecular opacities are still apossible option.

We are left with the problem of finding an adequatevalue of the mixing-length parameter to obtain a descrip-tion of the vertical temperature run in the superadiabaticlayers. In the present case we find a mixing-length param-eter αMLT(∆s)=2.1 which gives a match to the entropyjump and the temperature gradient of the deeper atmo-spheric layers (see Fig. 13). Despite perhaps the best valueto be employed in stellar atmosphere calculations, forglobal stellar structure models a value of αMLT(∆s)=1.5would be more appropriate since it ensures to find the cor-rect asymptotic entropy in the convective envelope whichis important for obtaining the correct stellar radius. Tocomplete the “zoo” of mixing-length parameters, we get avalue of αMLT(v)=3.5 when matching the convective ve-locities predicted in our hydrodynamical models. We re-iterate that all values are given with reference to the for-mulation of MLT by Mihalas (1978).

The various values of the mixing-length parameterpoint towards the deeper rooted problem that MLT cangive a reasonable but not exact description of the averageconvective properties. Even calibrating one aspect doesnot ensure the overall correct functional form of, say, thetemperature profile. We have seen that different formu-lations of MLT can give quite different functional depen-dencies. They offer the possibility to improve fits beyondthe quality limited by fitting the mixing-length parame-ter only. For late M-dwarfs all this does not matter muchsince the differences of the atmospheric structure for var-ious values of αMLT are small. However, we stress thatthis statement refers to cooler M-dwarfs on or close tothe main-sequence. For pre-main-sequence (PMS) objectsthe situation is markedly different (Baraffe et al. 2002).There the specific choice of αMLT has a large impact onthe resulting atmospheric structure. Work is underway toextend the present study into this regime which may alsoallow us to find the most suitable MLT formulation.

A result beyond the scope of classical model atmo-spheres is the derivation of a proxy of atmospheric mixing-time scales (see Fig 16) due to convective overshoot. Wefind an exponential “leaking” of the convective velocityfield into the formally stably stratified layers. Dependingon the exact criterion, overshooting extends the efficientlymixed regions about 2 pressure scale heights beyondthe Schwarzschild boundary. We suggest that the mix-ing found in the Teff ≈ 2800K model studied here, takesplace in an analogous fashion in brown dwarfs, and pro-vides the mixing which counteracts dust sedimentation.Hydrodynamical models can be used to address this prob-lem more directly by performing simulations including theformation and transport of dust which we consider as aninteresting challenge for the future.

Last but not least we would like to point out the twoweakest points of our investigation. While the precisionof the radiative transfer in the hydrodynamical calcula-tions is sufficient to address the questions discussed herethere is certainly room for improvement of the OBM toget an even closer agreement with detailed spectral synthe-sis calculations. Secondly, the models presented here arerather shallow. Improvements in the formulation of thelower boundary condition would perhaps allow the use ofdeeper computational domains, and would reduce the in-fluence of the specific formulation of the lower boundaryconditions.

Acknowledgements. The authors are indebted to IsabelleBaraffe and Gilles Chabrier for their supportive enthusiasmduring the course of the project, and their scientific input dur-ing many discussions. HGL would like to thank Ake Nordlundand Robert Stein for making available a version of their hy-drodynamical atmosphere code. PHH acknowledges support by“Pole Scientifique de Modelisation Numerique” (PSMN) at theEcole Normale Superieure de Lyon, NSF grants AST-9720704and AST-0086246, NASA grants NAG5-8425, NAG5-9222, aswell as NASA/JPL grant 961582 to the University of Georgia(UGA), Athens. The calculations presented in this paper wereperformed on machines operated by the PSMN , on the NEC-


SX5 at the “Institut du Developpement et des Ressources enInformatique Scientifique” (IDRIS), Paris, on the IBM SP2of the UGA UCNS, on the IBM SP “Blue Horizon” of theSan Diego Supercomputer Center (SDSC) with support fromthe National Science Foundation, and on the IBM SP of theNERSC with support from the US Department of Energy. Wethank all these institutions for a generous allocation of com-puter time.

Appendix A: Computation of the characteristic

time scales

The radiative time scales shown in Fig. 4 were evaluatedin Eddington approximation (see e.g. Edwards 1990) ac-cording

trad =cp

16σχT 3

(

1 +3χ2ρ2

k2

)

(A.1)

(cp denotes the specific heat at constant pressure, σStefan-Boltzmann’s constant, χ opacity, T temperature,ρ mass density, k wavenumber of the disturbance). We as-sumed that the geometrical size of a thermal disturbanceis the local pressure scale height HP, hence

k =2π

HP

. (A.2)

Introducing the optical size of the disturbance τdis ≡χρHP Eq. (A.1) can be written as

trad =cp

16σχT 3

(

1 +3

4π2τ2dis

)

. (A.3)

Relation (A.3) shows that in the optically thin limit τdis ≪1 the radiative relaxation time is independent of τdis, andin the optically thick limit τdis ≫ 1 scales quadraticallywith τdis. Equations (A.1) and (A.3) describe the decayof a Fourier mode in an infinite isothermal atmosphere,i.e. make very idealizing assumptions. Our choice of of thelocal pressure scale height as size of the perturbation issomewhat arbitrary. Hence, the reader should take trad asdepicted in Fig. 4 as order of magnitude estimates.

The (adiabatic) Brunt-Vaisala period tBV was calcu-lated according

tBV =2π

√

|ω2BV|

, with ω2BV =

δg

HP

(∇ad −∇) . (A.4)

(δ ≡ −(

∂ ln ρ∂ lnT

)

Pdenotes the thermal expansion coefficient

at constant pressure, g gravitational acceleration, ∇ad theadiabatic, and ∇ the actual temperature gradient of theatmosphere). The local Kelvin-Helmholtz time scale tKH

was evaluated according

tKH =PcpT

gσT 4eff

(A.5)

(P denotes the gas pressure) under the assumption thatthe overall energy content of a mass column is dominatedby its thermal energy content.

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