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The Stagger-grid: A grid of 3D stellar atmosphere models - III. The relation to mixing-length convection theory

We investigate the relation between 1D atmosphere models that rely on the mixing length theory and models based on full 3D radiative hydrodynamic (RHD) calculations to describe convection in the envelopes of late-type stars. The adiabatic entropy value of the deep convection zone, s_bot, and the entropy jump, {\Delta}s, determined from the 3D RHD models, are matched with the mixing length parameter, {\alpha}_MLT, from 1D hydrostatic atmosphere models with identical microphysics (opacities and equation-of-state). We also derive the mass mixing length, {\alpha}_m, and the vertical correlation length of the vertical velocity, C[v_z,v_z], directly from the 3D hydrodynamical simulations of stellar subsurface convection. The calibrated mixing length parameter for the Sun is {\alpha}_MLT (s_bot) = 1.98. For different stellar parameters, {\alpha}_MLT varies systematically in the range of 1.7 - 2.4. In particular, {\alpha}_MLT decreases towards higher effective temperature, lower surface gravity and higher metallicity. We find equivalent results for {\alpha}_MLT ({\Delta}s). Also, we find a tight correlation between the mixing length parameter and the inverse entropy jump. We derive an analytical expression from the hydrodynamic mean field equations that motivates the relation to the mass mixing length, {\alpha}_m, and find that it exhibits qualitatively a similar variation with stellar parameter (between 1.6 and 2.4) with a solar value of {\alpha}_m = 1.83. The vertical correlation length scaled with the pressure scale height yields for the Sun 1.71, but displays only a small systematic variation with stellar parameters, the correlation length slightly increasing with Teff. We derive mixing length parameters for various stellar parameters that can be used to replace a constant value. Within any convective envelope, {\alpha}_m and related quantities vary a lot.


Introduction
In the past century insights in various fields of physics led to a substantially more accurate interpretation and understanding of the processes taking place in the interior of celestial bodies. With theoretical stellar atmosphere models astronomers can parameterize the conditions on the surface of stars, and additionally with the theory of stellar structure and evolution, they are capable to predict the complex development of stars.
The radiated energy of cool stars, originating from the deeper interior due to nuclear burning in the center, is advected to the surface by convective motions in the envelope, driven by negative buoyancy acceleration. At the thin photospheric transition region the large mean free path of photons allows them to escape into space, and the convective energy flux is released abruptly. To model theoretically this superadiabatic boundary domain of stars is challenging due to the non-linear and non-local nature of turbulent sub-surface convection and radiative transfer, and an analytical solution is a long-standing unresolved issue.
(MLT), which was initially proposed by Ludwig Prandtl for the Earths atmosphere in analogy to the concept of the mean free path in the kinetic gas theory. In the framework of MLT, it is assumed that the heat flux is carried by convective elements for a typical distance, before they dissolve instantaneously into the background. This distance is the so-called mixing length, l, usually expressed in units of the pressure scale height, α MLT = l/H P . The mixing length parameter α MLT is a priori unknown, hence it has to be "calibrated", usually by matching the current radius and luminosity of the Sun by a standard solar model with a single depth-independent α ⊙ MLT . This calibrated value for the Sun is then used for all stellar parameters. It must be recalled that α ⊙ MLT , in fact, corrects for all other shortcomings of the solar model, due to, e.g., deficits in the equation-of-state (EOS), the opacities, or the solar composition. It therefore is no wonder that its numerical value (typically around 1.7 to 1.9; e.g., see Magic et al. 2010) varies with progress in the named aspects and from code to code. In addition, MLT is a local and time-independent theory that contains effectively three additional, free parameters, and assumes symmetry in the up-and downflows, hence also in the vertical and horizontal direction. The actual formulation of MLT can vary slightly too (e.g., see Henyey et al. 1965;Mihalas 1970;Ludwig et al. 1999).
Many attempts have been made to improve MLT, a substantial one being the derivation of a non-local mixing length theory (Gough 1977;Unno et al. 1985;Deng et al. 2006;Grossman et al. 1993). The standard MLT is a local theory, i.e. the convective energy flux is derived purely from local thermodynamical properties, ignoring thus any non-local properties (e.g. overshooting) of the flow. Usually, the non-local models are derived from the hydrodynamic equations, which are a set of non-linear moment equations including higher order moments. For their solution, closure approximations are considered (e.g. diffusion approximation, anelatistic approximations or introducing a diffusion length). Also, further aspects have been studied: the asymmetry of the flow by a two stream MLT model (Nordlund 1976), the anisotropy of the eddies (Canuto 1989), the time-dependence (Xiong et al. 1997) and the depth-dependence of α MLT (Schlattl et al. 1997). While standard MLT accounts for only a single eddy size (being l), Canuto & Mazzitelli (1991) extended the picture to a larger spectrum of eddy sizes by including the non-local second order moment (Canuto et al. 1996, see also). The original Canuto-Mazzitelli-theory -also known as the Full Spectrum Turbulence model -used the distance to the convective region border, as a proxy for the mixing length, a later version (Canuto & Mazzitelli 1992) re-introduced a free parameter resembling α MLT .
These approaches are often non-trivial, but so far the standard MLT is still widely in use, and a breakthrough has been absent, despite all the attempts for improvements. In 1D atmosphere modeling, the current procedure is to assume for the mixing length parameter α MLT a universal value of 1.5 (Gustafsson et al. 2008;Castelli & Kurucz 2004, see). For full stellar evolution models, the solar "calibration" yields values around ∼ 1.7 − 1.9 (Magic et al. 2010, see, e.g. ). Since the value of the mixing length parameter sets the convective efficiency and therefore changes the superadiabatic structure of stellar models, an accurate knowledge of α MLT for different stellar parameters would be a first step in improving models in that respect. However, besides the Sun, other calibrating objects are rare and data are much less accurate (see Sect. 3.7 for an example), such as binary stars with well determined stellar parameters.
The mixing length parameter can be deduced from multidimensional radiative hydrodynamic (RHD) simulations, where convection emerges from first principles (e.g., see Ludwig et al. 1999). Over the past decades, the computational power has increased and the steady development of 3D RHD simulations of stellar atmospheres has established their undoubted liability by manifold successful comparisons with observations (Nordlund 1982;Steffen et al. 1989;Ludwig et al. 1994;Freytag et al. 1996;Stein & Nordlund 1998;Nordlund & Dravins 1990;Nordlund et al. 2009). The 3D RHD models have demonstrated that the basic picture of MLT is incorrect, namely, the convective bubbles are not present, instead highly asymmetric convective motions are found. Nonetheless, an equivalent mixing length parameter has been calibrated by Ludwig et al. (1999) based on 2D hydrodynamic models by matching the resulting adiabats with 1D MLT models (see , for the metal-poor cases). They had shown that α MLT varies significantly with the stellar parameters (from 1.3 to 1.8), and they also studied the impact of a variable α MLT on globular cluster (Freytag & Salaris 1999). Also, Trampedach (2007) applied a grid of 3D atmosphere models with solar metallicity for the calibration of the mixing length parameter (from 1.6 to 2.0), and the so-called mass mixing length (Trampedach & Stein 2011).
In the present work we calibrate the mixing length parameter with a 1D atmosphere code that consistently employs the identical EOS and opacity as used in the 3D RHD simulations (Sect. 2). We present the resulting mixing length parameter in Sect. 3. We also determine the mass mixing length -the inverse gradient of the vertical mass flux -in Sect. 4, and the vertical correlation length of the vertical velocity (Sect. 5) directly from the 3D atmosphere models. For the former quantity, we derive a relation from the hydrodynamic mean field equations that demonstrates the relation to α MLT , which is further substantiated by our numerical results. Finally, we conclude our findings in Sect. 6.

3D atmosphere models
We have computed the Stagger-grid, a large grid of 3D RHD atmosphere models covering a wide range in stellar parameter space (see Magic et al. 2013a, hereafter Paper I). The 3D atmosphere models are computed with the Stagger-code, which solves the 3D hydrodynamic equations for conservation of mass, momentum and energy, coupled with a realistic treatment of the radiative transfer. We employ the EOS by Mihalas et al. (1988), and up-to-date continuum and lines opacities (Gustafsson et al. 2008). For the solar chemical abundances, we use the values by Asplund et al. (2009, hereafter AGS09). Our simulations are of the so-called "box-in-a-star" type, i.e. we compute only a small, statistically representative volume that includes typically ten granules. Also, our (shallow) simulations cover only a small fraction of the total depth of the convective envelope. Due to the adiabaticity of the gas in the lower parts of the simulation box, the asymptotic entropy value of the convective zone, s ad , is matched by the fixed entropy at the bottom of the simulation domain, s bot , which is one of the simulation parameters. The effective temperature is therefore a result in our 3D simulations, and is actually a temporally averaged quantity. In 1D models T eff is an actual fixed input value in addition without fluctuations.
We determine the entropy jump, ∆s, as the difference between the entropy minimum and the constant entropy value of the adiabatic convection zone with ∆s = s min − s bot . In Magic et al. (2013b, hereafter Paper II), we studied in detail the differences between mean 3D models resulting from different reference depth scales. In the present work, we show and discuss only averages on constant geometrical height 3D z , since these fulfill the hydrodynamic equilibrium and extend over the entire vertical depth of the simulations. The Stagger-grid encompasses ∼ 220 models ranging in effective temperature, T eff , from 4000 to 7000 K in steps of approximately 500 K (recall that T eff is the result of the input quantity s ad , and the intended T eff grid point values are adjusted within a margin below 100 K). Surface gravity, log g, ranges from 1.5 to 5.0 in steps of 0.5 dex, and metallicity, [Fe/H], from −4.0 to +0.5 in steps of 0.5 and 1.0 dex. We refer the interested reader to Paper I for detailed information on the actual methods for computing the grid models, their global properties and mean stratifications.

1D atmosphere models
For the Stagger-grid, a 1D MLT atmosphere code has been developed, which uses exactly the same opacities and EOS as the 3D models (Paper I). Therefore, the chemical compositions are identical. The code uses the MLT formulation by Henyey et al. (1965) (see App. C.1 for details), similar to the MARCS code (Gustafsson et al. 2008). Furthermore, we note that for consistency, the 1D models are computed with exactly the same T eff as the 3D models. In the deeper layers, we extended the 1D models (dashed lines) with the aid of the entropy gradient from the 3D models. The calibration of the mixing length parameter α MLT is illustrated by the smaller insets, which depict the relative differences between the 1D and 3D models (δs = s 1D /s 3D − 1) for s bot (solid) and the entropy jump ∆s (dashed). For the solar model the two approaches result in α MLT = 1.98 and 2.09, respectively.
The actual implementation of MLT differs slightly depending on the considered code (e.g. Ludwig et al. 1999). In the standard MLT formulation there are in total four parameters. The mixing length parameter, α MLT = l/H p , sets the convective efficiency, while one assumes for the temperature distribution y = 3/(4π 2 ) ≃ 0.076, and for the turbulent viscosity ν = 8 (see App. C.2 for their discussion). We considered only the mixing length parameter α MLT for the calibration, while the additional parameters were kept fixed to their default values and the turbulent pressure is entirely neglected.

Matching the mixing length parameter
We calibrated α MLT by matching either the asymptotic entropy value of the deep convection zone, s bot , or the entropy jump, ∆s, from the 1D and 3D models. Subsequently, we refer to these with α MLT (s bot ) and α MLT (∆s), respectively. The value of s bot is an input parameter in our 3D simulations, and represents the adiabatic entropy of the incoming upflows at the bottom of the box that are replenishing the outflows. The horizontally and temporally averaged entropy at the bottom, s bot , considers in contrast both the up-and downflows, and is thus slightly lower than s bot due to the entropy-deficient downflows. However, in our simulations the deeper layers are very close to adiabatic conditions. The entropy contrast at the bottom is extremely small, s bot − s bot ≪ 1 %.
For the calibration, we computed 1D models with α MLT from 1.0 to 2.5 in steps of 0.1 and determined α MLT by minimizing the difference δs = s 1D bot − s 3D bot or the difference in the entropy jumps δs = ∆s 1D − ∆s 3D . We remark that some 1D atmosphere models had convergence issues, when extended to the same depth as the 3D models. Therefore, we had to calculate slightly shallower 1D models. However, we extended the 1D entropy stratifications with the entropy gradients of the 3D model (see Fig. 1). Tests showed that the missing depth in the 1D entropy run leads to only minor uncertainties in the resulting α MLT . We fitted the differences, δs, with a second order polynom to get the value of α MLT . We emphasize that the calibration of α MLT is more meaningful for identical EOS, and the entropy is consistently computed. For the calibration, we neglected the turbulent pressure in the 1D models entirely (i.e. β = 0).
In Fig. 1, we illustrate the calibration of the solar model and for a cool metal-poor dwarf with the mean entropy, s, in the convection zone. For the solar simulation, we determined a mixing length parameter of α MLT = 1.98 and 2.09 from matching either the adiabatic entropy value (left panel) or the entropy jump (right panel). Note how s converges asymptotically against s bot . Furthermore, it is also evident from Fig. 1, that for a higher α MLT the adiabat (s bot ) of the 1D models is decreasing in the convection zone. The entropy minimum of the 3D z on geometrical height is slightly mismatched by the 1D models, a fact, which is reflected by slightly different calibrated α MLT (∆s) values. In the 1D models s min varies only little for different α MLT , and the differences, ∆α MLT , are between ∼ 10 −4 and 10 −3 (cf. also the right panel). Since the entropy jumps are in general much larger than the variation of s min , their influence is very small, and only for very cool metal-poor models with very small entropy jumps, differences in s min might influence the calibration slightly (right panel).
We note that we find in general very similar results for α MLT by employing a 1D envelope code, which solves the stellar structure equations down to the radiative interior by including the same EOS and opacities (Christensen-Dalsgaard 2008). This is in particular true for solar metallicity. The 1D envelope code  relies on an assumed T (τ) relation in the (Eddington gray) atmosphere, which obviously influences the thermal stratification at the outer boundary of the convective envelope. In particular, metal-poor 1D convective interior models with a fixed T (τ) relation are affected by this, and will return different mixing length parameters. The 1D atmosphere code manages without the need for any T (τ) relation, since it solves the radiative transfer by itself. We therefore present and discuss only the mixing length parameters matched by the 1D atmosphere code. Furthermore, we have performed functional fits for the calibrated mixing length parameters, i.e α MLT = f T eff , log g, [Fe/H] , with the same functional basis, as used by Ludwig et al. (1999). For more details see App. B, and the resulting coefficients are provided in Table B.1.

Calibrations with the adiabatic entropy value
In Fig. 2, we show an overview of the variation of the α MLT values calibrated with s bot for different stellar parameters in the Kiel-diagram, in particular for two illustrating metallicities ([Fe/H] = 0 and −2). The mixing length parameter varies rather systematically in the range between ∼ 1.7 and ∼ 2.3: α MLT increases for lower T eff and [Fe/H] and higher log g (see also Fig. 4). Some minor deviations (from a linear run) towards cooler T eff for metal-poor models can be attributed to the differences in the outer boundary condition of the 1D models. A larger α MLT relates to a higher convective efficiency, which implies that a smaller entropy jump is necessary to carry the same convective energy flux. Indeed, we find the entropy jump to increase for higher T eff , lower log g and higher [Fe/H] (see Paper I ); we find that α MLT varies qualitatively inversely to the entropy jump. The mixing length parameter is inversely proportional to the variation of the logarithmic values of the entropy jump, the peak in the entropy contrast and vertical rms-velocity (see Sect. 3.4). This is in agreement with the fact that both the entropy jump and the mixing length parameter are related to the convective efficiency (see Sect. 3.4).

Calibrations with the entropy jump
We also calibrated the mixing length parameter with the 1D MLT atmosphere code by matching the entropy jump ∆s. The resulting values are summarized for two metallicities in Fig. 3, showing a similar behavior as the results of the previous section (see also Fig. 4). We find that the α MLT values based on ∆s are systematically larger by ∼ 0.1 (between ∼ 1.8 and ∼ 2.4) than the values based on s bot (Fig. 5), but the range in α MLT (∆s) is with ∆α MLT ≈ 0.6 very similar to that for α MLT (s bot ). The differences arise from the minimum of the entropy s min around the optical surface, which is lower for the 1D models compared to the 3D model (see Fig. 1), therefore leading to larger mixing length parameters. The metal-poor simulations exhibit larger deviations between α MLT (∆s) and α MLT (s bot ), since the boundary effect, induced by the differences in ∆s, is increasing for lower [Fe/H]. We note that the entropy jump is a relative value, and consequently the matching is less prone to outer boundary effects.

Comparison with global properties
We searched for systematic correlations between the mixing length parameter and mean thermodynamic properties. The inverse of the entropy jump correlates well with α MLT . In Fig. 6 we demonstrate this by comparing the mixing length parameter α MLT (∆s) with the logarithm of the inverse of the entropy jump. Convection is driven by radiative cooling in the surface layers. The entropy jump results from the radiative losses at the optical surface, therefore, the correlation of α MLT roots in the interplay of the opacity, κ λ , radiative cooling rates, q rad , and vertical velocity, v z,rms . The vertical velocity results from buoyancy forces, f b = g∆ρ, acting on the overturning, overdense flows at the optical surface. Hence, a larger entropy jump will entail larger contrast in the entropy and density (δs rms and δρ rms ), which will induce a larger downward acceleration. We illustrate this in Fig. 7, where the peak values for δs rms and δρ rms in the superadiabatic region are plotted against the peak vertical rms-velocity. Evidently, the entropy and density contrast correlate well with the vertical velocity, and this is the underlying reason for the tight  (inverse) correlation between mixing length parameter and entropy jump. In Paper I we have already discussed the correlation of the entropy jump with the peak vertical velocity and the density at the same location, and we deduced the reason for this in the convective energy flux, which essentially contains these quantities.  Top panel: calibration of α MLT with 2D ) and 3D simulations in comparison (solid and dashed lines, respectively). The surface gravity is indicated and color-coded. The solar values are indicated. Bottom panel: relative differences (δ = 2D/3D−1).

Comparison with 2D calibrations
We compare the differences between our inferred mixing length parameters with those of Ludwig et al. (1999) based on similar, but 2D hydrodynamical surface convection simulations. Also, they matched the resulting 2D-based s bot by varying α MLT of a 1D envelope code that uses the same EOS and opacity. However, these are not identical to those used by us, and there are further differences in the models, such as, most importantly, the solar composition. Therefore, these facts should be kept in mind, when interpreting the comparison.
We also remark that Ludwig et al. (1999) derived T (τ)relations from the 2D models, and used them for the 1D models as boundary conditions to render the entropy minimum of the 2D simulations more closely. In Paper I we noticed that s bot resulting from the Stagger-grid is very similar to values from the 2D grid, while the entropy jump ∆s exhibits slight differences.
In Fig. 8 we show the calibrated mixing length parameter from both studies in comparison. The results of Ludwig et al. (1999) also show a clear T eff -dependence, while surface gravity has only very little influence on α MLT . While the 3D-calibrated mixing length parameter decreases with lower surface gravity, its 2D equivalent is moderately increasing. Their solar mixing length parameter is α MLT = 1.59, which is lower by 0.39 (∼ 20%) compared to our mixing length parameter, but comparable to the solar model value of that time, as is ours for the present generation of solar models. The α MLT values for dwarf models (log g = 4.5) are in general around 20 % lower than in our case. Towards giants the difference is diminishing, since the 3D values are decreasing with log g. In the case of 3D convection simulations, it is known that convection is more efficient in comparison to the 2D case. Therefore, the mixing length parameters derived from the 3D models are in general systematically larger. Taking into account the aforementioned "model generation effect", the comparison is, with the exception of the discrepant log g-dependence, quite satisfactory.

Impact on stellar evolutionary tracks
When one considers the variation of α MLT along typical stellar evolutionary tracks, α MLT ranges from 1.6 to 2.4 from higher to lower mass (see Fig. 9), and deviates by up to ±20 % from the solar value (α ⊙ MLT ). Note that in that figure we show α MLT along tracks calculated with a constant value of the mixing length parameter (1.78) obtained from the usual solar model calibration (see Magic et al. 2010). The figure is therefore not showing the actual, self-consistent changes in α MLT along the evolution, however, significant differences are hardly to be expected. During the main-sequence evolution α MLT varies only little and is almost constant, in particular for the lower masses without a convective core. The variable mixing length parameter has a larger influence during later evolutionary stages, the TO and the RGB ascent; α MLT increases first towards values around ∼ 1.9 − 2.1, and then drops sharply down to values of ∼ 1.7 for all masses, which is the consequence of the narrow range in red giant temperature and surface gravity.
The mixing length parameter not only determines T eff of the stellar models, but also influences the adiabatic stratification of the 1D models in the deeper convection zone. In particular, for a larger α MLT the lower boundary of the convection zone is located deeper in the interior. Therefore, for stars with lower (higher) masses, a variable mixing length parameter with stellar parameter will increase (decrease) the depth of the convection zone. As a consequence one can expect that the convective mixing will be enhanced (reduced) for less (more) massive stars in stellar evolutionary calculations. This may influence, for example, the depletion and burning of Li in low-mass stars.

Comparison with observations
Observations provide an opportunity to constrain free parameters in theoretical models. Bonaca et al. (2012) attempted to calibrate the mixing length parameter parameter from Keplerobservations of dwarfs and sub-giants (90 stars). Employing the usual scaling relations for the frequency of the maximal oscillation mode power, ν max , and the large frequency separation, ∆ν (see, for example, Huber et al. 2011), in connection with T eff and [Fe/H] from spectroscopic observations, they estimate mass and radius of the observed objects. Then, from a grid of stellar evolutionary tracks computed with different α MLT values, they selected the one matching the inferred stellar parameters. Bonaca et al. (2012) derived an average mixing length parameter of 1.60 from the observations, being in general lower than their solar-calibrated value with 1.69, which resulted from the 1D models without the comparison with observations. We compare the (linear) functional fit of α MLT derived in Bonaca et al. (2012), with stellar parameters to our own results in Fig. 10. We compare the calibration resulting from their complete data set. They also derive a fit for a subset of dwarfs, which, however, is quite restricted in the range of stellar parameters and quite different from the fit for the full sample. Their determined solar mixing length parameter is α ⊙ MLT = 1.59, which is 20 % smaller than our result of 1.98. However, we remark that differences between our and their EOS and opacity will have an important impact on α MLT , therefore, the comparison between absolute values of α MLT are limited. Interestingly, the variation with T eff for a given log g and [Fe/H] is rather similar apart from an almost constant offset. For different log g and [Fe/H], we find significant systematical differences (see Fig. 10). The values for dwarfs are in general smaller by up to ∼ 20 − 40 % depending on gravity and metallicity, while the giants are greater by the similar amount. The comparison is made more difficult by the fact that even the full sample of Bonaca et al. (2012) is rather limited in log g, and biased towards dwarfs. Additionally, the input physics (EOS and opacity) of their models deviates from ours. The authors themselves mention the absence of strong correlations with log g, their restricted range in [Fe/H], the discrepancies to the results by Ludwig et al. (1999) and Trampedach (2007), and the fact that α MLT effectively compensates for everything else that influences T eff .
Our mixing length parameters differ also significantly from the spectroscopical findings by Fuhrmann et al. (1993), who concluded that one would need an α MLT with very low values with ∼ 0.5, in order to properly fit hydrogen lines for various stars with the resulting temperature stratifications. This, however, can be explained completely by the fact that here only the outermost convective layers are traced, which are not tested with our method for inferring α MLT from the adiabatic structure at the bottom of the convection zone, and that the mixing length parameter is indeed depth-dependent (see Sect. 4.2). This was already verified by Schlattl et al. (1997).

Deriving the mass mixing length
In the following, we denote the temporal and spatial averaged thermodynamic quantities with . . . , which depict only the zdependence. Then, the momentum equation for a stationary system yields This equation states that a given mass stratification (ρg) has to be supported by the joint thermodynamic (p th ) and turbulent pressure (p turb = ρv 2 z ) forces, in order to sustain equilibrium. Since the vertical velocity, v z , appears here, we solve for the latter and get v z = g − p th / ρ ∂ z ln p th ∂ z ln ρ + 2∂ z ln v z .
Then, similar to the temperature gradient, ∇ = d ln T /d ln p tot , we introduce the notation for the gradient for a value X, however, instead of the total pressure it is scaled by the thermodynamic pressure scale height, and we can rewrite the vertical velocity to This analytical exact equation depicts the correlation of the vertical velocity with the gravity and pressure stratification, as well as the gradient of the density and the gradient of the vertical velocity itself in the hydrodynamic equilibrium. Now, we consider the gradient of the absolute vertical mass flux, j z = ρv z , for the up-or downflows (due to conservation of mass, the mass flux of the upflows, j ↑ z , equals the mass flux of the downflows, which indicates the length, over which the up-or downflow has changed by the e-fold, where the length scale is expressed in pressure scale heights. Trampedach & Stein (2011) introduced the mass mixing length as the inverse vertical mass flux scale height, i.e. l m = ∂ z ln | j ↑↓ z | −1 , which is in concordance with the gradient of the vertical mass flux with l m = H P /∇ j z . Furthermore, we define the mass mixing length as the inverse gradient of the vertical mass flux, and we can decompose the gradient of the vertical mass flux into its components and find which states that the mass mixing length is the inverse sum of the changes in the density and vertical velocity gradients. We note that this definition is the same as introduced by Trampedach & Stein (2011). Finally, we can identify now the mass mixing length in the denominator of the vertical velocity (Eq. 1) and get the following expression This illustrates why the vertical velocity depends on the mass mixing length, similar to the MLT velocity v MLT that depends on mixing length parameter with v MLT ∝ α MLT (see Eq. C.2).
To complete the comparison of the mass mixing length with the (MLT) mixing length parameter, we derive its dependence with the convective energy flux. The mean convective energy flux consists of the fluctuations of the total energy (ε tot = ε + p th /ρ + v 2 /2), which we depicted with f , and is carried by the mean vertical mass flux, i.e. where we assume that v z is the hydrodynamic velocity given in Eq. 1. We determine the divergence of the convective energy flux, i.e. ∂ z F conv , and solve for the total energy fluctuations and get Then, we can substitute the convective energy losses, ∂ z F conv , with the radiative cooling rate, − q rad , due to conservation of total energy, and we can identify the mass mixing length in the convective energy flux as well and yield This equation is basically the expression for the conservation of energy. Both of these equations for the velocity and the convective energy flux are just reformulations of the hydrodynamic mean field equations. To close this set of equations one still would need information about the gradient of the velocity and total energy fluctuation, as well as the radiative cooling rates.

Depth-dependence of the mass mixing length
In Fig. 11, we illustrate the horizontally and temporally averaged, depth-dependent mass mixing length for different stellar parameters, which we have derived from our 3D RHD simulations. In the convection zone, the mass mixing length has values around ∼ 2, while above the optical surface, α m exhibits lower values around ∼ 0.5. Fuhrmann et al. (1993) found that similar low values for the mixing length parameter α MLT yield better fits for Balmer lines, however, they also use large values for the temperature distribution parameter with y = 0.5 (see also App. C.2), and also the influence of α MLT becomes negligible towards the optical surface, where the Balmer lines form (see Fig. 1). Therefore, the agreement of the depth-dependent α m with their low values for α MLT might be just a coincidence. Furthermore, just below the optical surface (log p tot /p surf = 0) at the photospheric transition region, α m features a peak, which depends on the stellar parameters, in particular, for higher T eff , the peak in between increases, while in the convection zone it is the flatter. We remark that the peak in α m coincides with the location of the peak in the v z,rms . We included also the inverse gradient of density in the same figure with α m , demonstrating that the adiabatic value of α m in the convection zone is mainly contributed by the density gradient. We show also the gradients of the density and vertical velocity in Fig. 11, which are the both components of α m . One can depict that the variation of α m in the convection zone arises mainly due the different velocity gradients, since the density gradient converges always against very similar adiabatic values (γ ad ≃ ∇ −1 ρ ). For an monoatomic ideal gas with radiation pressure the adiabatic exponent is given by γ ad = (1 − ∇ ad ) −1 , and with ∇ ad = 1/4 one gets γ ad ∼ 4/3 (Kippenhahn et al. 2013, see). When one considers ∇ −1 ρ (see Fig. 11), it is close to 1.2. For a non-ideal gas differences due to non-ideal effects are to be expected. On the other hand, ∇ ρ is close to ∼ 0.8, therefore, similar to a value for an ideal gas with 3/4, while ∇ v z is between −0.4 and −0.15 (see also Fig. 14).
At the vicinity of the optical surface, the cooling rates are imprinted in the gradients for the density and velocity with a sharp transition. Towards the interior, the density is increasing due to the stratification and hydrostatic equilibrium, hence the gradient is ∇ ρ > 0, while the velocity is decreasing, and therefore ∇ v z < 0. The signs of ∇ ρ and ∇ v z are opposite due to the conservation of mass. In the interior, the stellar fluid gets compressed, and the velocity slows down, i.e. the convective energy is carried with slower, thicker mass flux. For higher T eff , the (negative) velocity gradient has a lower amplitude and therefore closer to zero, and Article number, page 9 of 15  a smaller amplitude of ∇ v z implies a steeper drop of the vertical velocity towards the interior, which also entails a larger maximum of the vertical velocity (see Fig. 14). The velocity gradient is reducing the density gradient, however, a lower sum of ∇ ρ and ∇ v z relates to a higher α m due to the inverse relation (see Fig. 12). Since the density gradient is very similar for different stellar parameters, the variation in α m arises mainly from the differences in the velocity gradient, therefore, we can relate the variation of the entropy jump with the variation of the velocity gradient, i.e. ∆s ∼ e ∇ vz , which was also concluded by Trampedach & Stein (2011) for the mass mixing length in an extended solar simulation.

Mean mass mixing length in the convection zone
We determined the mean mass mixing length of the convection zone below the optical surface between the location of the peak in the density scale height, i.e. max (∂ z ln ρ) τ>1 , and the bottom, however, avoiding bottom boundary effects on the vertical velocity. We performed linear fits of the density and vertical rms-velocity gradients by considering all snapshots, and from both gradients we determined the mean value of α m . The convection zones in the 3D simulations have to be extended enough, so that lower boundary effects on the vertical velocities are minimized, which is the case for most models, except some metal-poor giants that are slightly too shallow for matching α m properly.
The results for α m are displayed in Fig. 12, while in Fig. 14 we depict the mean values of the density and velocity gradients. From the solar simulation we determined α ⊙ m = 1.83, which is close to the solar mass mixing length by Trampedach & Stein (2011) with 1.76. Furthermore, the mass mixing length depicts qualitatively very similar systematic variations with stellar parameter, as we found for α MLT above. In particular, it decreases for higher T eff and [Fe/H], and lower log g, and the range in α m between ∼ 1.7 and ∼ 2.3 is qualitatively similar to the that of α MLT (see also Fig. 13). In general, we find qualitatively similar values for α m as found by Trampedach & Stein (2011), in particular, the dwarf models (log g = 4.5) have a similar slope with T eff . The variation of α m is also similar to the logarithmic inverse variation of the entropy jump. In Fig. 15 we compare α m with the logarithmic inverse entropy jump, and we find a similar tight correlation between the two, as we have found for the mixing length parameter α MLT (∆s) above (Sect. 3.3). The stronger deviations for the metal-poor giants originate from the fact that these models are slightly shallower, therefore, the match of the mass mixing length is perturbed due to the lower boundary effects on the velocity. We illustrate also the tight anti-correlation of the peak vertical rms-velocity with the mass mixing length in Fig. 15.
A comparison of the mass mixing length with the mixing length parameter calibrated with the entropy of the deep adiabatic convection zone and the entropy jump is shown in Fig. 16, and these correlate also well. The mixing length parameters are slightly higher than α m with a systematic offset around ∼ 0.1 and ∼ 0.2, which is in the case of α MLT (s bot ) smaller than for α MLT (∆s). This illustrates that the mixing length parameter in the framework of MLT has a physical background rooted in the mass mixing length (or inverse vertical mass flux gradient). However, since the MLT is incomplete a one-to-one correspondence between α MLT and α m would be hardly to be expected, nonetheless, the good agreement between the two is an interesting result.

Velocity correlation length
The physical interpretation of the mixing length parameter is conceptually the mean free path of a convective eddy, over which it can preserve its identity, before it resolves into its environment. In a real stratified hydrodynamic fluid the spatial twopoint (auto)correlation function of the vertical velocity can be regarded as the 3D analog of the mixing length parameter α MLT as proposed by Chan & Sofia (1987). The two-point correlation  C v z , v z , vs. the difference in the thermodynamic pressure, ∆ log P th , for the solar simulation. The different heights are indicated with a blue color-coding. Note the convergence of the correlation width in the convection zone against an adiabatic value. function for the values q 1 and q 2 is given by with σ i being the the standard deviation of q i and . . . depicts the spatial horizontal average. To derive the vertical correlation function of the convective velocity field, we consider the vertical component of the velocity field, v z , of a single fixed layer z 0 and derived the correlation functions for all other layers z i , i.e. C v z 0 , v z i , which is performed for twenty equidistant layers covering the whole vertical depth scale of the simulation box.
In Fig. 17 we show the two-point correlation function of the vertical velocity field, C v z , v z , derived for the solar simulation for the individual snapshots and then temporally av- eraged. For convenience, the correlation function is shown in differences of logarithmic pressure to the considered layer, ∆ log P th = log P th (z 0 ) − log P th (z i ). Then, the correlation function reaches always unity for z i = z 0 and exhibits a Gaussianlike shape. Furthermore, it is broader above the optical surface (p tot /p surf = 1), which is due to the rapid decline of the pressure scale height; while below the latter the width seems to converge against a certain adiabatic value (see Fig. 18). When one considers the width of the correlation function in geometrical depth, instead of pressure, then W (v z ) is constant around ∼ 0.6 Mm from the top down to ∼ 0.5 Mm and increases then with a fixed multiple (1.71) of the pressure scale height (see Fig. 18), which is the same as Robinson et al. (2003) found. The larger values for W (v z ) /H P above 0.5 Mm result from the lower H P .
The full-width-half-maximum (FWHM) of the two-point correlation function of the vertical velocity, C v z , v z , which we denote with W (v z ), gives an estimate on the size or length scale of the coherent vertical structures. The characteristic local length scale for the turbulent convective eddies can be determined with W (v z ). With the term vertical correlation length we refer to W (v z ). Similar to the mixing length, it is preferable to scale the correlation length by the pressure scale height, i.e. W (v z ) /H P , since the the latter is increasing towards deeper layers. Then, for the solar simulation (see Fig. 18) the convergent value for the width is W (v z ) /H P = 1.71. This means that the coherent vertical structures are extending 1.71H P in the convection zone, and this value is comparable to the mixing length parameter (α MLT = 1.94). Chan & Sofia (1987, 1989 found also a similar scaling of C v z , v z with pressure scale height in a 3D simulation for the Sun. For different stellar parameters we find a rather similar convergence of the correlation length of the vertical velocity in the convection zone (see Fig. 19).  We determined also the mean value of correlation length in the convection zone below log p tot / log p surf > 1 and close to the bottom boundary, the correlation function will increasingly overturn due to missing information in the deeper layers. Therefore, for the consideration of a mean correlation length we applied a cut at the bottom, where W (v z ) /H P starts to turn towards lower values (see Fig. 18).
The resulting mean values of W (v z ) /H P for different stellar parameters are depicted in Fig. 20, which are distributed between ∼ 1.5 and ∼ 1.8. This is an interesting result, since it confirms, to a certain extent, the physical motivation for the mix-ing length parameter, α MLT , namely the vertical velocity field, hence the vertical mass flux, correlates similarly with the pressure scale heights in the convection zone. However, the variation of W (v z ) /H P with stellar parameters (Fig. 20) is not as clear and systematical, as we found above for α MLT and α m (see Sect. 4).
Furthermore, in contrast to the mixing length parameter (α MLT and α m ) the correlation length seems to increase for higher T eff . The reason for this might be due to the horizontal granule size, which we found to decrease slightly for lower effective temperatures, since the pressure scale height is decreasing (see Paper I). Moreover, we find the granular cells, which can be highlighted with the temperature excess from the background, to feature distinct regular flat cylindric or pillar-like topologies.
Finally, we considered the correlation length of other variables, and we find that the horizontal velocity is rather similar, but with slightly lower correlation length with ∼ 1.4. Also, we find for the entropy, temperature and pressure values around ∼ 1.3, while for the density it is close to unity.

Conclusions
We have calibrated the mixing length parameter using realistic 3D RHD simulations of stellar surface convection by employing a 1D MLT stellar atmosphere code with identical microphysics. The calibration was achieved by varying the mixing length parameter and matching the adiabatic entropy value of the deeper convection zone, s bot , or alternatively matching the entropy jump, ∆s. In both ways we find the mixing length to decrease for higher T eff and [Fe/H], and lower log g. The mixing length varies in the range of 1.7 − 2.3 for α MLT (s bot ) and ∼ 1.8 − 2.4 for α MLT (∆s), and will lead to differences by up to ±20 % in α MLT depending on the stellar mass. This changes the stellar interior structure by extending or shortening the depth of the convection zone and thus the stellar evolution; we intend to return to how in detail a realistic α MLT will impact basic stellar evolution predictions in future studies.
Furthermore, we derived from the hydrodynamic mean field equations (for the first-time) a physical motivated connection of the mass mixing length, which is the inverse of the vertical mass flux gradient, with the mixing length. We determined the mass mixing length, and find that it varies qualitatively similar to the mixing length in the range of 1.6 − 2.3. The mass mixing length exhibits also a depth-dependence and drops above the surface to lower values around ∼ 0.5, which is in concordance to previous findings from observations. Finally, in both cases, the mass mixing length and mixing length, we find a strong correlation with the logarithmic inverse of the entropy jump for different stellar parameters, i.e. α MLT ∼ − ln ∆s. Finally, we derived also the vertical velocity correlation length, which features similar values to the mixing length with approximately ∼ 1.6 − 1.8 of pressure scale height, however, the dependence with T eff is inverted, i.e. the correlation length decreases with T eff .
To summarize the importance of our work: we can finally remove the free parameters inherent in MLT and also avoid having to use solar calibrations for other stars.   The entropy and superadiabatic gradient vs. depth (left and right panel, respectively) illustrating the influence of the additional MLT parameters ν, y and β (top, middle and bottom panel, respectively), the latter with the depth-independent v turb = 1 km/s. The mixing length is kept fixed at α MLT = 1.5. We included also the standard values of β = 0, ν = 8 and y = 0.076 (dashed lines). Shown is the case for solar parameters. reduction in α MLT , i.e. ν ∝ s bot . On the other hand, y enters in the (nonlinear) convective efficiency factor, Γ, for the superadiabatic excess (see Eq. C.4), and therefore y is correlated with α MLT in a more complex way.
Considering a variation of the three additional parameters in the computation of the solar 1D model, we notice that the adiabatic entropy value of the deep convection zone is altered significantly (see Fig. C.1). Furthermore, both parameters ν and y also change the entropy jump and the superadiabatic temperature gradient, ∇ sad , and in particular, its maximum of ∇ sad . The effect of the variation of y on the entropy stratification is similar to that by α MLT (see Fig. 1). However, the entropy of the deep convection zone exhibits a more nonlinear dependence with the y parameter. The increasing turbulent pressure with higher β changes the stratification only slightly, but shifts the location of the maximum of ∇ sad to the deeper interior. Towards the optical surface the influence of the MLT parameters is diminishing, as should be expected due to decreasing convective flux. A fine-tuning of β, ν and y is only useful, when these parameters introduce an independent influence to the mixing length, since otherwise its effects can be summarized in α MLT solely.