The role of convection, overshoot, and gravity waves for the transport of dust in M dwarf and brown dwarf atmospheres

Introduction

Attempts have been made to account for atmospheric dynamics in planetary atmospheres, whose models however cat not describe the convection cells and the resulting gravity waves (Marley et al. 2007; Fortney et al. 2006). However, local radiation hydrodynamics (RHD) models of the surface layers of the solar convection have been very successful in reproducing and analyzing the properties of the granulation (Nordlund 1982). In the meantime, various groups have developed similar codes to investigate the atmospheric flows on the sun and other stars (Steffen et al. 1989; Asplund et al. 2000; Skartlien et al. 2000; Stein & Nordlund 2000; Gadun et al. 2000; Robinson et al. 2003; Vögler 2004). Amongst others, these models can describe self-consistently the mixing of material beyond the classical boundaries of a convection zone, as demonstrated for instance for main-sequence A-type stars (Freytag et al. 1996) or for M dwarfs (Ludwig et al. 2002, 2006). A treatment of dust within a 3D simulation of the envelope of an AGB star was included by Freytag & Höfner (2008).

The aim of the current work is to extend the latter simulations into the regime of brown dwarfs, where dust clouds have a strong influence on the photospheric temperature structure, and to quantify the overshoot from the surface convection zone into the atmosphere.

Simulations with CO5BOLD

We computed a sequence of 2D RHD models for a gravity of $10^{5}$ cm s-2 (log $g$=5) and a range of effective temperatures from 900 K to 2800 K. The models have about 400$\times$300 grid points (see Table 1 for details). Most of them are restricted to two dimensions because we are unable to cover the prohibitively long sedimentation and mixing timescales in 3D: a 2D simulation in itself takes about one to three CPU-months to complete. However, a shorter run covering only several dynamical timescales and not trying to cover the longer mixing timescales is feasible in 3D (mt15g50mm00n06).

The code solves the coupled equations of compressible hydrodynamics and non-local radiation transport on a Cartesian grid with a time-explicit scheme. The tabulated equation of state accounts for the ionization of hydrogen and helium, and the formation of H2 molecules. The 1D hydrodynamics fluxes are computed with an approximate Riemann solver of Roe-type. Because the conditions in the cool objects are almost incompressible, the fluxes are combined non-split, i.e., the fluxes in both the vertical and horizontal directions are computed from the same state (and not after each other) and their contributions are added. In this way, the generation of spurious pressure waves is avoided, which may be produced by a split scheme in regions with large gradients but small divergence in the mass flux.

Condensation and evaporation are modeled as in Höfner et al. (2003), parameters and saturation vapor curve adapted to forsterite.

In the hydrodynamics module, monomers and dust densities are advected with the gas density. However, according to the terminal velocities given by the low-Reynolds-number case of Eq. (19) in Rossow (1978), a settling speed is added to the vertical advection velocity of dust grains, assuming instantaneous equilibrium between gravitational and viscous forces that act onto the grains.

One problem with modeling the dynamics of dust clouds is the span in timescales (short for dust formation and the wave period, long for dust settling and thermal relaxation) and spatial scales (small-scale dust clouds and possible global flows caused by rapid rotation). This is quite similar to simulations of weather patterns on Earth, where global wind systems and local cloud formation interact.

Another problem is the poorly known complex microphysics: a complicated chemical network of molecules with space- and time-dependent abundances can form dust by means of various processes, producing grains with different structures. The dynamical behavior and optical properties both depend on the grain type. Furthermore, depletion leads to a change in the gas composition that affects the equation of state and gas opacities. The current dust model in CO5BOLD is designed to reproduce the essential processes, but cannot account for all details that might possibly play a role.

1.2 Equation of state and opacities

The equation of state accounts for the ionization of hydrogen and helium, and the formation of molecular hydrogen. CO5BOLD can deal with the effects of ionization but not with an element composition that depends on space and time. Therefore, the depletion of elements is ignored for the equation of state: the formation of molecules has only a minor effect on e.g., the heat capacity as long as hydrogen exists in the form of $H_{2}$. However, molecules play a major role for the opacity, and the formation of molecules depends both on the abundance and depletion of elements. To take this into account, we derived the CO5BOLD gas phase opacities from monochromatic opacity tables, $\kappa(T,P,\nu)$, generated from detailed radiation transfer calculations with the general stellar atmosphere code PHOENIX (Hauschildt et al. 1997). We assume full sedimentation of dust from the gas phase: the removal of condensable material from the gas phase is considered assuming a solar elemental composition in full phase equilibrium at each temperature and pressure point (see Allard et al. 2001; Ferguson et al. 2005). This approximation is close to the conditions prevailing in: i) the lower atmospheric layers that are too hot for dust condensation, ii) the uppermost layers where the gravitational settling depletion is partially compensated by dynamical upwelling of monomers, and iii) in the cloud-forming layers as confirmed by observations as stated above. The monochromatic gas opacity table was averaged into 5 bins to minimize the computing time but retain the radiative equilibrium properties of the gas.

In contrast to the sophisticated treatment of the gas opacities, we use a simple formula for the dust opacities, which assumes that the large particle limit is valid for all grain sizes and treats scattering as true absorption. The dust opacity [cm-1] is

computed dynamically from the quantities as described for Eq. (1) in each cell of the simulated atmosphere and added to the gas opacity. We concentrate on forsterite grains (Mg2SiO4, 3.3 g/cm3) that are relatively abundant and provide the greatest contribution to the total dust opacities.

1.3 Boundary conditions

The top boundary is closed as well, partly to keep material inside. It has a damping zone of about 8 grid points where a strong drag force is applied. Damping at an open boundary did not appear sufficient to keep gravity waves with moderate Mach number (with peak values close to 1) from achieving additional growth to implausibly large amplitudes.

1.4 Initial conditions

The thermal structure of a start model is based on a classical 1D stationary stellar atmosphere model produced with PHOENIX assuming hydrostatic equilibrium and radiative plus convective (using the Mixing-Length Theory, Böhm-Vitense 1958) flux equilibrium. We preferred the dust-rich Dusty over the dust-free Cond models even for lower temperatures where the Cond models represent dust-free photospheres, because the resulting effective temperature of the CO5BOLD models agrees very well with the effective temperature of the Dusty models (CO5BOLD and PHOENIX models have per construction the same entropy in the deeper layers – not necessarily the same effective temperature). We interpolated the Dusty grid points to a finer grid with small or no variation in the grid spacing. To allow sufficient volume for the surface granules to form, we added several points at the bottom by integrating a hydrostatic stratification with constant entropy, taken from the bottom point of the Dusty model. At the top, we attached a few points with the internal energy value of the top point in the Dusty model, to maintain a sufficient distance between the top of the cloud layers and the top boundary of the computational box.

We enlarged the model in one horizontal dimension to 400 points, and imposed small random velocity fluctuations as seeds for convective instability. Initially, we set a constant fraction of the monomers plus dust mass density divided by the gas density, but which we reduced somewhat empirically in the uppermost layers to account for the partial depletion of material. The relative amount of material in the monomer bin is then determined by the saturation pressure of forsterite. Although the Dusty models assume hydrostatic equilibrium, there are small deviations from exact numerical equilibrium in the initial CO5BOLD models. These cause unwanted plane-parallel oscillations that we suppressed by a drag force in the initial phase of each simulation. To dampen these, we applied a strong drag force acting only on plane-parallel motions within the first 100 sec. In the following 9900 sec, we reduced the drag force to remove remaining plane-parallel residuals. For the remainder of the run (including the interval where we take averages from), we still have a very small but non-zero drag force that dampens plane-parallel vertical and horizontal motions on a timescale of 15 000 sec to suppress some modes that grew in early models over very long timescales.

Results of the simulations

To check the transition from the starting conditions to a quasi-stationary state, we consider time sequences of spatial-averaged quantities, such as temperature, rms velocities, and dust concentration. Starting from random initial fluctuations, the onset of convection takes a few 100 seconds, longer at lower effective temperatures. A statistically stable pattern develops after a few 1000 seconds. Wave amplitudes relax on somewhat longer timescales. In contrast, the thermal relaxation time – particularly of the deeper convective layers – is much longer. However, the thermal structure of these layers – an adiabat – remains essentially the same as in the initial model due to our choice of treatment of the lower boundary (keeping the entropy constant instead of imposing a certain flux). In this way, there is no need to cover the complete thermal relaxation time. The longest timescale to be covered is the relaxation time of the dust concentration that has an effect onto the temperature structure. Therefore, each simulation covers a few days of stellar time. The hydrodynamic time step is about 0.03 s and because of the relatively long radiative relaxation time, we perform multiple (typically 6) hydrodynamical sub-steps per radiation transport step.

Snapshots from our atmosphere simulations are presented in Figs. 1 through 8 while the complete videos are provided as supporting materialhttp://phoenix.ens-lyon.fr/papers/FreytagEtAl2009/. Figures 1 and 5 use pseudo-streamlines to visualize the flow field. Figures 2 (for a 1800 K model) and 6 (for a 1000 K model) display sequences of the typical granulation pattern, cool downdrafts occurring in a warmer environment. It is clearly separated from the atmosphere in the upper half of the box that shows inhomogeneities induced by gravity waves. The downdrafts are relatively narrower than in solar granulation (Ludwig et al. 2002, 2006). In the image sequences, the first pair is 20 s apart whereas the last snapshot is taken several minutes later.

The entropy profiles – averaged horizontally over constant height and in time — in Fig. 9 (top left) show a strong increase in the upper atmosphere, with only a minor drop at the top of the convection zone, and an almost flat distribution inside the convection zone. This is indicative of very efficient convection resembling typical conditions in the stellar interior.

2 Exponential overshoot

The typical magnitude of the velocity fields can be inferred from the plot of the rms of the vertical velocity versus pressure for various effective temperatures in Fig. 9 (middle panels). The convective velocities fall significantly from the peak value inside the convection zone (on the right) until the top of the unstable layers, and even further into the overshoot region. The scale height of exponentially decreasing overshoot velocities (Freytag et al. 1996; Ludwig et al. 2006) is so small that they do not induce significant mixing in the cloud layers about two pressure scale heights further up. Nevertheless, they are able to mix material across the boundary between stable and unstable layers.

3 Gravity waves

It is instead gravity waves that dominate the mixing of the atmospheric layers (the upper half of the models in Figs. 1 to 8) with periods of about 30 to 100 seconds and amplitudes that increase with height (Fig. 9). Most prominent is the fundamental g-mode, visible particularly in Fig. 6 as a significant brightening between heights of 0 and 20 km. In addition, there are several modes with larger horizontal and vertical wave number. These waves show up together with the first surface granules, well before the downdrafts “hit” the lower boundary. This indicates that the granular flow as such is responsible for the wave excitation, and not artifacts related to the way flows at the lower boundary are handled.

Figures 1 and 3 demonstrate the location of the dust clouds and the effect of the thermal inhomogeneities induced by the gravity waves onto the dust concentration. The generated small amount of vertical mixing (the wave motion is mostly reversible) is sufficient to balance gravitational settling of dust grains and allow dust clouds to form in the hotter models. In addition, dust concentration and cloud thickness are modulated by the waves because of the induced temperature fluctuations.

Atmospheric gravity waves are a common phenomenon. On Earth, they are known to form clouds over e.g., the US midwest plainsMesonet, I. E. 2007, Gravity Wave Movie: http://mesonet.agron.iastate.edu/cool/. Their energy release is involved in heating the exospheres of Jovian planets as observed for Jupiter from Galileo probe results (Young 1998).

Simulations of convection producing gravity waves in stellar conditions have a long tradition (Hurlburt et al. 1986). However, the quantitative estimate of the amplitude and the true detection of internal gravity waves can be a difficult task, even for the well-studied solar case (Belkacem et al. 2009). The detection of gravity modes that probe the solar core was announced by García et al. (2007).

The initial phases of simulation indicate that gravity waves are generated near the top of the convection zone (see e.g., Dintrans et al. 2005). The gravity waves are produced by non-stationary downdrafts “sucking” at the stable photospheric layers. In this way, the downdrafts are able to inject kinetic energy into the photosphere and to transport some material from there into the deeper convection zone. However, there are no obvious “events” of wave generation as for p-modes in the sun (Goode et al. 1998; Stein & Nordlund 2001) or in the simulations of gravity waves generated by an idealized convection zone embedded between stable layers by Dintrans et al. (2005).

The mixing efficiency of the waves increases rapidly with height – steeper than expected from the mere growths in amplitude caused by the increasing non-linearity. This could be the dynamical updraft mechanism responsible for the upwelling of N2 and CO gas observed via the enrichment of CO and depletion of CH4 and NH3 absorption bands in the spectra of T dwarfs (Saumon et al. 2006; Stephens et al. 2009; Geballe et al. 2009).

4 Convection within dust clouds

The fluctuations in the dust concentration in the 1800 K model in Fig. 1 are mainly induced by up and down motions of gravity waves that provide an inefficient mixing that balances the settling of dust grains. However, when the optical thickness of the dust clouds becomes sufficiently high, convective motions within the dust clouds start to develop and provide more efficient mixing of material (cf. the dust concentration of the 1000 K model in Fig. 7). However, in the snapshots the fluctuations and flows due to the waves somewhat obscure the dust cloud convection, whereas the overturning motions are clearly visible in movies and have a different signal in a $k$-$\omega$ diagram.

There are different intermittent processes: occasionally, material from the dust layers is dredged up to the layers with relatively low dust concentrations above the clouds. The grains quickly fall back. But monomers can remain a while, until they condense into dust at the top of the cloud deck. The cloud layer thickness varies not only with the wave on a timescale below one minute but also in irregular cycles on timescales of hours. The irregularity and amplitude increases with decreasing effective temperature.

During the initial phases of a simulations, a violent thin cloud convection zone develops for a limited time until the model is relaxed. This phenomenon relates to differences between our start model and the final outcome. However, on actual brown dwarfs large-scale flows might cause an imbalance in the local dust concentration that leads to a similar localized enhanced cloud activity.

5 Dust and stratification

In the top left panel in Fig.10, we show the location of the dust clouds (circles connected by vertical lines) relative to the underlying gas convection zone (located below the crosses).

The mixing processes within the dust cloud layers have different height regimes. At the bottom of the dust clouds, the temperature varies around the condensation value but there is little mixing. With our dust scheme, which assumes the presence of nuclei everywhere where dust or monomers are present, dust forms and evaporates during these temperature cycles (see the dust concentration at $z$$\sim$10 km in Fig. 7). The dust formation would be more difficult if new dust grains had to nucleate, because that would require some level of supersaturation. Within the clouds, material is mixed by gravity waves and/or convection (depending on effective temperature). The top of the clouds is sharp but inhomogeneous due to (sometimes braking) waves and cloud convection. Above the cloud layers, there are still mixing flows that try to equalize the concentration of monomers with height. The concentration value depends on the efficiency of mixing, dust formation, and dust settling in the cloud layers below (Fig. 9, bottom panels).

Dust clouds have a strong effect on the thermal structure (Fig. 9, top right panel): there is a fairly shallow temperature slope beneath the cloud layers with values of about 1600 K because of the greenhouse effect, a rapid drop within the clouds due to the large dust opacities, that can even drive cloud convection, low temperatures (with values around 1000 K and small variations) in the mostly dust-free upper atmosphere, and in some cases a small increase at the top of the models of about 100 K because of the dissipation of kinetic wave energy. At some height above the cloud, gravitational settling of dust grains becomes more efficient than mixing. The dust density drops rapidly and with it dust opacity and temperature, causing a rather sharp (but variable in space and time) upper boundary of the clouds. The concentration of dust and monomers (material that potentially can form dust) in Figs. 4 and 8 shows complete mixing in the convection zone, depleted layers at the top of the atmosphere (due to gravitational settling), and a partially mixed region in-between.

6 Effective temperature dependency

and for the logarithmic ratio of maximum convective velocity to wave amplitude extrapolated to this layer

For even cooler models, the amplitudes increase again. This increase is not because of the velocities in the underlying gas convection zone that decline steadily as effective temperature decreases. Instead, below 2000 K, the clouds have grown to such a large vertical thickness and density that cloud convection begins – with effects onto the atmospheric velocities and temperature structure that increase with further decreasing effective temperature. We attribute the rise in velocity for low-temperature models mainly to the emergence and growth of cloud convection.

The convective-radiative boundary becomes steeper, hence harder, with lower effective temperature, easing the gravity wave generation. In addition, there is a slight change in the topology of granules: in the hotter models, the downdrafts that delimit the granules are of roughly similar strength and merge occasionally, while in the cooler models just a few (2 or 3) “super downdrafts” dominate and absorb the smaller ones that form on top of the granules. This process occurs with higher frequency than would be expected if the merging type were the same as in the hot models – with possible consequences for the interaction between convection and waves.

Discussion

An early version of the models showed (in addition to the “normal” spectrum of gravity waves that occur as soon as convection sets in) after the simulation had run for a long time a slowly exponentially growing gravity wave in the fundamental mode. It grew until the code crashed because of too steep velocity gradients at the top of the box. It had relatively little effect on mixing, but induced temperature fluctuations modulating the dust concentration. Limiting the model depth, and using both a finer vertical grid and a smaller Courant number prevented an exponential growth of the mode. However, the mode itself is still present and quite prominent in the cooler models.

For a 1500 K model, we decreased our standard horizontal resolution by going from 400 to 300 horizontal grid points and found no noticeable difference in the mean structures, although the thin convective downdrafts and some small-scale cloud structures are somewhat less well resolved.

Simulations that are not yet complete and that will be presented in future publications include a sequence with other gravity values that shows no qualitative change in the outcome, although convective velocities, wave amplitudes, and dust-formation rate equations noticeably depend on gravity: the dependence of the flow field and the cloud thickness on effective temperature will be different at other gravities.

The exploration of other parameters such as grain size and other types of dust awaits further simulations, using a more detailed cloud model (multi-size-bin scheme, in preparation).

2 Comparison with previous simulations

The present calculations cover the actual parameter regime of dust harbouring atmospheres. They show that – in contrast to expectations motivated by hotter models – convective overshoot alone is not capable of keeping dust grains in the atmosphere. Overshooting motions decline more rapidly towards lower effective temperatures (Fig. 10). Our thorough investigation of the influence of the boundary conditions on the excitation of the gravity waves indicate that they are indeed intrinsic to the flow evolution proper and not a numerical artifact. The gravity waves’s ability to mix is indeed low but the waves remain in our hotter local models nevertheless the most efficient process, accompanied by dust convection in the case of heavy dust formation. All in all, we consider our present results consistent with the findings of Ludwig et al. (2002, 2006), but reassign the importance to the mixing by waves.

A complementary approach to ours is pursued by Helling et al. (2004): rather than on macroscopic scales (pressure scale heights, depth of the atmosphere, granular diameter), they concentrate on mesoscopic scales. Their 2D model is about as large (500$\times$500 m2) as one of our grid cells. They investigate the influence of driven turbulence, represented by a set of imposed pressure waves, on the formation of dust, particularly in regions where the temperature is slightly too high (T$=$2100 K) to allow nucleation in an undisturbed atmosphere. We agree with their findings that fluctuations in the thermodynamic quantities can have an influence on the dust formation process. However, we identify gravity waves and not pressure waves as important contributors to the mixing in BD and M dwarfs, in addition to with convection within thick clouds and convective overshoot very close to the underlying gas convection zone. An important parameter in their simulations is the Mach number of the induced acoustic waves, for which they assume values of about 0.1 in 1D models and 1 in 2D models. However, peak convective Mach numbers (taking vertical and horizontal velocities into account) in our models are between 0.1 and 0.01, and rapidly decrease in the overshoot regions where high-temperature dust might form. The amplitude of turbulent structures on the grid cell scale and below – that we obviously cannot resolve in our models – would be even smaller. And only a tiny fraction of the energy can be expected to be transformed into pressure waves under these nearly incompressible low-Mach-number conditions. Therefore, based on our simulations we cannot justify the assumption of almost sonic pressure waves in the atmospheres of brown dwarfs as made in Helling et al. (2004).

3 Diffusion coefficient estimate

One can model the mixing of material by macroscopic flows – on average – as a diffusion process. However, in the hydrodynamical models there is a correlation between the sign of the vertical motions (upward of downward) and the grain growth. At the same velocity amplitude, the mixing efficiency of convective overturning flows is also much higher than that of (nearly reversible) wave motions, causing errors in the translation from the rms velocities to the actual mixing efficiency. The mixing efficiency can however be estimated from the rms vertical velocity of our model sequence as in Eq. (5). And the diffusion coefficient can be estimated from the local vertical velocity and the pressure scale height $H_{p}$ as typical length scale via

Profiles for the diffusion coefficients according to the Eqs. (6) — (8) (replacing “$\propto$” by “$=$”) are displayed in Fig. 13. Additional crosses mark estimates based on the horizontal and temporal averages of vertical flux and density profiles of dust plus monomers. Because the flux is divided by the vertical derivative of the concentration, which can be very small, these values are not well-behaved everywhere.

The diffusion coefficient in brown dwarfs is not a “constant of nature” but depends on the physical process driving the mixing, the height in the atmosphere, and the effective temperature. A lower gravity would lead (via increased convective velocities and larger time and spatial scales) to high diffusion coefficients.

Scaling relations such as Eqs. (7) and (8) can serve as a first step in describing the diffusion properties in ultracool atmospheres in greater detail. In addition, they can easily be translated into classical atmosphere codes. In an earlier study, we have implemented a non-equilibrium chemistry model in the PHOENIX BT-Settl code, using the diffusion coefficients derived from the overshoot contribution of the RHD simulations only. In atmosphere models of the T1 brown dwarf $\epsilon$ Indi Ba based on this approach, we obtain characteristic values of 10${}^{7}\leq D\leq 10^{8}$ cm2 s-1 for the transition region from CO- to CH4-dominated carbon chemistry. We found the resulting non-equilibrium abundances of CO in the line-forming region, using an updated and more efficient reaction model than Saumon et al. (2006), to just slightly underestimate the observed CO line strengths in this benchmark transition dwarf (King et al. 2009). This indicates that somewhat more efficient mixing than provided by overshoot alone is required, thus supporting an additional contribution from gravity waves. The latest revision of the BT-Settl models beeing tested to include the effect of both dust mixing and CE departures, also find that Eq. (7) provides a close match to observational constraints.

In a more phenomenological approach Saumon et al. (2006) and Stephens et al. (2009) explored the effects of a constant eddy diffusion coefficient above the Schwarzschild boundary on observable departures from nitrogen and carbon equilibrium chemistry. They found best-fit values ranging from 102 to 106 cm2 s-1, although based on slower reaction rates (see above). Allowing for the differences in the reaction scheme, their results thus agree with the range of values that we find for the diffusion coefficient (crosses in Fig. 13) in the region spanning the top of the overshoot layers, the gravity-wave region, and the base of the cloud layers ($7.2>\log P>5.5$). Within the clouds, we find larger values.

4 Brightness variations

Our simulations show temporal intensity variations for a wide range of timescales from half a minute to hours. We find low-amplitude (less than 1%, see Fig. 15) and short-period variability (1 min) due to the gravity waves producing temperature fluctuations and modulating the dust density and the vertical thickness of the clouds. Relatively short-lived (several minute) phenomena are the occasional dredge-up or outburst of material above the clouds, where dust falls back rapidly while monomers remain much longer. In the coolest models, the gravity waves are less visible in the intensity fluctuations, which are dominated instead by aperiodic variations on the scale of hours with an amplitude of a few per cent. But these results barely reach the scales of the observed variability of L-type brown dwarfs, which is often aperiodic (scales of hours to days) and of low amplitude (mmags) as reported by Gelino et al. (2002). Still, the spatial intensity contrast found in the models (Fig. 15) with significant dust layers is significantly higher than the contrast that would be induced by granulation alone (the small contrast for models above about 2500 K).

Our results are only those for a patch of the surface and the variability amplitudes obtained will average over the rest of the surface. To determine the brown dwarf surface distribution of clouds, one must go beyond the present simulations to 3D models that are as large as possible and include rotation effects.

We have neglected the effects of rotation despite the rapid rotational periods of brown dwarfs (P $\leq 4$ hrs). The convective turnover time in the box is several minutes, which is short in comparison. Neither the surface granules, nor our rms velocities, should therefore be severely affected. But granules could move with a global meridional flow. Other global flows caused by rapid rotation may exist that could move the dust around.

A cloud cover disruption has indeed been suggested as a possible additional cause – the cloud layers sinking relative to the line forming layers – of the L-T spectral transition (Ackerman & Marley 2001) that could lead to weather phenomena and spectroscopic variability.

Conclusions

We have performed radiation hydrodynamics simulations with CO5BOLD of a sequence of brown dwarf atmospheres extending previous studies to lower temperatures. The numerical model includes a simple treatment of the formation and destruction of dust, as well as its gravitational settling and advection, and also the interaction with the radiation field.

We provide a fit to the rms velocity in the atmosphere that can be used to estimate the mixing. The convective velocities fall significantly from the peak value inside the convection zone to the top of the unstable layers, and even further into the overshooting region. However, the scale height of exponentially decreasing overshoot velocities is so small that they do not induce significant mixing in the cloud layers. Above a local minimum in the vertical velocities, gravity waves dominate in the hotter models with an amplitude and mixing efficiency that increase rapidly with height, enough to balance the gravitational settling of dust. The wave amplitude decreases with decreasing effective temperature. In the cooler models, the dust layers are thick enough to cause convection within the clouds leading to efficient mixing within the cloud layers.

References