| Abstract
| - In this paper, first solutions of the dust moment equations developed in (Woitke & Helling [CITE]) for the description of dust formation and precipitation in brown dwarf and giant gas planet atmospheres are presented. We consider the special case of a static brown dwarf atmosphere, where dust particles continuously nucleate from the gas phase, grow by the accretion of molecules, settle gravitationally and re-evaporate thermally. Mixing by convective overshoot is assumed to replenish the atmosphere with condensable elements, which is necessary to counterbalance the loss of condensable elements by dust formation and gravitational settling (no dust without mixing). Applying a kinetic description of the relevant microphysical and chemical processes for TiO 2-grains, the model makes predictions about the large-scale stratification of dust in the atmosphere, the depletion of molecules from the gas phase, the supersaturation of the gas in the atmosphere as well as the mean size and the mass fraction of dust grains as function of depth. Our results suggest that the presence of relevant amounts of dust is restricted to a layer, where the upper boundary (cloud deck) is related to the requirement of a minimum mixing activity (mixing time-scale $\tau_{\rm mix}\approx 10^{\,6\,}$s) and the lower boundary (cloud base) is determined by the thermodynamical stability of the grains. The nucleation occurs around the cloud deck where the gas is cool, strongly depleted, but nevertheless highly supersaturated ( $S \gg 1$). These particles settle gravitationally and populate the warmer layers below, where the in situ formation (nucleation) is ineffective or even not possible. During their descent, the particles grow and reach mean radii of ≈$30\,\mu{\rm m}\,...\,400\,\mu{\rm m}$ at the cloud base, but the majority of the particles in the cloud layer remains much smaller. Finally, the dust grains sink into layers which are sufficiently hot to cause their thermal evaporation. Hence, an effective transport mechanism for condensable elements exists in brown dwarfs, which depletes the gas above and enriches the gas below the cloud base of a considered solid/liquid material. The dust-to-gas mass fraction in the cloud layer results to be approximately given by the mass fraction of condensable elements in the gas being mixed up. Only for artificially reduced mixing we find a self-regulation mechanism that approximately installs phase equilibrium ( $S \approx 1$) in a limited region around the cloud base.
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