TY - GEN A1 - Woitke, Peter A1 - Helling, Christiane T1 - Dust in brown dwarfs II. The coupled problem of dust formation and sedimentation N2 - In this paper, we quantify and discuss the physical and surface chemical processes leading to the formation, temporal evolution and sedimentation of dust grains in brown dwarf and giant gas planet atmospheres: nucleation, growth, evaporation and gravitational settling. Considering dust particles of arbitrary sizes in the different hydrodynamical regimes (free molecular flow, laminar flow, turbulent flow), we evaluate the equilibrium drift velocities (final fall speeds) and the growth rates of the particles due to accretion of molecules. We show that a depth-dependent maximum size of the order of $a_{\rm max}\!\approx\!1\,\mu{\rm m\,(upper\ regions)} \ldots 100\,\mu{\rm m\,(lower\ regions)}$ exists, which depends on the condensate and the stellar parameters, beyond which gravitational settling is faster than growth. Larger particles can probably not be formed and sustained in brown dwarf atmospheres. We furthermore argue that the acceleration towards equilibrium drift is always very fast and that the temperature increase of the grains due to the release of latent heat during the growth process is negligible. Based on these findings, we formulate the problem of dust formation coupled to the local element depletion/enrichment of the gas in brown dwarf atmospheres by means of a system of partial differential equations. These equations state an extension of the moment method developed by Gail\plus Sedlmayr\,(1988) with an additional advective term to account for the effect of size-dependent drift velocities of the grains. A dimensionless analysis of the new equations reveals a hierarchy of nucleation $\to$ growth $\to$ drift $\to$ evaporation, which characterises the life cycle of dust grains in brown dwarf atmospheres. The developed moment equations can be included into hydrodynamics or classical stellar atmosphere models. Applications of this description will be presented in a forthcoming paper of this series. T3 - ZIB-Report - 02-51 KW - dust formation KW - precipitation KW - brown dwarfs Y1 - 2002 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-7186 ER - TY - GEN A1 - Woitke, Peter A1 - Helling, Christiane T1 - Formation and structure of quasi-static cloud-layers in brown dwarf atmospheres N2 - In this paper, first solutions of the dust moment equations developed in [{\sl Woitke & Helling 2002}] 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. 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 nucleation occu in the upper atmosphere 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 up to radii $\approx\!0.3\,\mu{\rm m}\,...\,150\,\mu{\rm m}$, depending gas around the cloud base. The particles finally sink into layers which are sufficiently hot to cause their thermal evaporation. Hence, an effective transport mechanism for condensable elements exi considered solid/liquid material. In the stationary case studied here, this downward directed element transport by precipitating dust grains is balanced by an upward directed flux of condensable elements from the deep interior of the star via convective mixing (no dust without mixing). We find a self-regulation mechanism which leads to an approximate phase equilibrium ($S\!\approx\!1$) around the cloud base. The mass fraction of dust present in the atmosphere results be to approximately given by the mass fraction of condensable elements in the gas being mixed up. T3 - ZIB-Report - 03-11 KW - dust KW - precipitation KW - clouds KW - atmosphere Y1 - 2003 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-7338 ER - TY - GEN A1 - Helling, Christiane A1 - Klein, Rupert A1 - Woitke, Peter A1 - Sedlmayr, Erwin T1 - Dust formation in brown dwarf atmospheres under conditions of driven turbulence N2 - Based on the knowledge gained from direct numerical simulations which are only possible in the microscale regime, a concept of driven turbulence is presented which allows to enter the mesoscopic scale regime. Here, dust formation under stochastic hydro- and thermodynamic conditions is studied: constructively superimposed stochastic waves initiate dust formation by the creation of singular nucleation events. It, hence, results a varying mean grain size and dust density in space and time. The newly formed dust changes the thermodynamic behavior from almost isotherm to adiabatic and chemically depletes the gas phase. T3 - ZIB-Report - 03-07 KW - dust formation KW - turbulence modelling KW - stochastic boundaries Y1 - 2003 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-7294 ER - TY - GEN A1 - Helling, Christiane A1 - Klein, Rupert A1 - Woitke, Peter A1 - Nowak, Ulrich A1 - Sedlmayr, Erwin T1 - Dust in Brown Dwarfs IV. Dust formation and driven turbulence on mesoscopic scales N2 - Dust formation {in brown dwarf atmospheres} is studied by utilizing a model for driven turbulence in the mesoscopic scale regime. We apply a pseudo-spectral method where waves are created and superimposed {within} a {limited} wavenumber interval. The turbulent kinetic energy distribution follows the Kolmogoroff spectrum which is assumed to be the most likely value. Such superimposed, stochastic waves may occur in a convectively active environment. They cause nucleation fronts and nucleation events and thereby initiate the dust formation process which { continues until} all condensible material is consumed. Small disturbances {are found to} have a large impact on the dust forming system. An initially dust-hostile region, which may originally be optically thin, becomes optically thick in a patchy way showing considerable variations in the dust properties during the formation process. The dust appears in lanes and curls as a result of the interaction with waves, i.e. turbulence, which form larger and larger structures with time. Aiming on a physical understanding of the variability of brown dwarfs, related to structure formation in substellar atmospheres, we work out first necessary criteria for small-scale closure models to be applied in macroscopic simulations of dust forming astrophysical systems. T3 - ZIB-Report - 03-33 KW - substellar atmospheres KW - dust formation KW - turbulence Y1 - 2003 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-7555 ER -