  Observationally, spectra of brown dwarfs indicate the presence of dust in
their atmospheres while theoretically it is not clear what prevents the dust
from settling and disappearing from the regions of spectrum formation.
Consequently, standard models have to rely on ad hoc assumptions about the
mechanism that keeps dust grains aloft in the atmosphere. We apply
hydrodynamical simulations to develop an improved physical understanding of the
mixing properties of macroscopic flows in M dwarf and brown dwarf atmospheres,
in particular of the influence of the underlying convection zone. We performed
2D radiation hydrodynamics simulations including a description of dust grain
formation and transport with the CO5BOLD code. The simulations cover the very
top of the convection zone and the photosphere including the dust layers for
effective temperatures between 900K and 2800K, all with logg=5 assuming solar
chemical composition. Convective overshoot occurs in the form of exponentially
declining velocities with small scale heights, so that it affects only the
region immediately above the almost adiabatic convective layers. From there on,
mixing is provided by gravity waves that are strong enough to maintain thin
dust clouds in the hotter models. With decreasing effective temperature, the
amplitudes of the waves become smaller but the clouds become thicker and
develop internal convective flows that are more efficient in mixing material
than gravity waves. The presence of clouds leads to a highly structured
appearance of the stellar surface on short temporal and small spatial scales.
We identify convectively excited gravity waves as an essential mixing process
in M dwarf and brown dwarf atmospheres. Under conditions of strong cloud
formation, dust convection is the dominant self-sustaining mixing component.
