Sub-stellar objects exhibit photometric variability, which is believed to be caused by a number of processes, such as magnetically-driven spots or inhomogeneous cloud coverage. Recent models have shown that turbulent flows and waves, including internal gravity waves, may play an important role in cloud evolution. The aim of this paper is to investigate the effect of IGW on dust nucleation and dust growth, and whether observations of the resulting cloud structures could be used to recover atmospheric density information. For a simplified atmosphere in two dimensions, we numerically solved the governing fluid equations to simulate the effect on dust nucleation and mantle growth as a result of the passage of an IGW. Furthermore, we derived an expression that relates the properties of the wave-induced cloud structures to observable parameters in order to deduce the atmospheric density. Numerical simulations show that the $rho, p, T$ variations caused by gravity waves lead to an increase of the nucleation rate by up to a factor 20, and an increase of the mantle growth rate by up to a factor 1.6, compared to their equilibrium values. An exploration of the wider parameter space shows that in absolute terms, the increase in nucleation due to IGW is stronger in cooler (T dwarfs) and TiO2-rich sub-stellar atmospheres. The relative increase, however, is greater in warmer (L dwarf) and TiO2-poor atmospheres due to conditions less suited for efficient nucleation at equilibrium. These variations lead to banded areas in which dust formation is much more pronounced, similar to the cloud structures observed on Earth. We show that IGW in the atmosphere of sub-stellar objects can produce banded clouds structures similar to that observed on Earth. We propose a method with which potential observations of banded clouds could be used to estimate the atmospheric density of sub-stellar objects.
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