Beyond monoculture: polydisperse moment methods for sub-stellar atmosphere cloud microphysics I. Examining properties of the exponential distribution

Observational data provided by JWST instruments continue to challenge theories and models of cloud formation in sub-stellar atmospheres, requiring more sophisticated approaches in an effort to understand their spatial complexity. However, to date, most cloud microphysical models using the moment method for sub-stellar atmospheres have assumed a monodisperse size distribution, neglecting polydisperse properties. We aim to extend beyond the common assumption of a monodisperse size distribution and analyse cloud microphysical processes assuming an exponential distribution. We derive expressions for the zeroth and first moments of condensation/evaporation and collisional growth processes under the assumption of an exponential size distribution. We then compare the differences between monodisperse and exponential distribution microphysics using a simple one-dimensional (1D) column model applied to a Y-dwarf KCl cloud scenario. We find that adopting an exponential distribution modifies condensation/evaporation rates by a factor of $\approx$0.9 and collisional growth rates by factors of $>$1.1 (Kn $\ll$ 1) and $\approx$1.37 (Kn $\gg$ 1) for Brownian coagulation and $\approx$0.85 for gravitational coalescence, compared to the monodisperse case. In our specific test cases, we find maximal relative differences of $>$200\% in total number density and $>$40\% in mean radius of the cloud particles between the monodisperse and exponential distributions. Our framework offer a simple way to take into account polydispersity with an assumed exponential size distribution for sub-stellar atmospheric cloud microphysics using a two-moment method. In follow up studies, we will examine more complex distributions, such as the log-normal and gamma distributions, that require more than two moments to characterise self-consistently.