Beyond monoculture: Polydisperse moment methods for sub-stellar atmosphere cloud microphysics

Context. Observational data provided by JWST instruments continue to challenge theories and models of cloud formation in substellar atmospheres, requiring more sophisticated approaches in the 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. Aims. We aim to extend beyond the common assumption of a monodisperse size distribution and analyse cloud microphysical processes assuming an exponential distribution. Methods. We derive expressions for the zeroth and first moments of condensation or 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 1D column model applied to a Y-dwarf KCl cloud scenario. Results. We find that adopting an exponential distribution modifies condensation or evaporation rates for different Knudsen number (Kn) regimes by a factor of ≈0.9 and collisional growth rates by factors of >1.1(Kn ≪ 1) and ≈ 1.37 (Kn ≫ 1) for Brownian coagulation and ≈0.85 for gravitational coalescence, compared to the monodisperse case. In our specific test cases, we find maximal relative differences of >200% in the total number density and >40% in the mean radius of the cloud particles between the monodisperse and exponential distributions. Conclusions. Our framework offers 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 shall examine more complex distributions, such as the log-normal and gamma distributions, which require more than two moments to be able to characterise self-consistently.