Beyond monoculture: Polydisperse moment methods for sub-stellar atmosphere cloud microphysics II. A three-moment gamma distribution formulation for GCM applications

TL;DR

This work extends sub-stellar cloud microphysics by introducing a three-moment framework based on a gamma size-distribution to capture polydispersity in cloud particles. It derives gamma-distribution–specific expressions for condensation, evaporation, Brownian coagulation, kinetic coagulation, and gravitational coalescence, including regime-appropriate Knudsen-number interpolation and a treatment of mathematical singularities via incomplete gamma functions. A 1D Y-dwarf KCl test demonstrates that polydispersity can significantly alter particle sizes and number densities, with effects depending on atmospheric gravity, and shows how the gamma framework can reproduce monodisperse limits as the distribution narrows. The methodology provides a self-consistent pathway to include distribution width evolution in GCMs, enabling more accurate predictions of cloud radiative properties in sub-stellar atmospheres. The study also highlights the potential for improved agreement with bin-resolved models and suggests future extensions to multi-component grains and alternative distributions.

Abstract

Context. Understanding how the shape of cloud particle size distributions affects the atmospheric properties of sub-stellar atmospheres is a key area to explore, particularly in the JWST era of broad wavelength coverage, where observations are sensitive to particle size distributions. It is therefore important to elucidate how underlying cloud microphysical processes influence the size distribution, in order to better understand how clouds affect observed atmospheric properties. Aims. In this follow-up paper, we aim to extend our sub-stellar atmosphere microphysical cloud formation framework from Paper I to include effects of assuming a polydisperse gamma particle size distribution, requiring a three-moment solution set of equations. Methods. We develop a three-moment framework for sub-stellar mineral cloud particle microphysical nucleation, condensation, evaporation and collisional growth assuming a gamma distribution. As in the previous paper, we demonstrate the effects of polydispersity using a simple one-dimensional Y-dwarf KCl cloud formation scenario, and compare the results with the monodisperse case. Results. Our three-moment scheme provides a generalised framework applicable to any size distribution with a defined moment generation expression. In our test case, we show that the gamma distribution evolves with altitude, initially broad at the cloud base and narrowing at lower pressures. We find that differences between the gamma and monodisperse cloud structures can be significant, depending on the surface gravity of the atmosphere. Conclusions. We present a self-consistent framework for including the effects of polydispersity for sub-stellar microphysical cloud studies using the moment method.

Figures (6)

• Figure 1: Gamma size distribution compared to the exponential, Rayleigh and monodisperse distributions for various values of $\nu$. The representative particle size is $r_{\rm c}$ = 1 $\mu$m and total number density is $N_{\rm c}$ = 1 cm$^{-3}$.
• Figure 2: Ratio of the population averaged Knudsen numbers with the monodisperse Knudsen number. Due to the gamma function properties, a pole occurs at values of $\nu$$\leq$ 1/3 for the number-weighted Knudsen number using Eq. \ref{['eq:Kn_N']} (blue solid line), leading to singularities at $\nu$$\leq$ 1/3. The incomplete gamma function formulation using Eq. \ref{['eq:Kn_N_fix']}, which accounts for a cutoff at small particle sizes, avoids the singularities in the gamma function (dashed blue line) and is valid across the $\nu$ range.
• Figure 3: Relative factors between the gamma distribution and monodisperse distribution for the condensation and evaporation rates (left) and collisional growth rates (right). The vertical dotted line shows the $\nu$ = 1 values, denoting an exponential distribution (Paper I). As $\nu$$\rightarrow$$\infty$, the rates tend towards the monodisperse limit. However, for coagulation in the Kn $\gg$ 1 regime, assuming $H$ = 0.85 does not recover the monodisperse limit, even for large $\nu$, while a value of $H$ = 1/$\sqrt{2}$ does recover the monodisperse limit
• Figure 4: KCl cloud structures using the log g = 3.25 (left) and log g = 4.25 (right) Y-dwarf temperature-pressure profiles from Gao2018 and assuming a constant $K_{\rm zz}$ = 10$^{8}$ cm$^{2}$ s$^{-1}$. The dashed lines show the monodisperse size distribution, the dotted lines the exponential distribution, and the solid lines the exponential size distribution results. The top panel shows the temperature-pressure profiles with the mass mixing ratio of the condensate, $q_{\rm 1}$, vapour, $q_{\rm v}$ and saturation point, $q_{\rm s}$ (pink dash-dot line). The middle panel shows the representative particle sizes, $r_{\rm c}$ and $r_{\rm p}$, and total number density, $N_{\rm c}$. The bottom panel shows the relative difference in $r_{\rm c}$ and $N_{\rm c}$ between the monodisperse and gamma cloud structure.
• Figure 5: Reconstructed gamma particle size distributions from the simulations performed in Section \ref{['sec:1D']} for the log g = 3.25 (left) and log g = 4.25 (right) cases. This shows the size distribution properties with pressure (colour bar) in the atmosphere. In general, the size distribution narrows with decreasing pressure, with the most broad distributions occurring near the cloud base.