Effects of stochastic coalescence and air turbulence on the size distribution of cloud droplets

An open question in warm rain process and precipitation formation is how rain forms in warm cumulus as rapidly as it has sometimes been observed. In general, the rapid growth of cloud droplets across the size gap from 10 to 50 μm in radius has not been fully explained. Three aspects related to the air turbulence and stochastic coalescence are considered here in an attempt to resolve this open question. The first is the enhanced geometric collision rates caused by air turbulence. The second is the effect of air turbulence on collision efficiencies. The third is stochastic fluctuations and correlations in the collision–coalescence process. Rigorous approaches are developed to address these issues. Preliminary results indicate that turbulence could shorten the time for drizzle formation to about a half of the time needed for the same growth process based on hydrodynamic–gravitational mechanism alone. To address the effect of stochastic correlations, we derive and validate a true stochastic coalescence equation. It is hoped that this new mean field equation will be useful in the future to improve the deterministic kinetic collection equation.