The Stochastic Growth of a Rain Droplet

Abstract

Two approaches are used in this theoretical analysis of the stochastic growth by coalescence of a single, large cloud droplet (rain droplet) as it falls through cloud droplets whose sizes and spatial locations are specified statistically. In the first approach, moments are taken of a linearized, difference-kernel, coalescence equation. The first and second moments are used to predict the mean and variance of the statistical density function for the mass of the rain droplet. The results are compared with a recent numerical prediction. In the second approach, an appropriate Fokker-Planck equation is obtained. This equation is derived both from the linearized coalescence equation and from an elementary statistical model. An exact solution to the Fokker-Planck equation is derived and plots of the evolution of the statistical density function for the mass of the rain droplet are presented. These results are compared with those based on a Gaussian distribution.