The Stochastic Coalescence Model for Cloud Droplet Growth

Abstract

The stochastic coalescence model for droplet growth in warm clouds is analyzed, with a view to clarifying the theoretical foundations and significance of the well-known stochastic coalescence equation. It is suggested that the analysis of the model is most logically carried out in terms of a function P (n, m; t which is defined as the probability that the number of cloud droplets consisting of m molecules at time t will be n. A time-evolution equation for P (n, m; t is derived, and under certain stated assumptions it is deduced that: 1) the mean value of P (n, m; t with respect to n satisfies the stochastic coalescence equation; and 2) regardless of the initial conditions, the graph of P (n, m; t vs n will approach the Poisson shape as t ?∞ to with an estimable ?relaxation tirne.? The implications of these results for the stochastic fluctuations in the number of cloud droplets are examined. It is found that a distinction must be made between fluctuations in droplet concentration arising from the assumed stochasticity of the coalescence process, and fluctuations in droplet concentration arising from the hypothesis that the droplets are randomly positioned in the cloud; the former fluctuations are normally very much smaller than the latter.