The Stochastic Coalescence Model for Cloud Droplet Growth

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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 ... 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 ...

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