Validity of the Finite-Difference Droplet Collection Equation

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Abstract
The validity of the finite-difference form of the droplet collection equation is examined. A derivation of this equation is presented in which emphasis is placed on the assumptions and approximations underlying the equation. The proper method of examining the validity of the equation is then discussed. This method involves comparing the dispersion in the behavior of the size distribution of a population with the predictions of the equation. The validity of the equation, when applied to a particular population subject to a particular type of collection probability, is quantitatively examined. The inferences drawn for this particular case are extrapolated to more general cases. It is shown, when the finite-difference collection equation is applied to populations with the naturally occurring “skewed-bell” type of size distribution, that its predictions at each time step are more valid for the small and large droplet sizes, at which there are, respectively, the greatest and least numbers of droplets. For intermediate droplet sizes, the equation is somewhat less valid. In addition, it is shown that the predictions of the collection equation at each time step are increasingly more valid, for the small droplet sizes, as the total number of droplets of all sizes in a population increases and, for the large droplet sizes, as the total number of droplets of all sizes decreases. It is concluded that the predictions of the equation are quite valid for small droplet sizes in atmospheric clouds, either for a single tune step or for longer time spans of integration. For laboratory clouds, the predictions of the equation are quite satisfactory for. large droplet sizes, and, perhaps, equally so for small droplet sizes.

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