Assessment of hydrometer collection rates from exact and approximate equations. Part I: A new approximate scheme

The collection equation is analyzed for the case of two spherical hydrometeors with collection efficiency unity and exponential size distributions. When the fall velocities are significantly different a more general form of the conventional Wisner approximation can be formulated. The accuracy of the new formula exceeds that of the Wisner approximation for all cases considered, except for the collection of a faster species by a slower species if the amount of the faster species is relatively small compared with that of the slower species. The exact solution of the collection equation is then rederived and cast into the form of a power series involving the ratio of the two characteristic fall velocities. It is shown that the new formulation is a first-order correction to the continuous collection equation for hydrometeors with finite diameters and fall velocities. Based on the analysis, the implications for the behavior of both the exact collection equation and its representation in numerical models are discussed.