Analytical solutions to the collection growth equation: comparison with approximate methods and application to cloud microphysics parameterization schemes

A closed form solution for the collection growth equation as used in bulk microphysical parameterizations is derived. Although the general form is mathematically complex, it can serve as a benchmark for testing a variety of approximations. Two special cases that can immediately be implemented in existing cloud models are also presented. This solution is used to evaluate two commonly used approximations. The effect of the selection of different basis functions is also investigated. -Authors