An Approximate Analytical Model of the Vapor Deposition and Aggregation Growth of Snowflakes

Abstract

It is well known that snowflakes tend to distribute exponentially with respect to their melted diameter. This fact is used to formulate an approximate analytical model of the deposition and aggregation growth of snow in stratiform clouds. The model predicts the height evolution in a steady-state, vertically heterogeneous cloud of the slope and intercept parameters, N(h) and λ (h), of the size distribution of snowflakes which is assumed to be given by n(D,h)=N(h) exp[−λ(h)D], where h is the height in the cloud and D the snowflake diameter. Solutions for N(t) and λ(t) for a time-dependent spatially homogeneous cloud are also presented. Results from this technique compare well with numerical integrations for the case of perfect geometric coalescence of raindrops. This stratiform snow model predicts the existence of radar reflectivity-snowfall rate relations although, for this first-order model, there is fair agreement between theoretical and observed values. The model suggests that “equilibrium” snow size spectra owe their existence to the counteracting effects of deposition and aggregation growth.