A consistent rainfall parameterization based on the exponential raindrop size distribution

There exists an impressive body of experimental evidence confirming the existence of power law relationships between various rainfall related variables. Many of these variables (such as rain rate, radar reflectivity factor and kinetic energy flux density) have a direct relevance for hydrology and related disciplines (hydrometeorology, soil erosion). There is one fundamental property of rainfall which ties all these variables together, namely the raindrop size distribution. It is the purpose of this article to explain (1) that there exist two fundamentally different forms of the raindrop size distribution, (2) how various hydrologically relevant rainfall variables are related to both these forms, and (3) how the coefficients of power law relationships between such rainfall variables are determined by the parameters of these two forms of the raindrop size distribution. The classical exponential raindrop size distribution is used as an example of a family of raindrop size distributions. Three groups of rainfall related variables are considered, namely properties of individual raindrops (size, speed, volume, mass, momentum and kinetic energy), rainfall integral variables (raindrop concentration, raindrop arrival rate, liquid rainwater content, rain rate, rainfall pressure, rainfall power and radar reflectivity factor) and characteristic sizes (median-volume diameter, volume-weighted mean diameter and mean-volume diameter). Six different consistent sets of power law relationships between these rainfall related variables and rain rate are presented, based on different assumptions regarding the rain rate dependence of the parameters of the raindrop size distribution.