The Modified Gamma Size Distribution Applied to Inhomogeneous and Nonspherical Particles: Key Relationships and Conversions

The four-parameter modified gamma distribution (MGD) is the most general mathematically convenient model for size distributions of particle types ranging from aerosols and cloud droplets or ice particles to liquid and frozen precipitation. The common three-parameter gamma distribution, the exponential distribution (e.g., Marshall–Palmer), and power-law distribution (e.g., Junge) are all special cases. Depending on the context, the particle “size” used in a given formulation may be the actual geometric diameter, the volume- or area-equivalent spherical diameter, the actual or equivalent radius, the projected or surface area, or the mass. For microphysical and radiative transfer calculations, it is often necessary to convert from one size representation to another, especially when comparing or utilizing distribution parameters obtained from a variety of sources. Furthermore, when the mass scales with Db, with b < 3, as is typical for snow and ice and other particles having a quasi-fractal structure, an exponential or gamma distribution expressed in terms of one size parameter becomes an MGD when expressed in terms of another. The MGD model is therefore more fundamentally relevant to size distributions of nonspherical particles than is often appreciated. The central purpose of this paper is to serve as a concise single-source reference for the mathematical properties of, and conversions between, atmospheric particle size distributions that can expressed as MGDs, including exponential and gamma distributions as special cases. For illustrative purposes, snow particle size distributions published by Sekhon and Srivastava, Braham, and Field et al. are converted to a common representation and directly compared for identical snow water content, allowing large differences in their properties to be discerned and quantified in a way that is not as easily achieved without such conversion.