An Investigation of the Slope–Shape Relation for Gamma Raindrop Size Distribution

Abstract

The gamma drop size distribution (DSD) has been widely used in the meteorological community for years to model observed DSD. It has been found that the relation between the slope (?) and shape (?) parameters of the gamma DSD can be empirically described by a polynomial of second degree. In this article, on the basis of disdrometer-measured DSDs from seven independent precipitation events associated with different weather systems, an empirical ??? relation that is slightly different from those reported by other scientists is obtained by best fitting a quadratic polynomial to observed data. In addition to the empirical relation, a ??? relation is derived based on theoretical relations between gamma DSD moments and ? and ?. It is shown that the derived ??? relation is independent of the order of the moment of the gamma DSD. The key factor dominating the ??? relation is the ratio of the number density parameter N(Dm) to total number density of the raindrop M0, where Dm is the mean diameter of the DSD. It is further shown that the skewness and the variance of the DSD determine the magnitude of the ratio N(Dm)/M0 that governs the slope of the ??? relation. A comparison between the derived and the empirical ??? relations shows that their behaviors are very similar, especially for large rainfall rates characterized by smaller ? and ? values. Moreover, the ratio N(Dm)/M0 bears a weak relation to the rainfall rate R. Nevertheless, the square of the ratio M0/N(Dm) is closely related to the ratio R/M0 and their relation can also be described by a second-degree polynomial. Considering this property, the authors examine the validity of the various ??? relations by simulating the relations between R/M0 and [M0/N(Dm)]2. A comparison between observed and simulated results shows that the relation generated from the derived ??? relation bears the best resemblance to the observed one in both magnitude and shape.