Evaluating errors in gamma-function representations of the raindrop size distribution: A method for determining the optimal parameter set for use in bulk microphysics schemes

Significant uncertainty lies in representing the rain droplet size distribution (DSD) in bulk cloud microphysics schemes and in the derivation of parameters of the function fit to the spectrum from the varying moments of a DSD. Here we evaluate the suitability of gamma distribution functions (GDFs) for fitting rain DSDs against observed disdrometer data. Results illustrate that double-parameter GDFs with prescribed or diagnosed positive spectral shape parameters m fit rain DSDs better than the Marshall–Palmer distribution function (with m 5 0). The relative errors of fitting the spectrum moments (especially high-order moments) decrease by an order of magnitude [from O(10) to O(10)]. Moreover, introduction of a triple-parameter GDF with mathematically solved m decreases the relative errors to O(10). Based on further investigation of potential combinations of the three prognostic moments for triple-moment cloud microphysical schemes, it is found that the GDF with parameters determined from predictions of the zeroth, third, and fourth moments (the 034 GDF) exhibits the best fit to rain DSDs compared to other moment combinations. Therefore, we suggest that the 034 prognostic moment group should replace the widely accepted 036 group to represent rain DSDs in triple-moment cloud microphysics schemes. An evaluation of the capability of GDFs to represent rain DSDs demonstrates that 034 GDF exhibits accurate fits to all observed DSDs except for rarely occurring extremely wide spectra from heavy precipitation and extremely narrow spectra from drizzle. The knowledge gained from this assessment can also be used to improve cloud microphysics retrieval schemes and data assimilation.