Three-moment bulk microphysics schemes are attractive since they can determine the shape parameter a of the gamma particle size distributions (PSDs) from the predicted moments, but they are more computationally expensive compared with two-moment schemes. In this study, new diagnostic relations between a and the mean-mass diameter are developed for rain, graupel and hail hydrometeor categories based on the simulations for a severe hailstorm over eastern China using the Milbrandt and Yau (MY) three-moment scheme.
The new relations are introduced into the MY two-moment scheme and applied to the simulation of the same hailstorm at 1 km grid spacing. Different configurations of the MY schemes, including one-moment, two-moment with fixed a, with the original and new a  diagnostic relations, and with the new a  relation applied to hail only, and three-moment schemes, are also used to simulate the hailstorm. The new fully diagnostic-a two-moment scheme is found to improve the simulation of the general storm structure compared with the other two-moment schemes. It also reproduces the maximum estimated size of hail closer to that from the three-moment scheme while still retaining computational efficiency. Explicit prediction of hail by the new diagnostic-a two-moment scheme, including the surface accumulated hail mass, number and distribution is the most consistent with that of the three-moment scheme. The experiment with the new a  relation applied to hail only predicts a larger number of smaller hail and higher radar reflectivity, consistent with rain and graupel size over-estimation from excessive size sorting. Detailed microphysical budget analyses for hail, graupel and rain indicate that the new diagnostic relations yield substantial improvements in hydrometeor mass growth processes and size sorting representation.
 

Two-moment microphysics scheme,explicit hail prediction,diagnostic shape parameter