Newly Developed Multiparameter Bulk Cloud Schemes. Part II: A New Triple-Moment Analytical Warm Rain Scheme

The formation mechanism of warm rain remains poorly represented in existing bulk microphysical parameterization schemes (BMPs), primarily due to persistent uncertainties in autoconversion and accretion rates, as well as oversimplified treatments of cloud droplet and raindrop spectra. Hence, a novel triple-moment analytical warm rain scheme is developed in this study to address the aforementioned issues. A core advancement of the proposed scheme lies in the derivation of general analytical solutions for conversion rates of the key warm rain processes based on the stochastic collection equation (SCE) combined with a parameterized collection kernel. The cloud droplet and raindrop spectra are separately described by gamma distributions, with both unimodal and bimodal cloud droplet spectrum forms considered in the scheme. The scheme prognoses three moments: number concentration, cloud water content, and reflectivity factor. Their temporal variations analytically determine the parameter changes of gamma distributions and thus characterize spectral evolution. Evaluations of the new scheme against high-resolution bin simulations confirm its ability to accurately calculate conversion rates and naturally capture the abrupt threshold behavior during autoconversion, without reliance on empirical step functions. Sensitivity experiments indicate that autoconversion rates decrease with increasing separation radius and are strongly modulated by the spectral shape parameter. Further implementation of the new scheme into a 1.5D cumulus model for an observed marine shallow cumulus case shows that the scheme accurately simulates warm rain processes, and the use of a bimodal droplet spectrum significantly accelerates raindrop formation. Overall, the new scheme exhibits superior performance and physical consistency in representing warm rain formation compared with existing BMPs.