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To improve the modeling of warm rain initiation, a two-moment bulk parameterization of the drop collection growth in warm clouds is developed by two steps: (i) its prototype is first derived based on the analytic solution of the stochastic collection equation (SCE) with the Golovin kernel, and (ii) the prototype is then revamped empirically to fit the numerical solution of SCE with the real hydrodynamic collection kernel, reaching the final version of the parameterization. Since the final version represents the self-collection of cloud drops explicitly, it replicates warm rain initiation well even when liquid water content (cloud-drop number concentration) is very low (high). It also replicates the autoconversion threshold and time delay of rain initiation via a small autoconversion rate.

A Two-Moment Bulk Parameterization of the Drop Collection Growth in Warm Clouds