Parameterization of the Autoconversion Process.Part I: Analytical Formulation of the Kessler-Type Parameterizations

Abstract

Various commonly used Kessler-type parameterizations of the autoconversion of cloud droplets to embryonic raindrops are theoretically derived from the same formalism by applying the generalized mean value theorem for integrals to the general collection equation. The new formalism clearly reveals the approximations and assumptions that are implicitly embedded in these different parameterizations. A new Kessler-type parameterization is further derived by eliminating the incorrect and/or unnecessary assumptions inherent in the existing Kessler-type parameterizations. The new parameterization exhibits a different dependence on liquid water content and droplet concentration, and provides theoretical explanations for the multitude of values assigned to the tunable coefficients associated with the commonly used parameterizations. Relative dispersion of the cloud droplet size distribution (defined as the ratio of the standard deviation to the mean radius of the cloud droplet size distribution) is explicitly included in the new parameterization, allowing for investigation of the influences of the relative dispersion on the autoconversion rate and, hence, on the second indirect aerosol effect. The new analytical parameterization compares favorably with those parameterizations empirically obtained by curve-fitting results from simulations of detailed microphysical models.