Theoretical expression for the autoconversion rate of the cloud droplet number concentration

[1] Accurate parameterization of the autoconversion rate of the cloud droplet concentration (number autoconversion rate in cm(-3) s(-1)) is critical for evaluating aerosol indirect effects using climate models; however, existing parameterizations are empirical at best. A theoretical expression is presented in this contribution that analytically relates the number autoconversion rate to the liquid water content, droplet concentration and relative dispersion of the cloud droplet size distribution. The analytical expression is theoretically derived by generalizing the analytical formulation previously developed for the autoconversion rate of the cloud liquid water content (mass autoconversion rate in g cm(-3) s(-1)). Further examination of the theoretical number and mass autoconversion rates reveals that the former is not linearly proportional to the latter as commonly assumed in existing parameterizations. The formulation forms a solid theoretical basis for developing multi-moment representation of the autoconversion process in atmospheric models in general.