Analytical Solutions of the Supersaturation Equation for a Warm Cloud

Abstract

pproximate solutions of the supersaturation equation are derived for a warm cloud. These solutions take account of the growth of the droplet radius but are only valid for small times. However, the validity of the solutions extends far enough to obtain reasonable estimates of the maximum supersaturation smax. It is shown that when the initial droplet radius and supersaturation are sufficiently small the scaling relation smax ? w3/4N?1/2 is obtained, where w is the vertical velocity and N is the droplet number density, in agreement with previous results. The range of validity of this result is discussed and other analytical expressions are derived when this result is not valid. It is shown that these analytical expressions generally agree well with numerical solutions.