Parameterization of Cloud Drop Activation Based on Analytical Asymptotic Solutions to the Supersaturation Equation

Toward improving parameterization of cloud droplet activation in cloud and climate models, the integro–differential equation for supersaturation is solved analytically for the algebraic size spectrum of the cloud condensation nuclei (CCN) that is equivalent to the lognormal spectrum. The analytical solutions are obtained for four limiting cases that are combinations of two different values of the updraft vertical velocity (small and large) and two different values of the condensation coefficient that correspond to pure and polluted cloud drops. The characteristics of the CCN can vary within each limit. Thus, these four limits and interpolation among them cover the vast majority of cloudy conditions. Analytical expressions are obtained for the time of CCN activation, maximum supersaturation, and the concentration of activated droplets. For small updraft vertical velocities, these quantities are the products of the power laws by six variables: CCN concentration, mean radius, soluble fraction, verti...