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

Abstract

Abstract 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, vertical velocities, surface tension, and condensation coefficient. At large updraft vertical velocities, the activation time and maximum supersaturation are the products of the power laws of only two variables—CCN concentration and vertical velocity—and are independent of the CCN physicochemical properties. The first limit is a generalization of the Twomey power laws, with Twomey’s coefficient CT and index k expressed via CCN physicochemical properties; the other three limits are new. The accuracy and regions of validity of these limits are determined by comparison with the exact numerical solution to the supersaturation equation. These solutions can be used for parameterization of drop activation in cloud and climate models and for control of numerical solutions. An advantage of this method is that it does not require running parcel models, and the drop concentrations can be obtained from lookup tables or as simple interpolation among the limiting solutions for the instantaneous model parameters.