Supersaturation and diffusional droplet growth in liquid clouds

The process of collective diffusional growth of droplets in an adiabatic parcel ascending or descending with the constant vertical velocity is analyzed in the frame of the regular condensation approach. Closed equations for the evolution of liquid water content, droplet radius, and supersaturation are derived from the mass balance equation centered with respect to the adiabatic water content. The analytical expression for the maximumsupersaturation S_max formed near the cloud base is obtained here. Similar analytical expressions for the height z_max and liquid water mixing ratio q_max corresponding to the level where S_max occurs have also been obtained. It is shown that all three variables S_max, q_max, and z_max are linearly related to each other and all are proportional to w^3/4N^-1/2, where w is the vertical velocity and N is the droplet number concentration. Universal solutions for supersaturation and liquid water mixing ratio are found here, which incorporates the dependence on vertical velocity, droplet concentration, temperature, and pressure into one dimensionless parameter. The actual solutions for S and q can be obtained from the universal solutions with the help of appropriate scaling factors described in this study. The results obtained in the frame of this study provide a new look at the nature of supersaturation formation in liquid clouds. Despite the fact that the study does not include a detailed treatment of the activation process, it is shown that this work can be useful for the parameterization of cloud microphysical processes in cloud models, especially for the parameterization of cloud condensation nuclei (CCN) activation.