Mixed-phase clouds in a turbulent environment. Part 2: Analytic treatment

The coexistence of supercooled liquid and ice is thermodynamically unstable. When no external forcing is applied, the ice will grow at the expense of the supercooled liquid water, eventually removing it completely. When dynamical forcing is applied, it is possible to activate supercooled liquid water in the presence of ice. Analytic solutions are presented that provide an estimate of the fraction and amount of supercooled liquid water that can be found in pre-existing ice cloud affected by turbulence. The approach used is to simplify the equation representing the evolution of supersaturation to a stochastic differential equation. The solution to this equation provides an expression for the variance of the supersaturation with respect to ice as a function of the parameters describing the turbulence and the ice cloud. Given a supersaturation distribution with respect to ice, the mixed-phase cloud properties can be predicted, including mixed-phase cloud fraction, mean liquid water content and liquid water distribution. Large Eddy Simulation results from decametre-resolution simulations of mixed-phase cloud in a turbulent environment are in favourable agreement with the analytic estimates.