Cloud Droplet Size Distributions Governed by Condensation, Collision–Coalescence, and Sedimentation. Part II: Analytical Expressions for Size Distribution Shape and Asymptotic Behavior for Large-Droplet Tails

Abstract Droplet growth via condensation and collision–coalescence has been investigated in the limit of very weak coalescence growth: a collector-mode approximation emerges for coalescence growth that consists solely of droplets in the tail collecting droplets near the main mode of the cloud droplet size distribution and that does not deplete the mode. In this second part, the approximation is formalized, leading to a simplified form of the collision–coalescence term in the equation governing the evolution of the droplet size distribution. This form allows for analytical solutions for transient growth and for steady-state distributions representing a balance between condensation, collision–coalescence growth, and removal by sedimentation. The solutions lead to an expression for the drizzle growth time, formally, the finite time in which a droplet approaches infinite size. The drizzle growth time is related to a transition radius at which droplet growth switches from condensation-dominated to coalescence-dominated. The solutions also provide a dimensionless group that governs the steady-state distribution shape, which is referred to as the drizzle number. The solutions are checked by comparing to a Monte Carlo model developed in part 1, and it is found that the analytical solution is excellent when the drizzle number is much greater than unity, which is the typical situation for the microphysical conditions achievable in a convection–cloud chamber. The steady-state size distribution has the form of a modified gamma distribution, with a transition to a power-law tail at a sufficiently large radius. The solutions are not only formally valid for a convection–cloud chamber but can also be extended to shallow, mixed-layer clouds such as mixing fogs.