Stochastic coalescence in Lagrangian cloud microphysics

Stochasticity of the collisional growth of cloud droplets is studied using the super-droplet method (SDM) of Shima et al. (2009). Statistics are calculated from ensembles of simulations of collision-coalescence in a single well-mixed cell. The SDM is compared with direct numerical simulations and the master equation. It is argued that SDM simulations in which one computational droplet represents one real droplet are at the same level of precision as the master equation. Such simulations are used to study fluctuations in the autoconversion time, the sol-gel transition and the growth rate of lucky droplets, which is compared with a theoretical prediction. The size of the coalescence cell is found to strongly affect system behavior. In small cells, correlations in droplet sizes and droplet depletion slow down rain formation. In large cells, collisions between raindrops are more frequent and this can also slow down rain formation. The increase in the rate of collision between raindrops may be an artifact caused by assuming an overly large well-mixed volume. The highest ratio of rain water to cloud water is found in cells of intermediate sizes. Next, we use these precise simulations to determine the validity of more approximate methods: the Smoluchowski equation and the SDM with multiplicities greater than 1. In the latter, we determine how many computational droplets are necessary to correctly model the expected number and the standard deviation of the autoconversion time. The maximal size of a volume that is turbulently well mixed with respect to coalescence is estimated at V-mix = 1.5 x 10(-2) cm(3). The Smoluchowski equation is not valid in such small volumes. It is argued that larger volumes can be considered approximately well mixed, but such approximation needs to be supported by a comparison with fine-grid simulations that resolve droplet motion.