Stochastic Coalescence in Lagrangian Cloud Microphysics

Coalescence of hydrometeors is commonly modeled using the Smoluchowski equation. It is an approximate equation that can be derived from the more fundamental stochastic description by neglecting correlations in the number of droplets of different sizes. In the last decade, number of Lagrangian methods for modeling cloud microphysics have been developed. Out of these methods, the super-droplet method (SDM) has proven to give the most accurate description of coalescence. It also has the advantage of capturing the stochastic nature of coalescence. We present a comparison of the coalescence algorithm of the SDM with the Smoluchowski equation and two more detailed methods: direct numerical simulations (DNS) and master equation. First, assumptions and approximations made in each method are compared. We show that a special type of the SDM (“one-to-one” simulations, in which each real droplet is represented by a single computational droplet) is at the same level of precision as the master equation. Then, results of the Smoluchowski equation are compared with the reference “one-to-one” simulations. It is shown that the Smoluchowski equation becomes valid for coalescence cells containing more than 10⁷ cloud droplets. It is not obvious if cells of this size can be considered well-mixed, which is one of the assumptions made in the Smoluchowski equation. Next, results of typical SDM simulations, in which a single computational droplet represents multitude of real droplets, are compared with “one-to-one” simulations. We assess the minimal number of computational droplets required in SDM to reproduce correct mean value and standard deviation of the autoconversion time.