Journal cover Journal topic
Atmospheric Chemistry and Physics An interactive open-access journal of the European Geosciences Union
Atmos. Chem. Phys., 17, 13509-13520, 2017
https://doi.org/10.5194/acp-17-13509-2017
© Author(s) 2017. This work is distributed under
the Creative Commons Attribution 3.0 License.
Research article
14 Nov 2017
Stochastic coalescence in Lagrangian cloud microphysics
Piotr Dziekan and Hanna Pawlowska Institute of Geophysics, Faculty of Physics, University of Warsaw, Warsaw, Poland
Abstract. 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 Vmix  =  1.5  ×  10−2 cm3. 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.

Citation: Dziekan, P. and Pawlowska, H.: Stochastic coalescence in Lagrangian cloud microphysics, Atmos. Chem. Phys., 17, 13509-13520, https://doi.org/10.5194/acp-17-13509-2017, 2017.
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Short summary
Raindrops form when small cloud droplets collide with each other. In most computer models of clouds, this process is described using the Smoluchowski equation. We compare the Smoluchowski equation with computer simulations in which each droplet within a small part of the cloud is modeled. We show, depending on the simulation setup, that the Smoluchowski equation can give overly slow or fast rain formation. This implies that many cloud models used do not correctly represent rain formation.
Raindrops form when small cloud droplets collide with each other. In most computer models of...
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