The impact of fluctuations and correlations in droplet growth by collision–coalescence revisited – Part 1: Numerical calculation of post-gel droplet size distribution

摘要

The impact of stochastic fluctuations in cloud droplet growth isa matter of broad interest, since stochastic effects are one of the possibleexplanations of how cloud droplets cross the size gap and form the raindropembryos that trigger warm rain development in cumulus clouds. Mosttheoretical studies on this topic rely on the use of the kinetic collectionequation, or the Gillespie stochastic simulation algorithm. However, thekinetic collection equation is a deterministic equation with no stochasticfluctuations. Moreover, the traditional calculations using the kineticcollection equation are not valid when the system undergoes a transition froma continuous distribution to a distribution plus a runaway raindrop embryo(known as the sol–gel transition). On the other hand, the stochasticsimulation algorithm, although intrinsically stochastic, fails to adequately reproducethe large end of the droplet size distribution due to the hugenumber of realizations required. Therefore, the full stochastic descriptionof cloud droplet growth must be obtained from the solution of the masterequation for stochastic coalescence.In this study the master equation is used to calculate the evolution of thedroplet size distribution after the sol–gel transition. These calculationsshow that after the formation of the raindrop embryo, the expected dropletmass distribution strongly differs from the results obtained with thekinetic collection equation. Furthermore, the low-mass bins and bins fromthe gel fraction are strongly anticorrelated in the vicinity of thecritical time, this being one of the possible explanations for thedifferences between the kinetic and stochastic approaches after the sol–geltransition. Calculations performed within the stochastic framework provideinsight into the inability of explicit microphysics cloud models to explainthe droplet spectral broadening observed in small, warm clouds.