The validity of the kinetic collection equation revisited – Part 3: Sol–gel transition under turbulent conditions

Abstract. Warm rain in real clouds is produced by the collision and coalescence of an initial population of small droplets. The production of rain in warm cumulus clouds is still one of the open problems in cloud physics, and although several mechanisms have been proposed in the past, at present there is no complete explanation for the rapid growth of cloud droplets within the size range of diameters from 10 to 50 μm. By using a collection kernel enhanced by turbulence and a fully stochastic simulation method, the formation of a runaway droplet is modeled through the turbulent collection process. When the runaway droplet forms, the traditional calculation using the kinetic collection equation is no longer valid, since the assumption of a continuous distribution breaks down. There is in essence a phase transition in the system from a continuous distribution to a continuous distribution plus a runaway droplet. This transition can be associated to gelation (also called sol–gel transition) and is proposed here as a mechanism for the formation of large droplets required to trigger warm rain development in cumulus clouds. The fully stochastic turbulent model reveals gelation and the formation of a droplet with mass comparable to the mass of the initial system. The time when the sol–gel transition occurs is estimated with a Monte Carlo method when the parameter ρ (the ratio of the standard deviation for the largest droplet mass over all the realizations to the averaged value) reaches its maximum value. Moreover, we show that the non-turbulent case does not exhibit the sol–gel transition that can account for the impossibility of producing raindrop embryos in such a system. In the context of cloud physics theory, gelation can be interpreted as the formation of the "lucky droplet" that grows at a much faster rate than the rest of the population and becomes the embryo for runaway raindrops.