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

Abstract

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 When the runaway droplet forms, the traditional calculation using the kinetic collection equation is no longer valid, since the assumption of a continuous distribution breaks down.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.

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

“…multiplicative and hydrodynamic). Our results confirm the findings of Bayewitz et al (1974) that in systems of small populations the results of the kinetic deterministic equations approach may differ substantially from the stochastic means at the large end of the droplet size distribution.…”

“…In their pioneering studies using the stochastic framework, Marcus (1968) and Bayewitz et al (1974) solved the stochas- tic master equation (Eq. 2) for a constant collection kernel and a monodisperse initial droplet distribution.…”

“…The sum of the first two terms is the gain due to transition from other states, and the sum of the last two terms is the loss due to transitions into other states. This formulation was introduced in the pioneer works of Marcus (1968) and Bayewitz et al (1974), and was studied in detailed by Lushnikov (1978Lushnikov ( , 2004 and Tanaka and Nakazawa (1993). However, these studies only offer analytical results for a limited number of cases (with constant, sum and product kernels), for monodisperse initial conditions.…”

An algorithm for the numerical solution of the multivariate master equation for stochastic coalescence

“…After this moment, the true averages calculated from the master equation will differ from the averages obtained from Eq. (1) and there is a transition from a system with a continuous droplet distribution to one with a continuous distribution plus a giant cluster (Alfonso et al, 2013). After the sol-gel transition the KCE breaks down: the second moment of the size distribution diverges at the gel point and, as was remarked, the first moment decays, i.e., mass is not conserved.…”

The impact of fluctuations and correlations in droplet growth by collision–coalescence revisited – Part 2: Observational evidence of gel formation in warm clouds

“…Other research points to the supersaturation fluctuations resulting from homogeneous (Warner, 1969) and inhomogeneous mixing (Baker et al, 1980), which allow some droplets to grow faster by condensation in areas with larger supersaturation. Cooper (1989) found evidence of faster growth of the larger droplets due to the variability that results from mixing and random positioning of droplets during cloud formation.…”