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

Abstract

<p><strong>Abstract.</strong> The impact of stochastic fluctuations in cloud droplet growth is a matter of broad interest, since stochastic effects are one of the possible explanations of how cloud droplets cross the size-gap and form the raindrop embryos that trigger warm rain development in cumulus clouds. Most theoretical studies in this topic rely on the use of the kinetic collection equation, or the stochastic simulation algorithm (Gillespie 1975). However, the kinetic collection equation is a deterministic equation with no stochastic fluctuations. Moreover, the traditional calculations using the kinetic collection equation are not valid when the system undergoes a transition from a continuous distribution to a distribution plus a runaway raindrop embryo (known as the sol-gel transition). On the other hand, the stochastic simulation algorithm, although intrinsically stochastic, fails to reproduce adequately the large end of the droplet size distribution due to the huge number of realizations required. Therefore, the full stochastic description of cloud droplet growth must be obtained from the solution of the master equation for stochastic coalescence. <br><br> In this study the master equation is used to calculate the evolution of the droplet size distribution after the sol-gel transition. These calculations show that after the formation of the raindrop embryo, the expected droplet mass distribution strongly differs from the results obtained with the kinetic collection equation. Furthermore, the low mass bins and bins from the gel fraction are strongly anti-correlated in the vicinity of the critical time, this being one of the possible explanations for the differences between the kinetic and stochastic approaches after the sol-gel transition. Calculations performed within the stochastic framework provide insight into the inability of explicit microphysics cloud models to explain the droplet spectral broadening observed in small, warm clouds.</p>

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

“…Within this approach there is no mass loss, and the phase transition is manifested in the emergence of a giant particle that contains a finite fraction of the total mass of the system. Solutions in the post-gel regime were obtained analytically by Lushnikov (2004), Nakazawa (1993, 1994) and Matsoukas (2015), and numerically by Alfonso and Raga (2017).…”

“…As the coagulation process continues, fluctuations and correlations between droplets increase and the system reaches a critical point (Alfonso and Raga, 2017), where the largest droplets are no longer independent random variables, the Limit Theorem for Extremal Variables no longer applies, and the largest droplet distribution is no longer described by a Gumbel distribution. At later times, away from 190 the pseudo-critical region, the Gaussian contribution is the most important part of the largest droplet mass distribution.…”

“…the mass of the largest particle and maxMits ensemble mean over all the realizations.Even though the second moment of the distribution 2 ()M  diverges (see Eq.3) for the infinite system,210there is no divergence of the second moment for a finite system (with no critical behavior). For that case, the standard deviation for the largest particle mass, in estimating gelT in Inaba (1999),Alfonso et al (2008Alfonso et al ( , 2010Alfonso et al ( , and 2013 andAlfonso and Raga (2017). 215 example, for the system defined at the beginning of this section.…”

“…The recent studies that address this topic in cloud physics (Alfonso et al, 2013;Alfonso and Raga, 2017) analyze the problem from the theoretical and simulation point of view. The aim of the study reported here is to find observational evidence of gel formation, taking as a reference recent studies in 65 percolation theory and nuclear physics, which can shed some light on the gel (largest droplet) size distribution during the initial stage of precipitation formation.…”