Abstract
Abstract. 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 on this topic rely on the use of the kinetic collection equation, or the Gillespie stochastic simulation algorithm. 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 adequately reproduce 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. 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 anticorrelated 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.
Highlights
Rain has been observed to form in warm cumulus clouds within about 20 min, calculations that represent condensation and coalescence accurately in such clouds have had difficulty producing rainfall in such a short time except via processes involving giant cloud condensation nuclei
In order to study the droplet size distribution after the formation of raindrop embryos, for systems with kernels relevant to cloud physics and arbitrary initial conditions, we must rely on numerical methods that are capable of solving the master equation (Eq 2)
Lushnikov (2004) demonstrated that right after the sol–gel transition, the particle mass distribution splits into two parts: the thermodynamically populated one with behavior described by the kinetic collection equation, and a narrow peak with a mass very close to the gel mass
Summary
Rain has been observed to form in warm cumulus clouds within about 20 min, calculations that represent condensation and coalescence accurately in such clouds have had difficulty producing rainfall in such a short time except via processes involving giant cloud condensation nuclei (with diameters larger than 2 μm). In order to study the droplet size distribution after the formation of raindrop embryos (sol–gel transition), for systems with kernels relevant to cloud physics and arbitrary initial conditions, we must rely on numerical methods that are capable of solving the master equation (Eq 2). We can address this problem through a detailed comparison of the droplet size distributions obtained from the stochastic description for a finite system with the master equation (Eq 2), and the deterministic approach for an infinite system by using the KCE (Eq 1), using the numerical algorithm reported in Alfonso (2015).
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