Stochastic Collection Equation: Invariant Evolution of the Cloud Droplet–Size Distribution

Abstract. The present paper presents a symmetry analysis of the stochastic collection equation (SCE), which describes the evolution of cloud droplets by their coalescence into larger sizes, and eventually to the size of rain droplets. As the main conclusion, the rain formation is not a simple consequence of the growth of the size of the droplets. When the given kernel is invariant under a scale transformation, the size distribution of the cloud droplets simply shift towards the larger sizes with time, asymptotically preserving its shape (i.e., invariance), and no separate peak in distribution identified as rain emerges. Findings with the symmetry analysis are supported further by numerical experiments.