Stochastic Theory of the Size Distribution of Raindrops

Abstract: For over a century, raindrop size distributions have been a subject of extensive scientific study, typically described by models including the Marshall-Palmer exponential equation, gamma, Weibull, lognormal, and other mathematical functions. In this work, we present a theory that integrates deterministic principles from thermodynamics and fluid dynamics with stochastic elements to predict expected raindrop diameters and describe ground-level drop-size distributions. Importantly, our approach avoids assuming specific drop-size dispersion processes or relying on multi-variable empirical data fitting. We derive analytical equations for key raindrop parameters (e.g., median diameter, expected minimum and maximum diameters) and drop-size distributions as functions of rainfall intensity. Our theoretical predictions align well with extensive published experimental data, covering rainfall intensities from 0.4 to 40 mm/h across diverse global locations. Additionally, consistent with observations of super-large raindrops, our theory suggests a maximum ground-level raindrop diameter limit of around 10 mm. We establish analytical expressions for meteorological parameters like the Marshall-Palmer constant and total raindrop concentration, validated against empirical data. The theoretical approach presented here can find broad applications in climate modeling, sprays and aerosol dynamics, bubbles and particles, extraterrestrial rainfall, and paleoclimatology.