A Theory of the Size Distribution of Raindrops*

Rain, as a natural precipitation phenomenon, has captivated human curiosity for millennia. To gain a deeper understanding of rain characteristics, extensive research has been conducted on raindrop diameters through systematic experimental and theoretical studies. In this work (an advancement over an earlier model we shared in 2021) we combine a deterministic approach based on thermodynamics and fluid dynamics with a stochastic methodology. A key aspect of our approach involves utilizing energy balance, dimensional analysis, and scale analysis principles to ascertain the possible, impossible, and expected raindrop diameters as a function of rain intensity. By considering the first law of thermodynamics, accounting for the work of fluid dynamic forces, and augmenting it with a stochastic description of random disturbances originating from the raindrop ensemble and atmospheric flow, we can predict the functional form of ground-level size distributions of raindrops. Our results exhibit good agreement with numerous published measurements for various rain intensities across diverse global locations. The versatility of our approach holds promise for broad applications in climate models, sprays, aerosols, bubbles, particles, and even rainfall on other planets.