Size distribution of drop diameters in natural rain*

For more than 100 years scientists have systematically studied diameters of raindrops and discussed which type of distribution function is best able to reflect the physical picture of natural rain. Currently the size distributions of raindrops are fitted to the experimental data using different laws, including Marshall–Palmer's equation, gamma, beta, Weibull, lognormal, exponential, and other distribution functions. Here we show from first principles that in equilibrium the mass and number fractions of raindrop diameters follow lognormal distribution law. From our theory we derived analytical equations describing the probability density function of raindrop diameters and the dependence of mass and count median diameters of raindrops on the rain intensity. Furthermore, we developed a map for the graphical determination of the range of possible diameters and expected median diameter of raindrops at different values of rain intensity. Our results demonstrate good agreement with experimental data obtained by different research groups at various ground locations around the globe for convective and stratiform rains with overall range of rain intensities 0.4-40 mm/h.