A gravity driven inverse cascade controls the size distribution of raindrops

In the last century, scientists have failed to explain the drop-size distribution of natural rain and in particular the increase of the mean raindrop diameter with the rain intensity. This robust dependence implies a control of raindrops polydispersity by collective effects, and not by the instability nor by the stochastic evolution of individual drops. Here we show from first hydrodynamics principles that gravity driven drop coalescence controls the size distribution of raindrops. Our theory adapts the concept of energy cascade across scales in turbulence to the distribution of drop mass. We derive the steady state distribution reached when drops nucleate at a constant rate by solving the condition of a constant water mass flux across scales and compare it to existing experimental data. A key aspect of the model is the collision efficiency of gravity settling drops — large droplets fall faster than small ones and coalesce with them in their path. We compute the collision efficiency, including the lubrication pressure between drops, whose divergence as the air gap vanishes is regularized by both non-continuum effects in the air film, and flow inside the drops. It opens the possibility to improve the description of cloud micro-physics in atmospheric simulations.