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A gravity driven inverse cascade controls the size distribution of raindrops
The size distribution of raindrops follows a particularly simple form, measured over the last century. It is exponential, and the average diameter of raindrops increases with the intensity of the rain. However, this relationship is still poorly explained. The dependence of the mean diameter on intensity implies that the polydispersity of raindrops is controlled by collective effects, and not by the instability or stochastic evolution of individual drops. Here we show, based on first principles of hydrodynamics, that drop coalescence by gravity controls the size distribution of raindrops. Our theory adapts the concept of energy cascading through the scales of turbulence to the drop mass distribution. We derive the steady-state distribution attained when drops nucleate at a constant rate by solving for the condition of constant water mass flow through the scales, and compare it to existing experimental data.
A key aspect of the model is the effective collision cross section of the sedimenting drops : large drops fall faster than small ones and coalesce with them on their way. Paradoxically, the lubrication pressure due to the air film trapped between the drops diverges rapidly, preventing any collision. Here we improve the hydrodynamic description of collisions by considering both the long-range hydrodynamic interaction and the lubrication film that forms between the drops immediately before the collision. Two different mechanisms regulate the contact pressure divergence : the transition to a dilute gas regime in the lubrication film, when the gap is comparable to the mean free path of air, and the formation of a flow inside the drops by shear at their surface. We show that lubrication is responsible for a decrease in collision efficiency at the scale for which inertia and viscous dissipation are of comparable magnitude. Lubrication thus explains the relative stability of fogs and non-precipitating clouds formed of micrometer droplets. This opens the possibility to improve the description of cloud microphysics in atmospheric simulations.