Gap in drop collision rate between diffusive and inertial regimes explains the stability of fogs and non-precipitating clouds

Abstract

Rain drops form in clouds by collision of submillimetric droplets falling under gravity: larger drops fall faster than smaller ones and collect them on their path. The puzzling stability of fogs and non-precipitating warm clouds with respect to this avalanche mechanism has been a longstanding problem. How can droplets of diameter around 10 $\mathrm {\mu }$ m have a low collision probability, inhibiting the cascade towards larger and larger drops? Here we review the dynamical mechanisms that have been proposed in the literature and quantitatively investigate the frequency of drop collisions induced by Brownian diffusion, electrostatics and gravity, using an open-source Monte Carlo code taking all of them into account. Inertia dominates over aerodynamic forces for large drops, when the Stokes number is larger than $1$ . Thermal diffusion dominates over aerodynamic forces for small drops, when the Péclet number is smaller than $1$ . We show that there exists a range of size (typically 3–30 $\mathrm {\mu }$ m for water drops in air) where neither inertia nor Brownian diffusion are significant, leading to a gap in the collision rate. The effect is particularly important, due to the lubrication film forming between the drops immediately before collision, and secondarily to the long-range aerodynamic interaction. Two different mechanisms regularise the divergence of the lubrication force at vanishing separation: the transition to a non-continuum regime in the lubrication film, when the separation is comparable to the mean free path of air, and the induction of a flow inside the drops due to shear at their surfaces. In the gap between inertia-dominated and diffusion-dominated regimes, dipole–dipole electrostatic interactions becomes the major effect controlling the efficiency of drop collisions.