Turbulence effects on the collision kernel. II: increase of the swept volume of colliding drops

An equation is deduced which gives the relative velocity between drops falling in three-dimensional turbulent flow. The turbulence is assumed to be homogeneous and isotropic. In describing the flow, both the inertial and viscous turbulent ranges are taken into account. It is demonstrated that the inertia of drops leads to the formation of a significant relative velocity between them, which for small droplets can be greater than the gravity-driven velocity difference. For a turbulence dissipation rate of ε = 100 cm²s^-3, typical of small early cumulus clouds, the turbulence-induced relative velocity of drops with radii up to 30 μm is nearly twice the difference in their terminal velocities induced by gravity. At ε = 400 cm²s^-3, turbulence driven relative drop velocity for drop radii up to 50 μm can be five times as great as their terminal-velocity differences. Factors giving rise to the relative drop velocity are discussed. It is indicated that, for small drops, the inertial acceleration term dominates the terms for the vertical shear and the vertical shear of the inertial acceleration. Calculations of the evolution of the drop spectrum using the stochastic coalescence equation were carried out. In the expression widely used for collision kernels, the difference in terminal velocities was replaced by the root-mean-square difference in drop velocities. The results indicate that cloud turbulence influences the rate of generation of large drops significantly. For example, the formation of rain drops with radii of 100 μm starting from an initial narrow droplet-spectrum centred at about 10 μm takes 12 minutes for ε = 200 cm²s^-3 and about 24 minutes for ε = 50 cm²s^-3. When, during this period, turbulence is put to zero, no rain drops form.