The Collisional Problem of Cloud Droplets Moving in a Turbulent Environinent–Part I: A Method of Solution

Abstract

In the theoretical study of the evolution of a cloud droplet spectrum, the collision kernel function in the stochastic collection growth equation plays an important role in defining its characteristics. To evaluate this kernel function the collisional behavior of isolated pain of cloud droplets is investigated. This constitutes the collisional problem. This problem has been solved in the literature for the case of droplets moving under the action of deterministic forcing fields, such as gravity and electrical forces. Here, we will introduce a new approach to incorporate the case of droplets moving under the action of probabilistic forcing fields, such as turbulence. This has resulted in a more general definition of the collision kernel function. This new collision kernel is shown to be capable of representing not only the turbulent case, but to reduce to the deterministic definition of the kernel function given in the literature. The method of solution includes the numerical solution of the equations of motion for droplet pairs moving in a viscous turbulent environment and the sampling procedure to determine the probability curves of the kernel function as defined in this study. Errors committed in the evaluation of the probability curves, or for that matter, of the kernel function for both deterministic and probabilistic problems are analyzed. The general solutions are outlined.