Vertical Variation of the Steady-State Drop Spectrum in a One-Dimensional Rain Shaft

Abstract

Past work has provided thorough analysis of the coalescence/breakup process in a ?box model? setting in which the drop size distribution is assumed invariant with height. In this work, the analysis is extended to examine the coalescence/breakup process in a one-dimensional shaft model setting that allows vertical variation of the drop size distribution due to sedimentation. The objectives are to gain a better understanding of the rain process and to acquire the knowledge necessary to parameterize the shaft model solutions. When vertical variation is taken into account, the steady-state form of the model equation describes the rate of change in the drop size distribution with fall distance for a fixed input condition at the shaft top. This equation is formally quite similar to the box model equation describing temporal evolution of the drop spectrum, and many characteristics of the box model solutions carry over to the steady-state, shaft model solutions, but with fall distance replacing time as the independent variable. Both solutions, for example, approach the same trimodal equilibrium form. Some important differences do exist, however. Analysis of the box model and shaft model equations reveals that the roles of coalescence and breakup are reversed in determining the rate at which the solutions approach equilibrium, and that the final adjustment to equilibrium is slightly different in the two cases. Further comparison of the box and shaft models shows that the water mass and water mass flux reverse roles as conserved quantities. In spite of these differences, the strong similarities in the equations allow direct adaptation of a box model parameterization to describe the steady-state, shaft model solutions.