Mass Conservation Considerations in Analytic Representation of Raindrop Fragment Distributions

Abstract

Model formulation of drop breakup requires a set of analytic functions to describe the size distribution of water fragments that result from the collision of two raindrops of arbitrary diameter. The set of fragment distribution functions derived by Low and List in 1982 has provided the foundation for most of the recent modeling studies of raindrop collision. The formulas provide reasonably accurate approximations to histogram representations of laboratory data but produce distributions of drop fragments whose masses do not sum to the masses of the colliding drops. To correct the problem, new analytic expressions are derived using least squares fits with constraints on the total water mass content of the fragments and on the number of fragments produced by collision. Introduction of the mass conservation constraint reveals that, for drop collisions in which the mass of the larger colliding drop greatly exceeds that of the smaller drop, a certain feature is missing from both the histograms and the fits of Low and List. For such large-drop?small-drop collisions, fragmentation occurs only as filament-type breakup that results in very small satellites and leaves the large-drop mass nearly intact. From a theoretical point of view, then, the distribution of large-drop remnants must resemble a delta function rather than the broader distribution indicated by the Low and List histogram representation of the data. Mass-conserving fragment distributions derived for sheet- and disk-type breakup also differ from the distributions of Low and List but to a lesser degree. New fragment distribution functions for breakup have been computed, but only for collisions involving drops of widely disparate size. Preliminary tests show that when the new formulation is incorporated in the coalescence/breakup equation, numerical solutions produce equilibrium raindrop-size distributions that have a greater number of large drops but a 6% lower rainfall rate than have equilibrium distributions calculated in past studies.