Drag-Induced Transfer of Horizontal Momentum between Air and Raindrops

Abstract

A theoretical model for unsteady drag-induced transfer of horizontal momentum between air and raindrops in moderate to heavy rainfall is presented. The model accounts for a two-way coupling in which the relative horizontal motion between air and raindrops appears as a drag forcing in both the air and raindrop equations of motion. Analytical solutions of these coupled equations are obtained for the case of rain falling through (i) an initial step change in environmental wind, (ii) a uniform shear profile, and (iii) periodically varying vertical shears of various wavenumbers (a crude proxy for turbulent eddies). Formulas for the propagation (descent) speeds of the shear zones are obtained for (ii), (iii), and for the later stage of (i). However, these speeds are generally quite small?on the order of a few centimeters per second even for heavy rainfall. More importantly, the solutions of (i) and (iii) indicate that the drag interaction leads to a decay of the velocity gradients. A formula for the e-folding decay time of the periodically varying shear profiles indicates that at small wavelengths, the smallest decay times are found for the smaller drops, but at large wavelengths, the smallest decay times are found for the larger drops. The decay times decrease with decreasing wavelength, and approach a value equal to the reciprocal of the product of the rainwater mixing ratio and a drag parameter in the limit of vanishing wavelength. For parameters typical of moderate to heavy rainfall, the small-scale decay times are on the order of a few minutes.