A New Representation of Collision-Induced Breakup of Raindrops and Its Implications for the Shapes of Raindrop Size Distributions

Abstract

Using results of laboratory experiments on collision-induced raindrop breakup, Low and List (LL) developed a parameterization describing the fragment size distribution (FSD) produced by collisions of raindrops. An equilibrium raindrop size distribution (ED) is approached when this parameterization is used in numerical models of steady rain. Since scant observational evidence of such EDs exists, the need for a careful examination of the parameterization's foundation is evident. Using LL's experimental observations, an alternate parameterization is developed that alleviates three shortcomings of the original scheme, namely, ensuring mass conservation, the use of adequate uncertainty analysis, and the use of a more physical basis for deriving parameterized relationships. FSDs generated by raindrop collisions are represented by combinations of lognormal, Gaussian, and modified delta distributions for each of the three breakup types (filament, sheet, and disk) observed. The mode, width, and height of these distributions are calculated for the 10 colliding drop size combinations used in the LL experiments; uncertainty estimates for these parameters are determined using a bootstrap method, a technique that randomly chooses results of individual collisions. Relations giving the mode, width, and height in terms of the diameters of arbitrary large and small colliding raindrops are then determined so that the FSD from any raindrop collision can be predicted. Simulations with a time-dependent box model are conducted using an exponential Marshall–Palmer distribution, with a rain rate R of 54 mm h−1 as input. A bimodal ED, with peaks at 0.26 and 2.3 mm, is approached numerically from the opposing forces of coalescence and breakup. The nature of the ED differs from that found in previous studies using the LL parameterization, which had three peaks at diameters of 0.26, 0.91, and 1.8 mm. Simulations that produced EDs from consideration of only specific breakup types showed that filament breakups were mainly responsible for the production of the peak at 0.26 mm, and that the small drop peaks associated with sheet and disk breakup had too small of an amplitude compared to the small drop peak produced by filament breakup and occurred at diameters separated too far from each other to produce a third peak in the ED generated from all breakup types acting simultaneously. There are substantial differences in total number of raindrops, N, for the ED associated with this parameterization (8.6 × 103 m−3) to the N associated with that of LL (4.1 × 103 m−3); however, differences in R (51.1– 50.5 mm h−1) and radar reflectivity factor, Z (4.7 × 104 cf. 3.5 × 104 mm6 m−3), are less significant. Large differences between Z and N values associated with the Marshall–Palmer distribution (8.0 × 104 mm6 m−3 and 4.0 × 103 m−3) also exist. Bimodal distributions with peaks at 0.26 and 2.3 mm are consistently realized in a series of Monte Carlo simulations, where fit coefficients are randomly chosen from the surface of equally realizable solutions, but the height of the small drop peak can vary from 5.1 × 103 to 1.2 × 104 m−3 per logarithmic coordinate, and that of the large drop peak from 4.2 × 102 to 6.7 × 102 m−3 per logarithmic coordinate. Although N can vary from 6.2 × 103 to 1.1 × 104 m−3 between simulations, variations in R (48.4 to 53.1 mm h−1) and Z (4.1 × 104 to 5.5 × 104 mm6 m−3) are less significant. These results hence place a limit on the certainty with which the collision-induced breakup of raindrops can be predicted.