VARIATION WITH ALTITUDE OF THE DROP-SIZE DISTRIBUTION IN STEADY LIGHT RAIN

By application of the stochastic coalescence equation to evolving populations of cloud droplets and raindrops, we show that in light rain there is a range of drop sizes for which, to a good approximation, raindrops grow only by cloud accretion and are neither created by autoconversion nor destroyed by breakup. In these conditions, the differential equation that describes the one-dimensional steady-state form of the drop-size distribution may be solved analytically, using plausible assumptions about the cloud water content and the raindrop collection efficiency. The solution gives the drop-size distribution as a function of distance below a reference level, from which we can calculate radar-measurable quantities such as the reflectivity factor and the Doppler spectrum. Recent radar data from the Hawaiian Rainband Project provide the opportunity for a close comparison of the steady-state accretion theory with observations. An example is presented of a steady situation in which, over a limited range of altitude, the observed vertical profiles of reflectivity and Doppler velocity do agree reasonably well with the theory. Other observations in equally steady-appearing conditions, however, are not in accord with the theoretical predictions. If the parameters of the model are chosen to fit the velocity profile, the theory then tends to predict reflectivity gradients stronger than those actually observed. Similarly, if the parameters are chosen to fit the reflectivity profile, the predicted increase of Doppler velocity with distance fallen is less than what is observed. Effects of evaporation can be included in the steady-state model, and are shown to explain the sense of the discrepancy but not its magnitude. The conclusion is that the one-dimensional model, possibly because of its inability to account for horizontal advection, is usually inadequate to explain the observations.