Simple Analytical Expressions for Steady-State Vapor Growth and Collision-Coalescence Particle Size Distribution Parameter Profiles

Abstract

This study derives simple analytical expressions for the theoretical height profiles of particle number concentrations (Nt) and mean volume diameters (Dm) during the steady-state balance of vapor growth and collision-coalescence with sedimentation. These equations are general for both rain and snow gamma size distributions with size-dependent power-law functions that dictate particle fall speeds and masses. For collision-coalescence only, Nt (Dm) decreases (increases) as an exponential function of the radar reflectivity difference between two height layers. For vapor deposition only, Dm increases as a generalized power law of this reflectivity difference. Simultaneous vapor deposition and collision-coalescence under steady-state conditions with conservation of number, mass, and reflectivity fluxes lead to a coupled set of first-order, nonlinear ordinary differential equations for Nt and Dm. The solutions to these coupled equations are generalized power-law functions of height z for Dm(z) and Nt(z) whereby each variable is related to one another with an exponent that is independent of collision-coalescence efficiency. Compared to observed profiles derived from descending in-situ aircraft Lagrangian spiral profiles from the CRYSTAL-FACE field campaign, these analytical solutions can on average capture the height profiles of Nt and Dm within 8% and 4% of observations, respectively. Steady-state model projections of radar retrievals aloft are shown to produce the correct rapid enhancement of surface snowfall compared to the lowest-available radar retrievals from 500 m MSL. Future studies can utilize these equations alongside radar measurements to estimate Nt and Dm below radar tilt elevations and to estimate uncertain microphysical parameters such as collision-coalescence efficiencies.