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

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.