# Analytical solutions for precipitation size distributions at steady-state

## TL;DR

This paper derives analytical steady-state size distribution solutions for rain and snow particles, linking microphysical and dynamic cloud properties to observed particle size distributions.

## Contribution

It introduces a gamma distribution model for precipitation size spectra based on the Liouville equation, connecting cloud parameters to particle size distributions.

## Key findings

- Size distribution shape depends on cloud liquid water path and updraft speed.
- Stronger updrafts increase large particle concentrations.
- The model relates ground and airborne measurements to cloud microphysics.

## Abstract

Analytical solutions are derived for the steady-state size distributions of precipitating rain and snow particles assuming growth via collection of suspended cloud particles. Application of the Liouville equation to the transfer of precipitating mass through size bins in a cascade yields a characteristic gamma distribution with a "Marshall-Palmer" exponential tail with respect to particle diameter. For rain, the principle parameters controlling size distribution shape are cloud droplet liquid water path and cloud updraft speed. Stronger updrafts lead to greater concentrations of large precipitating drops and a peak in the size distribution. The solutions provide a means for relating size distributions measured in the air or on the ground to cloud bulk microphysical and dynamic properties.