Dynamics of precipitation zones

Abstract

A simple model is formulated to represent the dynamics of freely falling zones of thermodynamically inert and non-interacting precipitation sized particles in a constant density fluid. The time dependent Navier-Stokes equations of motion and continuity equations for the air and precipitation are numerically solved for two-dimensional flow in both slab and radial symmetry. Appropriate initial-boundary conditions are specified for the system. The zones' mean vertical motion, horizontal spreading, pressure support and effects of particle size distributions are investigated. Formulae are derived for radially symmetric zones of single sized particles relating the zones' vertical convective velocity and horizontal spreading velocity to an internal Froude number. Also, a formula for the zones' momentum range is derived in terms of the internal Froude number. Onset and duration of precipitation times are studied for four different initial conditions for multi-sized particle zones and were found to vary considerably from those calculated by Kessler (1967). It was noted in one particular case that particle separation for multi-sized particle zones was as large in the horizontal direction as in the vertical direction. An example calculation for an atmosphere with an exponential density profile indicates that a mean layer density for the air and a mean layer terminal velocity for the single sized particles are reasonable assumptions to make in studies of this type.