Application of Quadrature Method of Moments for Sedimentation and Coagulation of Raindrops

Abstract

The formation and sedimentation of raindrops is described by an integro-differential equation for the size distribution function (”population balance equation”). The numerical solution of this equation by discretization in spatial and property coordinates (spectral, i.e. diameter or mass coordinates) may be straightforward in principle, but is prohibitively expensive in practice. For applications such as numerical weather prediction (NWP), efficient approximations must be formulated. One promising approach is the quadrature method of moments (QMoM), which is quite well-known, e.g., in chemical engineering. With QMoM, a population of particles is described approximately by a few moments of its size distribution, for which transport equations are formulated. Moment transport velocities as well as source terms (due to evaporation/condensation or break-up/coagulation, etc.) are evaluated approximately by Gaussian quadrature. The method is quite flexible and does not make any particular assumptions on the functional form of the size distribution or the break-up/coagulation kernels. In this paper, QMoM is applied to sedimentation of raindrops, taking simultaneously the effects of coagulation into account. Only simple 1D (vertical ”rainshaft”) configurations are considered. This allows the computation of solutions of the population dynamics in a spectral resolution, which serves as a reference solution for validation studies. Results obtained with different coagulation kernel functions show that QMoM properly captures the essential physical behaviour, in general with good quantitative accuracy at very low computational costs.