Evolution of rain water profiles resulting from pure sedimentation: Spectral vs. parameterized description

The paper focuses on a comparison of solutions of the budget equations for specific moments of a hydrometeor size distribution, simplified to the problem of pure drop sedimentation, as following from models with a one-moment or a two-moment parameterization of microphysical processes and from a reference model based on spectral treatment of sedimentation. The solutions for the transient vertical profiles of liquid water content show remarkable differences in their spatial structure: starting from an idealized, discontinuous initial profile in form of a square wave, shock and rarefaction waves evolve in the parameterized models, while the reference solution describes a smooth distribution. These principle differences follow from the fact, that the budget equations for the moments in the parameterized models are of the form of quasi-linear advection equations, with the decisive nonlinearity arising from the parameterization relation for the sedimentation flux, whereas the reference solution follows from a linear partial differential equation. Hence, the quasi-linear shock effects have to be interpreted as a mathematical artifact of the parameterization assumptions. The differing evolution of the liquid water profiles finds expression in remarkable differences in the time series of surface precipitation rate for the spectral and parameterized models.