Analytic upscaling of a local microphysics scheme. Part I: Derivation

In large-scale simulations of the atmosphere, microphysical processes occur on smaller scales than the grid-box size. If the processes are nonlinear, and the variability within a grid box is large, then ignoring the subgrid variability leads to inaccuracy. To avoid this inaccuracy, local microphysical formulas, valid at a point in space, may be upscaled to a large grid box. This may be done by integrating the local microphysical formula over the probability density function (PDF) that represents spatial subgrid variability. This paper will upscale local formulas proposed by Khairoutdinov and Kogan, which constitute a complete microphysics scheme for drizzle in marine stratocumulus. It is tractable to upscale this scheme because all formulas in it are power laws. The marginal PDFs will be assumed to have either a normal mixture or log-normal functional form. In Part II of this pair of papers, the upscaled formulas are implemented interactively in a single-column model and tested for a drizzling marine stratocumulus case. Copyright (c) 2012 Royal Meteorological Society