Prognostic Equations for Cloud Fraction and Liquid Water, and Their Relation to Filtered Density Functions

Abstract

This paper derives prognostic equations for cloud fraction and specific liquid water content. Such equations are carried by host models in order to provide more accurate forcing of microphysical and radiative parameterizations. The starting point of the derivations is a prognostic equation for the filtered density function (FDF) of a moisture variable. The FDF is the probability density function (PDF) of subgrid (i.e., filtered) fluctuations. The resulting equations for cloud fraction and liquid water contain unclosed terms. Many of these terms can be closed if the relevant joint FDF is known. In addition, there emerge from the derivation two dissipation terms that are not closed even if the FDF of moisture is known. These dissipation terms do not appear in equations for conserved variables such as total water content. The paper then compares various closures in the literature for turbulent flux, source, and dissipative mixing terms. Intuition suggests that dissipative mixing cannot usually increase liquid water content but that dissipative mixing can sometimes increase cloud fraction. The paper discusses two closures for dissipative mixing: one is based on the linear mean-square estimation (LMSE) model, and the other is due to Tiedtke. The LSME model permits cloud fraction to increase, as expected, but the Tiedtke model does not.