Abstract
Assumed-PDF methods for the parameterization of subgrid-scale processes in atmospheric models provide many benefits. Many currently used assumed-PDF schemes reconcile the high number of required PDF parameters with the relative paucity of input moments by employing simplifying assumptions that are difficult to test. This paper explores the possibility of constructing a trivariate double-Gaussian PDF from the first three orders of moments without simplifying assumptions and proves that no unique solution exists. In an effort to provide a path for future improvement of current assumed-PDF schemes, the expectation maximization (EM) algorithm for Gaussian mixture models is used with LES output of shallow cumulus, stratocumulus, and deep convection cases to determine “best fit” PDFs using from one through four Gaussian clusters. The EM PDFs are evaluated using PDF-diagnosed higher-order moments, PDF-diagnosed cloud statistics, and the Akaike information criterion. It was found that two Gaussian clusters were almost always adequate to represent both higher-order moments and cloud statistics like cloud fraction, water content, and vertical fluxes of cloud water and buoyancy in layered clouds such as stratocumulus and deep convective anvils. However, higher-order moments and higher-order cloud statistics were only properly represented when three or four Gaussians were used in the upper regions of shallow cumulus layers and throughout the active portion of deep convection. Evidence is also provided that several common assumptions employed to diagnose trivariate double-Gaussian PDFs from a minimum number of input moments are weak.
