Prognostic assumed-probability-density-function (distribution density function) approach: further generalization and demonstrations

Abstract

A methodology for directly predicting the time evolution of the assumed parameters for distribution densities based on the Liouville equation, as proposed earlier, is extended to multidimensional cases and to cases in which the systems are constrained by integrals over a part of the variable range. The general formulation developed here is applicable to a wide range of problems, including the frequency distributions of subgrid-scale variables, hydrometeor size distributions, and probability distributions characterizing data uncertainties. The extended methodology is tested against a convective energy-cycle system and the Lorenz strange attractor. As a general tendency, the variance tends to collapse to a vanishing value over a finite time, regardless of the chosen assumed distribution form. This general tendency is likely due to a common cause, as the collapse of the variance is commonly found in ensemble-based data assimilation due to low dimensionality.