Prognostic Assumed-PDF (DDF) Approach: Further Generalization and Demonstrations

Abstract. A methodology for directly predicting the time evolution of the assumed parameters for the distribution densities based on the Liouville equation, as proposed earlier, is extended to multi–dimensional cases as well as when the systems are constrained by integrals over a part of the variable range. The extended methodology is tested against a convective energy cycle system as well as the Lorenz’s stranger 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 the common cause as collapse of the variance commonly found in ensemble–based data assimilation.