Technical Note: General Formulation For the Distribution Problem: Prognostic Assumed PDF Approach Based on The Maximum–Entropy Principle and The Liouville Equation

Abstract. A general formulation for the distribution problem is presented, which is applicable to frequency distributions of subgrid-scale variables, hydrometeor size distributions, as well as to probability distributions characterizing data uncertainties. The general formulation is presented based upon two well-known basic principles: the maximum-entropy principle and Liouville equation. The maximum-entropy principle defines the most likely general distribution form, if necessary constraints are specified. This paper proposes to specify these constraints as the output variables to be used in a host model. Once a general distribution form is defined, the problem of temporal evolution of the distribution reduces to that of predicting a small number of parameters characterizing it. This paper derives prognostic equations for these parameters from the Liouville equation. The developed formulation, which is applicable to a wide range of atmospheric modelling problems, is specifically applied to condensation growth of cloud droplets as a demonstration.