THE LIOUVILLE EQUATION AND ITS POTENTIAL USEFULNESS FOR THE PREDICTION OF FORECAST SKILL .1. THEORY

The Liouville equation provides the framework for the consistent and c omprehensive treatment of the uncertainty inherent in meteorological f orecasts. This equation expresses the conservation of the phase-space integral of the number density of realizations of a dynamical system o riginating at the same time instant from different initial conditions, in a way completely analogous to the continuity equation for mass in fluid mechanics. Its solution describes the temporal development of th e probability density function of the state vector of a given dynamica l model, Consideration of the Liouville equation ostensibly avoids in a natural way the problems inherent to more standard methodology for p redicting forecast skill, such as the need for higher-moment closure w ithin stochastic-dynamic prediction, or the need to generate a large n umber of realizations within ensemble forecasting. These benefits, how ever, are obtained only at the expense of considering high-dimensional problems. The purpose of this work, presented in two parts, is to inv estigate the potential usefulness of the Liouville equation in the con text of predicting forecast skill. After a review of the basic form of the Liouville equation, for the case that the dynamical system consid ered is represented by a set of coupled ordinary nonlinear first-order (nonstochastic) differential equations that are generic for meteorolo gically relevant situations, the general analytical solution of the Li ouville equation is presented in this first part. This explicit soluti on allows one, at least in principle, to express in analytical terms t he time evolution of the probability density function of the state vec tor of a given meteorological model. Several properties of the general solution are discussed. As an illustration, the general solution is u sed to solve the Liouville equation relevant for a one-dimensional non linear dynamical system. The fundamental role of the Liouville equatio n in the context of predicting forecast skill is emphasized.