The Liouville Equation and Its Potential Usefulness for the Prediction of Forecast Skill. Part I: Theory

The Liouville equation provides the framework for the consistent and comprehensive treatment of the uncertainty inherent in meteorological forecasts. This equation expresses the conservation of the phase-space integral of the number density of realizations of a dynamical system originating 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 the probability density function of the state vector of a given dynamical model. Consideration of the Liouville equation ostensibly avoids in a natural way the problems inherent to more standard methodology for predicting forecast skill, such as the need for higher-moment closure within stochastic-dynamic prediction, or the need to generate a large number of realizations within ensemble forecasting. These benefits, however, are obtained only at the expense of considering high-dimensional problems. The purpose of this work, presented in two pans, is to investigate the potential usefulness of the Liouville equation in the context of predicting forecast skill. After a review of the basic form of the Liouville equation, for the case that the dynamical system considered is represented by a set of coupled ordinary nonlinear first-order (nonstochastic) differential equations that are generic for meteorologically relevant situations, the general analytical solution of the Liouville equation is presented in this first part. This explicit solution allows one, at least in principle, to express in analytical terms the time evolution of the probability density function of the state vector of a given meteorological model. Several properties of the general solution are discussed. As an illustration, the general solution is used to solve the Liouville equation relevant for a one-dimensional nonlinear dynamical system. The fundamental role of the Liouville equation in the context of predicting forecast skill is emphasized.