The Liouville Equation and Prediction of Forecast Skill

The Liouville equation represents the consistent and comprehensive framework for dealing with uncertainty arising in meteorological forecasts due to uncertainty in the initial condition. 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 evolution of the probability density function of the state vector of a given dynamical model. The main purposes of this paper are (i) to review the basic form of the Liouville equation, (ii) to present the explicit general solution of the Liouville equation for a large class of dynamical systems in analytical terms, and (iii) to investigate the potential usefulness of the Liouville equation in the context of the prediction of forecast skill. As an illustration, the general analytical solution of the Liouville equation is used to obtain the solution of the Liouville equation relevant for a low-dimensional chaotic dynamical system. The information contained in this solution is compared with results obtained by application of the method of ensemble forecasting. It is found that a large number of ensemble integrations is required in order to obtain estimates of statistics, such as means, variances, and covariances, with the accuracy that is obtained by integration of the solution of the LE over phase space, even in the low-dimensional situation considered. The paper is concluded with a discussion of the fundamental role of the Liouville equation in dealing with initial state uncertainties in dynamical models, and of the problems that arise in this context. Even though some of these problems may be difficult to deal with in situations more realistic than considered here, the argument is made that the Liouville equation must be considered as an extremely valuable and useful guideline in the process of studying, developing and refining methods for the prediction of forecast skill.