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

Abstract

The Liouville equation represents the consistent and comprehensive framework for the treatment of the uncertainty inherent in meteorological forecasts. By its very nature, consideration of the Liouville equation avoids problems that are inherent to commonly used methods for predicting forecast skill, such as the need for higher-moment closure within stochastic-dynamic prediction, or the need for the generation of large ensemble sizes within ensemble forecasting. The general analytical solution of the Liouville equation presented in the first part of this work is used here to find the solution of the Liouville equation relevant for two low-dimensional nonlinear dynamical systems and to investigate the potential usefulness of the Liouville equation in the context of predicting forecast skill. The analytical solution of the Liouville equation in these examples, namely, the time-dependent probability density function, is discussed and compared with corresponding results obtained by stochastic-dynamic prediction and ensemble forecasting. The negative effect of the unavoidable higher-moment discard within stochastic-dynamic prediction is quantitatively demonstrated. It is also shown that a large number of ensemble members is necessary to obtain accurate estimates of statistics, such as means and (co)variances, even in the low-dimensional situations considered. It is concluded that, due to its fundamental role in dealing with initial state uncertainty in dynamical models, the Liouville equation must be considered as a highly valuable and useful guideline during the process of developing a coherent methodology for forecasting forecast skill that minimizes deficiencies of currently used methods.