The Liouville equation and atmospheric predictability

Abstract

Introduction and motivation It is widely recognised that weather forecasts made with dynamical models of the atmosphere are inherently uncertain. Such uncertainty of forecasts produced with numerical weather prediction (NWP) models arises primarily from two sources: from imperfect knowledge of the initial model conditions and from imperfections in the model formulation itself. The recognition of the potential importance of accurate initial model conditions and an accurate model formulation dates back to times even prior to operational NWP (Bjerknes, 1904; Thompson, 1957). In the context of NWP, the importance of these error sources in degrading the quality of forecasts was demonstrated to arise because errors introduced in atmospheric models are, in general, growing (Lorenz, 1982, 1963, 1993, this volume), which at the same time implies that the predictability of the atmosphere is subject to limitations (Errico et al ., 2002). An example of the amplification of small errors in the initial conditions, or, equivalently, the divergence of initially nearby trajectories is given in Figure 4.1, for the system discussed by Lorenz (1984). The uncertainty introduced into forecasts through uncertain initial model conditions, and uncertainties in model formulations, has been the subject of numerous studies carried out in parallel with the continuous development of NWP models (e.g. Leith, 1974; Epstein, 1969; Palmer, 2000, this volume, Buizza, this volume).