Application of generalised stability theory to deterministic and statistical prediction

Understanding of the stability of deterministic and stochastic dynamical systems has evolved recently from a traditional grounding in the system’s normal modes to a more comprehensive foundation in the system’s propagator and especially in an appreciation of the role of non-normality of the dynamical operator in determining the system’s stability as revealed through the propagator. This set of ideas, which approach stability analysis from a non-modal perspective, will be referred to as generalised stability theory (GST). Some applications of GST to deterministic and statistical forecast are discussed in this review. Perhaps the most familiar of these applications is identifying initial perturbations resulting in greatest error in deterministic error systems, which is in use for ensemble and targeting applications. But of increasing importance is elucidating the role of temporally distributed forcing along the forecast trajectory and obtaining a more comprehensive understanding of the prediction of statistical quantities beyond the horizon of deterministic prediction. The optimal growth concept can be extended to address error growth in non-autonomous systems in which the fundamental mechanism producing error growth can be identified with the necessary non-normality of the system. The influence of model error in both the forcing and the system is examined using the methods of stochastic dynamical systems theory. In this review deterministic and statistical prediction, i.e. forecast and climate prediction, are separately discussed.