Abstract:

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 deter-
mining 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 determinis-
tic 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 appli-
cations. But of increasing importance is elucidating the role of temporally dis-
tributed forcing along the forecast trajectory and obtaining a more comprehensive 
understanding of the prediction of statistical quantities beyond the horizon of deter-
ministic prediction. The optimal growth concept can be extended to address error 
growth in non-autonomous systems in which the fundamental mechanism produc-
ing error growth can be identified with the necessary non-normality of the sys-
tem. 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 determin-
istic and statistical prediction, i.e. forecast and climate prediction, are separately 
discussed.