Spectral peaks, non-normality, and predictability in simple coupled systems

The predictability of simple coupled linear systems is explored with an emphasis on the relationship between predictability and spectral peaks, as well as the non-normality of the system. A general predictability analysis is presented based on stochastic calculus. This analysis is applied to two linear stochastic models. The first is a stochastically driven coupled oscillatory system. The second system is a simple coupled model of Tropical Atlantic Variability. This model has nontrivial coupling interactions, making the governing linear system non-normal.

The predictability analysis of these systems reveals that 1) there is no general relationship between spectral peaks and predictability of a linear stochastic system; 2) under a unitary noise forcing, the non-normal dynamics always enhances the predictability of the system for short lead times in terms of error growth; 3) the predictability is influenced not only by the deterministic dynamics of the system but also by the spatial structure of the noise forcing. These findings have important implications for the predictability of a weakly coupled climate system whose dynamics can be described by a linear stochastic model. For such a system, although the dominant normal mode can exhibit a damped oscillation, the oscillation contributes little to the predictability of the system. The enhanced predictability comes from the non-normality introduced by the air-sea feedback or other processes. Therefore, in order to fully understand predictability of a weakly coupled climate system, it is necessary to analyze both the deterministic dynamics and the structure of stochastic processes.