Predictability of Linear Coupled Systems. Part I: Theoretical Analyses

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Abstract
The predictability of stochastically forced linear systems is investigated under the condition that an ensemble of forecasts are each initialized at the true state but driven by different realizations of white noise. Some important issues of predictability are brought out by analytically investigating a stochastically driven, damped inertial oscillator. These issues are then studied in a generic context without reference to any specific linear stochastic system. The predictability is measured primarily by the mean-square forecast error normalized by the mean climatological variance. If the dynamical operator is normal, then this metric depends only on the real eigenvalues (i.e., the normal mode damping rates) and is independent of the existence of spectral peaks at nonzero frequency caused by oscillatory eigenmodes. It is shown that a nonnormal system is more predictable by this measure than a normal system with identical eigenvalues for short lead times. The noise structure that optimizes this metric can be calculated analytically and shown to approach the stochastic optimals in the limit of short lead times; this calculation gives bounds on predictability that depend only on the dynamical operator. The most predictable components, or spatial structures, can also be determined analytically and are shown to approach the empirical orthogonal functions (EOFs) of the variance in the limit of short lead times. In a companion paper, these concepts are applied to the predictability analysis of a simple coupled climate model of tropical Atlantic variability.