Dynamics of Model Error: The Role of Unresolved Scales Revisited

Abstract

The dynamical and probabilistic aspects of model error arising from the neglect of unresolved scales and the concomitant reduction of phase space dimensionality of a reference system are analyzed. A general expression of the mean quadratic error involving the time correlation function of the excess phase space velocity is derived. In the short time regime, this expression reduces to the sum of a t2 contribution reflecting the structure of the invariant distribution of the reference system and of a t3 part bearing the signature of the Lyapunov exponents of both the reduced and the reference models. The approach is illustrated on two classes of systems involving, successively, multiple time and space scales. A comparison between the purely deterministic analysis and the one in which the model system is augmented by error sources assimilated to Gaussian random noises is carried out. It is shown that such a representation leads to a deterioration of the predictive skill of the model as far as mean values are concerned, but may enhance its variability properties, bringing them closer to the variability of the reference system.