Short-Term Dynamics of Model Errors

Abstract

The natural instability of the atmosphere is at the origin of the rapid amplification of errors coming from the uncertainty on the initial conditions and from the imperfect representation (the model) of the atmospheric dynamics. In this paper, the short-term dynamics of model error is examined in the context of low-order chaotic dynamical systems. A mathematical model describing the dynamics of this error is first derived where some of the key ingredients of its erratic behavior are incorporated, namely, the variability of the local Lyapunov exponents and of the model error source term along the dominant unstable direction. The analysis of this simplified equation indicates that depending on the nature of the model error sources (here limited to a white noise or an Ornstein?Uhlenbeck process), the mean square error initially follows either a linear or a quadratic evolution, the latter being generic. The numerical analysis of the Lorenz 1984 low-order atmospheric system, in which the model error source is associated with the inaccurate estimate of one of its parameters, supports the main features demonstrated by the simplified mathematical model. However, it also reveals a more involved behavior of the mean square error, which can be traced back to some intrinsic properties of the underlying dynamics not incorporated in the simplified model. The role of the truncation of the small scales of the flow on the dynamics of the larger scales is also studied in two spatially distributed systems. In this context, the mean square error closely follows a quadratic evolution for short times. In the light of these results, the classical view of the linear evolution of the mean square error advanced thus far in the literature should be reassessed.