The predictability problem in systems with an uncertainty in the evolution law

The problem of unpredictability in a physical system due to the incomplete knowledge of the evolution laws is addressed. Major interest is devoted to the analysis of error amplification in chaotic systems with many characteristic times and scales when the fastest scales are not resolved. The parametrization of the unresolved scales introduces a non-infinitesimal uncertainty (with respect to the true evolution laws) which affects the forecasting ability on the large resolved scales. The evolution of non-infinitesimal errors from the unresolved scales up to the large scales is analysed by means of the finite-size Lyapunov exponent. It is shown that proper parametrization of the unresolved scales allows one to recover the maximal predictability of the system.