Predictability in systems with many characteristic times: The case of turbulence

In chaotic dynamical systems, an infinitesimal perturbation is exponentially amplified at a rate given by the inverse of the maximum Lyapunov exponent. In fully developed turbulence, grows as a power of the Reynolds number. This result could seem to be in contrast to phenomenological arguments suggesting that, as a consequence of ``physical'' perturbations, the predictability time is roughly given by the characteristic lifetime of the large scale structures, and hence is independent of the Reynolds number. We show that such a situation is present in generic systems with many degrees of freedom, since the growth of a noninfinitesimal perturbation is determined by the cumulative effects of many different characteristic times and is unrelated to the maximum Lyapunov exponent. Our results are illustrated in a chain of coupled maps and in a shell model for the energy cascade in turbulence. 1996 The American Physical Society.