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 λ. In fully developed turbulence, λ 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.