Predictability in Systems with Many Characteristic Times: The Case of Turbulence

Abstract

In chaotic dynamical systems, an infinitesimal perturbation is exponentially amplified at a time-rate given by the inverse of the maximum Lyapunov exponent $\lambda$. In fully developed turbulence, $\lambda$ grows as a power of the Reynolds number. This result could seem in contrast with phenomenological arguments suggesting that, as a consequence of `physical' perturbations, the predictability time is roughly given by the characteristic life-time of the large scale structures, and hence 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 non-infinitesimal perturbation is determined by 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.

24 pages, 10 Postscript figures (included), RevTeX 3.0, files packed with uufiles

References in corpus (2)

• Transition to Chaos in a Shell Model of Turbulence
• Statistical Mechanics of Shell Models for 2D-Turbulence

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