We investigate the predictability problem in dynamical systems with many
degrees of freedom and a wide spectrum of temporal scales. In particular, we
study the case of 3D turbulence at high Reynolds numbers by introducing a
finite-size Lyapunov exponent which measures the growth rate of finite-size
perturbations. For sufficiently small perturbations this quantity coincides
with the usual Lyapunov exponent. When the perturbation is still small compared
to large-scale fluctuations, but large compared to fluctuations at the smallest
dynamically active scales, the finite-size Lyapunov exponent is inversely
proportional to the square of the perturbation size. Our results are supported
by numerical experiments on shell models. We find that intermittency
corrections do not change the scaling law of predictability. We also discuss
the relation between finite-size Lyapunov exponent and information entropy.Comment: 4 pages, 2 Postscript figures (included), RevTeX 3.0, files packed
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