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We investigate the predictability problem in dynamical systems with manydegrees of freedom and a wide spectrum of temporal scales. In particular, westudy the case of $3D$ turbulence at high Reynolds numbers by introducing afinite-size Lyapunov exponent which measures the growth rate of finite-sizeperturbations. For sufficiently small perturbations this quantity coincideswith the usual Lyapunov exponent. When the perturbation is still small comparedto large-scale fluctuations, but large compared to fluctuations at the smallestdynamically active scales, the finite-size Lyapunov exponent is inverselyproportional to the square of the perturbation size. Our results are supportedby numerical experiments on shell models. We find that intermittencycorrections do not change the scaling law of predictability. We also discussthe relation between finite-size Lyapunov exponent and information entropy.Comment: 4 pages, 2 Postscript figures (included), RevTeX 3.0, files packed  with uufile

Predictability in the large: an extension of the concept of Lyapunov exponent