  We study the chaoticity and the predictability of a turbulent flow on the
basis of high-resolution direct numerical simulations at different Reynolds
numbers. We find that the Lyapunov exponent of turbulence, which measures the
exponential separation of two initially close solution of the Navier-Stokes
equations, grows with the Reynolds number of the flow, with an anomalous
scaling exponent, larger than the one obtained on dimensional grounds. For
large perturbations, the error is transferred to larger, slower scales where it
grows algebraically generating an "inverse cascade" of perturbations in the
inertial range. In this regime our simulations confirm the classical
predictions based on closure models of turbulence. We show how to link
chaoticity and predictability of a turbulent flow in terms of a finite size
extension of the Lyapunov exponent.
