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The Navier-Stokes equations generate an infinite set of generalized Lyapunovexponents defined by different ways of measuring the distance betweenexponentially diverging perturbed and unperturbed solutions. This set isdemonstrated to be similar, yet different, from the generalized Lyapunovexponent that provides moments of distance between two fluid particles belowthe Kolmogorov scale. We derive rigorous upper bounds on dimensionless Lyapunovexponent of the fluid particles that demonstrate the exponent's decay withReynolds number $Re$ in accord with previous studies. In contrast, terms ofcumulant series for exponents of the moments have power-law growth with $Re$.We demonstrate as an application that the growth of small fluctuations ofmagnetic field in ideal conducting turbulence is hyper-intermittent, beingexponential in both time and Reynolds number. We resolve the existingcontradiction between the theory, that predicts slow decrease of dimensionlessLyapunov exponent of turbulence with $Re$, and observations exhibiting quitefast growth. We demonstrate that it is highly plausible that a pointwise limitfor the growth of small perturbations of the Navier-Stokes equations exists.

Reynolds number dependence of Lyapunov exponents of turbulence and fluid particles