Reynolds number dependence of Lyapunov exponents of turbulence and fluid particles

Abstract

The Navier-Stokes equations generate an infinite set of generalized Lyapunov exponents defined by different ways of measuring the distance between exponentially diverging perturbed and unperturbed solutions. This set is demonstrated to be similar, yet different, from the generalized Lyapunov exponent that provides moments of distance between two fluid particles below the Kolmogorov scale. We derive rigorous upper bounds on dimensionless Lyapunov exponent of the fluid particles that demonstrate the exponent's decay with Reynolds number $Re$ in accord with previous studies. In contrast, terms of cumulant series for exponents of the moments have power-law growth with $Re$. We demonstrate as an application that the growth of small fluctuations of magnetic field in ideal conducting turbulence is hyper-intermittent, being exponential in both time and Reynolds number. We resolve the existing contradiction between the theory, that predicts slow decrease of dimensionless Lyapunov exponent of turbulence with $Re$, and observations exhibiting quite fast growth. We demonstrate that it is highly plausible that a pointwise limit for the growth of small perturbations of the Navier-Stokes equations exists.