Reynolds number dependence of Lyapunov exponents of turbulence and fluid particles
Abstract
The NavierStokes 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 powerlaw growth with Re. We demonstrate as an application that the growth of small fluctuations of magnetic field in ideal conducting turbulence is hyperintermittent, 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 NavierStokes equations exists.
 Publication:

Physical Review E
 Pub Date:
 March 2021
 DOI:
 10.1103/PhysRevE.103.033110
 arXiv:
 arXiv:2104.01235
 Bibcode:
 2021PhRvE.103c3110F
 Keywords:

 Physics  Fluid Dynamics;
 Astrophysics  Astrophysics of Galaxies;
 Condensed Matter  Statistical Mechanics;
 Nonlinear Sciences  Chaotic Dynamics
 EPrint:
 15 pages