Fouxon Itzhak, Feinberg Joshua, Käpylä Petri, Mond Michael
Department of Mechanical Engineering, Ben-Gurion University of the Negev, Beer Sheva 84105, Israel.
Department of Mathematics and Haifa Research Center for Theoretical Physics and Astrophysics, University of Haifa, Haifa 31905, Israel.
Phys Rev E. 2021 Mar;103(3-1):033110. doi: 10.1103/PhysRevE.103.033110.
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 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 Navier-Stokes equations exists.
