Scaling of Lyapunov Exponents in Homogeneous Isotropic Turbulence

TL;DR

This paper investigates how Lyapunov exponents, which quantify chaos and predictability, scale with Reynolds number in homogeneous isotropic turbulence, revealing that instabilities operate on scales smaller than Kolmogorov scales.

Contribution

It provides the first detailed analysis of Lyapunov exponent scaling in turbulence across various Reynolds numbers, highlighting the role of small-scale instabilities.

Findings

Maximum Lyapunov exponent increases faster than inverse Kolmogorov time scale with Reynolds number.

Instabilities predominantly act on the smallest eddies in the turbulence.

Multiple local sites of instability are active at any given time.

Abstract

Lyapunov exponents measure the average exponential growth rate of typical linear perturbations in a chaotic system, and the inverse of the largest exponent is a measure of the time horizon over which the evolution of the system can be predicted. Here, Lyapunov exponents are determined in forced homogeneous isotropic turbulence for a range of Reynolds numbers. Results show that the maximum exponent increases with Reynolds number faster than the inverse Kolmogorov time scale, suggesting that the instability processes may be acting on length and time scales smaller than Kolmogorov scales. Analysis of the linear disturbance used to compute the Lyapunov exponent, and its instantaneous growth, show that the instabilities do, as expected, act on the smallest eddies, and that at any time, there are many sites of local instabilities.