Covariant Lyapunov Analysis of Chaotic Kolmogorov Flows and Time-correlation Function

We study a hyperbolic/non-hyperbolic transition of the flows on two-dimensional torus governed by the incompressible Navier-Stokes equation (Kolmogorov flows) using the method of covariant Lyapunov analysis developed by Ginelli et al.(2007) [1], . As the Reynolds number is increased, chaotic Kolmogorov flows become non-hyperbolic at a certain Reynolds number, where some new physical property is expected to appear in the long-time statistics of the fluid motion. Here we focus our attention on behaviors of the time-correlation function of vorticity across the transition point, and that the long-time asymptotic form of the correlation function changes at the Reynolds number close to that of the hyperbolic/non-hyperbolic transition, which suggests that the time-correlation function reflects the transition to non-hyperbolicity [3].