Lagrangian chaos and the enstrophy cascade in Ekman-Navier-Stokes two-dimensional turbulence

Two-dimensional turbulence with linear (Ekman) friction exhibits spectral properties that deviate from the classical Kraichnan prediction for the direct enstrophy cascade. In particular, for sufficiently small viscosity and large friction, the enstrophy flux is suppressed in the cascade and, as a consequence, the small-scale vorticity field becomes passively transported by the large-scale chaotic flow. We numerically address this problem by investigating how the statistics of the Lagrangian finite time Lyapunov exponent (FTLE) in 2D Ekman-Navier-Stokes simulations are affected by the friction coefficient and by the other parameters of the flow. We derive a simple phenomenological model that interpolates the dependence of the Lyapunov exponent on the flow statistics from the large friction limit, where analytical predictions are available, to the small friction region. We find that the distribution of the FTLE around this mean value is always close to a Gaussian, and this allows us to make a simple prediction for the correction of the spectral slope of the direct cascade, which is in very good agreement with the numerical results.