Ergodic Boundary in Numerical Simulations of Two-Dimensional Turbulence

Abstract

In numerical calculations, we observe a dichotomy in the time evolution of solutions of the high-Reynolds-number, two-dimensional, incompressible Navier-Stokes equations using power-law initial modal energy spectra $E(\ensuremath{\kappa},0)\ensuremath{\sim}{\ensuremath{\kappa}}^{\ensuremath{-}{\ensuremath{\mu}}_{0}}$. The boundary is expressed in terms of critical values of viscosity ${\ensuremath{\nu}}_{\mathrm{cr}}$ and microscale ${\ensuremath{\lambda}}_{\mathrm{cr}}$. For both parameters initially above critical, $\ensuremath{\mu}$ approaches 4. For both parameters initially below critical, $\ensuremath{\mu}$ approaches 1 at large $\ensuremath{\kappa}$, consistent with equipartition of the vorticity spectrum. In the former case, large-scale vortex states form after a long time; and, in both cases, energy flows to the lowest wave numbers.