Ergodic Boundary in Numerical Simulations of Two-Dimensional Turbulence

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 $E(\kappa ,0)\sim {\kappa }^{-{\mu }_0}$. The boundary is expressed in terms of critical values of viscosity ${\nu }_{\mathrm{cr}}$ and microscale ${\lambda }_{\mathrm{cr}}$. For both parameters initially above critical, $\mu$ approaches 4. For both parameters initially below critical, $\mu$ approaches 1 at large $\kappa$, 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.