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(κ,0)~κ^-μ₀. The boundary is expressed in terms of critical values of viscosity νcr and microscale λcr. For both parameters initially above critical, μ approaches 4. For both parameters initially below critical, μ approaches 1 at large κ, 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.