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Hyperviscosity and statistical equilibria of Euler turbulence on the torus and the sphere

Coherent structures such as jets and vortices appear in two-dimensional (2D) turbulence. To gain insight into both numerical simulation and equilibrium statistical mechanical descriptions of 2D Euler flows, the Euler equation with added hyperviscosity is integrated forward in time on the square torus and on the sphere. Coherent structures that form are compared against a hierarchy of truncated Miller-Robert-Sommeria equilibria. The energy-circulation-enstrophy MRS-2 description produces a complete condensation of energy to the largest scales, and in the absence of rotation correctly predicts the number and polarity of coherent vortices. Perturbative imposition of the quartic Casimir constraint improves agreement with numerical simulation by sharpening the cores and transferring some energy to smaller-scale modes. MRS-2 cannot explain qualitative changes due to rotation, but descriptions that conserve higher Casimirs beyond enstrophy have the potential to do so. The result is in agreement with the somewhat paradoxical observation that hyperviscosity helps to remedy the non-conservation of the third and higher Casimirs in numerical simulation. For a rotating sphere, numerical simulation also demonstrates that coherent structures found at late times depend on initial conditions, limiting the usefulness of statistical mechanics.