  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.
