Statistical Equilibria of Turbulence on Surfaces of Different Symmetry

We test the validity of statistical descriptions of freely decaying 2D turbulence by performing direct numerical simulations (DNS) of the Euler equation with hyperviscosity on a square torus and on a sphere. DNS shows, at long times, a dipolar coherent structure in the vorticity field on the torus but a quadrapole on the sphere\footnote{J. Y-K. Cho and L. Polvani, Phys. Fluids {\bf 8}, 1531 (1996).}. A truncated Miller-Robert-Sommeria theory\footnote{A. J. Majda and X. Wang, \emph{Nonlinear Dynamics and Statistical Theories for Basic Geophysical Flows} (Cambridge University Press, 2006).} can explain the difference. The theory conserves up to the second-order Casimir, while also respecting conservation laws that reflect the symmetry of the domain. We further show that it is equivalent to the phenomenological minimum-enstrophy principle by generalizing the work by Naso et al.\footnote{A. Naso, P. H. Chavanis, and B. Dubrulle, Eur. Phys. J. B {\bf 77}, 284 (2010).} to the sphere. To explain finer structures of the coherent states seen in DNS, especially the phenomenon of confinement, we investigate the perturbative inclusion of the higher Casimir constraints.