New numerical methods for calculating statistical equilibria of two-dimensional turbulent flows, strictly based on the Miller–Robert–Sommeria theory

New numerical methods are proposed for the mixing entropy maximization problem in the context of Miller–Robert–Sommeria’s (MRS) statistical mechanics theory of two-dimensional turbulence, particularly in the case of spherical geometry. Two of the methods are for the canonical problem; the other is for the microcanonical problem. The methods are based on the original MRS theory and thus take into account all Casimir invariants. Compared to the methods proposed in previous studies, our new methods make it easier to detect multiple statistical equilibria and to search for solutions with broken zonal symmetry. The methods are applied to a zonally symmetric initial vorticity distribution which is barotropically unstable. Two statistical equilibria are obtained, one of which has a wave-like structure with zonal wavenumber 1, and the other has a wave-like structure with zonal wavenumber 2. While the former is the maximum point of the mixing entropy, the wavenumber 2 structure of the latter is nearly the same as the structure that appears in the end state of the time integration of the vorticity equation. The new methods allow for efficient computation of statistical equilibria for initial vorticity distributions consisting of many levels of vorticity patches without losing information about all the conserved quantities. This means that the statistical equilibria can be obtained from an arbitrary initial vorticity distribution, which allows for the application of statistical mechanics to interpret a wide variety of flow patterns appearing in geophysical fluids.