Coarse-grain entropy in two-dimensional turbulence

A generalized coarse-grain entropy for a simple two-dimensional system is derived and used to explain the emergence of coherent structures, which are the most probable states, and their equilibrium with random fluctuations. The relationship between the canonical distribution function and the most probable states, which appear at critical values of the inverse temperature β, is elucidated. Under some conditions, the coherent structures are localized and intermittent, i.e., occupy only a small fraction of the system. For small fluctuation amplitudes, the macrostate entropy is equal to minus the coarse-grain enstrophy, in agreement with the selective decay and minimum enstrophy theories. Maximizing the macrostate entropy leads to vortex coalescing, and in some cases, a process of concentration occurs in which vorticity and energy are concentrated into vortices of decreasing area