Statistical Mechanics of Quasi-geostrophic Vortices

Abstract

The statistical mechanics of quasi-geostrophic vortices is investigated numerically and theoretically. Direct numerical simulations of a point vortex system of mixed sign under periodic boundary conditions are performed using a fast special-purpose computer for molecular dynamics (GRAPE9). Clustering of point vortices of like sign is observed and a columnar dipole structure appears as an equilibrium state. These numerical results are explained from the viewpoint of the classical statistical mechanics. A three-dimensional mean field equation is derived based on the maximum entropy theory. The numerically obtained end states are shown to be the two-dimensional sn-sn dipole solutions of the mean field equation (i.e., the sinh-Poisson equation). We present other branches of two- and three-dimensional solution of the mean field equation. The entropy of these solution branches is found to be smaller than that of the two-dimensional sn-sn dipole branch. The stability of the maximum entropy states is studied theoretically and numerically. The two-dimensional (sn-sn dipole and zonal) solutions are stable against disturbances of finite amplitude, whereas the three-dimensional solutions are shown to be unstable. These findings explain the reason why only the two-dimensional sn-sn dipole states are found in the numerical simulations of point vortices. The influence of the aspect ratio of periodic unit box on the maximum entropy states and their stability is investigated. When the horizontal aspect ratio ($L_y/L_x$) is less than unity, the entropy of the zonal flow solution becomes larger than that of the dipole solution if the energy is less than a certain critical value. This critical energy increases as the aspect ratio is decreased. In contrast, the dipole solution in a box of square cross section ($L_y/L_x=1$) has the largest entropy, even if $L_z/L_x$ is changed.