Clustering and Entropy Growth of Quasi-geostrophic Point Vortices

Abstract eng:
The statistical mechanics of quasi-geostrophic point vortices of mixed sign is investigated numerically and theoretically. Direct numerical simulations of a point vortex system 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 two tall columnar vortices appear in the course of time. These numerical results are analyzed quantitatively using a density-based algorithm. The number of clusters decreases as about t−1 , which is significantly slower than t−1.25 found in the previous spectral simulations of geostrophic turbulence (e.g. McWilliams et al., 1999). The equilibrium states are identified as the sn-sn dipole solutions of the two-dimensional mean field equation (the sinh-Poisson equation). A three-dimensional mean field equation is derived based on the maximum entropy theory, and several branches of two- and three-dimensional solutions are obtained. It is shown that the two-dimensional sn-sn dipole branch has the largest entropy.

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