Statistical mechanics of two-dimensional vortices

Equilibrium statistics of a cluster of a large number of positive two-dimensional point vortices in an infinite region and the associated thermodynamic functions, exhibiting negative temperatures, are evaluated analytically and numerically from a microcanonical ensemble. Extensive numerical simulations of vortex motion are performed to verify the predicted equilibrium configurations. An application of Kubo's linear response theory is used to study the nonequilibrium situation that results from placing a cluster, of vortices in a weak external velocity field, such as that produced by a distant vortex cluster. The weak field causes the cluster to grow in size as if there were an effective positive eddy viscosity. When a number of clusters interact, the effect is for each to grow while the distances between them decrease with time. The latter effect is an exhibit of negative viscosity. The application of this to the motion of the atmosphere is discussed.Supported in part by National Science Foundation Grant GK-40263.Advanced Study Program, National Center for Atmospheric Research (sponsored by the National Science Foundation).