Abstract
Nonequilibrium statistical models of point vortex systems are constructed using an optimal closure method, and these models are employed to approximate the relaxation toward equilibrium of systems governed by the two-dimensional Euler equations, as well as the quasi-geostrophic equations for either single-layer or two-layer flows. Optimal closure refers to a general method of reduction for Hamiltonian systems, in which macroscopic states are required to belong to a parametric family of distributions on phase space. In the case of point vortex ensembles, the macroscopic variables describe the spatially coarse-grained vorticity. Dynamical closure in terms of those macrostates is obtained by optimizing over paths in the parameter space of the reduced model subject to the constraints imposed by conserved quantities. This optimization minimizes a cost functional that quantifies the rate of information loss due to model reduction, meaning that an optimal path represents a macroscopic evolution that is most compatible with the microscopic dynamics in an information-theoretic sense. A near-equilibrium linearization of this method is used to derive dissipative equations for the low-order spatial moments of ensembles of point vortices in the plane. These severely reduced models describe the late-stage evolution of isolated coherent structures in two-dimensional and geostrophic turbulence. For single-layer dynamics, they approximate the relaxation of initially distorted structures toward axisymmetric equilibrium states. For two-layer dynamics, they predict the rate of energy transfer in baroclinically perturbed structures returning to stable barotropic states. Comparisons against direct numerical simulations of the fully-resolved many-vortex dynamics validate the predictive capacity of these reduced models.
Highlights
The equilibrium statistical mechanics of point vortex systems exists as a mature theory.Onsager provided the first insight that equilibrium theory is able to explain the organization of many-vortex systems into coherent structures [1]
We address the nonequilibrium statistical mechanics of point vortex systems by means of a less traditional method of model reduction, which we call “optimal closure” [21,22]
The problem addressed in the present paper, offers a useful test of the optimal closure method, since it concerns a severe coarse-graining of point vortex dynamics
Summary
The equilibrium statistical mechanics of point vortex systems exists as a mature theory. From the point of view of fluid mechanical outcomes, these more intricate theories are similar to the point vortex theory, in that they derive special mean-field equations satisfied by their equilibrium states These theories, when extended to include quasi-geostrophic dynamics, have been shown to realize interesting coherent structures such as the zonal jets and embedded Great Red Spot on Jupiter [7,17,18]. The problem addressed in the present paper, offers a useful test of the optimal closure method, since it concerns a severe coarse-graining of point vortex dynamics For this reason the paper first develops the reduced model theoretically, and proceeds to validate it against direct numerical simulations of ensembles. The useful outcome of such severe model reduction and closure is to approximate gross features and functional dependences with much greater computational efficiency than is required by direct numerical studies of the full dynamics
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