  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 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.
