Statistical mechanics, Euler’s equation, and Jupiter’s Red Spot

Abstract

We construct a statistical-mechanical treatment of equilibrium flows in the two-dimensional Euler fluid which respects all conservation laws. The vorticity field is fundamental, and its long-range Coulomb interactions lead to an exact set of nonlinear mean-field equations for the equilibrium state. The equations depend on an infinite set of parameters, in one-to-one correspondence with the infinite set of conserved variables. We illustrate the equations by solving them numerically in simple cases. In more complicated cases we use Monte Carlo techniques, with the eventual aim of detailed comparison with the Red Spot dynamical simulations of Marcus: Preliminary efforts show good agreement. We review previous literature on two-dimensional (2D) Euler flow in light of our theory. In particular, we derive the Kraichnan energy-enstrophy theory and the Lundgren and Pointin point-vortex mean-field theory as special cases of our own. Our techniques may be generalized to a number of other Coulomb-like Hamiltonian systems with an infinite number of conservation laws, including some in higher dimensions. For example, we rederive Lynden-Bell's theory of stellar-cluster formation, as well as the Debye-Huckel theory of electrolytes. Our results may also be applicable to cylindrically bound guiding-center plasmas, which under idealized conditions provide another realization of 2D Euler flow.