Statistical equilibrium principles in 2D fluid flow: from geophysical fluids to the solar tachocline

TL;DR

This paper reviews the principles of statistical equilibrium in effectively 2D fluid flows, highlighting diverse phenomena constrained by conservation laws, and discusses the complex interplay of energy, entropy, and large-scale structures across various physical systems.

Contribution

It provides a comprehensive overview of the statistical mechanics of 2D fluid equilibria, connecting classical theories to complex geophysical and astrophysical phenomena, and discusses potential extensions to non-equilibrium systems.

Findings

Large scale jet and eddy structures emerge from conserved variable cascades.

Balance between structure and fluctuations is governed by energy-entropy competition.

The statistical description is mathematically rich but sensitive to assumptions like ergodicity.

Abstract

An overview is presented of several diverse branches of work in the area of effectively 2D fluid equilibria which have in common that they are constrained by an infinite number of conservation laws. Broad concepts, and the enormous variety of physical phenomena that can be explored, are highlighted. These span, roughly in order of increasing complexity, Euler flow, nonlinear Rossby waves, 3D axisymmetric flow, shallow water dynamics, and 2D magnetohydrodynamics. The classical field theories describing these systems bear some resemblance to perhaps more familiar fluctuating membrane and continuous spin models, but the fluid physics drives these models into unconventional regimes exhibiting large scale jet and eddy structures. From a dynamical point of view these structures are the end result of various conserved variable forward and inverse cascades. The resulting balance between large scale structure and small scale fluctuations is controlled by the competition between energy and entropy in the system free energy, in turn highly tunable through setting the values of the conserved integrals. Although the statistical mechanical description of such systems is fully self-consistent, with remarkable mathematical structure and diversity of solutions, great care must be taken because the underlying assumptions, especially ergodicity, can be violated or at minimum lead to exceedingly long equilibration times. Generalization of the theory to include weak driving and dissipation (e.g., non-equilibrium statistical mechanics and associated linear response formalism) could provide additional insights, but has yet to be properly explored.