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

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.

arXiv comment25 pages, 9 figures

References in corpus (23)

• Statistical mechanics of the shallow water system
• A Hexagon in Saturn's Northern Stratosphere Surrounding the Emerging Summertime Polar Vortex
• Dynamics and thermodynamics of axisymmetric flows: I. Theory
• A Casimir preserving scheme for long-time simulation of spherical ideal hydrodynamics
• Statistical mechanics of Beltrami flows in axisymmetric geometry: Theory reexamined
• Langevin dynamics, large deviations and instantons for the quasi-geostrophic model and two-dimensional Euler equations
• On the Late-Time Behaviour of a Bounded, Inviscid Two-Dimensional Flow
• Vortex-density fluctuations, energy spectra and vortical regions in superfluid turbulence
• Computation of rare transitions in the barotropic quasi-geostrophic equations
• Hyperviscosity and statistical equilibria of Euler turbulence on the torus and the sphere
• Statistical mechanics of the 3D axi-symmetric Euler equations in a Taylor-Couette geometry
• Canonical scale separation in two-dimensional incompressible hydrodynamics
• Equilibrium solutions of the shallow water equations
• Statistical mechanics of Beltrami flows in axisymmetric geometry: Equilibria and bifurcations
• Phase transitions and marginal ensemble equivalence for freely evolving flows on a rotating sphere
• Statistical mechanics of quasi-geostrophic flows on a rotating sphere
• On maximally mixed equilibria of two-dimensional perfect fluids
• Thermodynamics of MHD flows with axial symmetry
• Rotating Shallow Water Dynamics: Extra Invariant and the Formation of Zonal Jets
• Long-range correlations and coherent structures in magnetohydrodynamic equilibria
• Equilibrium statistical mechanics and energy partition for the shallow water model
• Joint Instability and Abrupt Nonlinear Transitions in a Differentially Rotating Plasma
• Strong vorticity fluctuations and antiferromagnetic correlations in axisymmetric fluid equilibria