Simpler Variational Problem for Statistical Equilibria of the 2D Euler Equation and Other Systems with Long Range Interactions

Abstract

The Robert-Sommeria-Miller equilibrium statistical mechanics predicts the final organization of two dimensional flows. This powerful theory is difficult to handle practically, due to the complexity associated with an infinite number of constraints. Several alternative simpler variational problems, based on Casimir's or stream function functionals, have been considered recently. We establish the relations between all these variational problems, justifying the use of simpler formulations.

5 pages, submited to Physica D

References in corpus (4)

• Generalized thermodynamics and Fokker-Planck equations. Applications to stellar dynamics, two-dimensional turbulence and Jupiter's great red spot
• Classification of phase transitions and ensemble inequivalence, in systems with long range interactions
• Brownian theory of 2D turbulence and generalized thermodynamics
• Emergence of Jupiter's vortices and jets from random initial conditions

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• Large time behavior and asymptotic stability of the two-dimensional Euler and linearized Euler equations
• Solvable phase diagrams and ensemble inequivalence for two-dimensional and geophysical turbulent flows
• Statistical mechanics of two-dimensional Euler flows and minimum enstrophy states
• Statistical mechanics of Beltrami flows in axisymmetric geometry: Theory reexamined
• Dynamical stability criterion for inhomogeneous quasi-stationary states in long-range systems
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• Langevin dynamics, large deviations and instantons for the quasi-geostrophic model and two-dimensional Euler equations
• Vortex-density fluctuations, energy spectra and vortical regions in superfluid turbulence
• Computation of rare transitions in the barotropic quasi-geostrophic equations
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• Hyperviscosity and statistical equilibria of Euler turbulence on the torus and the sphere
• Restricted Equilibrium and the Energy Cascade in Rotating and Stratified Flows
• Oceanic rings and jets as statistical equilibrium states
• Kinetic theory of collisionless relaxation for systems with long-range interactions
• Invariant measures of the 2D Euler and Vlasov equations
• The catalytic role of beta effect in barotropization processes
• Additional Invariants and Statistical Equilibria for the 2D Euler Equations on a spherical domain
• Phase transitions and marginal ensemble equivalence for freely evolving flows on a rotating sphere
• Statistical mechanics of Fofonoff flows in an oceanic basin
• On maximally mixed equilibria of two-dimensional perfect fluids
• Relaxation equations for two-dimensional turbulent flows with a prior vorticity distribution
• Equilibrium statistical mechanics and energy partition for the shallow water model
• Nonlinear energy transfers and phase diagrams for geostrophically balanced rotating--stratified flows
• A New Interpretation of Vortex-Split Sudden Stratospheric Warmings in Terms of Equilibrium Statistical Mechanics
• Sampling microcanonical measures of the 2D Euler equations through Creutz's algorithm: a phase transition from disorder to order when energy is increased
• A complete theory of low-energy phase diagrams for two-dimensional turbulence steady states and equilibria
• Direct Statistical Simulation of a Jet
• New numerical methods for calculating statistical equilibria of two-dimensional turbulent flows, strictly based on the Miller-Robert-Sommeria theory
• An Introduction to Large Deviations and Equilibrium Statistical Mechanics for Turbulent Flows
• Statistical equilibria of two-dimensional turbulent flows for generic initial vorticity fields on a sphere, calculated on the basis of the original Miller-Robert-Sommeria theory