Statistical mechanics of two-dimensional Euler flows and minimum enstrophy states

Abstract

Dated: To be included later)A simplified thermodynamic approach of the incompressible 2D Euler equation is considered based on the conservation of energy, circulation and microscopic enstrophy. Statistical equilibrium states are obtained by maximizing the Miller-Robert-Sommeria (MRS) entropy under these sole constraints. We assume that these constraints are selected by properties of forcing and dissipation. We find that the vorticity fluctuations are Gaussian while the mean flow is characterized by a linear ω − ψ relationship. Furthermore, we prove that the maximization of entropy at fixed energy, circulation and microscopic enstrophy is equivalent to the minimization of macroscopic enstrophy at fixed energy and circulation. This provides a justification of the minimum enstrophy principle from statistical mechanics when only the microscopic enstrophy is conserved among the infinite class of Casimir constraints. Relaxation equations towards the statistical equilibrium state are derived. These equations can serve as numerical algorithms to determine maximum entropy or minimum enstrophy states. We use these relaxation equations to study geometry induced phase transitions in rectangular domains. In particular, we illustrate with the relaxation equations the transition between monopoles and dipoles predicted by Chavanis & Sommeria [J. Fluid. Mech. 314, 267 (1996)]. We take into account stable as well as metastable states and show that metastable states are robust and have negative specific heats. This is the first evidence of negative specific heats in that context. We also argue that saddle points of entropy can be long-lived and play a role in the dynamics because the system may not spontaneously generate the perturbations that destabilize them. PACS numbers: 05.20.-y Classical statistical mechanics -05.45.-a Nonlinear dynamics and chaos -05.90.+m Other topics in statistical physics, thermodynamics, and nonlinear dynamical systems -47.10.-g General theory in fluid dynamics -47.15.ki Inviscid flows with vorticity -47.20.-k Flow instabilities -47.32.-y Vortex dynamics; rotating fluids

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“…However, treating dissipation in this way may not work well, because during the relaxation to equilibrium, viscosity can significantly alter the integrals of motion, especially the high-order moments. Recently, Aurore Naso, Pierre-Henry Chavanis, and Berengere Dubrulle [13] showed that maximizing the MRS entropy, holding only the energy, circulation, and enstrophy fixed, is equivalent to minimizing the coarse-grained enstrophy (the mean square ofω), as in the earlier minimum enstrophy theory. The result uses the statistical MRS theory to nicely justify the intuitive minimum enstrophy variational principle.…”

Universal statistics of point vortex turbulence

“…the condensed state), because in this case there will be a strong tendency for the system to remain either in a positive polarity state with Ω 1 > 0 or in a negative polarity state with Ω 1 < 0. Passage between the two states requires the system to evolve through a relatively improbable configuration with Ω 1 = 0 (similar transitions between positive and negative polarity equilibrium solutions of the Euler equations have been studied by Naso et al 2010). …”

“…One might in fact anticipate that the expected time of transition is inversely proportional to p ε (ω 1 = 0), this is in fact the case with stochastic models of bistable systems (e.g. Naso et al 2010), which might be used as a simple phenomenological models for the dynamic evolution of Ω 1 (t)…”

The catalytic role of the beta effect in barotropization processes

“…This is a direct consequence of a more general result of Bouchet (2008), see also Majda & Wang (2006); Naso et al (2010). Here bottom topography is omitted to simplify the presentation, but taking into account this additional parameter is straightforward.…”