Statistical mechanics of two-dimensional Euler flows and minimum enstrophy states |
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Authors: | A Naso P H Chavanis and B Dubrulle |
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Institution: | (1) INFM, Unitá di Palermo and Dipartimento di Fisica e Tecnologie Relative - Universitá di Palermo, Viale delle Scienze, pad.18, 90128 Palermo, Italy;(2) Radiophysics Department, Nizhny Novgorod State University, 23 Gagarin Ave., 603950 Nizhny Novgorod, Russia |
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Abstract: | 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 `(w)]-y\overline{\omega}-\psi 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 and
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. |
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Keywords: | |
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