Statistical mechanics of Fofonoff flows in an oceanic basin

Abstract

We study the minimization of potential enstrophy at fixed circulation and energy in an oceanic basin with arbitrary topography. For illustration, we consider a rectangular basin and a linear topography h=by which represents either a real bottom topography or the beta-effect appropriate to oceanic situations. Our minimum enstrophy principle is motivated by different arguments of statistical mechanics reviewed in the article. It leads to steady states of the quasigeostrophic (QG) equations characterized by a linear relationship between potential vorticity q and stream function psi. For low values of the energy, we recover Fofonoff flows [J. Mar. Res. 13, 254 (1954)] that display a strong westward jet. For large values of the energy, we obtain geometry induced phase transitions between monopoles and dipoles similar to those found by Chavanis and Sommeria [J. Fluid Mech. 314, 267 (1996)] in the absence of topography. In the presence of topography, we recover and confirm the results obtained by Venaille and Bouchet [Phys. Rev. Lett. 102, 104501 (2009)] using a different formalism. In addition, we introduce relaxation equations towards minimum potential enstrophy states and perform numerical simulations to illustrate the phase transitions in a rectangular oceanic basin with linear topography (or beta-effect).

arXiv comment26 pages, 28 figures

References in corpus (9)

• Classification of phase transitions and ensemble inequivalence, in systems with long range interactions
• Coarse-grained distributions and superstatistics
• Ensemble inequivalence, bicritical points and azeotropy for generalized Fofonoff flows
• Simpler Variational Problem for Statistical Equilibria of the 2D Euler Equation and Other Systems with Long Range Interactions
• Quasilinear theory of the 2D Euler equation
• Statistical mechanics of two-dimensional Euler flows and minimum enstrophy states
• Brownian theory of 2D turbulence and generalized thermodynamics
• Statistical mechanics of 2D turbulence with a prior vorticity distribution
• Relaxation equations for two-dimensional turbulent flows with a prior vorticity distribution

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