Eur. Phys. J. B 80, 493-517 (2011)
https://doi.org/10.1140/epjb/e2011-10440-8

Statistical mechanics of Fofonoff flows in an oceanic basin

¹ Laboratoire de Physique, École Normale Supérieure de Lyon and CNRS (UMR 5672), 46 allée d'Italie, 69007 Lyon, France
² LMFA, Université de Lyon, École Centrale de Lyon, and CNRS (UMR 5509), 69134 Ecully Cedex, France
³ Laboratoire de Physique Théorique (IRSAMC), CNRS and UPS, Université de Toulouse, 31062 Toulouse, France
⁴ SPEC/IRAMIS/CEA Saclay, and CNRS (URA 2464), 91191 Gif-sur-Yvette Cedex, France

Corresponding author: ^a aurore.naso@ec-lyon.fr

Received: 5 June 2010
Revised: 10 January 2011
Published online: 30 March 2011

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 β-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 ψ. 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 β-effect).