Eur. Phys. J. B 77, 167-186 (2010)
https://doi.org/10.1140/epjb/e2010-00264-5

Relaxation equations for two-dimensional turbulent flows with a prior vorticity distribution

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

Corresponding author: ^a chavanis@irsamc.ups-tlse.fr

Received: 16 October 2009
Revised: 18 March 2010
Published online: 16 September 2010

Abstract

Using a Maximum Entropy Production Principle (MEPP), we derive a new type of relaxation equations for two-dimensional turbulent flows in the case where a prior vorticity distribution is prescribed instead of the Casimir constraints [R. Ellis, K. Haven, B. Turkington, Nonlinearity 15, 239 (2002)]. The particular case of a Gaussian prior is specifically treated in connection to minimum enstrophy states and Fofonoff flows. These relaxation equations are compared with other relaxation equations proposed by Robert and Sommeria [Phys. Rev. Lett. 69, 2776 (1992)] and Chavanis [Physica D 237, 1998 (2008)]. They can serve as numerical algorithms to compute maximum entropy states and minimum enstrophy states with appropriate constraints. We perform numerical simulations of these relaxation equations in order to illustrate geometry induced phase transitions in geophysical flows.