Field theoretical formulation of the asymptotic relaxation states of two-dimensional ideal fluids

Abstract

The ideal incompressible fluid in two dimensions (Euler fluid) evolves at relaxation from turbulent states to highly coherent states of flow. For the case of double spatial periodicity and zero total vorticity it is known that the streamfunction verifies the \emph{sinh}-Poisson equation. These exceptional states can only be identified in a description based on the extremum of an action functional. Starting from the discrete model of interacting point-like vortices it was possible to write a Lagrangian in terms of a matter function and a gauge potential. They provide a dual representation of the same physical object, the vorticity. This classical field theory identifies the stationary, coherent, states of the $2D$ Euler fluid as derived from the self-duality. We first provide a more detailed analysis of this model, including a comparison with the approach based on the statistical physics of point-like vortices. The second main objective is the study of the dynamics in close proximity of the stationary self-dual state, \emph{i.e.} before the system has reached the absolute extremum of the action functional. Finally, limitations and possible extensions of this field theoretical model for the $2D$ fluids model are discussed and some possible applications are mentioned.