  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.
