2 00 2 Self - duality of the asymptotic relaxation states of fluid and plasmas

The states of asymptotic relaxation of 2-dimensional fluids and plasma present a high degree of regularity and obey to the sinh-Poisson equation. We find that embedding the classical fluid description into a field-theoretical framework, the same equation appears as a manifestation of the self-duality. The states generated by externally driving (stirring) an ideal fluid can have very irregular form. It is however known from experiments and numerical simulations that after suppressing the drive the system evolves to states with high degree of order, essentially consisting of few large vortices, with very regular geometry. These states are attained after long time evolution and are not due to the residual dissipation. The process consists of vortex merging, which is an essentially topological event where the weak dissipation only allows the reconnection of the field lines but does not produce significant energy loss from the fluid motion. Inferring from results of numerical simulations, Montgomery et al. [1], [2] have proved that the scalar stream function ψ describing the motion in two-dimensional space obeys in the far asymtotic regime (where the regular structures are dominant) the sinh-Poisson equation ∆ψ + γ sinh (βψ) = 0 (1) where γ and β are positive constants. The relations of ψ to the velocity and vorticity are v = ∇ψ × e z , ω = ∇ × v = −∇ 2 ψ e z where e z is the unitary vector perpendicular to the plane. With these variables, the Euler equations for the two dimensional ideal incompressible fluid are ∂ω ∂t + (v · ∇) ω = 0 (2) ∇ · v = 0 We will try to develop a field-theoretical model of the stationary asymptotic relaxed states, i.e. we look for a model that could provide a derivation of Eq.(1). This will be done progressively, examining models ellaborated for closely related problems and collecting the relevant suggestions that could allow us to write a Lagrangian density. In the study of the two-dimensional Euler fluids, and in particular in explaining the origin of Eq.(1), an important model consists of a system of N discrete vorticity filaments perpendicular on plane, having circular transversal section of radius a and carrying the 1