Self-organization of the vorticity field in two-dimensional quasi-ideal fluids: The statistical and field-theoretical formulations

•The objective is to understand the self-organization of the vorticity in 2D fluids.•We compare two approaches: statistical and field theoretical.•The Field Theoretical formulation offers a wide range of developments.•We suggest a connection between self-duality and statistical equilibrium.

The natural tendency of the quasi-ideal two-dimensional fluid to evolve by self-organization to highly coherent flow patterns can be formulated as a statistical and as a field theoretical problem. We show that both can derive the asymptotic ordered flows as solutions of the sinh-Poisson equation but the two approaches are different in their possibilities to describe the dynamic phase of the vorticity self-organization. This comparison suggests that, at least for relaxation phenomena, the statistical equilibrium and the geometric-algebraic property of self-duality are two aspects of aspects of the same reality.