Physical aspects of the field-theoretical description of two-dimensional ideal fluids

TL;DR

This paper explores the field-theoretical description of two-dimensional ideal fluids, focusing on understanding the physical meaning of the variables and operations within the formalism to enhance its interpretability.

Contribution

It investigates the physical interpretation of the variables and operations in the field theory model of 2D ideal fluids, aiming to deepen understanding of the formalism.

Findings

Derived the sinh-Poisson equation for stationary structures.

Clarified the physical meaning of variables in the field theory.

Analyzed the operations used in the field-theoretical description.

Abstract

The two-dimensional ideal (Euler) fluids can be described by the classical fields of streamfunction, velocity and vorticity and, in an equivalent manner, by a model of discrete point-like vortices interacting in plane by a self-generated long-range potential. This latter model can be formalized, in the continuum limit, as a field theory of scalar matter in interaction with a gauge field, in the $su(2) $ algebra. This description has already offered the analytical derivation of the \emph{sinh}-Poisson equation, which was known to govern the stationary coherent structures reached by the Euler fluid at relaxation. In order this formalism to become a familiar theoretical instrument it is necessary to have a better understanding of the physical meaning of the variables and of the operations used by the field theory. Several problems will be investigated below in this respect.