Pointed vortex loops in ideal 2D fluids

TL;DR

This paper investigates a novel class of singular vorticities called pointed vortex loops in ideal 2D fluids, analyzing their mathematical structure and the associated configuration space.

Contribution

It introduces pointed vortex loops as a new type of singular vorticity and characterizes their coadjoint orbits and symmetry properties.

Findings

Configuration space is the set of loops enclosing a fixed area.

Diffeomorphisms preserving the loop form the polarization subgroup.

The structure of these vortex loops relates to the symmetry group of the fluid.

Abstract

We study a special kind of singular vorticities in ideal 2D fluids that combine features of point vortices and vortex sheets, namely pointed vortex loops. We focus on the coadjoint orbits of the area-preserving diffeomorphism group of ${\mathbb R}^2$ determined by them. We show that a polarization subgroup consists of diffeomorphisms that preserve the loop as a set, thus the configuration space is the space of loops that enclose a fixed area, without information on vorticity distribution and attached points.