Symplectic structures and dynamics on vortex membranes

TL;DR

This paper develops a Hamiltonian framework for higher-dimensional vortex membranes and sheets, generalizing classical vortex filament dynamics and introducing symplectic structures on these singular vorticity elements.

Contribution

It introduces a unified Hamiltonian approach for vortex membranes and sheets, extending the classical filament equations to higher dimensions and defining associated symplectic structures.

Findings

LIA describes skew-mean-curvature flow on vortex membranes of any codimension.

Framework generalizes classical vortex filament equations to higher dimensions.

Symplectic structures are constructed on spaces of vortex sheets.

Abstract

We present a Hamiltonian framework for higher-dimensional vortex filaments (or membranes) and vortex sheets as singular 2-forms with support of codimensions 2 and 1, respectively, i.e. singular elements of the dual to the Lie algebra of divergence-free vector fields. It turns out that the localized induction approximation (LIA) of the hydrodynamical Euler equation describes the skew-mean-curvature flow on vortex membranes of codimension 2 in any dimension, which generalizes the classical binormal, or vortex filament, equation in 3D. This framework also allows one to define the symplectic structures on the spaces of vortex sheets, which interpolate between the corresponding structures on vortex filaments and smooth vorticities.