Vortex dynamics in $R^4$

TL;DR

This paper explores vortex dynamics in four-dimensional space, focusing on membrane-like vorticity distributions, their regularized velocities related to curvature, and the Hamiltonian structure, extending classical fluid dynamics concepts to higher dimensions.

Contribution

It introduces a novel membrane model for vortex dynamics in R^4, linking regularized velocities to mean curvature and deriving the symplectic structure from vorticity distributions.

Findings

Regularized velocity proportional to mean curvature vector

Membrane dynamics extend classical vortex filament models

Ertel's vorticity theorem is a special case in R^4

Abstract

The vortex dynamics of Euler's equations for a constant density fluid flow in $R^4$ is studied. Most of the paper focuses on singular Dirac delta distributions of the vorticity two-form $\omega$ in $R^4$. These distributions are supported on two-dimensional surfaces termed {\it membranes} and are the analogs of vortex filaments in $R^3$ and point vortices in $R^2$. The self-induced velocity field of a membrane is shown to be unbounded and is regularized using a local induction approximation (LIA). The regularized self-induced velocity field is then shown to be proportional to the mean curvature vector field of the membrane but rotated by 90 degrees in the plane of normals. Next, the Hamiltonian membrane model is presented. The symplectic structure for this model is derived from a general formula for vorticity distributions due to Marsden and Weinstein (1983). Finally, the dynamics of the four-form $\omega \wedge \omega$ is examined. It is shown that Ertel's vorticity theorem in $R^3$, for the constant density case, can be viewed as a special case of the dynamics of this four-form.