Vortex dynamics 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.