Vortex dynamics in $\mathbb {R}^4$R4

Abstract

The vortex dynamics of Euler's equations for a constant density fluid flow in \documentclass[12pt]{minimal}\begin{document}$\mathbb {R}^4$\end{document}R4 is studied. Most of the paper focuses on singular Dirac delta distributions of the vorticity two-form ω in \documentclass[12pt]{minimal}\begin{document}$\mathbb {R}^4$\end{document}R4. These distributions are supported on two-dimensional surfaces termed membranes and are the analogs of vortex filaments in \documentclass[12pt]{minimal}\begin{document}$\mathbb {R}^3$\end{document}R3 and point vortices in \documentclass[12pt]{minimal}\begin{document}$\mathbb {R}^2$\end{document}R2. The self-induced velocity field of a membrane is shown to be unbounded and is regularized using a local induction approximation. 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° 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 [“Coadjoint orbits, vortices and Clebsch variables for incompressible fluids,” Physica D 7, 305–323 (1983)10.1016/0167-2789(83)90134-3]. Finally, the dynamics of the four-form ω ∧ ω is examined. It is shown that Ertel's vorticity theorem in \documentclass[12pt]{minimal}\begin{document}$\mathbb {R}^3$\end{document}R3, for the constant density case, can be viewed as a special case of the dynamics of this four-form.