Evolution of toroidal magnetic eddies in an ideal fluid

Abstract

The magnetohydrodynamic evolution of axisymmetric magnetic eddies within which the magnetic field is purely toroidal with $B_\theta/r$ piecewise-constant, and the velocity field is poloidal, is studied both analytically and numerically. A family of exact solutions, generalizing Hill's spherical vortex to the case of non-zero magnetic field, is found. These exact solutions are (like Hill's vortex) unstable, so that, under weak disturbance, a narrow spike of vorticity is shed from the neighbourhood of the rear stagnation point. Numerical simulation using a contour-dynamics formulation shows that, for general initial contour shape, a contour singularity appears at a finite time $t^*$, like that which appears on a disturbed vortex sheet. Techniques of regularization and sample-point redistribution are used so that the eddy contours can be tracked well beyond $t^*$. When the fluid is initially at rest, the magnetic eddy first contracts towards the axis of symmetry under the action of its Lorentz force distribution; then two spherical fronts form, which propagate in the two opposite directions along the axis of symmetry, in a manner captured well by the exact solution. The magnetic energy remains bounded away from zero despite the fact that there is no topological barrier to its further decrease. Magnetic eddy evolution and the possible existence of steady states under a uniform compressive strain field is also numerically investigated.