Exact spherical vortex-type equilibrium flows in fluids and plasmas

Abstract

The famous Hill’s solution describing a spherical vortex with nested toroidal pressure surfaces, bounded by a sphere, propelling itself in an ideal Eulerian fluid, is re-derived using Galilei symmetry and the Bragg–Hawthorne equations in spherical coordinates. The correspondence between equilibrium Euler equations of fluid dynamics and static magnetohydrodynamic equations is used to derive a generalized vortex type solution that corresponds to dynamic fluid equilibria and static plasma equilibria with a nonzero azimuthal vector field component, satisfying physical boundary conditions. Separation of variables in Bragg–Hawthorne equation in spherical coordinates is used to construct further new fluid and plasma equilibria with nested toroidal flux surfaces, featuring respectively boundary vorticity sheets and current sheets. Finally, the instability of the original Hill’s vortex with respect to certain radial perturbations of the spherical flux surface is proven analytically and illustrated numerically.

Keywords

• Plasma vortex
• Fluid vortex
• Euler equations
• MHD equations
• Exact solution
• Axial symmetry