Hill's Spherical Vortex in a Rotating Fluid

TL;DR

This paper derives an exact solution for Hill's spherical vortex in a rotating ideal fluid, revealing how background rotation influences vortex structure, induces inertial waves, and leads to the formation of concentric vortex rings at higher rotation rates.

Contribution

It provides the first exact analytical solution for a spherical vortex in a rotating fluid, detailing the vortex's behavior and structure changes with varying rotation rates.

Findings

Vortex swirls to cancel background rotation

Closed streamlines form at critical rotation rates

Multiple concentric vortex rings emerge with increased rotation

Abstract

A popular model for a generic fat-cored vortex ring or eddy is Hill's spherical vortex (Phil. Trans. Roy. Soc. A vol. 185, 1894, p. 213). Here we find an exact solution for such a spherical vortex steadily propagating along the axis of a rotating ideal fluid. The spherical vortex swirls in such a way as to exactly cancel out the background rotation of the system. The flow external to the spherical vortex exhibits fully nonlinear inertial wave motion. We show that above a critical rotation rate, closed streamlines form in this outer fluid region and hence fluid is carried along with the spherical vortex. As the rotation rate is further increased, further concentric "sibling" vortex rings are formed, counter rotating with respect to the original spherical vortex.