Vortex interaction on curved surfaces

TL;DR

This paper investigates vortex energies and interactions on curved surfaces, specifically on spheres and negatively curved surfaces, revealing linear relationships between energy and vortex size or separation.

Contribution

It introduces a method using stereographic projection to calculate vortex energies on curved surfaces and provides explicit numerical vortex configurations.

Findings

Vortex excitation energy on a sphere can be derived via stereographic projection.

Vortex energy on negatively curved surfaces is linearly proportional to vortex radius.

Vortex interaction energy varies linearly with separation distance.

Abstract

The vortex-excitation energy on a sphere can be obtained by using the stereographic projection. By applying this method, we calculate the energy needed to create a vortex on a surface with a constant negative curvature. It is found that the energy is a linear function of the radius of the vortex. In accordance with this result, the interaction energy between a pair of vortices is also found to change linearly with the vortex separation distance. Explicit vortex configurations are obtained numerically with this interaction.