Hydrodynamics of Quantum Vortices on a Closed Surface

TL;DR

This paper develops a vortex fluid theory on closed surfaces, revealing how curvature influences quantum vortex dynamics and deriving an analytical stationary flow analogous to classical Rossby waves.

Contribution

It introduces a covariant vortex fluid equation on curved surfaces, incorporating curvature effects and providing analytical solutions for vortex flows on a sphere.

Findings

Curvature couples to vortex dynamics via an additional term in the fluid equation.

Analytical stationary vortex flow on a sphere analogous to Rossby-Haurwitz wave.

Vortex velocity differences driven solely by curvature, vanishing on a plane.

Abstract

We develop a neutral vortex fluid theory on closed surfaces with zero genus. The theory describes collective dynamics of many well-separated quantum vortices in a superfluid confined on a closed surface. Comparing to the case on a plane, the covariant vortex fluid equation on a curved surface contains an additional term proportional to Gaussian curvature multiplying the circulation quantum. This term describes the coupling between topological defects and curvature in the macroscopic level. For a sphere, the simplest nontrivial stationary vortex flow is obtained analytically and this flow is analogous to the celebrated zonal Rossby-Haurwitz wave in classical fluids on a nonrotating sphere. For this flow the difference between the coarse-grained vortex velocity field and the fluid velocity field generated by vortices is solely driven by curvature and vanishes in the corresponding vortex flow on a plane when the radius of the sphere goes to infinity.