Co-rotating Vortices on Surfaces of Variable Negative Curvature: Hamiltonian Structure and Curvature-Induced Drift

TL;DR

This paper investigates the Hamiltonian dynamics of co-rotating vortices on a negatively curved surface, deriving explicit equations, analyzing stability, and revealing curvature-induced vortex drift with implications for collective vortex behavior.

Contribution

It provides explicit equations of motion for vortices on a catenoid, an analysis of their stability, and insights into curvature-induced vortex drift, extending vortex dynamics to curved geometries.

Findings

Exact analytic solution for a rotating vortex pair at fixed latitude.

The vortex pair's angular velocity depends on the curvature gradient.

Numerical simulations confirm curvature-induced azimuthal drift.

Abstract

Vortices in fluids and superfluids are fundamental to phenomena ranging from Bose-Einstein condensates and superfluid films to neutron stars and hydrodynamic micro-rotors, where background geometry often plays an important role. Curvature can induce vortex motion distinct from planar domains. We study Hamiltonian vortex motion on a catenoid, a minimal surface of variable negative curvature, and derive explicit equations of motion and conserved quantities for co-rotating vortex pairs. For two identical vortices we find an exact analytic solution in which the pair rotates rigidly at fixed latitude, with angular velocity $\Omega=(\Gamma/16\pi)\,K'(V)/\sqrt{-K(V)}$, where $K(V)$ is the Gaussian curvature. Thus the motion is governed by the curvature gradient rather than the curvature itself. This state is linearly unstable, with growth rate $\lambda=\sqrt{3}|\Omega|$, in agreement with numerical simulations. For generic co-rotating pairs, conservation of the Hamiltonian and rotational momentum reduces the nonlinear dynamics to a single quadrature, yielding bounded relative oscillations together with a secular azimuthal drift. Simulations of the full equations confirm this and reveal the same curvature-induced azimuthal drift in a localized many-vortex cluster, motivating a broader theory of collective vortex drift on curved surfaces.