Co-rotating Vortices on Surfaces of Variable Negative Curvature: Hamiltonian Structure and Curvature-Induced 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 $\Omega =(\Gamma /16\pi )K^{\prime }(V)/\sqrt{-K(V)}$, 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.

Prepared for submission to Transport Phenomena, De Gruyter Brill, Berlin

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