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.

arXiv commentPrepared for submission to Transport Phenomena, De Gruyter Brill, Berlin

References in corpus (14)

• Observation of vortex dipoles in an oblate Bose-Einstein condensate
• Giant Vortex Clusters in a Two-Dimensional Quantum Fluid
• Vortices in Superfluid Films on Curved Surfaces
• Vortices and turbulence in trapped atomic condensates
• Suppression of Kelvon-induced decay of quantized vortices in oblate Bose-Einstein Condensates
• Creation and characterization of vortex clusters in atomic Bose-Einstein condensates
• Superfluid vortex dynamics on an ellipsoid and other surfaces of revolution
• Superfluid vortex dynamics on a spherical film
• Ultra-quantum turbulence in a quenched homogeneous Bose gas
• Vortex Flows and Streamline Topology in Curved Biological Membranes
• Singular vortex pairs follow magnetic geodesics
• Vortex Dynamics in Tubular Fluid Membranes
• Dynamics of Vortex Clusters on a Torus
• A Self Propelled Vortex Dipole Model on Surfaces of Variable Negative Curvature