Co-rotating vortices on surfaces of variable negative curvature: Hamiltonian structure and curvature-induced drift

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 that in 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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mi mathvariant="normal">Ω</m:mi> <m:mo>=</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi mathvariant="normal">Γ</m:mi> <m:mo>/</m:mo> <m:mn>16</m:mn> <m:mi>π</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mspace width="0.17em"/> <m:msup> <m:mrow> <m:mi>K</m:mi> </m:mrow> <m:mrow> <m:mo>′</m:mo> </m:mrow> </m:msup> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>V</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo>/</m:mo> <m:msqrt> <m:mrow> <m:mo>−</m:mo> <m:mi>K</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>V</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:msqrt> </m:math> Ω=(Γ/16π ) K(V)/√-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 a growth rate <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mi>λ</m:mi> <m:mo>=</m:mo> <m:msqrt> <m:mrow> <m:mn>3</m:mn> </m:mrow> </m:msqrt> <m:mo stretchy="false">|</m:mo> <m:mi mathvariant="normal">Ω</m:mi> <m:mo stretchy="false">|</m:mo> </m:math> λ =√3 Ω , 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.