Point vortex dynamics in quasi-periodic channels: transporting trajectories and ergodic distribution of equilibria

Abstract

We study the dynamics of a single point vortex in an unbounded planar channel whose interfaces are quasi-periodic in the longitudinal direction. The motion is governed by the Robin function. We introduce a hull formulation that lifts the quasi-periodic geometry to a periodic problem on a finite-dimensional torus and yields an exact quasi-periodic representation of the Robin function. Under a natural transversality condition, we construct global quasi-periodic invariant graphs describing transporting vortex trajectories and show that a full neighborhood of each boundary component is foliated by such graphs. Under a Diophantine condition on the spatial frequencies, the dynamics along each graph can be straightened to a constant drift, so that the vortex motion is quasi-periodic modulo translation. A central contribution concerns the distribution of vortex equilibria. In the quasi-periodic setting, where no fundamental spatial cell exists, we identify critical points of the Robin function with crossings of a hypersurface by a Kronecker flow on the hull torus. Exploiting unique ergodicity, we establish a general zero-counting theorem, allowing finite-order tangencies, which yields an explicit geometric flux formula for the asymptotic density of critical points and their limiting phase distribution. The result is nonperturbative once the critical hull is constructed. Explicit periodic and quasi-periodic models reveal bifurcations and phase transitions in the critical-point distribution, while numerical computations provide quantitative validation of the analytical results.