Simply Connected Topology in Perturbed Vortices and Field-Reversed Configurations
Zero-helicity vortices, such as Hill's vortex and field-reversed configurations (FRCs), have long been assumed to be toroidal in topology. This paper proves this assumption false: under arbitrarily small odd-parity (with respect to the symmetry axis) transverse field perturbations, interior flux surfaces become simply connected. The previous topological categorization--open and closed field lines separated by an ellipsoid separatrix--is updated to three distinct categories: open field lines in the outermost region, closed field lines on torus flux surfaces in an intermediate region, and closed field lines on simply connected flux surfaces in the innermost region. In addition to a shifted ellipsoid outer separatrix separating closed and open field lines, a new crescent-shaped inner separatrix separates the torus and simply connected surfaces. The simply connected region is significant even for small perturbations; e.g., in a spherical vortex with a perturbation 10% of the background field strength, it occupies 40% of the outer separatrix. The analysis also proves the conjecture regarding field line closure under odd-parity perturbation in the full three-dimensional context. Preliminary numerical simulations of charged particle trajectories in FRC magnetic confinement under odd-parity perturbation were also conducted; crescent-like simply connected volumes were observed even when gyro-radii were small compared to the system size. Since FRCs are sustained by a rotating magnetic field with odd parity, these results motivate a revision of FRC-related fusion confinement physics. Given the mathematical equivalence to Hill's vortex, this also updates our topological understanding of fluid flow in a wide array of phenomena.