A theory for attractors of microplastic particles in the resonant structures of a 3D eddy

Recent laboratory and numerical investigations have revealed the presence of a variety of attractors, usually in the form of closed loops, for small, rigid spheres in recirculating fluid flows. We present a theory that predicts the presence of such attractors and sets down criteria for their existence in swirling, 3D vortex flows that serve as idealizations of ocean eddies. The three-dimensional fluid circulation in the eddy consists of a horizontal azimuthal flow along with an overturning component, and when this circulation is steady and axially symmetric, the fluid trajectories are confined to a set of nested tori that foliate the container. In this "background" state, a single attractor for slightly buoyant rigid spheres may exist close to the center of the nested tori. When the axisymmetric background flow is perturbed by a non-symmetric, and possibly time-dependent, disturbance, additional attractors can arise within new Kolmogorov-Arnold-Moser tori that appear in the resonant structures created by the disturbance. The tori appear as "islands" in the stroboscopic sections for fluid parcels. Under conditions laid out in the theory, an attracting orbit for slightly buoyant rigid spheres can form near the center of an island. The criteria are tested against numerical simulations of rigid particle trajectories using the Maxey-Riley equations.