Transport structures in a 3D periodic flow

The linearized 3D Euler equations on an f-plane with constant stratification admit a family of analytical wave solutions. Here, we investigate the Lagrangian properties of one such solution, a standing wave quadrupole, whose simplicity and symmetry make it an ideal time-varying 3D testbed for developing dynamical systems methods. In spite of its simplicity, the Eulerian solution gives rise to highly complex transport structures. Particle trajectories wind around tori-like surfaces with varying cross-sections. They are generally governed by the internal wave frequency plus subinertial frequencies, which depend on starting locations. The spatial variation in this subinertial period produces mixing in the periodic wave motion, a process completely distinct from diapycnal mixing typically associated with internal waves. Nonetheless, finite-time Lyapunov exponents, calculated from the 3D velocity field, clearly delineate transport barriers. These barriers identify five types of coherent Lagrangian structures, which oscillate at the super-inertial internal wave frequency. Two of these types are solely located near the surface, extending to depths unassociated with any Eulerian flow characteristic. The discovery of such shallow structures in the absence of a related Eulerian signal raises the interesting question whether similar structures may be hiding in the real ocean.