Particle Trajectories in Topography-Induced Currents: Looking for Chaos

Abstract

Tidal waves are experienced all over the world. In shallow tidal areas, the tidal current over a bottom-topography leads to induced residual currents. To study the effect of these residual currents in combination with the tidal current on the trajectories of passive particles, previous studies used a simplified Hamiltonian model. To compensate for the neglected terms, the relative strength of the residual currents was overestimated. It was shown that under certain circumstances the dynamics leads to chaotic particle trajectories in phase space. In this study it is shown that, up to first order, the chaotic behaviour, as previously found, is not as commonly present in these systems. Furthermore, this study tries to investigate the effect of adding the Coriolis-term, a vorticity-harmonic term and a divergent term, which were previously neglected, to the dynamics. The effects of adding these terms in the dynamical equations did not trigger the onset of chaotic particle trajectories. In order to compute these particle trajectories, a splitting method is adapted to fit the time-periodic forcing that is present in these tidal systems. In contrast to previously used methods, this numerical method preserves the symplectic structure of the time-evolution of a Hamiltonian system. In the order of hundreds of tidal periods, this method produces similar results to Runge-Kutta methods for the majority of the cases. For longer periods of time, the Runge-Kutta methods showed divergence of the particle trajectories, even when this is not expected by the dynamics equations. A special case in phase space was found for choosing typical parameters that link to the Bessel functions (Beerens, Ridderinkhof, and Zimmerman 1994). In these special cases the shape of the particle trajectories changes significantly. For future studies, this behaviour may be of interest. Future studies may also look at adding even more tidal constituents to the dynamics or investigating the effect of choosing a bottom-topography with different spatial form. The numerical method presented here, may be applied in different problems, where the system of interest (largely) obeys a symplectic nature and allows for a splitting method.