Sensitivity of modeled tracer motion in tidal areas to numerics and to non-Hamiltonian perturbations

Abstract

This study focuses on the motion of passive tracers induced by the joint action of tidal and residual currents in shallow seas with an irregular bottom topography. Interest in this problem has rapidly increased in recent years, because of the detection of large-scale pollution of marine waters by plastics. Early simplified models considered advection of tracers by a two-dimensional depth-averaged velocity field that is solenoidal, thereby resulting in a system that is Hamiltonian and nonintegrable. Here, two new aspects are considered. First, the sensitivity of solutions to three different numerical schemes is investigated. To quantify the behavior of orbits, both the largest Lyapunov exponent and the K-coefficient of the zero-one test for chaos were calculated. It turns out that a new scheme, which extends a known symplectic scheme to systems that also contain non-Hamiltonian terms, performs best. The second aspect concerns the fact that a depth-averaged velocity field is actually divergent, thereby rendering the model of tracer motion to be non-Hamiltonian. It is demonstrated that the divergent velocity components, no matter how small, cause the appearance of attractors in the system and thus they have a strong impact on the fate of tracers. Interpretation of the numerical results is given by deriving and analyzing approximate analytical solutions of the system.