Yingshuo ``Poppy'' Shen

Nonlinear Periodic Flow Over Isolated Topography: Eulerian and Lagrangian Perspectives

Thesis Approved September 1992

The time-varying current field associated with tidal flow over an isolated topographic feature is relatively well understood in the Eulerian frame of reference. This study examines the flow in a Lagrangian frame of reference. The ultimate goal is to characterize the particle motion in terms of parameters such as the Rossby number and the ratio of the tidal frequency to the inertial frequency. The underlying question is straightforward: under what condition does such a bump act as a "stirring rod", causing strong dispersion of particles, and under what condition does it act to trap particles over the bump? The approach is to use simple analytical models wherever possible and then to extend the analysis with a numerical model, implemented with a radiation condition on the open boundary.

In order to obtain particle motion with confidence, the Eulerian flow field is calculated, analyzed and compared with previous studies. Topographic Rossby waves are excited by the sloshing of the tidal flow over the bump. The analytical model predicts a linear relation between the fractional depth change and the resonant frequency of the first mode topographic Rossby wave, in agreement with Rhines (1969). The tidally- rectified mean Eulerian flow and elevation slope are also predicted for the rectilinear tidal flow. The results agree with those of Loder (1980) only for high frequency flow (e.g. M_2 tide). For weakly nonlinear flow, it is also shown that Eulerian mean effectively cancels the Stokes drift. Long-term particle movements are described in terms of "tidal Poincaré maps" which show the net displacement of a grid of particles over one tidal period. For high frequency and weak advection, particles are retained over the bump and exhibit small net displacements over a tidal period. Even with increased advection, there are always some "islands" surrounding elliptic points where particles stay together and exhibit weak mixing and dispersion. For low frequency and weak advection, the particles are still trapped over the bump but are subject to strong mixing.

Returning to the underlying question posed above, I conclude that there is no simple parameterization of mixing associated with tidal flow over a bump. The tidal Poincaré maps are highly complex, with many elliptic and hyperbolic points whose location, and even existence, are strongly dependent on the Rossby number, the forcing frequency, and finally the frictional spin- down time in pendulum days. I speculate that near resonant topographic Rossby wave may make a major contribution to the complexity of tidal Poincaré maps. However, the question as to whether such resonances determine the onset of chaotic particle motion (Pratte and Hart, 1991) remains open.