# 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.