Surf zone vortices near piecewise flat topography.

The motion of ideal shallow water vortices near piecewise flat topography is studied without background rotation for two choices of bottom topography. First, topography is chosen in the form of a rectilinear step change in depth. Finite area monopolar vortices which propagate steadily, without change in shape, parallel to the step are computed numerically. Next, the general motion of a pair of point vortices of arbitrary circulation near an escarpment is found explicitly using Hamiltonian techniques. Some vortex paths are periodic and for specific initial conditions each vortex comprising the pair translates parallel to the step. Comparisons to point vortex trajectories are made with vortex patch trajectories computed using contour dynamics. Agreement between the two trajectories is close provided a vortex patch is sufficiently away from the escarpment. Then, the scattering at a rectilinear step change in depth of a shallow water vortex pair consisting of two patches of equal, but oppositely signed vorticity is studied. Using the constants of motion an explicit relationship is derived relating the angle of incidence to the refracted angle after crossing. It is found that for certain initial conditions a pair can be totally internally reflected by the escarpment. For large depth changes numerical computations show the coherence of the vortex pair is lost on encountering the escarpment. The second part of this work concerns the dynamics of shallow water vortices near circular topography. Finite area monopolar vortices which translate without change in shape around the topography arc computed near a seamount or well including the limiting cases of each: an island or deep well. The behaviour of a vortex pair propagating toward circular topography is examined. Using Hamiltonian techniques, trajectories of point vortices exterior to the topography are found and are compared to trajectories of vortex patches computed using contour dynamics. Point vortex trajectories can be periodic and, for specific initial conditions, each vortex orbits the topography with the same frequency. Finally, laboratory experiments arc performed to find the behaviour of a dipole propagating toward a step change in height. Dipoles approaching the step from either deep or shallow water at normal and oblique incidence are considered. Qualitative observations agree well with theoretical predictions: a dipole increases its separation when crossing from deep water and decreases for a dipole crossing from shallow water. Furthermore, for dipoles approaching from shallow water with a sufficiently large incident angle the dipole was observed to perform total internal reflection