Vortex interactions with topographic features in geophysical fluid dynamics

There are regions in the abyssal ocean where sharp topographic gradients occur, for example escarpments, canyons or seamounts. In such regions the contribution of the topography to the ambient potential vorticity dominates over the ubiquitous effects of planetary curvature, and may play an important part in steering abyssal eddies, such as those affecting the dispersal of newly formed bottom water. This thesis studies some models of vortex motion near a topographic escarpment. The topography produces a restoring mechanism for wave generation, and acts as a wave guide, i.e. the topographic wave phase and energy travels parallel to the isobaths with shallow water on the right in the Northern hemisphere. The ratio. S', of the time scale for topographic wave generation to the time scale for the vortex circulation, is a measure of vortex intensity. If the two scales are well separated, i.e. S ≫ 1 (a weak vortex) or S ≪ 1 (an intense vortex), analytical progress is made. For a moderate intensity vortex (S ≈ 1) the wave-vortex interaction is nonlinear and the contour dynamics algorithm is adopted to study the vortex motion in this regime. In Chapter 1 some examples of geophysical vortices are described, along with their significance. Chapter 2 gives a brief summary of the mathematical preliminaries. Chapter 3 constitutes a review of the relevant work, namely vortex motion over varying topography in quasigeostrophic dynamics. Of interest is vortex motion on the β-plane, since the methods employed in such studies can be adapted for the present work. In Chapter 4 the results of McDonald (1998) for the motion of an intense singular vortex near an escarpment are extended to cover the full range of vortex intensities. Analytic results indicate that a weak singular vortex moves parallel to the escarpment in the sense of its image in the escarpment. The vortices which travel in the same direction the phase of the topographic waves radiate waves and experience motion perpendicular to the isobath as a result of energy loss. Numerical results for moderate intensity singular vortices show that the motion is characterised by dipole formation. The primary vortex pairs up with an opposite signed patch of relative vorticity which has been produced as a result of cross escarpment advection. An anticyclone located on the shallow side of the escarpment or a cyclone located on the deep side cross the escarpment as a result. A cyclone located on the shallow side of the escarpment or an anticyclone located on the deep side are reflected away from the escarpment. Chapter 5 is an investigation into the motion of an initially circular vortex patch near an escarpment. It is found that weak vortex patches behave as if the escarpment were a wall. At large times weak vortices which travel in the same direction as the topographic wave phase radiate wave energy, and are destroyed as a result of topographic wave radiation. Analytical results show that an intense vortex patch moves in the same manner as an intense β-plane vortex, i.e. cyclones move along curved northwest trajectories and anticyclones move southwest. Numerical studies for moderate intensity patches show that the motion is again characterised by dipole formation. Finally, Chapter 6 considers the motion of a singular vortex near a coastal ridge, i.e. an escarpment running parallel to a wall. It is shown that weak vortices behave as if the escarpment were a plane wall. In the cases where the vortex travels in the direction of the topographic wave phase radiate wave energy, and the vortex drifts towards the escarpment as a result. Numerical studies are presented in the cases of intense and moderate vortices