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Thesis (Ph.D.(Exploration Geophysics))--University of Pretoria, 2007. / Abstract in English. Includes bibliographical references (leaves 72-76).

<p> Oceanic motion outside thin boundary layers is primarily a mixture of quasi-geostrophic flow and internal waves with either near-inertial frequencies or the frequency of the semidiurnal lunar tide. This dissertation seeks a deeper understanding of waves and flow through reduced models that isolate their nonlinear and coupled evolution from the Boussinesq equations. Three physical-space models are developed: an equation that describes quasi-geostrophic evolution in an arbitrary and prescribed field of hydrostatic internal waves; a three-component model that couples quasi-geostrophic flow to both near-inertial waves and the near-inertial second harmonic; and a model for the slow evolution of hydrostatic internal tides in quasi-geostrophic flow of near-arbitrary scale. This slow internal tide equation opens the path to a coupled model for the energetic interaction of quasi-geostrophic flow and oceanic internal tides. </p><p> Four results emerge. First, the wave-averaged quasi-geostrophic equation reveals that finite-amplitude waves give rise to a mean flow that advects quasi-geostrophic potential vorticity. Second is the definition of a new material invariant: Available Potential Vorticity, or APV. APV isolates the part of Ertel potential vorticity available for balanced-flow evolution in Eulerian frames and proves necessary in the separating waves and quasi-geostrophic flow. The third result, hashed out for near-inertial waves and quasi-geostrophic flow, is that wave-flow interaction leads to energy exchange even under conditions of weak nonlinearity. For storm-forced oceanic near-inertial waves the interaction often energizes waves at the expense of flow. We call this extraction of balanced quasi-geostrophic energy 'stimulated generation' since it requires externally-forced rather than spontaneously-generated waves. The fourth result is that quasi-geostrophic flow can encourage or 'catalyze' a nonlinear interaction between a near-inertial wave field and its second harmonic that transfers energy to the small near-inertial vertical scales of wave breaking and mixing. </p>

The main focus of this work was to investigate the nature of unsteady boundary-layer development over finite domains, with the behaviour of the boundary layer on a rotating sphere in an unbounded, rotating fluid used as a prototype. The sphere and its surrounding fluid are assumed to be initially rotating as a solid body, and the evolution of a boundary layer on the sphere is analysed in cases where the sphere has been smoothly slowed, or brought to a state of rotation in an opposite sense to its initial conditions. It may be seen that a characteristic property of this flow is that the boundary layer is bi-directional; over most of the streamwise domain for the flow, whether the flow is positive or negative in the streamwise coordinate direction depends on the transverse location being considered. This fact leads to challenges in the numerical evaluation of the flow field due to the parabolic nature of the boundary-layer equations. A further consideration is the implication that these regions of reversed flow cause the flow field to contain minima and maxima in the streamwise velocity component. This has been shown in a little-known study by Cowley et al. (1985) to cause the boundary layer to become susceptible to asymptotically short-scale perturbations with large frequencies. The unsteady boundary layer on a rotating sphere under these conditions is consequently shown to be extremely challenging to compute numerically. It is also found that using local approximations at the ends of the finite domain, which in the case of the sphere are the pole and equator, to investigate the two-dimensional boundary layer can cause difficulties, as in some cases there exist steady, spatial perturbations to a boundary-layer state which introduce short spatial scales. The instabilities and other features analysed in this work are framed largely in the context of the rotating sphere, but the causes of the phenomena are found to be sufficiently generic that they may be observed in other physical contexts. To demonstrate this, the shallow katabatic flow down a cooled slope is briefly investigated, and the above mathematical features are again uncovered.