Dynamics of large - amplitude internal waves in stratified flows over topography

The present research is comprised of theoretical investigations of three problems in the mechanics of internal waves of large amplitude, using analysis and numerical methods. In the first problem, the flow of a Boussinesq density-stratified fluid of large depth past the algebraic mountain ('Witch of Agnesi') is studied in the hydrostatic limit using the asymptotic theory of Kantzios & Akylas (1993). The upstream conditions are those of constant velocity and Brunt-Viislhli frequency. On the further assumptions that the flow is steady and there is no permanent alteration of the upstream flow conditions (no upstream influence), Long's model (Long 1953) predicts a critical amplitude of the mountain (E = 0.85) above which local density inversions occur, leading to convective overturning. Linear stability analysis demonstrates that Long's steady flow is in fact unstable to infinitesimal modulations at topography amplitudes below this critical value, 0.65 < E < 0.85. This instability grows at the expense of the mean flow and may be attributed to a discrete spectrum of modes that become trapped over the mountain in the streamwise direction. The transient problem is also solved numerically, mimicking impulsive startup conditions. In the absence of instability, Long's steady flow is reached. For topography amplitudes in the unstable range 0.65 < e < 0.85, however, the flow fluctuates about Long's steady state over a long timescale; there is no significant upstream influence and no evidence of transient wave breaking is found for e < 0.75. In the second problem, the phenomenon of shelf generation by nonlinear waves in twodimensional stratified flows is investigated. The case of a uniformly stratified, Boussinesq fluid of finite depth is of primary interest; it is shown that the use of asymptotically matched (streamwise) regions becomes necessary. The 'inner region' is described by the fully nonlinear theory of Grimshaw & Yi (1991), while the 'outer region' consists of linear, downstreampropagating fronts, the cumulative effect of which is to give the appearance of a shelf that carries mass but no energy. A similar shelf is found to exist in the corresponding infinitedepth problem. The case of weakly nonlinear waves in an arbitrarily stratified fluid is also examined, where it is found that a shelf of fourth order in wave amplitude is generated. Moreover, the shelf extends both upstream and downstream in general and could thus lead to an upstream influence of a type that has not been previously considered. The mechanism of shelf generation in all cases is shown to be a self-interaction of the nonlinear wave, where transience is an essential ingredient. In the third problem, a theory is developed for the resonant generation by submerged topography of weakly three-dimensional internal waves in a fluid with a linearly varying density distribution. The flow is shown to be governed by an integro-differential equation, which is capable of describing finite-amplitude waves and is valid until incipient density inversions take place. In addition to the nonlinearity caused by the presence of a topographic forcing, it is found that three-dimensional effects are also manifested as nonlinear terms in this evolution equation. The theory is observed to break down in the far-field, owing to the formation of an infinite downstream shelf, which results in a flux of both mass and energy from the resonant wave. As in the two-dimensional problem, matched asymptotic expansions are used to resolve the difficulties caused by the shelf. Numerical solutions of the nonlinear evolution equation for waves in a channel are presented; the parameter space consists of a resonance detuning and a relative blockage, which measures three-dimensional effects. Wave breaking is found to occur over a finite range of detuning for a given relative blockage. The scaling of the breaking time is also investigated. Thesis Supervisor: Triantaphyllos R. Akylas Title: Professor of Mechanical Engineering