Surface tides are the periodic motions of our world's ocean due to the gravitational attraction of the Sun and Moon. When these deep tidal flows encounter variable bottom topography, the stratified ocean interior is forced to move up and down periodically, leading to the generation of internal waves of tidal frequency - the internal tide. These internal tides radiate away from the topography and can break in the ocean interior, leading to vertical mixing that plays a role in the ocean potential energy budget. The energy conversion to internal tides is thus of great interest. While global satellite observations estimate a tidal conversion rate of ~1 TW, there remains considerable uncertainty in this estimate, and its split between open-ocean generation (seamounts, mid-ocean ridges, etc.) and coastal generation (continental slopes).
In this thesis, we perform mathematical and numerical modelling of internal-tide generation at continental slopes, which is less well-studied than for open-ocean topographic features. We use a two-layer linear shallow-water model to study the forcing of internal tides by a barotropic Kelvin wave, which is typical of many semi-diurnal coastal tides. Using a modal decomposition of the barotropic and internal modes above variable topography, we focus on two main problems.
The first problem is internal-tide generation along a uniform continental slope. Here, the internal tides are forced by a weak cross-shore flow that is induced in the barotropic Kelvin wave as it moves over variable bottom topography, the strength of which can be estimated asymptotically. This is a three-dimensional internal-tide generation problem, which we solve analytically when the continental slope is a step, and numerically for more realistic slopes of finite width.
The second problem is internal-tide generation along a continental slope that is incised by submarine canyons, which are a ubiquitous feature of continental slopes. Whilst there have been modelling studies of internal-tide generation at particular canyons, our emphasis is on understanding how the amplitude and direction of the radiating internal tides depend upon the canyon geometry more generally, given the diversity of canyons that exist across the globe. To do this, we study idealised canyon configurations incising idealised continental slopes, enabling us to define and then explore a wide parameter space. This is possible due to the development of a high-accuracy discontinuous Galerkin finite element numerical scheme, which enables us to focus the numerical resolution on the narrow canyons and continental slope while still permitting wave radiation in the far field. We conclude that, despite the larger topographic gradients, canyons decrease the strength of internal-tide generation, because the canyons strongly modify the local form of the barotropic tide.
