Generation of internal waves by ocean tides

ides are the oceanic response to well-known gravitational forcing by the Sun and Moon. These hugely energetic oscillations influence various parts of the Earth system, such as the history of the lunar orbit, the transport of heat by the large-scale ocean circulation, and ice sheets flowing into the polar oceans.

As tidal currents flow over the bottom topography of the ocean, they generate internal gravity waves of tidal frequency in the density-stratified ocean interior. Although almost invisible from the ocean surface, these internal tides play an important role in a range of coastal and open-ocean dynamics. Of particular interest is the amount of energy converted from the (surface) tide to internal tides (along with the magnitude of the implied internal tide drag on the depth-averaged flow), and how this energy is transported within the ocean interior.

In this PhD project, simple mathematical models of internal tide generation and propagation will be developed in idealised three-dimensional geometries, building on previous two-dimensional studies. This involves making some combination of simplifying assumptions about the motion (e.g., linear, weakly nonlinear, time-periodic), density stratification (e.g., uniform, piecewise constant), and sea-floor topography (e.g., small-amplitude, stepped), so that the nonlinear partial differential equations of motion can be reduced to a more tractable system. Solutions can then be sought analytically or via simple numerical computations, allowing a wide range of parameter space to be explored. This gives a better understanding of the energy transfer from (surface) tides to internal tides in different forcing scenarios, which is known to have a non-trivial dependence upon topographic width and height.

A particular focus would be on extending the solutions and methods of [7], which accounted for arbitrarily large topographic variations but only in two dimensions, to three-dimensional settings. The main targets are to

1. Understand internal tides generated by a three-dimensional surface tide (in particular, the Kelvin wave) above a two-dimensional topography (representing a continental slope). The Kelvin wave is the dominant form of the surface tide in many coastal zones, and the implied energy transfers are significant on a global scale.

2. Understand the role of topographic undulations in the along-shore direction, i.e., for a three-dimensional tide above a fully three-dimensional topography. It is known from observations that canyons (for exam-ple) lead to enhanced internal tide generation (and dissipation).