Internal Wave Dynamics at Sloping Boundaries

The stably stratified interior of the ocean is never still due to the ubiquitous presence of internal inertia-gravity waves, referred to here simply as internal waves. These waves exist in a stratified fluid where motions are subject to both gravity and Coriolis forces. The dispersion relation for freely propagating internal waves is specified for frequencies, $\omega$ , between the Coriolis frequency $f$ and the Brunt-Vaisala frequency $N$ . The spectrum of these internal waves is similar throughout most of the ocean and is generally well described by the Garrett and Munk (1979) spectrum. Internal waves propagate at an angle, $\theta$ , relative to the vertical, that depends on the wave frequency, the Brunt-Vaisala frequency and the Coriolis frequency such that $\omega^2=N^2cos^2\theta+f^2sin^2\theta$ . A peculiar aspect of internal wave dynamics arises upon reflection at a sloping boundary. Unlike light waves which undergo specular reflection at a smooth boundary, internal waves reflect so that the incident and reflected angle are maintained relative to the vertical, rather than to the normal angle of the boundary. In the extreme case where the topography and the wave propagation have the same slope, critical reflection occurs and linear theory predicts that the reflected wave has an infinite amplitude, an infinitesimal wavelength and a near-zero group velocity (Phillips, 1977).

In general, the reflected wave is steeper and produces greater shear than the incident wave, particularly around the critical frequency (Eriksen, 1985; Eriksen, 1982). Eriksen (1982) observes an enhancement in horizontal kinetic energy around the critical frequency within $100 m$ of the bottom, at a variety of island, seamount and continental slope sites, and which was absent $1000 m$ above the bottom. These enhanced levels of kinetic energy near the bottom can lead to enhanced turbulent dissipation and mixing.

The mixing associated with critical reflection on a non-planar slope can depend on the slope configuration above and below the critical region. Analytical solutions have shown that critical reflection produces more mixing in the case of a convex slope (Müller and Liu, 2000; Gilbert and Garrett, 1989) than in the case of a concave slope. In another study, Legg and Adcroft (2003) used a non-hydrostatic general circulation model to study the critical reflection of internal waves and concluded that the enhanced mixing associated with critical reflection did not depend on the curvature of the slope, but only on the critical character of the slope. To our knowledge, modeling studies have not been made of internal wave reflections on complex, finite amplitude, 3D bathymetry, such as at our study site.

Internal waves can also approach a sloping boundary obliquely, when the vertical plane containing the wave group velocity vector does not intersect the boundary along the line of greatest slope. Upon such a reflection, density fronts can form and propagate up the slope (Thorpe, 1999). These fronts are most likely to form when the incident wave is near critical and its obliqueness is limited to within $30^\circ$ of the normal incidence.

Most of the studies on internal wave reflection consider internal waves coming from deep water and reflecting off a slope. Internal waves can also propagate down along a near-critical slope from their generation site, especially along continental shelves (Pingree and New, 1989). In the Bay of Biscay, Gemmrich and van Haren (2001) observed rapid temperature drops near the bottom occurring at semidiurnal periods. Gemmrich and van Haren (2001) attributed these temperature drops to fronts generated by the flow field of an obliquely downward propagating internal waves. We will consider this process further in section 6.1.2