Instability and focusing of internal tides in the deep ocean

The interaction of tidal currents with sea-floor topography results in the radiation of internal gravity waves into the ocean interior. These waves are called internal tides. Their dissipation due to nonlinear wave breaking and concomitant three-dimensional turbulence is currently believed to play an important role in the mixing of the abyssal ocean and hence in controlling the large-scale ocean circulation.

As part of an on-going effort to develop a theory for the vertical distribution of wave breaking over sea-floor topography, we investigate the instability of internal tides in a very simple linear model that allows us to relate the formation of unstable regions to simple features in the sea-floor topography. We concentrate on a real-space representation of the internal tides using Green's functions rather than on the usual spectral representation using Fourier modes. This is because we want to investigate the formation of localized high-amplitude regions in the real-space wave field and for this aim the Fourier representation is not very helpful.

We present results for both two-dimensional and three-dimensional flows over idealized sea-floor topography. Our main aim is to obtain a clear understanding of where the linear wave field is unstable and of how this can be related to topography features such as localized singularities or more extended features that may lead to focusing in the ocean interior. A particular and recurring question will be to understand under what conditions instability in the ocean interior occurs without there being instability directly at the topography. This question has clear implications for the distribution of the wave breaking regions throughout the water column.

For two-dimensional tides over one-dimensional topography we find that the formation of instabilities is closely linked to singularities in the topography shape and that it is possible to have stable waves at the sea floor and unstable waves in the ocean interior above. For three-dimensional tides over two-dimensional topography there is in addition an effect of geometric focusing of wave energy into localized regions of high wave amplitude, and we investigate this focusing effect in simple examples. Geometric focusing is a genuinely three-dimensional flow phenomenon, i.e., it does not occur in the most commonly studied case of two-dimensional flow over one-dimensional topography. Overall, we find that the distribution of unstable wave breaking regions can be highly non-uniform even for very simple idealized topography shapes.