Surface and internal tides above sea-floor topography

Waves of long spatial scale (meaning wavelengths of hundreds or thousands of kilometres) and long time scale (meaning periods of hours or days) are important in a range
of dynamical phenomena in the ocean. For example, these waves are fundamental to the dynamics of ocean tides, which are the focus of this study. Here we are concerned with both barotropic waves and internal waves, and the forcing of internal waves of tidal frequency (internal tides) by barotropic tides.

After an introduction to the background and physical significance of this subject, the governing equations for long barotropic linear waves are set out and the underlying assumptions are discussed. We then turn to the issue of coastally-trapped barotropic waves, and review some simple solutions for the three main classes of such waves (the Kelvin wave, edge waves, and topographic Rossby waves). Detailed solutions are derived for these waves above a simple step topography, based upon an analytically derived dispersion relation, and these solutions are compared with numerical solutions above a smooth topography. A detailed solution is also derived for a family of topographic Rossby waves above a smooth slope in an unbounded domain, and the frequencies of these waves are shown to be in good agreement with the frequencies determined by numerical solutions with a coastline. Throughout, there is a focus on waves of tidal period.

A simple solution of internal tide generation is also presented, in a two-layer fluid with a step topography (and no background rotation or coastline). Explicit analytical expressions are derived for the outgoing internal wave energy fluxes in this model, and are compared with estimates of energy fluxes in the real ocean.