Notes on a 1-D model of continental shelf resonances

Abstract

This paper reviews earlier work and provides new results from a one-dimensional model of the scattering of a tidal wave by a continental shelf. When the continental shelf is many wavelengths wide, a large fraction of the incident tidal energy can propagate from the deep ocean onto the shelf. In contrast when there is a nearby coastline most of the energy is reflected unless the shelf approximately 1/4, 3/4, 5/4 etc wavelengths wide and has a suitable amount of frictional damping. The properties of the reflection coefficient are investigated. It is shown that it can be treated as a function of complex angular velocity, strong absorption of tidal energy being associated with nearby poles in the complex angular velocity plane. Physically the poles correspond to decaying shelf modes, their real component depending primarily on the geometry of the region and their imaginary components to their rate of decay. The latter depends on both the bottom friction, acting on the shelf, and on the radiation of energy back into the deep ocean. It is also shown that the reflected wave can be zero when the two are equal and that this can occur for physically realistic values for the depths, shelf width and bottom friction coefficient. Shelf resonances thus provide a classic example of impedance matching. The amplitude and phase of the reflected wave are also investigated and it is found that when plotted in a suitable manner as a function of real angular velocity they produce a characteristic loop as each resonance is passed. Such loops can be be useful for identifying nearby resonances in model studies and in data from the real ocean.

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