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Resonant Interactions of Surface and Internal Waves with Seabed Topography


This dissertation provides a fundamental understanding of water-wave transformations over seabed corrugations in the homogeneous as well as in the stratified ocean. Contrary to a flat or mildly sloped seabed, over which water waves can travel long distances undisturbed, a seabed with small periodic variations can strongly affect the propagation of water waves due to resonant wave-seabed interactions--a phenomenon with many potential applications.

Here, we investigate theoretically and with direct simulations several new types of wave transformations within the context of inviscid fluid theory, which are different than the classical wave Bragg reflection. Specifically, we show that surface waves traveling over seabed corrugations can become trapped and amplified, or deflected at a large angle ($\sim 90\degree$) relative to the incident direction of propagation. Wave trapping is obtained between two sets of parallel corrugations, and we demonstrate that the amplification mechanism is akin to the Fabry-Perot resonance of light waves in optics. Wave deflection requires three-dimensional and bi-chromatic corrugations and is obtained when the surface and corrugation wavenumber vectors satisfy a newly found class I$_2$ Bragg resonance condition. Internal waves propagating over seabed topography in a stratified fluid can exhibit similar wave trapping and deflection behaviors, but more surprising and intricate internal wave dynamics can also be obtained. Unlike surface waves, internal waves interacting with monochromatic seabed corrugations can simultaneously generate many new waves with different wavenumbers and directions of propagation--a phenomenon which we call chain resonance. Here, we demonstrate that the chain resonance leads to significant energy transfer from long internal waves to short internal waves for almost all angles of the incident waves relative to the orientation of the oblique seabed corrugations. Since short internal waves are prone to breaking, the resonance mechanism, therefore, may have important consequences on the spatial variability of ocean mixing and energy dissipation. Potential applications of the theory of resonant wave-seabed interactions for wave energy extraction and coastal protection are also discussed.

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