Oblique internal-wave chain resonance over seabed corrugations

Here we show that monochromatic long-crested corrugations on an otherwise flat seafloor can coherently scatter the energy of an oblique incident internal wave to multiple multi-directional higher-mode internal waves via a series of resonant interactions. We demonstrate that a resonance between seabed corrugations and a normally or slightly oblique incident internal wave results in a series of follow-up resonant interactions, which take place between the same corrugations and successively resonated shorter waves. A chain resonance of internal waves that carries energy to small scales is thus obtained, and we find that the Richardson number decreases by several orders of magnitude over the corrugated patch. If the incidence angle is large, and the incident wave perfectly satisfies a resonance condition with the topography, it turns out that not many higher-mode resonance or near-resonance conditions can be satisfied, such that energy stays confined within the first few modes. Nevertheless, if the incident waves are sufficiently detuned from satisfying a perfect resonance condition with the seabed corrugations, then we show that this frequency detuning may balance off the large detuning due to oblique incidence, leading to a chain resonance that again carries energy to small scales. The evolution of the incident and resonated wave amplitudes is predicted from the envelope equation for internal waves over resonant seabed topography in a three-dimensional rotating fluid, which we derive considering the Boussinesq and $f$-plane approximations with $f$ the Coriolis frequency, linear density stratification and small-amplitude corrugations. Our results suggest that topographic features on the ocean floor with a well-defined dominant wavenumber vector, through the chain resonance mechanism elucidated here, may play a more important role than previously thought in the enhancement of diapycnal mixing and energy dissipation.