Royal Society Publishing

Subharmonic resonance of oblique interfacial waves by a progressive surface wave

D. F. Hill, M. A. Foda

Abstract

Experimental and theoretical investigations into the generation of internal gravity waves by monochromatic progressive surface waves are presented. Using the method of nonlinear resonant interactions, a triad consisting of a single surface wave and two oblique internal waves in a two–layer model is considered. A multiple scales analysis is adopted and the boundary value problem is expanded in a power series of the surface–wave steepness. At the leading order, the linear harmonics are obtained and the conditions for resonance are determined. A second–order analysis is then used to derive temporal evolution equations for the internal–wave amplitudes. As a consequence of having a single generating train of the surface waves, two oblique trains of internal waves of much shorter wavelength are found to be resonated exponentially in time. Both linear and nonlinear bounds on surface–wave frequency, density ratio and interaction angle are found, demonstrating that the instability is highly narrow banded. It is found that the internal waves grow most rapidly at the linear cut–off values. Experimental evidence is presented and demonstrates good agreement with the theoretical results. Discussion of an application of the theory to the nonlinear energy transfer between very–low–frequency waves in the deep ocean is then provided.

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