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Surface wave interaction with oblique internal waves

University of British Columbia

The results of a theoretical and experimental study of a resonant interaction between a surface wave and two internal waves are presented. It is shown that the motion of a surface wave in a horizontally infinite two-layer fluid can lead to generation of two oblique internal waves. The internal waves are short compared to the surface wave and have nearly opposite propagation directions. The frequencies of the internal waves are approximately half of the frequency of the surface wave. Two analytical models are developed. The first model is based on a three-dimensional formulation of weakly non-linear interaction of the waves in a two-layer medium. The problem is initially formulated assuming that both layers are inviscid, but after obtaining the evolution equations of the internal waves, the damping due to a viscous lower layer is incorporated in the analysis. A standard technique is employed to obtain the evolution equations of the internal waves. The second model has a two-dimensional viscous formulation and serves to explore the effects of the viscosity of the lower layer on both forcing and damping in the interaction. The model places no limitation on the viscosity of the lower layer. Using the three-dimensional model, the effects of different parameters of the system on the interaction are investigated by changing each parameter separately and observing the effects on the evolution of the internal waves. Of particular importance is the effect of the direction angle of the internal wave pair on the interaction. It is shown that the more oblique the internal waves are to the surface wave, the higher their growth rates are. In a medium without side constraints the internal waves are nearly perpendicular to the surface wave at the maximum growth rate. Besides producing further damping, higher viscosity may also enhance the generation of the internal waves by increasing the surface-wave induced shear at the interface. The two-dimensional viscous model is used to investigate this effect of viscosity along with the corresponding dissipation effect on the interaction. The results indicate that although the forcing increases with viscosity, the damping effect of viscosity is more significant. At large viscosity, the internal waves are heavily damped and hence can not grow. As the viscosity approaches zero, the results of the viscous model become asymptotic to those of the three-dimensional model for two-dimensional interactions. A series of experiments were carried out in a wave flume to test the theoretical results. Salt water was used as the lower layer and fresh water as the upper layer. The experiments led to generation of a three-dimensional internal wave pattern at the interface. It is shown that the pattern is created by the reflection of the oblique internal waves from the flume sidewalls. Wavelengths, frequencies, and amplitudes of the internal waves were measured for comparison with the theoretical values. The experiments confirm the theoretical results in general. In contrast with the past theoretical studies, it is shown that viscosity is not essential to the excitation and growth of the internal waves. Also, it is shown that contrary to Hill's (1997) theoretical results, there are no specific bounds for growth of the internal waves on the density ratio of the two layers, the frequency of the surface wave, and the direction angle of the internal waves. In particular, it is demonstrated that the instability of the internal waves is not a selective process, and it may occur over wide ranges of the parameters.

Thesis/Dissertation

application/pdf

2009-06-30

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http://hdl.handle.net/2429/9897

Civil Engineering

University of British Columbia

UBCV

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