On the nonlinear evolution of internal gravity waves

Yan, Liren (1993) On the nonlinear evolution of internal gravity waves. Masters thesis, Memorial University of Newfoundland.

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Abstract

The purpose of this thesis is to compare the nonlinear evolution of internal gravity waves obtained from a fully nonlinear primitive equation model with the evolution predicted by weakly nonlinear theories (i.e., the KdV and mKdV equations). In order to focus on the nonlinear evolution, we consider the idealized case of an inviscid, incompressible Boussinesq fluid of constant depth with simple stratifications. In such an environment, the evolution of internal waves is theoretically analyzed up to second order in amplitude by the method of asymptotic expansion following Lee & Beardsley [1974]. The resulting governing equation is the mKdV equation. Meanwhile, the evolution of the same system is simulated by a fully nonlinear, inviscid numerical model. The model is believed to be reliable (Lamb [1993 a,b]) and thus can be used to quantitatively verify the derived theory. The theory is compared against the model results. An initial depression is generated and the nonlinear steepening of the wave front and its subsequent evolution into an undular bore is investigated for different stratifications and wave amplitudes. It is found that the theory is in very good quantitative agreement with the model results. The mKdV equation generally improves the first-order KdV results for waves with nondimensional amplitude ∊ up to 0.07. For small waves with ∊ < 0.02, the second order nonlinearity is not crucial. The mKdV equation is not appropriate for waves with ∊ larger than 0.07. These results provide further evidence that the fully nonlinear primitive equation model is reliable and give some indication of when the KdV and mKdV equations correctly predict the wave evolution. It is also shown that after the undular bore begins to form, rotation has a minor effect on its subsequent evolution.

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