The Effects of the Earth's Rotation on Internal Wave Near-resonant Triads and Weakly Nonlinear Models

Abstract

This thesis investigates the effects of the earth's rotation on internal waves from two perspectives of nonlinear internal wave theory: near-resonant triads and weakly nonlinear models.

We apply perturbation theory (multiple scale analysis) to the governing equations of internal waves and develop a near-resonant internal wave triad theory. This theory explains a resonant-like phenomenon in the numerical results obtained from simulating internal waves generated by tide topography interaction. Furthermore, we find that the inclusion of the earth's rotation (nonzero $f$) in the numerical runs leads to a very special type of resonance: parametric subharmonic instability.

Through using perturbation expansion to solve separable solutions to the governing equations of internal waves, we derive a new rotation modified KdV equation (RMKdV). Of particular interest, the dispersion relation of the new equation obeys the exact dispersion relation for internal waves for both small and moderate wavenumbers ($k$). Thus this new RMKdV is able to model wea kly nonlinear internal waves with various wavenumbers ($k$), better than the Ostrovsky equation which fails at describing waves of small $k$.