Instabilities of finite-width internal wave beams

Abstract

Internal gravity waves are fundamental to the dynamics of density stratified fluids and the instability mechanisms by which these waves dissipate their energy are a potentially significant factor that underlies the distribution of energy and momentum in the natural environment. Recently, it has been recognized that internal waves in the oceans and atmosphere often take the form of beams: plane waves with locally confined spatial profile. While there is a large body of theoretical work concerning the instability of sinusoidal internal waves, instability mechanisms of beams are not yet fully understood. Although various nonlinear mechanisms have been proposed, it remains unclear which, if any, are dominant in the natural environment and under what circumstances. This thesis examines the instability of finite-width internal wave beams in order to extend the current understanding of internal wave instability into more realistic settings.

Part I of this thesis uses a combination of experimental and theoretical techniques to investigate finite-amplitude instabilities of beams. First, using a variant of the classical 'St. Andrew's Cross' experiment, whereby beams are generated using a harmonically oscillated horizontal cylinder, we present novel experimental observations of instability in large-amplitude internal wave beams. These results are compared against the predictions of linear stability analysis based on Floquet theory and reveal the competition between two- and three-dimensional instability mechanisms. Next, Floquet theory is used to investigate the well-known parametric subharmonic instability (PSI) for finite-width beams. Our findings show that frequency components typically ignored in standard analyses based on triad resonance are in fact crucial to the instability dynamics of fine-scale perturbations.

The Floquet stability analysis also reveals that PSI is restricted to a finite range of perturbation wavenumbers and that a broadband instability dominates at large perturbation wavenumber. Furthermore, in the nearly inviscid limit, this broadband instability persists for small-amplitude beams that are not typically susceptible to PSI. Part II focuses on the PSI of finite-width internal wave beams and investigates the role of background mean flows, which provide a more realistic setting for PSI in the natural environment. Using weakly nonlinear asymptotic theories, two types of internal wave beams are considered: nearly-monochromatic beams whose spatial profile consists of a sinusoidal carrier modulated by a locally confined envelope, and thin beams with general profile under the effects of Earth's rotation.

In both cases, the presence of a uniform background mean flow has a stabilizing effect on PSI for finite-width beams, in contrast to the PSI of a purely sinusoidal plane wave where the background mean flow has no effect.