Theoretical and experimental study of nonlinear internal gravity wave beams

Abstract

Continuously stratified fluids, like the atmosphere and the oceans, support internal gravity waves due to the effect of buoyancy. This type of wave motion is anisotropic since gravity provides a preferred direction. As a result, a localized source oscillating at a frequency below the buoyancy frequency in a uniformly stratified Boussinesq fluid, rather than cylindrical wavefronts, gives rise to elongated disturbances propagating along specific directions depending on the driving frequency. Such wave beams can be readily generated in the laboratory by oscillating a cylinder in a stratified fluid tank, and, according to recent numerical simulations and field observations, often arise in the atmosphere due to thunderstorms and may also be generated in the oceans by tidal flow over sea-floor topography. So far, internal wave beams have been studied mostly using the linearized equations of motion valid for small-amplitude disturbances. The present thesis examines theoretically and experimentally some aspects of non-linearity in the propagation, reflection and collision of internal gravity wave beams. An asymptotic theory is developed for the propagation of isolated two-dimensional or axisymmetric nonlinear beams, that also takes into account viscous as well as re- fraction effects due to the presence of a mean flow and non-uniform buoyancy frequency. In this instance, it turns out that non-linearity plays a secondary role even for a finite-amplitude beam, which explains why a linear approach has been useful in interpreting observations of isolated beams in the atmosphere. On the other hand, nonlinear effects play an important part in the reflection of wave beams from a sloping wall.

(cont.) Using small-amplitude expansions, it is shown that nonlinear interactions are confined solely in the vicinity of the sloping wall where the incident and reflected beams meet, and this overlap region acts as a source of additional reflected beams with higher-harmonic frequencies. In some flow geometries, higher-harmonic reflected beams are found on the opposite side to the vertical than the primary reflected beam. Similarly, when two obliquely propagating beams collide, nonlinear interactions in the overlap region induce secondary beams with frequencies equal to the sum and difference of those of the colliding beams, consistent with recent numerical simulations of oscillatory stratified flow of finite depth over a ridge. A singularity arises in the reflection of linear wave beams when the angle of incidence is close to the wall slope and the reflected beam propagates nearly parallel to the sloping wall. The near-critical reflection of weakly nonlinear wave beams is studied separately by a matched-asymptotics approach. Non-linearity alone is not capable of healing the singularity of linear theory, as the inviscid nonlinear response at the critical angle grows with time and most likely eventually overturns. Laboratory experiments are also performed for the reflection of two-dimensional wave beams from a sloping wall. Internal-wave disturbances are generated by oscillating a circular cylinder in salt-stratified water and visualized using the synthetic schlieren non-intrusive technique. Secondary reflected beams due to nonlinear effects can be quite strong under certain flow conditions, and are in excellent agreement with the theoretical predictions in regards to their propagation characteristics.