Three-Dimensional Nonlinear Internal Wave Beams: Mathematical Models and Laboratory Experiments

Density-stratified fluids support internal gravity waves due to the restoring force of buoyancy. Although not as familiar as free-surface waves, internal waves in fact are ubiquitous in oceans, lakes, and the atmosphere. In these natural settings, stratification is caused by temperature variations as well as changes in salinity (in oceans) and pressure (in the atmosphere). Internal wave motion is inherently anisotropic because gravity provides a preferred direction. As a result, energy is transported in the form of wave beams, which are the analogues of the familiar circular wave fronts in isotropic media. Internal wave beams are of great geophysical interest because they form the backbone of the oceanic internal tide; moreover, in the atmosphere, they frequently arise due to thunderstorms. In contrast to most prior studies, which assume strictly two-dimensional disturbances, this project is concerned with internal wave beams featuring three-dimensional variations, as expected in the field. Motivation comes from recent findings, by the investigator's group and others, that suggest that the propagation of internal wave beams in three dimensions differs fundamentally from its two-dimensional counterpart. Apart from making basic contributions to applied mathematics, this project is expected to also shed light on the role of the oceanic internal tide in momentum deposition, material transport, and deep-ocean mixing. These processes affect climate dynamics, pollutant dispersal, and nutrient cycles. A graduate student is included in the project.

This research effort advances theoretical modeling to gain a fundamental understanding of three-dimensional nonlinear internal wave beams, particularly the attendant resonant nonlinear interaction with induced mean flows that is central to three-dimensional wave beam dynamics. Using asymptotic methods, nonlinear evolution equations are derived for three-dimensional beam propagation under conditions of geophysical interest, taking into account the effects of forcing, non-uniform background, dissipation, and the Earth's rotation. These new mathematical models are analyzed by analytical and numerical means in order to understand beam-mean-flow nonlinear interactions from a mathematical and a physical perspective. The theoretical study is supported by companion laboratory experiments that exploit recent advances in three-dimensional flow visualization.