Nonlinear interactions among internal gravity waves

Abstract

This paper reviews the nonlinear interaction calculations for the internal gravity wave field in the deep ocean. The nonlinear interactions are a principal part of the dynamics of internal waves and are an important link in the overall energy cascade from large to small scales. Four approaches have been taken for their analysis: the evaluation of the transfer integral describing weakly and resonantly interacting waves, the application of closure hypotheses from turbulence theories to more strongly interacting waves, the integration of the eikonal or ray equations describing the propagation of small‐scale internal waves in a background of large‐scale internal waves, and the direct numerical simulation of the basic hydrodynamic equations of motion. The weak resonant interaction calculations have provided most of the conventional wisdom. Specific interaction processes and their role in shaping the internal wave spectrum have been unveiled and a comprehensive inertial range theory developed. The range of validity of the resonant interaction approximation, however, is not known and must be seriously doubted for high‐wave number, high‐frequency waves. The turbulence closure calculations and the direct numerical modeling are not yet in a state to be directly applicable to the oceanic internal wave field. The closure models are too complex and rest on conjectures that are not demonstrably justified. Numerical modeling can treat strongly interacting waves and buoyant turbulence, but is severely limited by finite computer resolutions. Extensive suites of experiments have only been carried out for two‐dimensional flows. The eikonal calculations provide an efficient and versatile tool to study the interaction of small‐scale internal waves, but it is not clear to what extent the scale‐separated interactions with larger‐scale internal waves compete with and might be overwhelmed by interactions among like scales. The major shortcoming of all four approaches is that they neglect the interaction with the vortical (=potential vorticity carrying) mode of motion that must be expected to exist in addition to internal waves at small scales. This interaction is intrinsically neglected in all Lagrangian‐based studies and in the non‐rotating two‐dimensional simulations. The most promising approach for the future that can handle both arbitrarily strong interactions and the interaction with the vortical mode is numerical modeling once the resolution problem is overcome.