Very large internal waves in the ocean — observations and nonlinear models

This chapter describes the observations and modelling of very large internal waves in the ocean. We begin with a brief description of the dead-water phenomenon explained by Nansen, and the internal tides discovered by Pettersson. We continue by describing the very large oceanic solitary internal waves of depression observed in the field from 1965 and onwards. The main point of the mathematical analysis is to model the formation and propagation properties of the waves. Tidal generation of internal undular bores and solitary waves of amplitude comparable to the surface layer thickness at rest are exemplified by numerical simulations. The fully nonlinear and fully dispersive mathematical and numerical modelling is found to reproduce the wave motion taking place in the ocean, including the excursions of the mixed upper layer as large as 4–5 times the level at rest, as in the COPE experiment. The interface method — derived in two and three dimensions — is found to compare well with laboratory measurements of the waves using Particle Image Velocimetry. A fully nonlinear theory that accounts for the continuously stratified motion within the pycnocline is derived. The method is used to support experiments on internal wave breaking governed by shear. The latter model is also useful to compute internal wave motion when the upper part of the ocean is linearly stratified. The fully nonlinear modelling is put into perspective by deriving in parallel the weakly nonlinear Korteweg-de Vries, Benjamin-Ono and intermediate long wave equations. Recent observations in deep water reveal significant internal wave motion and corresponding strong bottom currents of magnitude about 0.5 m/s where the pycnocline meets the shelf slope. Future directions of internal wave research are indicated.