Nonlinear Reflection of Internal Waves at a Density Discontinuityat the Base of the Mixed Layer

Internal waves propagating with an upward component of group velocity toward the ocean surface are reflected at the base of the mixed layer. A simple model is constructed to examine nonlinear aspects of the reflection. It consists of a uniform layer of depth h, representing the mixed layer, bounded above by a rigid surface and below by an interface, across which there is a density discontinuity Δρ and beneath which fluid is stably stratified with buoyancy frequency, N = const. Attention is given to the case in which an internal wave in the stratified layer, incident from below on the density interface, has frequency σ < N/2. In addition to a first-order wave of frequency σ that is reflected downward from the density discontinuity, a second-order wave is then generated with frequency 2σ and with horizontal wavenumber twice that of the incident wave, which also propagates downward away from the interface. The shape of the waves generated at the interface is investigated and a measure of their nonlinearity is defined. Highly nonlinear waves, with steeper slopes ahead of the wave crest than following it, are expected when the frequency of free interfacial waves with the same horizontal wavenumber as the incident wave is close or equal to σ and when the vertical wavelength of the incident waves is much greater than h. The results are used to describe the nature of forced waves in the thermocline as supercritical internal waves propagate up a sloping boundary. The large soliton waves observed in the Bay of Biscay, where internal tidal waves propagating from their source at the shelf break encounter the thermocline, may be a consequence of the effects of nonlinear reflection.