Nonlinear Properties of Reflecting Internal Waves
Over the past few years, increased attention on global ocean circulation patterns and their affect on global climate has caused intense research into internal waves. These waves propagate in the interior of any stably stratified fluid body, where the density decreases as a function of height. Internal waves are a significant mode of tidal energy input into the ocean, estimated to represent as much as 30 per cent of the tidal energy dissipation, i.e. conversion of tidal motion into other forms of energy. Internal wave beams in the ocean can propagate for long distances so that energy input in one region of the ocean may be dissipated much further away. However, there is evidence that the internal wave spectrum relaxes quickly back to the typical oceanic wave spectrum within about 100km or so of the generation region indicating that local wavebeams are modified quickly after being created. Current theory is that these changes occur primarily through nonlinear self-interactions, interactions with other wave beams, and by reflection from boundaries. Such processes may lead to mixing that is the source of the potential energy increase necessary to return deep, dense water to the surface[6, 7]. Such a return flow is required to complete the meridional overturning circulation, or the ocean would fill with cold, dense water that does not return to the surface.[8, 9].
The importance of internal waves motivates a better understanding of their basic properties which are not well understood, primarily because of their unusual dispersion relation. In a stratified fluid, any vertically displaced fluid parcel experiences restoring forces from buoyancy and gravity causing it to oscillate about its equilibrium height, and this oscillatory motion supports internal waves. Without the effects of rotation (Coriolis forces), the dispersion relation for plane internal gravity waves is:
where ω is the frequency of oscillation, kx and kz are, respectively, the horizontal and vertical wavenumbers, N is the buoyancy frequency:
and ϴ is the angle of propagation relative to the horizontal. In the definition of the buoyancy frequency, g is gravity, ρ-naught is a constant background density, ρ is the local density and z is the vertical coordinate, aligned anti-parallel to the gravity vector.
What is unusual in this dispersion relation is that frequency and wavenumber are only indirectly related, as seen in the last term of the equation, which shows that once the stratification is established, the wave frequency determines its propagation angle. Thus, a packet of these plane waves can only be constructed of components with the same frequency, otherwise different components would travel at different angles and separate. Conversely, because of the independence of frequency and wave number, a wave packet may have an arbitrary spectral composition in wavenumber space. An internal wave beam is an example of this type of wave packet with a single frequency and a spectrum of wavenumbers that determines the wave beam profile. The oscillation of stratified fluids over topography creates wave beams in regions where the slope of the topography matches the angle of an internal wave with the same frequency as the fluid oscillation. Such beams are common in the ocean, caused primarily by the diurnal tides [11, 12].
In the case of internal waves propagating in a linearly stratified fluid, the wave beam travels straight at the same angle relative to the horizontal before and after the reflection regardless of the angle of the topography from which it reflects. The following figure provides a schematic of the process in two dimensions:
The focus of our study is the nonlinear creation of second harmonic waves by internal waves reflecting from sloping boundaries. In particular, how does the intensity of the second harmonic depend on the boundary angle from which the internal wave beam reflects. We use experiments and two-dimensional numerical simulations to study this process and compare our results to theories by S.A. Thorpe and Tabaei et al. .
Our experiments are conducted in a laboratory tank using a wavemaker invented by Gostiaux et al., which produces a colimated wave beam. We use particle image velocimetry to determine the wave fields and the wave beam intensities.
Because obtaining experimental data for a wide range of parameters is difficult, numerical simulations solving the full equations of motion in the Boussinesq limit were also used. The numerical simulations show excellent agreement with the experiments (see Fig. 2), both in the instantaneous fields and in the results of each beam angle studies as shown in the next two figures.
Experiments and simulations agree, but we find that neither the theory by Thorpe nor the theory by Tabaei accurately predicts the boundary angle where maximum harmonic generation occurs, i.e. strongest nonlinearity. This boundary angle, however, is predicted by a geometric relationship we found between the incoming wave beam and the second harmonic wave as shown below
If the wavebeam energy and viscosity are reduced, the theory of Tabaei et al. can be fully recovered, but we also find an unexpected dependence on wave period, i.e. longer period waves were strongly nonlinear at much lower wave beam intensities.