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Harmonic generation by reflecting internal waves. B. E. Rodenborn, D. Kiefer, H. P. Zhang, and H. L. Swinney. Phys. Fluids, 23(2), 2011.

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.[2] 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[3] within about 100km or so of the generation region[4] 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[5]. 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 ? is the frequency of oscillation, kx and kz are, respectively, the horizontal and vertical wavenumbers, N is the buoyancy frequency:

and ? is the angle of propagation relative to the horizontal. In the definition of the buoyancy frequency, g is gravity, ?-naught is a constant background density, ? 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:

FIG. 1: Schematic of internal wave beam reflection from a boundary at an angle ? relative to the horizontal. The incoming and reflected wavebeams are at angles, ?i and ?r respectively, which are equal be- cause their frequencies, ?i and ?r, are the same and the dispersion relation requires sin?= ?/N. However, the width of the incoming wave beam, Wi, is narrowed upon reflection by a factor, Wr /Wi= sin(?r ' ? )/ sin(?i + ? ). Shown in dashed lines is a second harmonic wave beam created by nonlinear interactions between the incoming and reflected wave beams in the overlap (shaded) region. The second harmonic beam angle is given by sin ?h =2? /N. The group velocities, cx, indicate the direction of energy propagation.

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[11] and Tabaei et al. [12].

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.

Figure 2: The wavemaker consists of five acrylic plates held within a parallelepiped box (top right) driven by a camshaft (top left) using a computer-controlled stepper motor, so the plate oscillation frequency is known precisely. The camshaft discs are oriented with an angular difference, '?, that creates the desired profile. In this example, '?=45?, which creates a half-sine wave configuration. The wavemaker is mounted near the top of the tank and adjusted to the angle of the resulting wavebeam. The second image is of the experimental wave tank showing the setup and the laser illumination used for the particle image velocimetry.

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.

FIG. 7: Examples of instantaneous experimental and numerical wave fields showing the velocity field with vectors and a color map of the vorticity, both generated with a forcing frequency 0.628 rad/s (N=1.63 rad/s). In the experiment, the wave maker was configured with the plates as a the half-sine ('? = 45'?). The velocity and vorticity fields are found by performing the CIV algorithm between consecutive particle images. A resolution of 50 x 50 grid points corresponding to a 5 mm x 5 mm area per grid point was typical. CIV assigns each grid point a value for the velocity and the vorticity. Waves were usually sampled with a time separation between consecutive particle images of 1/20 of the wave period, resulting in 20 velocity values per period. The numerical results were sampled more frequently, typically 128 times over five wave periods for better resolution. Here, the results are interpolated to a regular grid to show the close correspondence with the experimental wave beam. The blue lines are the locations where the cross-sectional data (insets) are taken (bluepoints: instantaneous eld, redpoints: amplitude prole, at the indicated cross-section (blue lines)).

FIG. 7: Second harmonic generation from reflection of internal wave beams for different geometries. Experimental data is shown in panels (a) and (b) and and numerical results shown below in panels (c) and (d). Dashed lines are a cubic spline fit to the data points. The boundary angles where maximum kinetic energy occur for the ? = 15.1 '? experimental and numerical wave beams are ? = 7.3 ± 2.0 '? and ? = 8.6 ± 0.5 '? , respectively. For the ? = 22.7 '? wave beams, the boundary angles where peak harmonic generation occur are ? = 13.2 ± 0.5 '? and ? = 13.9±0.5'? for the experimental and numerical wave beams, respectively. The plate angles for peak harmonic generation agree within error, but the curve shapes are different, possibly because of slightly different beam profiles.

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

FIG. 2: Vorticity plots from two numerical simulations showing the incoming, reflected and second harmonic wave beams. The images have the same spatial and vorticity scales. and the size of the indicators in both images is the same. In panel (a), the boundary angle is ? = 2.5'?, which generates a relatively weak second harmonic. In panel (b), ? = 13.9'? which is where ray theory predicts that the width of the second harmonic is the same as the incoming wave beam, and where the strongest second harmonic wave beam is generated.

FIG. 3: Three predictions for the boundary angle where peak second harmonic generation will occur. The dash-dot line is the prediction by Tabaei et al.; the peak harmonic generation occcurs at the critical angle. The dashed line is the prediction by S.A. Thorpe derived using the resonant triad concept. The solid line is an empirical relationship derived using ray theory and the reflection geometry; the width of the incoming beam and the width of the second harmonic beams are the same. Experimental data (triangles) and numerical data (solid circles) show close agreement with the empirical curve.

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.