Tidally generated near-resonant internal wave triads at a shelf break



[1] Numerical simulations of internal wave generation by tidal flow over a shelf break using a linear background stratification show the presence of near-resonant internal wave triads with spatially modulated amplitudes propagating away from the shelf break. The near-resonant triad is comprised of mode-one and -three waves of tidal frequency, which are directly forced by the tide-topography interaction, and a second harmonic, mode-two wave generated by the nonlinear interaction of these waves. A theory for these near-resonant triads is presented and compared with the numerical simulations.

1. Introduction

[2] Internal wave generation by tide-topography interaction exhibits a rich range of behaviour. In this paper a new weakly-nonlinear process, near-resonant triads, is reported in this context for the first time. This mechanism may be important for transferring energy in the oceanic internal wave field.

[3] Exact phase-locked resonant triads occur between three waves whose wave vectors equation imagej = (kj, mj) and frequencies ωj satisfy the resonant conditions equation image1 + equation image2 + equation image3 = 0 and ω1 + ω2 + ω3 = 0 where the wave frequencies satisfy the dispersion relation appropriate for the type of wave in question. The nonlinearly-coupled evolution equations for the envelopes of the three waves, which vary slowly in time and space, have soliton solutions [Craik, 1985; Degasperis et al., 2006]. Resonant internal wave triads are of great interest to oceanographers because they are believed to play a fundamental role in nonlinear energy transfer across the oceanic internal wave spectrum [Hibiya et al., 2002; Furuichi et al., 2006; MacKinnon and Winters, 2005; Gerkema et al., 2006].

[4] Near-resonant triads occur when there is a slight frequency or wave number mismatch, i.e.,

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where the wave number and frequency mismatches are small. Evolution equations for the amplitudes of waves in a near-resonant triad were first derived in the context of light waves in nonlinear dielectrics [Armstrong et al., 1962] and were discussed in a general setting by Bretherton [1964]. They have since been described in other contexts including shoaling surface gravity waves in shallow water [Freilich and Guza, 1984], waves in plasma [Drysdale and Robinson, 2002] and nonlinear optics [Robinson and Drysdale, 1996]. The theory of near-resonant interactions is summarized by Craik [1985]. Near-resonant interactions among discrete internal wave modes have received little attention in the study of internal waves, one exception being a study of the stochasticities of two-triad interactions [Kim and West, 1996]. Near-resonant internal wave interactions among plane waves are also briefly mentioned by Koudella and Staquet [2006].

[5] In this Letter the occurrence of phase-locked, near-resonant internal wave triads are reported for the first time. The strength of the interaction is surprisingly strong which demonstrates that under certain conditions near-resonant interactions may be an important factor in determining the internal wave spectrum in the ocean. The waves are generated in numerical simulations of tide-topography interaction at a shelf break. The form of the forcing generates waves of all vertical modes at predominantly tidal frequency. The mode-one and -three waves are part of a near-resonant triad which includes a mode-two wave with twice the frequency. Because the mode-one and -three waves have a localized source from which they propagate, the amplitudes of the three waves in the near-resonant triads vary predominantly in space rather than in time.

2. Near-Resonant Triads in Fully Nonlinear Numerical Simulations

[6] The simulations were done using a two-dimensional, non-hydrostatic primitive equation model [Lamb, 2004] which uses the Boussinesq and rigid lid approximations. Explicit viscosity and diffusion terms are ignored. A shelf slope in the shape of a half Gaussian h(x) = 4000exp(−(x/13000)2) is used with a deep-water depth of 5 km. In the shallow water (x > 0) the water depth is 1 km. The horizontal resolution is 100 m in a central region 200 km in length beyond which a stretched grid is used. The total computational domain is 3400 km in length. The model uses a sigma-coordinate grid with 180 grid cells of equal height. The stratification is uniform with constant buoyancy frequency N = 0.001 s−1. The effects of the Earth's rotation are ignored in this Letter. Rotational effects will modify but not change the existence of near-resonant triads. The shelf slope is supercritical, with the maximum topographic slope being 1.9 times the slope of internal wave beams of tidal frequency. The internal waves are generated by a barotropic tidal current of amplitude 0.05 m s−1 in the shallow water oscillating with the M2 tidal frequency, approximated by image = 1.4075 × 10−4 s−1. The simulation is started at maximum on-shelf flow. The shape of the topography used is not crucial, although it is important that it is supercritical. For subcritical slopes the amplitude of the mode-three internal wave generated by tide-topography interaction would be greatly reduced, substantially reducing the strength of the nonlinear interaction between the mode-one and -three waves. Many topographic features in the world's ocean are supercritical.

[7] The horizontal velocity field after 20.25 tidal periods, which is at the end of on-shelf flow, is shown in Figure 1. The wave field on the shelf is very complicated due to the presence of waves of many modes and frequencies. The intense patches in the shallow water follow the tidal-frequency rays, with the beam losing its coherence after its first reflection off the bottom at x = 10 km. A subharmonic beam of lower slope can be seen emanating from the shelf break down to the left towards deep water.

Figure 1.

Horizontal velocity fields after 20.25 tidal periods. Deep water depth 5 km. Forced with barotropic current of image m s−1 at the left boundary. Velocities in cm s−1. Sloping black lines indicate slopes of internal wave beams of tidal (smaller slope) and twice tidal (larger slope) frequency. Only part of the computational domain is shown.

[8] The velocity field near the end of the run is close to periodic except near the wave front so a Fourier decomposition in time was done over a period of two tidal periods between t = 22τ and 24τ via

equation image

where ξ is the horizontal coordinate in a reference frame moving with the barotropic (vertically averaged) tide and the tilde denotes values in this reference frame. Use of this reference frame eliminates the spurious effects associated with advection of the internal waves by the barotropic tide in regions with a flat bottom, as considered here. The decomposition was done using data stored 64 times in each tidal period in order to minimize aliasing.

[9] The Fourier components were also separated into vertical modes via

equation image

The coefficients anp(ξ) and bnp(ξ) give the amplitudes of the mode-p waves of frequency image

[10] A striking feature of the resulting decomposition is the appearance of spatially modulated mode-two and -three wave packets of twice tidal and tidal frequency respectively. These are shown in Figure 2 along with the first-harmonic mode-one wave. These waves are all close to being hydrostatic with (ω/N)2 = 0.02 and 0.08 for the first and second harmonic waves respectively. In the hydrostatic (long-wave) limit (ω/N → 0, or, equivalently, k/m → 0) mode-two, 2nd harmonic and mode-three, 1st harmonic (tidal frequency) waves form a resonant triad with a mode-one, 1st harmonic wave. These three waves in the present simulation, because of small non-hydrostatic effects, do not form a resonant triad. The vertical wave numbers and frequencies of the three waves are

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From the dispersion relation the horizontal wave numbers are

equation image

where f is the Coriolis frequency. The sign of the kj are chosen such that the phase speeds are positive. It follows easily that in the hydrostatic limit (ω/N → 0) and in the absence of rotation (f = 0), the three waves form an exact triad. For the non-rotating, non-hydrostatic case considered here

equation image


equation image

using ω1/N ≪ 1. A measure of the strength of the detuning from exact resonance is given by the ratios ∣Δk/kj∣ which are 0.12, 0.03 and 0.04 for the mode-one, -two and -three waves respectively. In this non-rotating case the strength of the detuning is also a measure of non-hydrostaticity. Non-zero f would also contribute to the detuning.

Figure 2.

Contributions to the horizontal velocity at the surface after 22 tidal periods from the three components of the near-resonant triad. (a) Mode-1, first harmonic. (b) Mode-2, second harmonic. (c) Mode-3, first harmonic.

[11] Any of the three waves in the near-resonant triad belong to resonant triads consisting of three co-propagating waves of the same vertical modes. For example a non-hydrostatic resonant triad consists of a mode-one wave of tidal frequency and co-propagating mode-two and mode-three waves with frequencies (ω1, ω2, ω3) = image image image The resonant triads are easily distinguishable from the near-resonant triad. To illustrate this, Figure 3 compares the mode-two second-harmonic wave with analytic fits image where a = 0.013 m s−1, kp = 2π/80000 m−1 is the wave number of the observed amplitude modulation and image is taken to be the horizontal wave number for the fast oscillations. Four values for image are considered. In Figure 3aimage = k2 = −0.001843 m−1 is taken from the near-resonant triad. In Figure 3bimage = −k1k3 = k2 + Δk = 0.001787 m−1 where k1 and k3 are the horizontal wave numbers of the mode-one and -three waves in the near-resonant triad. In Figure 3cimage = −0.0013265 m−1 is the wave number one would obtain in the hydrostatic limit and in Figure 3dimage = −0.00200 m−1 is the horizontal wave number for the mode-2 wave in the nearby, non-hydrostatic, resonant triad. The analytic wave packet using the near-resonant triad horizontal wave number k2 gives the best fit.

Figure 3.

Comparison of analytic fits with the mode-two, second harmonic wave extracted from the tidal simulation (solid curved). Dashed curves are analytic fits of the form u = asin((π/40000)x)sin(kx) where (a) k = k2 from the near-resonant triad; (b) k = −k1k3 with k1 and k3 from the near-resonant triad; (c) k = kh2, the mode-two wave number from the exact, hydrostatic, resonant triad; and (d) k = −0.002 m−1 from the mode-two wave in the exact, nonhydrostatic, resonant triad with frequency image

3. Near-Resonant Triads: Theory

[12] A measure of the nonlinearity of the waves is the ratio of the horizontal velocity amplitude to the horizontal phase speed. For the case under consideration these values are 0.06, 0.09 and 0.13 for the modes 1–3 waves respectively. These values are comparable to the detuning ratios ∣Δk/kj∣. This allows the introduction of a small amplitude parameter ɛ and slow space and time scales (χ, τ) = ɛ (x, t) over which the wave amplitudes vary. In addition we set Δk = ɛequation image where equation image is comparable to the kj.

[13] Following standard procedures, at first order in ɛ we assume there are three waves. At this order the total streamfunction Ψ, defined by u = Ψz, can be written as

equation image

where the aj are the amplitudes of the horizontal velocity. The phases have the form

equation image

where the wave vectors and frequencies satisfy the dispersion relation. For convenience the phase shifts ϕj are chosen to sum to π/2, thus the sum of the phases is π/2 − equation imageχ, i.e., a function of the slow space variable.

[14] For a resonant triad (Δk = 0), at the next order resonant forcing terms appear from the sum interactions. As is well known, these are removed by letting the amplitudes aj be functions of the slow variables. For our near-resonant triad the resonant forcing is slightly detuned; however, it becomes resonant by treating the sum of the phases as a function of χ. This leads to the amplitude equations [Craik, 1985]

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is the group velocity and the nonlinear coupling coefficients are given by

equation image

[15] Consideration of the wave field at different times shows that behind the wave fronts the wave envelopes are stationary, hence the time dependence can be ignored. As the mode-two group velocity exceeds the mode-three group velocity the mode-two wave front propagates ahead of the mode-three wave front as can be detected in Figure 2.

[16] Figure 4 compares the solution of the time-independent near-resonant triad equations with the results from the numerical simulation. The amplitudes and phase of the mode-one and mode-three waves in the near-resonant triad amplitude equations were initialized based on values in the simulation at x = 0. The initial amplitude of the mode-two second-harmonic wave was taken as zero. The near-resonant triad solution captures the essential features, including the modulation length scale of the amplitude of all waves and the non-zero minimum amplitude of the mode-three wave. The wavelength of the fast oscillations do not match exactly and hence drift out of phase over several wavelengths. Comparisons with solutions of the exact, nonhydrostatic resonant triads were also made, showing significantly poorer agreement in the fast oscillations; however, the length-scale and size of the amplitude modulation was very similar. Solutions of the resonant triad equations in the hydrostatic limit showed a modulation length scale that was about 12% longer.

Figure 4.

Comparison of numerical simulations with waves generated by tidal flow across a shelf break (solid) to solution of near resonant triad model (dashed). (a) Mode 1. (b) Mode 2. (c) Mode 3.

4. Conclusions

[17] For the first time, the occurrence of phase-locked near-resonant internal gravity wave triads has been demonstrated. The form of the forcing, tide-topography interaction, generates mode-one and mode-three co-propagating waves of tidal frequency which are not co-members of the same resonant triad. Nevertheless their interaction generates a mode-two second harmonic wave with amplitude comparable to that of the mode-three wave. The interaction is comparable in strength to the interaction that would occur in a nearby exact, nonhydrostatic resonant triad with different frequencies for the mode-two and mode-three waves. For the case considered here the time scale of the interaction is about four tidal periods. This is the time taken for energy in the mode-two waves to propagate 20 km, the length of half a packet. Larger amplitude waves would have faster interaction times. These results have been duplicated in numerical simulations in which only mode-one and mode-three waves are directly forced by appropriate boundary conditions imposed at the left boundary, mimicking the laboratory experiments of Martin et al. [1972]. This method allows a more detailed exploration of parameter space. Results of this study, including the effects of rotation which increases the strength of the detuning, will be reported elsewhere.

[18] The stratification considered here is highly idealized, having constant buoyancy frequency. While this stratification is commonly used in laboratory experiments and theoretical work it is not common in the ocean. Resonant triad interactions have been demonstrated to occur in stratifications with a strong pycnocline [Davis and Acrivos, 1967]. It remains to be seen how strong near-resonant interactions can be under these conditions. They have been studied in the context of shallow-water surface gravity waves [Freilich and Guza, 1984] suggesting that near-resonant internal wave triads may occur in more realistic stratifications.


[19] This work is supported by a Research Network Grant (CLIVAR) funded by the Natural Sciences and Engineering Research Council and the Canadian Foundation for Climate and Atmospheric Science.