Nonlinear interactions among standing surface and internal gravity waves

Joyce, Terrence M.

doi:10.1575/1912/1269

Cited by 5 publications

(9 citation statements)

References 4 publications

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“…In set E1 of the experiments, two standing surface-mode waves are set up in a long tank which subsequently generate an internal-mode resonant third wave. Figure 20 compares Joyce (1974) experimental and approximate theoretical results to the present HOS simulation. The initial growth agrees well with the experiment and approximate theoretical results.…”

Section: Resultsmentioning

confidence: 92%

“…As a further validation, we compare our numerical results with experiments of Joyce (1974). In set E1 of the experiments, two standing surface-mode waves are set up in a long tank which subsequently generate an internal-mode resonant third wave.…”

Section: Resultsmentioning

confidence: 98%

“…For further validation, comparisons of the numerical results with theoretical predictions for the second-order two-layer fluid resonant interaction problem with a flat bottom (e.g. Ball 1964;Wen 1995) and experiments of Joyce (1974) are given in Appendix B.…”

Section: Numerical Resultsmentioning

confidence: 99%

“…Figure 20. Time evolution of the amplitudes of the resonant (standing) internal-mode wave corresponds to the experiment E1 of Joyce (1974) with two standing surface-mode waves of ω 1 = 18.20 rad s −1 and ω 2 = 17.72 rad s −1 in a 2 m long tank with R = 1.000/1.049 = 0.953 and h /h u = 0.6/0.3 = 2.0 and with initial amplitude of a 1 = 3.77 mm and a 2 = 2.68 mm. An internal-mode wave of frequency ω r = 0.484 rad s −1 will satisfy a triad resonance condition, hence its amplitude will grow with time (shown in this figure).…”

Section: Resultsmentioning

confidence: 99%

“…For the simulation the total length of the tank is assumed to be 18 m and NDX = 1024, M = 4 and T / t = 1024. Plotted are theoretical (--) and experimental (᭺) results of Joyce (1974) and numerical simulation with HOS (---). The discrepancy at large times comes from the assumption of constant surface wave amplitude in Joyce (1974) work.…”

Section: Resultsmentioning

confidence: 99%

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Bragg resonance of waves in a two-layer fluid propagating over bottom ripples. Part II. Numerical simulation

2009

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We develop a direct numerical method to study the general problem of nonlinear interactions of surface/interfacial waves with variable bottom topography in a two-layer density stratified fluid. We extend a powerful high-order spectral (HOS) method for nonlinear gravity wave dynamics in a homogeneous fluid to the case of a two-layer fluid over non-uniform bottom. The method is capable of capturing the nonlinear interactions among large number of surface/interfacial wave mode and bottom ripple components up to an arbitrary high order. The method preserves exponential convergence with respect to the number of modes of the original HOS and the (approximately) linear effort with respect to mode number and interaction order. The method is validated through systematic convergence tests and comparison to a semi-analytic solution we obtain for an exact nonlinear Stokes waves on a two-layer fluid (in uniform depth). We apply the numerical method to the three classes of generalized Bragg resonances studied in Alam, Liu & Yue (J. Fluid Mech., vol. 624, 2009, p. 225), and compare the perturbation predictions obtained there with the direct simulation results. An important finding is possibly the important effect of even higher-order nonlinear interactions not accounted for in the leading-order perturbation analyses. To illustrate the efficacy of the numerical method to the general problem, we consider a somewhat more complicated case involving two incident waves and three bottom ripple components with wavenumbers that lead to the possibility of multiple Bragg resonances. It is shown that the ensuing multiple (near) resonant interactions result in the generation of multiple new transmitted/reflected waves that fill a broad wavenumber band eventually leading to the loss of order and chaotic motion.

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Section: Resultsmentioning

confidence: 92%

Section: Resultsmentioning

confidence: 98%

Section: Numerical Resultsmentioning

confidence: 99%

Section: Resultsmentioning

confidence: 99%

Section: Resultsmentioning

confidence: 99%

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Bragg resonance of waves in a two-layer fluid propagating over bottom ripples. Part II. Numerical simulation

2009

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A new triad resonance between co-propagating surface and interfacial waves

Alam

2011

J. Fluid Mech.

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In a two-layer density-stratified fluid it is known, due to Ball (J. Fluid Mech., vol. 19, 1964, p. 465), that two oppositely travelling surface waves may form a triad resonance with an interfacial wave. Ball claims ‘there are no other interactions’ between two surface waves and one interfacial wave. Contrary to this, here we present a new class of triad resonance that occurs between two co-propagating surface waves and one interfacial wave. While in Ball’s resonance the interfacial wave has a wavelength of about half of two surface waves, in the new resonance presented here the interfacial wave has a much higher wavelength compared to those of surface waves. This, together with the unidirectionality of the participant triplet, makes the realization of the new resonance more likely in real ocean scenarios. We further show, via theoretical analysis and direct simulation, that, unique to this new class of resonance, the triad inevitably undergoes a cascade of (near-) resonance interaction that spreads the energy of initial waves to a number of lower and higher harmonics. The significance of the resonance studied here is, particularly, more emphasized in the littoral zones, where the spectrum refracts toward a unidirectional wave train.

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