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Four-wave resonant interactions in the classical quadratic Boussinesq equations

Published online by Cambridge University Press:  10 January 2009

M. ONORATO ,
A. R. OSBORNE ,
P. A. E. M. JANSSEN  and
D. RESIO
M. ONORATO*
Affiliation:
Dipartimento di Fisica Generale, Università di Torino, Via Pietro Giuria 1, 10125 Torino, Italy
A. R. OSBORNE
Affiliation:
Dipartimento di Fisica Generale, Università di Torino, Via Pietro Giuria 1, 10125 Torino, Italy
P. A. E. M. JANSSEN
Affiliation:
ECMWF, Shinfield Park, Reading, UK
D. RESIO
Affiliation:
US Army Engineer Research & Development Center, Vicksburg, 39180 MS, USA
*
†Email address for correspondence: [email protected]
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Abstract

We investigate theoretically the irreversibile energy transfer in flat bottom shallow water waves. Starting from the oldest weakly nonlinear dispersive wave model in shallow water, i.e. the original quadratic Boussinesq equations, and by developing a statistical theory (kinetic equation) of the aforementioned equations, we show that the four-wave resonant interactions are naturally part of the shallow water wave dynamics. These interactions are responsible for a constant flux of energy in the wave spectrum, i.e. an energy cascade towards high wavenumbers, leading to a power law in the wave spectrum of the form of k−3/4. The nonlinear time scale of the interaction is found to be of the order of (h/a)4 wave periods, with a the wave amplitude and h the water depth. We also compare the kinetic equation arising from the Boussinesq equations with the arbitrary-depth Hasselmann equation and show that, in the limit of shallow water, the two equations coincide. It is found that in the narrow band case, both in one-dimensional propagation and in the weakly two-dimensional case, there is no irreversible energy transfer because the coupling coefficient in the kinetic equation turns out to be identically zero on the resonant manifold.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 618 , 10 January 2009 , pp. 263 - 277
Copyright
Copyright © Cambridge University Press 2008

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References

REFERENCES

Cavaleri, L., Alves, J.-H. G. M., Ardhuin, F., Babanin, A., Banner, M., Belibassakis, K., Benoit, M., Donelan, M., Groeneweg, J., Herbers, T. H. C., Hwang, P., Janssen, P. A. E. M., Janssen, T., Lavrenov, I. V., Magne, R., Monbaliu, J., Onorato, M., Polnikov, V., Resio, D., Rogers, W. E., Sheremet, A., McKee Smith, J., Tolman, H. L., van Vledder, Wolf, J. & Young, I. 2007 Wave modelling—The state of the art, Prog. Oceanogr. 75 (4), 603–674.CrossRefGoogle Scholar
Eldeberky, Y. 1996 Nonlinear Transformation of Wave Spectra in the Nearshore Zone. Thesis Delft University of Technology. Also published in the series ‘Communications on Hydraulic and Geotechnical Engineering’ (ISSN 0169-6548), Report No. 96–4.Google Scholar
Gorman, R. M. 2003 The treatment of discontinuities in computing the nonlinear energy transfer for finite-depth gravity wave spectra. J. Atmos. Oceanic Technol. 20, 206–216.2.0.CO;2>CrossRefGoogle Scholar
Hasimoto, H. and Ono, H. 1972 Nonlinear modulation of gravity waves. J. Phys. Soc. Jpn. 33, 805–811.CrossRefGoogle Scholar
Herbers, T. H. C. & Burton, M. C. 1997 Nonlinear shoaling of directionally spread waves on a beach. J. Geophys. Res. 102 (C9), 21101–21114.CrossRefGoogle Scholar
Herterich, K. & Hasselmann, K. 1980 A similarity relation for the nonlinear energy transfer in a finite-depth gravity-wave spectrum. J. Fluid Mech. 97, 215–224.CrossRefGoogle Scholar
Janssen, T. T., Herbers, T. H. C. & Battjes, J. A. 2006 Generalized evolution equations for nonlinear surface gravity waves over two-dimensional topography. J. Fluid Mech. 552, 393–418.CrossRefGoogle Scholar
Janssen, P. A. E. M. & Onorato, M. 2007 The intermediate water depth limit of the Zakharov equation and consequences for wave. J. Phys. Ocean. 10, 2389–2400.CrossRefGoogle Scholar
Kaihatu, J. M., Veeramony, J., Edwards, K. L. & Kirby, J. T. 2007 Asymptotic behaviour of frequency and wave number spectra of nearshore shoaling and breaking waves. J. Geophys. Res. 112, C06016.CrossRefGoogle Scholar
Krasitskii, V. P. 1994 On reduced equations in the Hamiltonian theory of weakly nonlinear surface waves. J. Fluid Mech. 272, 1–20.CrossRefGoogle Scholar
Lin, R. Q. & Perrie, W. 1997 A new coastal wave model. Part III. Nonlinear wave-wave interaction. J. Phys. Ocean. 27, 1813–1826.2.0.CO;2>CrossRefGoogle Scholar
Madsen, P. A., Bingham, H. B. & Liu, H. 2002 A new Boussinesq method for fully nonlinear waves from shallow to deep water. J. Fluid Mech. 462, 1–20.CrossRefGoogle Scholar
Madsen, P. A. & Schaffer, H. A. 1998 Higher-order Boussinesq-type equations for surface gravity waves: derivation and analysis. Phil. Trans. R. Soc. Lond. A 356, 3123–3184.CrossRefGoogle Scholar
Mei, C. C. 2000 The Applied Dynamics of Ocean Surface Waves. John Wiley and Sons.Google Scholar
Smith, J. M. & Vincent, C. L. 2003 Equilibrium ranges in surf zone wave spectra. J. Geophys. Res.-Ocean 108, 3366, published 26 November 2003, doi:10.1029/2003JC001930.CrossRefGoogle Scholar
Stiassnie, M. & Shemer, L. 1984 On modifications of the Zakharov equation for surface gravity waves. J. Fluid Mech. (1984) 143, 47–67.CrossRefGoogle Scholar
Whitham, G. B. 1974 Linear and Nonlinear Waves, John Wiley and Sons, 636 pp.Google Scholar
Zakharov, V. 1968 Stability of periodic waves of finite amplitude on the surface of a deep fluid. J. Appl. Mech. Tech. Phys. 9, 190–194.CrossRefGoogle Scholar
Zakharov, V. E. 1998 Weakly nonlinear waves on the surface of an ideal finite depth fluid. In Nonlinear Waves and Weak Turbulence (ed. Zakharov, V. E.), American Mathematical Society.Google Scholar
Zakharov, V. E. 1999 Statistical theory of gravity and capillary waves on the surface of a finite-depth fluid. Eur. J. Mech. B/Fluids 18, 327–344.CrossRefGoogle Scholar
Zakharov, V. E., L'vov, V. S. & Falkovich, G. 1992 Kolmogorov Spectra of Turbulence. Springer.CrossRefGoogle Scholar
Zakharov, V. E. & Schulman, E. I. 1988 On additional motion invariants of classical Hamiltonian wave systems. Phys. D 29, 283–320.Google Scholar