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Resonant excitation of trapped waves by Poincaré waves in the coastal waveguides

Published online by Cambridge University Press:  24 February 2011

G. M. REZNIK  and
V. ZEITLIN
G. M. REZNIK
Affiliation:
P.P. Shirshov Institute of Oceanology, Russian Academy of Sciences, 36, Nahimovski Prospect, Moscow 117997, Russia
V. ZEITLIN*
Affiliation:
LMD–ENS, 24 Rue Lhomond, 75231 Paris CEDEX 05, 75005, France University of Pierre and Marie Curie, 4 Place Jussieu 75005, Paris, France
*
†Email address for correspondence: [email protected]
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Abstract

After having revisited the theory of linear waves in the rotating shallow-water model with a straight coast and arbitrary shelf/beach bathymetry, we undertake a detailed study of resonant interaction of free Poincaré waves with modes trapped in the coastal waveguide. We describe and quantify the mechanisms of resonant excitation of waveguide modes and their subsequent nonlinear saturation. We obtain the modulation equations for the amplitudes of excited waveguide modes in the absence and in the presence of spatial modulation and analyse their solutions. Different saturation regimes are exhibited, depending on the nature of the modes involved. The excitation is proved to be efficient, i.e. the saturated amplitudes of the excited waves considerably exceed the amplitude of the generator waves. Back-influence of the excited waveguide modes onto the open ocean results in a phase shift of the reflected Poincaré waves and possible energy redistribution between them. A comparison of rotating and non-rotating cases displays substantial differences in excitation mechanisms in the two cases.

JFM classification

Geophysical and Geological Flows: Ocean processes Geophysical and Geological Flows: Shallow water flows Geophysical and Geological Flows: Waves in rotating fluids
Type
Papers
Information
Journal of Fluid Mechanics , Volume 673 , 25 April 2011 , pp. 349 - 394
Copyright
Copyright © Cambridge University Press 2011

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References

REFERENCES

Akylas, T. R. 1983 Large-scale modulations of edge waves. J. Fluid Mech. 132, 197–208.CrossRefGoogle Scholar
Ball, F. K. 1967 Edge waves in the ocean of finite depth. Deep-Sea Res. 14, 79–88.Google Scholar
Dubinina, V. A., Kurkin, A. A., Pelinovsky, E. N. & Polukhina, O. E. 2006 Resonance three-wave interactions of Stokes edge waves. Izvestia. Atmos. Ocean. Phys. 42 (2), 254–261.CrossRefGoogle Scholar
Fedoryuk, M. V. 1983 Asymptotic Methods for Linear Ordinary Differential Equations. Nauka (in Russian).Google Scholar
Gallagher, B. 1971 Generation of surf beat by non-linear wave interactions. J. Fluid Mech. 49, 1–20.CrossRefGoogle Scholar
Gill, A. E. & Schumann, E. H. 1974 The generation of long shelf waves by the wind. J. Phys. Oceanogr. 4, 83–90.2.0.CO;2>CrossRefGoogle Scholar
Grimshaw, R. 1977 The stability of continental shelf waves. I. Side band instability and long wave resonance. J. Austral. Math. Soc. 20 (B), 13–30.CrossRefGoogle Scholar
Guza, R. T. & Bowen, A. J. 1975 The resonant instabilities of long waves obliquely incident on a beach. J. Geophys. Res. 80 (33), 4529–4534.CrossRefGoogle Scholar
Guza, R. T. & Bowen, A. J. 1976 Finite amplitude edge waves. J. Marine Res. 34, 270–293.Google Scholar
Hsieh, W. W. & Mysak, L. A. 1980 Resonant interactions between shelf waves, with applications to the Oregon shelf. J. Phys. Oceanogr. 10, 1729–1741.2.0.CO;2>CrossRefGoogle Scholar
Huthnance, J. M. 1975 On trapped waves over a continental shelf. J. Fluid Mech. 69, 689–704.CrossRefGoogle Scholar
LeBlond, P. H. & Mysak, L. A. 1978 Waves in the Ocean, chap. 4. Elsevier.Google Scholar
Mathew, J. & Akylas, T. R. 1990 On the radiation damping of finite-amplitude progressive edge waves. Proc. R. Soc. Lond. A 431, 419–431.Google Scholar
Mei, C. C. 1989 The Applied Dynamics of Ocean Surface Waves. World Scientific.Google Scholar
Miles, J. 1990 Parametrically excited standing edge waves. J. Fluid Mech. 214, 43–57.CrossRefGoogle Scholar
Miles, J. 1991 Nonlinear asymmetric excitation of edge waves. IMA J. Appl. Math. 46, 101–108.CrossRefGoogle Scholar
Minzoni, A. A. & Whitham, G. B. 1977 On the excitation of edge waves on beaches. J. Fluid Mech. 79, 273–278.CrossRefGoogle Scholar
Morse, P. M. & Feshbach, H. 1953 Methods of Theoretical Physics, vol. 1, chap. 5. McGraw-Hill.Google Scholar
Pierce, R. D. & Knobloch, E. 1994 Evolution equations for counterpropagating edge waves. J. Fluid Mech. 264, 137–163.CrossRefGoogle Scholar
Reznik, G. M. & Zeitlin, V. 2006 Resonant excitation of Rossby waves in the equatorial waveguide and their nonlinear evolution. Phys. Rev. Lett. 96, 034502 (1–5).CrossRefGoogle ScholarPubMed
Reznik, G. M. & Zeitlin, V. 2007 a Resonant excitation of waves in the equatorial waveguide with mean current and related nonlinear phenomena. Phys. Rev. Lett. 99, 064501 (1–4).CrossRefGoogle Scholar
Reznik, G. M. & Zeitlin, V. 2007 b Interaction of free Rossby waves with semi-transparent equatorial waveguide. Part I. Wave triads. Physica D 226, 55–79.CrossRefGoogle Scholar
Reznik, G. M. & Zeitlin, V. 2009 a Resonant excitation of Rossby and Yanai waves in the presence of the mean zonal current in the equatorial waveguide. Nonlinear Proc. Geophys. 16, 381–392.CrossRefGoogle Scholar
Reznik, G. M. & Zeitlin, V. 2009 b Resonant excitation of coastal Kelvin waves by inertia-gravity waves. Phys. Lett. A 373, 1019–1021.CrossRefGoogle Scholar
Tang, Y. M. & Grimshaw, R. 1995 A modal analysis of coastally trapped waves generated by tropical cyclones. J. Phys. Oceanogr. 25, 1577–1598.2.0.CO;2>CrossRefGoogle Scholar
Whitham, G. B. 1976 Nonlinear effects in edge waves. J. Fluid Mech. 74 (2), 353–368.CrossRefGoogle Scholar