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Amplitude variations of edge waves on a shelf slowly varying in the alongshore direction

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Abstract

Within the framework of the linear shallow-water theory, the dynamics of edge waves over a shelf characterized by a cylindrical bottom relief is investigated under the assumption that shelf parameters vary slowly in the alongshore direction. An asymptotic theory and an energy approach are used to calculate the amplitude of the edge wave. In the analytic form, the results are obtained for shelves of three different profiles with parameters varying along the shore: an infinite linear profile, a concave exponential profile, and a stepwise profile.

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References

  1. D. A. Huntley and A. J. Bowen, “Field Observations of Edge Waves,” Nature 243, 160–162 (1973).

    Article  Google Scholar 

  2. A. J. Bowen and D. A. Huntley, “Waves, Longwaves and Nearshore Morphology,” Mar. Geol. 60, 1–13 (1984).

    Article  Google Scholar 

  3. D. A. Huntley, R. T. Guza, and E. B. Thornton, “Field Observation of Surf Beat 1. Progressive Edge Waves,” J. Geophys. Res. 86, 6451–6466 (1981).

    Article  Google Scholar 

  4. K. P. Bryan, P. A. Hows, and A. J. Bowen, “Field Observations of Bar-Trapped Edge Waves,” J. Geophys. Res. 103, 1285–1305 (1998).

    Article  Google Scholar 

  5. A. B. Rabinovich, Long Gravity Waves in the Ocean: Trapping, Resonance, Emission (Gidrometeoizdat, St. Petersburg, 1993) [in Russian].

    Google Scholar 

  6. P. Komar, Beach Processes and Sedimentation (Prentice-Hall, New Jersey, 1998).

    Google Scholar 

  7. J. Fredsoe and R. Deigaard, Mechanics of Coastal Sediment Transport (World Science, London, 1992).

    Google Scholar 

  8. R. T. Guza and R. E. Davis, “Excitation of Edge Waves by Waves Incident on a Beach,” J. Geophys. Res. 79, 1285–1291 (1974).

    Google Scholar 

  9. M. A. Foda and C. C. Mei, “Nonlinear Excitation of Long Trapped Waves by a Group of Short Swell,” J. Fluid Mech. 111, 319–345 (1981).

    Article  Google Scholar 

  10. Y. Agnon and C. C. Mei, “Trapping and Resonance of Long Shelf Waves Due to Groups of Short Waves,” J. Fluid Mech. 195, 201–221 (1988).

    Article  Google Scholar 

  11. J. W. Miles, “Parametrically Excited Standing Edge Waves,” J. Fluid Mech. 214, 43–57 (1990).

    Article  Google Scholar 

  12. P. Blondeaux and G. Vittori, “The Nonlinear Excitation of Synchronous Edge Waves by a Monochromatic Wave Normally Approaching a Plane Beach,” J. Fluid Mech. 301, 251–268 (1995).

    Article  Google Scholar 

  13. Y. M. Tang and R. Grimshaw, “A Modal Analysis of Coastally Trapped Waves Generated by Tropical Cyclones,” J. Phys. Oceanogr. 25, 1577–1598 (1995).

    Article  Google Scholar 

  14. H. Ishii and K. Abe, “Propagation of Tsunami on a Linear Slope between Two Flat Regions. I. Eigenwave,” J. Phys. Earth 28, 531–541 (1980).

    Google Scholar 

  15. E. N. Pelinovsky, Hydrodynamics of Tsunamis (IPF RAN, Nizhni Novgorod, 1996) [in Russian].

    Google Scholar 

  16. K. Fujima, R. Dozono, and T. Shigemura, “Generation and Propagation of Tsunami Accompanying Edge Waves on a Uniform Sloping Shelf,” Coast. Eng. J. 42, 211–236 (2000).

    Google Scholar 

  17. I. V. Fain, G. V. Shevchenko, and E. A. Kulikov, “Study of Trapped Properties of the Kuril Shelf by Ray-Tracing Techniques,” Okeanologiya 23, 23–26 (1983).

    Google Scholar 

  18. F. K. Ball, “Edge Waves in an Ocean of Finite Depth,” Deep-Sea Res. 14, 79–88 (1967).

    Google Scholar 

  19. D. V. Evans and P. McIver, “Edge Waves over a Shelf: Full Linear Theory,” J. Fluid Mech. 142, 79–95 (1984).

    Article  Google Scholar 

  20. R. Grimshaw, “Edge Waves: A Long Wave Theory for Oceans of Finite Depth,” J. Fluid Mech. 62, 775–791 (1974).

    Article  Google Scholar 

  21. W. Munk, F. Snodgrass, and M. Wimbush, “Tides Off-Shore: Transition from California Coastal to Deep-Sea Waters,” Geophys. Fluid Dyn. 1, 161–235 (1970).

    Article  Google Scholar 

  22. F. Ursell, “Edge Waves on a Sloping Beach,” Proc. R. Soc. A, London 214, 79–97 (1955).

    Article  Google Scholar 

  23. P. Le Blond and L. Mysak, Waves in the Ocean (Elsevier, Amsterdam, 1978; Mir, Moscow, 1981).

    Google Scholar 

  24. A. Constantin, “Edge Waves along a Sloping Beach,” J. Phys. A: Math. Gen. 34, 9723–9731 (2001).

    Article  Google Scholar 

  25. R. Grimshaw, “Nonlinear Aspects of Long Shelf Waves,” Geophys. Astrophys. Fluid Dyn. 8, 3–16 (1977).

    Google Scholar 

  26. G. B. Whitham, “Nonlinear Effects in Edge Waves,” J. Fluid Mech. 74, 353–368 (1976).

    Article  Google Scholar 

  27. A. Minzoni, “Nonlinear Edge Waves and Shallow-Water Theory,” J. Fluid Mech. 74, 369–374 (1976).

    Article  Google Scholar 

  28. A. Minzoni and G. B. Whitham, “On the Excitation of Edge Waves on Beaches,” J. Fluid Mech. 79, 273–287 (1977).

    Article  Google Scholar 

  29. V. A. Dubinina, A. A. Kurkin, E. N. Pelinovsky, and O. E. Poloukhina, “Weakly Nonlinear Periodic Stokes Edge Waves,” Izv. Akad. Nauk, Fiz. Atmos. Okeana 40, 525–530 (2004) [Izv., Atmos. Ocean. Phys. 40, 464–469 (2004)].

    Google Scholar 

  30. T. R. Akylas, “Large-Scale Modulation of Edge Waves,” J. Fluid Mech. 132, 197–208 (1983).

    Article  Google Scholar 

  31. H. H. Yeh, “Nonlinear Progressive Edge Waves: Their Instability and Evolution,” J. Fluid Mech. 152, 479–499 (1985).

    Article  Google Scholar 

  32. J. Yang, “The Stability and Nonlinear Evolution of Edge Waves,” Stud. Appl. Math. 95, 229–246 (1995).

    Google Scholar 

  33. K. E. Kenyon, “A Note on Conservative Edge Wave Interactions,” Deep-Sea Res. 17, 197–201 (1970).

    Google Scholar 

  34. J. T. Kirby, U. Putrevu, and H. T. Ozkan-Haller, “Evolution Equations for Edge Waves and Shear Waves on Longshore Uniform Beaches,” in Proceedings of 26th International Conference on Coastal Engineering (1998), pp. 203–216.

  35. I. E. Kochergin and E. N. Pelinovsky, “Nonlinear Interaction of a Triad of Edge Waves,” Okeanologiya 29, 899–903 (1989).

    Google Scholar 

  36. V. A. Dubinina A. A. Kurkin, E. N. Pelinovsky, and O. E. Poloukhina, “Resonance Three-Wave Interactions of Stokes Edge Waves,” Izv. Akad. Nauk, Fiz. Atmos. Okeana 42, 277–284 (2006) [Izv., Atmos. Ocean. Phys. 42, 254–261 (2006)].

    Google Scholar 

  37. V. A. Dubinina, A. A. Kurkin, and O. E. Poloukhina, “Nonlinear Dynamics of Edge Waves over a Linearly Slopong Bottom,” Izv. Akad. Nauk, Fiz. Atmos. Okeana 41, 124–128 (2005) [Izv., Atmos. Ocean. Phys. 41, 242–246 (2005)].

    Google Scholar 

  38. A. A. Kurkin, “Dynamics of Nonstationary Stokes Edge Waves,” Okeanologiya 45, 325–331 (2005).

    Google Scholar 

  39. A. Sheremet and R. T. Guza, “A Weakly Dispersive Edge Wave Model,” Coast. Eng. 38, 47–52 (1999).

    Article  Google Scholar 

  40. T. F. Stoker and E. R. Johnson, “The Trapping and Scattering of Topographic Waves by Estuaries and Headlands,” J. Fluid Mech. 222, 501–524 (1991).

    Article  Google Scholar 

  41. A. Baquerizo, M. A. Losada, and I. J. Lozada, “Edge Wave Scattering by a Coastal Structure,” Fluid Dyn. Res. 31, 275–287 (2002).

    Article  Google Scholar 

  42. V. A. Dubinina, A. A. Kurkin, and O. E. Poloukhina, “Focusing of Edge Waves over the Sea Shelf,” Izv. Akad. Nauk, Fiz. Atmos. Okeana 39, 839–848 (2003) [Izv., Atmos. Ocean. Phys. 39, 758–767 (2003)].

    Google Scholar 

  43. F. E. Snodgrass, W. H. Munk, and G. R. Miller, “Long-Period Waves over California’s Continental Borderland,” J. Marine Res. 20, 3–30 (1962).

    Google Scholar 

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Authors and Affiliations

  1. Nizhni Novgorod State Technical University, ul. Minina 24, Nizhni Novgorod, 603600, Russia

    A. A. Kurkin & O. E. Poloukhina

  2. Institute of Applied Physics, Russian Academy of Sciences, ul. Ul’yanova 46, Nizhni Novgorod, 603950, Russia

    E. N. Pelinovsky

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  1. A. A. Kurkin
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  2. E. N. Pelinovsky
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  3. O. E. Poloukhina
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Additional information

Original Russian Text © A.A. Kurkin, E.N. Pelinovsky, O.E. Poloukhina, 2006, published in Izvestiya AN. Fizika Atmosfery i Okeana, 2006, Vol. 42, No. 3, pp. 384–392.

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Kurkin, A.A., Pelinovsky, E.N. & Poloukhina, O.E. Amplitude variations of edge waves on a shelf slowly varying in the alongshore direction. Izv. Atmos. Ocean. Phys. 42, 353–361 (2006). https://doi.org/10.1134/S000143380603008X

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  • DOI: https://doi.org/10.1134/S000143380603008X

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