Skip to main content
SpringerLink
Log in

An Exact Solution for Geophysical Edge Waves in the \({\beta}\)-Plane Approximation

  • Published:
Journal of Mathematical Fluid Mechanics Aims and scope Submit manuscript

Abstract

By taking into account the \({\beta}\)-plane effects, we provide an exact nonlinear solution to the geophysical edge-wave problem within the Lagrangian framework. This solution describes trapped waves propagating eastward or westward along a sloping beach with the shoreline parallel to the Equator.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Constantin A.: On the deep water wave motion. J. Phys. A 34, 1405–1417 (2001)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  2. Constantin A.: Edge waves along a sloping beach. J. Phys. A 34, 9723–9731 (2001)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  3. Constantin A.: The trajectories of particles in Stokes waves. Invent. Math. 166, 523–535 (2006)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  4. Constantin, A.: Nonlinear Water Waves with Applications to Wave–Current Interactions and Tsunamis, vol. 81. CBMS-NSF Conference Series in Applied Mathematics, vol. 2l. SIAM, Philadelphia (2011)

  5. Constantin A.: An exact solution for equatorial trapped waves. J. Geophys. Res. 117, C05029 (2012)

    ADS  Google Scholar 

  6. Constantin A.: On the modelling of Equatorial waves. Geophys. Res. Lett. 39, L05602 (2012)

    ADS  Google Scholar 

  7. Constantin A., Johnson, R.S.: The dynamics of waves interacting with the equatorial undercurrent. Geophys. Astrophys. Fluid Dyn. (2015). doi:10.1080/03091929.2015.1066785

  8. Constantin A., Strauss W.: Pressure beneath a Stokes wave. Commun. Pure Appl. Math. 63, 533–557 (2010)

    MATH  MathSciNet  Google Scholar 

  9. Cushman-Roisin B., Beckers J.-M.: Introduction to Geophysical Fluid Dynamics. Academic Press, San Diego (2011)

    MATH  Google Scholar 

  10. Ehrenmark U.: Oblique wave incidence on a plane beach: the classical problem revisited. J. Fluid Mech. 368, 291–319 (1998)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  11. Gallagher I., Saint-Raymond L.: On the influence of the Earth’s rotation on geophysical flows. Handb. Math. Fluid Mech. 4, 201–329 (2007)

    Google Scholar 

  12. Gerstner F.: Theorie der Wellen samt einer daraus abgeleiteten Theorie der Deichprofile. Ann. Phys. 2, 412–445 (1809)

    Article  Google Scholar 

  13. Henry D.: On Gerstner’s water wave. J. Nonlinear Math. Phys. 15, 87–95 (2008)

    Article  MathSciNet  ADS  Google Scholar 

  14. Henry, D.: The trajectories of particles in deep-water Stokes waves, Int. Math. Res. Not. 13 pp (2006). (Art. ID 23405)

  15. Howd P.A., Bowen A.J., Holman R.A.: Edge waves in the presence of strong longshore currents. J. Geophys. Res. 97, 11357–11371 (1992)

    Article  ADS  Google Scholar 

  16. Ionescu-Kruse D.: An exact solution for geophysical edge waves in the f-plane approximation. Nonlinear Anal. Real World Appl. 24, 190–195 (2015)

    Article  MathSciNet  Google Scholar 

  17. Johnson R.S.: Edge waves: theories past and present. Phil. Trans. R. Soc. A 365, 2359–2376 (2007)

    Article  MATH  ADS  Google Scholar 

  18. Lamb H.: Hydrodynamics. Cambridge University Press, Cambridge (1895)

    MATH  Google Scholar 

  19. Matioc, A.-V.: An explicit solution for deep water waves with Coriolis effects. J. Nonlinear Math. Phys. 19, 1240005 (8 pp) (2012)

  20. Matioc, A.-V.: An exact solution for geophysical equatorial edge waves over a sloping beach. J. Phys. A Math. Theor. 45, 365501 (10 pp) (2012)

  21. Mollo-Christensen E.: Edge waves in a rotating stratified fluid, an exact solution. J. Phys. Oceanogr. 9, 226–229 (1979)

    Article  ADS  Google Scholar 

  22. Mollo-Christensen E.: Allowable discontinuities in a Gerstner wave. Phys. Fluids 25, 586–587 (1982)

    Article  MathSciNet  ADS  Google Scholar 

  23. Pedlosky J.: Geophysical Fluid Dynamics. Springer, New York (1979)

    Book  MATH  Google Scholar 

  24. Stokes, G.G.: Report on recent researches in hydrodynamics. Rep. 16th Brit. Assoc. Adv. Sci. pp. 1–20 (1846). (See also Papers, vol. 1, 157–187, Cambridge University Press, 1880)

  25. Stuhlmeier R.: On edge waves in stratified water along a sloping beach. J. Nonlinear Math. Phys. 18, 127–137 (2011)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  26. Ursell F.: Edge waves on a sloping beach. Proc. R. Soc. A 214, 79–97 (1952)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  27. Weber, J.E.H.: A note on trapped Gerstner waves. J. Geophys. Res. 117, C03048 (2012). doi:10.1029/2011JC007776

  28. Verkley W.T.M.: On the beta-plane approximation. J. Atmos. Sci. 47, 2453–2460 (1990)

    Article  MathSciNet  ADS  Google Scholar 

  29. Whitham G.B.: Nonlinear effects in edge waves. J. Fluid Mech. 74, 353–368 (1976)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  30. Whitham, G.B.: Lecture on Wave Propagation. Springer, New York (1979). (Published for Tata Institute of Fundamental Research)

  31. Yih C.S.: Note on edge waves in a stratified fluid. J. Fluid Mech. 24, 765–767 (1966)

    Article  MATH  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Simion Stoilow Institute of Mathematics of the Romanian Academy, Research Unit No. 6, P.O. Box 1-764, 014700, Bucharest, Romania

    Delia Ionescu-Kruse

Authors
  1. Delia Ionescu-Kruse
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Delia Ionescu-Kruse.

Additional information

Communicated by A. Constantin

The paper was supported by the ERC Advanced Grant “Nonlinear studies of water flows with vorticity” (NWFV).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ionescu-Kruse, D. An Exact Solution for Geophysical Edge Waves in the \({\beta}\)-Plane Approximation. J. Math. Fluid Mech. 17, 699–706 (2015). https://doi.org/10.1007/s00021-015-0233-6

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00021-015-0233-6

Keywords

Search

Navigation