Skip to main content
Log in

An exact solution for equatorial waves

  • Published:
Monatshefte für Mathematik Aims and scope Submit manuscript

Abstract

This study presents an explicit exact solution for nonlinear geophysical equatorial waves in the \(f\)-plane approximation near the Equator. The solution describes in the Lagrangian framework equatorial waves propagating westward in a homogenous inviscid fluid.

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

Access this article

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

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Cushman-Roisin, B., Beckers, J.M.: Introduction to Geophysical Fluid Dynamics, p. 320. Academic Press, New York (2011)

    Google Scholar 

  2. Bennett, A.: Lagrangian Fluid Dynamics, p. 308. Cambridge University Press, Cambridge (2006)

    Book  MATH  Google Scholar 

  3. Aleman, A., Constantin, A.: Harmonic maps and ideal fluid flows. Arch. Ration. Mech. Anal. 204(2), 479–513 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  4. Hsu, H.C., Chen, Y.Y., Hsu, J.R.C.: Nonlinear water waves on uniform current in Lagrangian coordinates. J. Nonlinear Math. Phys. 16(1), 47–61 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  5. Hsu, H.C., Ng, C.O., Hwung, H.H.: A New Lagrangian asymptotic solution for gravity-capillary waves in water of finite depth. J. Math. Fluid Mech. 14(1), 79–94 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  6. Hsu, H.C.: Particle trajectories for wave on a linear shear current. Nonlinear Anal. Real World Appl. 14(5), 2013–2021 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  7. Gerstner, F.: Theorie der Wellen samt einer daraus abgeleiteten Theorie der Deichprofile (in German). Ann. Phys. 2, 412445 (1809)

    Google Scholar 

  8. Froude, W.: On the rolling of ships. Trans. Inst. Naval Arch. 3, 45–62 (1862)

    Google Scholar 

  9. Rankine, W.J.M.: On the exact form of waves near the surface of deep water. Philos. Trans. Roy. Soc. London A 153, 127–138 (1863)

    Article  Google Scholar 

  10. Reech, F.: Sur la theory des ondes liquids periodiques. C. R. Acad. Sci. Paris 68, 1099–1101 (1869)

    Google Scholar 

  11. Constantin, A.: Edge waves along a sloping beach. J. Phys. 34A, 97239731 (2001)

    Google Scholar 

  12. Henry, D.: On Gerstner’s water wave. J. Nonlinear Math. Phys. 15, 8795 (2008)

    Article  MathSciNet  Google Scholar 

  13. Yih, C.-S.: Note on edge waves in a stratified fluid. J. Fluid Mech. 24, 765767 (1966)

    Article  Google Scholar 

  14. Constantin, A.: On the deep water wave motion. J. Phys. 34A, 14051417 (2001)

    Google Scholar 

  15. Stuhlmeier, R.: On edge waves in stratified water along a sloping beach. J. Nonlinear Math. Phys. 18, 127137 (2011)

    Article  MathSciNet  Google Scholar 

  16. Matioc, A.V.: An exact solution for geophysical equatorial edge waves over a sloping beach. J. Phys. A 45, 365501 (2012)

    Article  MathSciNet  Google Scholar 

  17. Constantin, A.: An exact solution for equatorially trapped waves. J. Geophys. Res. 117, C05029 (2012). doi:10.1029/2012JC007879

    Google Scholar 

  18. Constantin, A., Germain, P.: Instability of some equatorially trapped waves. J. Geophys. Res. 118, 2802 (2013). doi:10.1002/jgrc.20219

    Article  Google Scholar 

  19. Constantin, A.: Some three-dimensional nonlinear equatorial flows. J. Phys. Oceanogr. 43, 165–175 (2013)

    Article  Google Scholar 

  20. Constantin, A.: Some nonlinear equatorially trapped, non-hydrostatic, internal geophysical waves. J. Phys. Oceanogr. (2014). doi:10.1175/JPO-D-13-0174.1

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

    Article  Google Scholar 

  22. Constantin, A.: Nonlinear Water Waves with Applications to Wave-Current Interactions and Tsunamis, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 81, p. 321. SIAM, Philadelphia (2011)

    Book  Google Scholar 

  23. Constantin, A.: The trajectories of particles in Stokes waves. Invent. Math. 166, 523535 (2006)

    Article  MathSciNet  Google Scholar 

  24. Henry, D.: On the deep-water Stokes flow. Int. Math. Res. Not. 22, (2008). doi:10.1093/imrn/rnn071

  25. Constantin, A., Strauss, W.: Pressure beneath a Stokes wave. Comm. Pure Appl. Math. 53, 533557 (2010)

    Google Scholar 

  26. Longuet-Higgins, M.S.: On the transport of mass by time-varying ocean currents. Deep-Sea Res. 16, 431447 (1969)

    Google Scholar 

Download references

Acknowledgments

The author would like to acknowledge the insightful critiquing of the two referees. The author acknowledges the support of the International Wave Dynamics Research Center in Taiwan (NSC 103-2911-I-006-302).

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Hung-Chu Hsu.

Additional information

Communicated by A. Constantin.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Hsu, HC. An exact solution for equatorial waves. Monatsh Math 176, 143–152 (2015). https://doi.org/10.1007/s00605-014-0618-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00605-014-0618-2

Keywords

Mathematics Subject Classification

Navigation