Skip to main content
Log in

On Some Background Flows for Tsunami Waves

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

Abstract

With the aim to describe the state of the sea in a coastal region prior to the arrival of a tsunami, we show the existence of background flow fields with a flat free surface which model isolated regions of vorticity outside of which the water is at rest.

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. Batchelor G.K.: An Introduction to Fluid Dynamics. Cambridge University Press, Cambridge (1967)

    MATH  Google Scholar 

  2. Bryant, E.: Tsunami: the underrated hazard. In: Springer Praxis Books. Springer, Berlin (2008)

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

    Article  MathSciNet  ADS  MATH  Google Scholar 

  4. Constantin A.: On the relevance of soliton theory to tsunami modelling. Wave Motion 46, 420–426 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  5. Constantin A.: On the propagation of tsunami waves, with emphasis on the tsunami of 2004. Discrete Contin. Dyn. Syst. Ser. B 12, 525–537 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  6. Constantin A.: A dynamical systems approach towards isolated vorticity regions for tsunami background states. Arch. Ration. Mech. Anal. 200, 239–253 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  7. Constantin A., Escher J.: Analyticity of periodic traveling free surface water waves with vorticity. Ann. Math. 173, 559–568 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  8. Constantin A., Henry D.: Solitons and tsunamis. Z. Naturforsch. 64a, 65–68 (2009)

    Google Scholar 

  9. Constantin A., Johnson R.S.: Modelling tsunamis. J. Phys. A 39, L215–L217 (2006)

    Article  MathSciNet  ADS  Google Scholar 

  10. Constantin A., Johnson R.S.: Propagation of very long water waves, with vorticity, over variable depth, with applications to tsunamis. Fluid Dyn. Res. 40, 175–211 (2008)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  11. Constantin A., Johnson R.S.: On the non-dimensionalisation, scaling and resulting interpretation of the classical governing equations for water waves. J. Nonlinear Math. Phys. 5, 58–73 (2008)

    Article  MathSciNet  Google Scholar 

  12. Constantin, A., Johnson, R.S.: Addendum: Propagation of very long water waves, with vorticity, over variable depth, with applications to tsunamis. Fluid Dyn. Res. 42, Art. No. 038901 (2010)

  13. Constantin A., Strauss W.: Exact steady periodic water waves with vorticity. Commun. Pure Appl. Math. 57, 481–527 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  14. Constantin A., Strauss W.: Pressure beneath a Stokes wave. Commun. Pure Appl. Math. 53, 533–557 (2010)

    MathSciNet  Google Scholar 

  15. Coppel W.A.: Stability and Asymptotic Behavior of Differential Equations. D. C. Heath and Co., Boston (1965)

    MATH  Google Scholar 

  16. Craig W.: Surface water waves and tsunamis. J. Dyn. Differ. Equations 18, 525–549 (2006)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  17. Dieudonné J.: Foundations of Modern Analysis. Academic Press, New York (1969)

    MATH  Google Scholar 

  18. Guillemin V., Pollack A.: Differential Topology. Prentice-Hall, New Jersey (1974)

    MATH  Google Scholar 

  19. Hammack J.L.: A note on tsunamis: their generation and propagation in an ocean of uniform depth. J. Fluid Mech. 60, 769–799 (1973)

    Article  ADS  MATH  Google Scholar 

  20. Johnson R.S.: A Modern Introduction to the Mathematical Theory of Water Waves. Cambridge University Press, Cambridge (1997)

    Book  MATH  Google Scholar 

  21. Lakshmanan M.: Integrable nonlinear wave equations and possible connections to tsunami dynamics. In: Kundu, A. (eds) Tsunami and Nonlinear Waves, pp. 31–49. Springer, Berlin (2007)

    Chapter  Google Scholar 

  22. Madsen, P.A., Fuhrman, D.R., Schaeffer, H.A.: On the solitary wave paradigm for tsunamis. J. Geophys. Res. Oceans 113, Art. No. C12012 (2008)

  23. Segur H.: Waves in shallow water with emphasis on the tsunami of 2004. In: Kundu, A. (eds) Tsunami and Nonlinear Waves, pp. 3–29. Springer, Berlin (2007)

    Chapter  Google Scholar 

  24. Segur, H.: Integrable models of waves in shallow water. In: Pinsky M. et al. (eds.) Probability, Geometry and Integrable Systems, pp. 345–371. Math. Sci. Res. Inst. Publ., Cambridge University Press, Cambridge (2008)

  25. Stuhlmeier R.: KdV theory and the Chilean tsunami of 1960. Discrete Contin. Dyn. Syst. Ser. B 12, 623–632 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  26. Zahibo N., Pelinovsky E., Talipova T., Kozelkov A., Kurkin A.: Analytical and numerical study of nonlinear effects at tsunami modeling. Appl. Math. Comput. 174, 795–809 (2006)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Anna Geyer.

Additional information

Communicated by A. Constantin

Rights and permissions

Reprints and permissions

About this article

Cite this article

Geyer, A. On Some Background Flows for Tsunami Waves. J. Math. Fluid Mech. 14, 141–158 (2012). https://doi.org/10.1007/s00021-011-0055-0

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00021-011-0055-0

Mathematics Subject Classification (2010)

Keywords

Navigation