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.
Similar content being viewed by others
References
Batchelor G.K.: An Introduction to Fluid Dynamics. Cambridge University Press, Cambridge (1967)
Bryant, E.: Tsunami: the underrated hazard. In: Springer Praxis Books. Springer, Berlin (2008)
Constantin A.: The trajectories of particles in Stokes waves. Invent. Math. 166, 523–535 (2006)
Constantin A.: On the relevance of soliton theory to tsunami modelling. Wave Motion 46, 420–426 (2009)
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)
Constantin A.: A dynamical systems approach towards isolated vorticity regions for tsunami background states. Arch. Ration. Mech. Anal. 200, 239–253 (2011)
Constantin A., Escher J.: Analyticity of periodic traveling free surface water waves with vorticity. Ann. Math. 173, 559–568 (2011)
Constantin A., Henry D.: Solitons and tsunamis. Z. Naturforsch. 64a, 65–68 (2009)
Constantin A., Johnson R.S.: Modelling tsunamis. J. Phys. A 39, L215–L217 (2006)
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)
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)
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)
Constantin A., Strauss W.: Exact steady periodic water waves with vorticity. Commun. Pure Appl. Math. 57, 481–527 (2004)
Constantin A., Strauss W.: Pressure beneath a Stokes wave. Commun. Pure Appl. Math. 53, 533–557 (2010)
Coppel W.A.: Stability and Asymptotic Behavior of Differential Equations. D. C. Heath and Co., Boston (1965)
Craig W.: Surface water waves and tsunamis. J. Dyn. Differ. Equations 18, 525–549 (2006)
Dieudonné J.: Foundations of Modern Analysis. Academic Press, New York (1969)
Guillemin V., Pollack A.: Differential Topology. Prentice-Hall, New Jersey (1974)
Hammack J.L.: A note on tsunamis: their generation and propagation in an ocean of uniform depth. J. Fluid Mech. 60, 769–799 (1973)
Johnson R.S.: A Modern Introduction to the Mathematical Theory of Water Waves. Cambridge University Press, Cambridge (1997)
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)
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)
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)
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)
Stuhlmeier R.: KdV theory and the Chilean tsunami of 1960. Discrete Contin. Dyn. Syst. Ser. B 12, 623–632 (2009)
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)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by A. Constantin
Rights 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
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00021-011-0055-0