Article Contents
Article Contents

# On isolated vorticity regions beneath the water surface

• We present a class of vorticity functions that will allow for isolated, circular vorticity regions in the background of still water, preceding the arrival of waves at the shoreline.
Mathematics Subject Classification: Primary: 37E35; Secondary: 37E45, 76B15, 76B47.

 Citation:

•  [1] E. Bryant, "Tsunami: The Underrated Hazard," Springer-Praxis books, Springer, Berlin, 2008. [2] A. Constantin, The trajectories of particles in Stokes waves, Invent. Math., 166 (2006), 523-535.doi: 10.1007/s00222-006-0002-5. [3] A. Constantin and R. S. Johnson, Propagation of very long water waves, with vorticity, over variable depth, with applications to tsunamis, Fluid Dynam. Res., 40 (2008), 175-211.doi: 10.1016/j.fluiddyn.2007.06.004. [4] A. Constantin and D. Henry, Solitons and tsunamis, Z. Naturforsch., 64a (2009), 65-68. [5] A. Constantin and R. S. Johnson, Addendum: Propagation of very long water waves, with vorticity, over variable depth, with applications to tsunamis, Fluid Dynam. Res., 42 (2010), Art. No. 038901.doi: PMid:20419082, PMCid:2857605. [6] A. Constantin and W. Strauss, Pressure beneath a Stokes wave, Comm. Pure Appl. Math., 63 (2010), 533-557. [7] A. Constantin and E. Varvaruca, Steady periodic water waves with constant vorticity: regularity and local bifurcation, Arch. Rational Mech. Anal., 199 (2011), 33-67.doi: 10.1007/s00205-010-0314-x. [8] A. Constantin, A dynamical systems approach towards isolated vorticity regions for tsunami background states, Arch. Rational Mech. Anal., 200 (2011), 239-253.doi: 10.1007/s00205-010-0347-1. [9] A. Constantin and J. Escher, Analyticity of periodic traveling free surface water waves with vorticity\/, Ann. Math., 173 (2011), 559-568.doi: 10.4007/annals.2011.173.1.12. [10] M. Ehrnström and G. Villari, Linear water waves with vorticity: Rotational features and particle paths, J. Differential Equations, 244 (2008), 1888-1909.doi: 10.1016/j.jde.2008.01.012. [11] A. Geyer, On some background flows for tsunami waves\/, J. Math. Fluid. Mech., (2011), on-line.doi: 10.1007/s00021-011-0055-0. [12] J. Ko and W. Strauss, Effect of vorticity on steady water waves, J. Fluid Mech., 608 (2008), 197-215.doi: 10.1017/S0022112008002371. [13] H. Segur, Waves in shallow water, with emphasis on the tsunami of 2004, in "Tsunami and Nonlinear Waves" (ed. A. Kundu), Springer, (2007), 3-29. [14] H. Segur, Integrable models of waves in shallow water, in "Probability, Geometry and Integrable Systems" (ed. M. Pinsky et al.), Cambridge University Press, (2008), 345-371. [15] R. Stuhlmeier, KdV theory and the Chilean tsunami of 1960, Discrete Contin. Dyn. Syst. Ser. B, 12 (2009), 623-632.doi: 10.3934/dcdsb.2009.12.623. [16] C. Swan, I. Cummins and R. James, An experimental study of two-dimensional surface water waves propagating on depth-varying currents, J. Fluid Mech., 428 (2001), 273-304.doi: 10.1017/S0022112000002457. [17] G. Thomas and G. Klopman, Wave-current interactions in the nearshore region. Gravity waves in water of finite depth, in "Advances in Fluid Mechanics," 10. WIT (Wessex Institute of Technology) (1997), Southhampton, UK, 215-319. [18] J. F. Tolland, Stokes waves, Topol. Methods Nonlinear Anal., 7 (1996), 1-48. [19] E. Wahlén, Steady water waves with a critical layer, J. Differential Equations, 246 (2009), 2468-2483.doi: 10.1016/j.jde.2008.10.005.