July  2012, 11(4): 1523-1535. doi: 10.3934/cpaa.2012.11.1523

On isolated vorticity regions beneath the water surface

1. 

University of Vienna, Faculty of Mathematics, Nordbergstrasse 15, A-1090 Vienna, Austria

Received  January 2011 Revised  June 2011 Published  January 2012

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.
Citation: Octavian G. Mustafa. On isolated vorticity regions beneath the water surface. Communications on Pure & Applied Analysis, 2012, 11 (4) : 1523-1535. doi: 10.3934/cpaa.2012.11.1523
References:
[1]

E. Bryant, "Tsunami: The Underrated Hazard,", Springer-Praxis books, (2008).   Google Scholar

[2]

A. Constantin, The trajectories of particles in Stokes waves,, Invent. Math., 166 (2006), 523.  doi: 10.1007/s00222-006-0002-5.  Google Scholar

[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.  doi: 10.1016/j.fluiddyn.2007.06.004.  Google Scholar

[4]

A. Constantin and D. Henry, Solitons and tsunamis,, Z. Naturforsch., 64a (2009), 65.   Google Scholar

[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).  doi: PMid:20419082, PMCid:2857605.  Google Scholar

[6]

A. Constantin and W. Strauss, Pressure beneath a Stokes wave,, Comm. Pure Appl. Math., 63 (2010), 533.   Google Scholar

[7]

A. Constantin and E. Varvaruca, Steady periodic water waves with constant vorticity: regularity and local bifurcation,, Arch. Rational Mech. Anal., 199 (2011), 33.  doi: 10.1007/s00205-010-0314-x.  Google Scholar

[8]

A. Constantin, A dynamical systems approach towards isolated vorticity regions for tsunami background states,, Arch. Rational Mech. Anal., 200 (2011), 239.  doi: 10.1007/s00205-010-0347-1.  Google Scholar

[9]

A. Constantin and J. Escher, Analyticity of periodic traveling free surface water waves with vorticity\/,, Ann. Math., 173 (2011), 559.  doi: 10.4007/annals.2011.173.1.12.  Google Scholar

[10]

M. Ehrnström and G. Villari, Linear water waves with vorticity: Rotational features and particle paths,, J. Differential Equations, 244 (2008), 1888.  doi: 10.1016/j.jde.2008.01.012.  Google Scholar

[11]

A. Geyer, On some background flows for tsunami waves\/,, J. Math. Fluid. Mech., (2011).  doi: 10.1007/s00021-011-0055-0.  Google Scholar

[12]

J. Ko and W. Strauss, Effect of vorticity on steady water waves,, J. Fluid Mech., 608 (2008), 197.  doi: 10.1017/S0022112008002371.  Google Scholar

[13]

H. Segur, Waves in shallow water, with emphasis on the tsunami of 2004,, in, (2007), 3.   Google Scholar

[14]

H. Segur, Integrable models of waves in shallow water,, in, (2008), 345.   Google Scholar

[15]

R. Stuhlmeier, KdV theory and the Chilean tsunami of 1960,, Discrete Contin. Dyn. Syst. Ser. B, 12 (2009), 623.  doi: 10.3934/dcdsb.2009.12.623.  Google Scholar

[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.  doi: 10.1017/S0022112000002457.  Google Scholar

[17]

G. Thomas and G. Klopman, Wave-current interactions in the nearshore region. Gravity waves in water of finite depth,, in, (1997), 215.   Google Scholar

[18]

J. F. Tolland, Stokes waves,, Topol. Methods Nonlinear Anal., 7 (1996), 1.   Google Scholar

[19]

E. Wahlén, Steady water waves with a critical layer,, J. Differential Equations, 246 (2009), 2468.  doi: 10.1016/j.jde.2008.10.005.  Google Scholar

show all references

References:
[1]

E. Bryant, "Tsunami: The Underrated Hazard,", Springer-Praxis books, (2008).   Google Scholar

[2]

A. Constantin, The trajectories of particles in Stokes waves,, Invent. Math., 166 (2006), 523.  doi: 10.1007/s00222-006-0002-5.  Google Scholar

[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.  doi: 10.1016/j.fluiddyn.2007.06.004.  Google Scholar

[4]

A. Constantin and D. Henry, Solitons and tsunamis,, Z. Naturforsch., 64a (2009), 65.   Google Scholar

[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).  doi: PMid:20419082, PMCid:2857605.  Google Scholar

[6]

A. Constantin and W. Strauss, Pressure beneath a Stokes wave,, Comm. Pure Appl. Math., 63 (2010), 533.   Google Scholar

[7]

A. Constantin and E. Varvaruca, Steady periodic water waves with constant vorticity: regularity and local bifurcation,, Arch. Rational Mech. Anal., 199 (2011), 33.  doi: 10.1007/s00205-010-0314-x.  Google Scholar

[8]

A. Constantin, A dynamical systems approach towards isolated vorticity regions for tsunami background states,, Arch. Rational Mech. Anal., 200 (2011), 239.  doi: 10.1007/s00205-010-0347-1.  Google Scholar

[9]

A. Constantin and J. Escher, Analyticity of periodic traveling free surface water waves with vorticity\/,, Ann. Math., 173 (2011), 559.  doi: 10.4007/annals.2011.173.1.12.  Google Scholar

[10]

M. Ehrnström and G. Villari, Linear water waves with vorticity: Rotational features and particle paths,, J. Differential Equations, 244 (2008), 1888.  doi: 10.1016/j.jde.2008.01.012.  Google Scholar

[11]

A. Geyer, On some background flows for tsunami waves\/,, J. Math. Fluid. Mech., (2011).  doi: 10.1007/s00021-011-0055-0.  Google Scholar

[12]

J. Ko and W. Strauss, Effect of vorticity on steady water waves,, J. Fluid Mech., 608 (2008), 197.  doi: 10.1017/S0022112008002371.  Google Scholar

[13]

H. Segur, Waves in shallow water, with emphasis on the tsunami of 2004,, in, (2007), 3.   Google Scholar

[14]

H. Segur, Integrable models of waves in shallow water,, in, (2008), 345.   Google Scholar

[15]

R. Stuhlmeier, KdV theory and the Chilean tsunami of 1960,, Discrete Contin. Dyn. Syst. Ser. B, 12 (2009), 623.  doi: 10.3934/dcdsb.2009.12.623.  Google Scholar

[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.  doi: 10.1017/S0022112000002457.  Google Scholar

[17]

G. Thomas and G. Klopman, Wave-current interactions in the nearshore region. Gravity waves in water of finite depth,, in, (1997), 215.   Google Scholar

[18]

J. F. Tolland, Stokes waves,, Topol. Methods Nonlinear Anal., 7 (1996), 1.   Google Scholar

[19]

E. Wahlén, Steady water waves with a critical layer,, J. Differential Equations, 246 (2009), 2468.  doi: 10.1016/j.jde.2008.10.005.  Google Scholar

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