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

[1]

Mikhaël Balabane, Mustapha Jazar, Philippe Souplet. Oscillatory blow-up in nonlinear second order ODE's: The critical case. Discrete & Continuous Dynamical Systems - A, 2003, 9 (3) : 577-584. doi: 10.3934/dcds.2003.9.577

[2]

Yuzo Hosono. Phase plane analysis of travelling waves for higher order autocatalytic reaction-diffusion systems. Discrete & Continuous Dynamical Systems - B, 2007, 8 (1) : 115-125. doi: 10.3934/dcdsb.2007.8.115

[3]

Tomáš Bárta. Exact rate of decay for solutions to damped second order ODE's with a degenerate potential. Evolution Equations & Control Theory, 2018, 7 (4) : 531-543. doi: 10.3934/eect.2018025

[4]

G. Caginalp, Christof Eck. Rapidly converging phase field models via second order asymptotics. Conference Publications, 2005, 2005 (Special) : 142-152. doi: 10.3934/proc.2005.2005.142

[5]

Antonio Marigonda. Second order conditions for the controllability of nonlinear systems with drift. Communications on Pure & Applied Analysis, 2006, 5 (4) : 861-885. doi: 10.3934/cpaa.2006.5.861

[6]

Annamaria Canino, Elisa De Giorgio, Berardino Sciunzi. Second order regularity for degenerate nonlinear elliptic equations. Discrete & Continuous Dynamical Systems - A, 2018, 38 (8) : 4231-4242. doi: 10.3934/dcds.2018184

[7]

Jae-Hong Pyo, Jie Shen. Normal mode analysis of second-order projection methods for incompressible flows. Discrete & Continuous Dynamical Systems - B, 2005, 5 (3) : 817-840. doi: 10.3934/dcdsb.2005.5.817

[8]

Florian Schneider. Second-order mixed-moment model with differentiable ansatz function in slab geometry. Kinetic & Related Models, 2018, 11 (5) : 1255-1276. doi: 10.3934/krm.2018049

[9]

Anurag Jayswala, Tadeusz Antczakb, Shalini Jha. Second order modified objective function method for twice differentiable vector optimization problems over cone constraints. Numerical Algebra, Control & Optimization, 2019, 9 (2) : 133-145. doi: 10.3934/naco.2019010

[10]

Ruyuan Zhang. Hopf bifurcations of ODE systems along the singular direction in the parameter plane. Communications on Pure & Applied Analysis, 2005, 4 (2) : 445-461. doi: 10.3934/cpaa.2005.4.445

[11]

Daniel Franco, Donal O'Regan. Existence of solutions to second order problems with nonlinear boundary conditions. Conference Publications, 2003, 2003 (Special) : 273-280. doi: 10.3934/proc.2003.2003.273

[12]

Alberto Boscaggin, Fabio Zanolin. Subharmonic solutions for nonlinear second order equations in presence of lower and upper solutions. Discrete & Continuous Dynamical Systems - A, 2013, 33 (1) : 89-110. doi: 10.3934/dcds.2013.33.89

[13]

Saroj Panigrahi, Rakhee Basu. Oscillation results for second order nonlinear neutral differential equations with delay. Conference Publications, 2015, 2015 (special) : 906-912. doi: 10.3934/proc.2015.0906

[14]

Yuri V. Rogovchenko, Fatoş Tuncay. Interval oscillation of a second order nonlinear differential equation with a damping term. Conference Publications, 2007, 2007 (Special) : 883-891. doi: 10.3934/proc.2007.2007.883

[15]

Inara Yermachenko, Felix Sadyrbaev. Types of solutions and multiplicity results for second order nonlinear boundary value problems. Conference Publications, 2007, 2007 (Special) : 1061-1069. doi: 10.3934/proc.2007.2007.1061

[16]

Raegan Higgins. Asymptotic behavior of second-order nonlinear dynamic equations on time scales. Discrete & Continuous Dynamical Systems - B, 2010, 13 (3) : 609-622. doi: 10.3934/dcdsb.2010.13.609

[17]

Naoufel Ben Abdallah, Yongyong Cai, Francois Castella, Florian Méhats. Second order averaging for the nonlinear Schrödinger equation with strongly anisotropic potential. Kinetic & Related Models, 2011, 4 (4) : 831-856. doi: 10.3934/krm.2011.4.831

[18]

Wenmin Sun, Jiguang Bao. New maximum principles for fully nonlinear ODEs of second order. Discrete & Continuous Dynamical Systems - A, 2007, 19 (4) : 813-823. doi: 10.3934/dcds.2007.19.813

[19]

Alessandro Fonda, Fabio Zanolin. Bounded solutions of nonlinear second order ordinary differential equations. Discrete & Continuous Dynamical Systems - A, 1998, 4 (1) : 91-98. doi: 10.3934/dcds.1998.4.91

[20]

Hongqiu Chen. Well-posedness for a higher-order, nonlinear, dispersive equation on a quarter plane. Discrete & Continuous Dynamical Systems - A, 2018, 38 (1) : 397-429. doi: 10.3934/dcds.2018019

2018 Impact Factor: 0.925

Metrics

  • PDF downloads (6)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]