Letters in Mathematical Physics

, Volume 79, Issue 3, pp 303–315

A Hamiltonian Formulation of Water Waves with Constant Vorticity

Article

Abstract

We show that the governing equations for two-dimensional water waves with constant vorticity can be formulated as a canonical Hamiltonian system, in which one of the canonical variables is the surface elevation. This generalizes the well-known formulation due to Zakharov [32] in the irrotational case.

Mathematics Subject Classification (2000)

35Q35 37K05 76B15 

Keywords

water waves constant vorticity hamiltonian formulation 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Benjamin T.B. and Olver P.J. (1982). Hamiltonian structures, symmetries and conservation laws for water waves. J. Fluid Mech. 125: 137–185 MATHCrossRefADSMathSciNetGoogle Scholar
  2. 2.
    Buffoni B. (2004). Existence and conditional energetic stability of capillary-gravity solitary water waves by minimisation. Arch. Ration. Mech. Anal. 173: 25–68 MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Buffoni B. (2005). Conditional energetic stability of gravity solitary waves in the presence of weak surface tension. Topol. Methods Nonlinear Anal. 25: 41–68 MATHMathSciNetGoogle Scholar
  4. 4.
    Constantin A. (2001). On the deep water wave motion. J. Phys. A 34: 1405–1417 MATHCrossRefADSMathSciNetGoogle Scholar
  5. 5.
    Constantin, A.: Wave-current interactions. In: EQUADIFF 2003, pp. 207-212. World Scientific, Hackensak, NJ (2005)Google Scholar
  6. 6.
    Constantin A. (2005). A Hamiltonian formulation for free surface water waves with non-vanishing vorticity. J. Nonl. Math. Phys. 12: 202–211 MathSciNetGoogle Scholar
  7. 7.
    Constantin A. (2006). The trajectories of particles in Stokes waves. Invent. Math. 166: 523–535 MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Constantin A. and Escher J. (2004). Symmetry of steady periodic surface water waves with. J. Fluid Mech. 498: 171–181 MATHCrossRefADSMathSciNetGoogle Scholar
  9. 9.
    Constantin A. and Escher J. (2004). Symmetry of steady deep-water waves with vorticity. Eur. J. Appl. Math. 15: 755–768 MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Constantin, A., Ivanov, R.I., Prodanov, E.M.: Nearly-Hamiltonian structure for water waves with constant vorticity. J. Math. Fluid Mech. (to appear)Google Scholar
  11. 11.
    Constantin A. and Strauss W. (2004). Exact steady periodic water waves with vorticity. Comm. Pure Appl. Math. 57: 481–527 MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Constantin A., Sattinger D.H. and Strauss W. (2006). Variational formulations for steady water waves with vorticity. J. Fluid Mech. 548: 151–163 CrossRefADSMathSciNetGoogle Scholar
  13. 13.
    Craig, W.: Water waves, Hamiltonian systems and Cauchy integrals. In: Microlocal Analysis and Nonlinear Waves (Minneapolis, MN, 1988–1989), pp. 37–45, IMA Volume Mathematical Applications, 30. Springer, New York (1991)Google Scholar
  14. 14.
    Craig W. and Groves M. (1994). Hamiltonian long-wave approximations to the water-wave problem. Wave Motion 19: 367–389 MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Ehrnström M. (2005). Uniqueness for steady water waves with vorticity. Int. Math. Res. Not. 2005: 3721–3726 MATHCrossRefGoogle Scholar
  16. 16.
    Gerstner F. (1809). Theorie der Wellen samt einer daraus abgeleiteten Theorie der Deichprofile. Ann. Phys. 2: 412–445 Google Scholar
  17. 17.
    Haragus M. and Scheel A. (2002). Finite-wavelength stability of capillary-gravity solitary waves. Comm. Math. Phys. 225: 487–521 MATHCrossRefADSMathSciNetGoogle Scholar
  18. 18.
    Kolev B. and Sattinger D.H. (2006). Variational principles for water waves. SIAM J. Math. Anal. 38: 906–920 CrossRefMathSciNetGoogle Scholar
  19. 19.
    Lewis D., Marsden J., Montgomery R. and Ratiu T. (1986). The Hamiltonian structure for dynamic free boundary problems. Physica D 18: 391–404 MATHCrossRefADSMathSciNetGoogle Scholar
  20. 20.
    Lighthill J. (1978). Waves in Fluids. Cambridge University Press, Cambridge MATHGoogle Scholar
  21. 21.
    Luke J.C. (1967). A variational principle for a fluid with a free surface. J. Fluid Mech. 27: 395–397 MATHCrossRefADSMathSciNetGoogle Scholar
  22. 22.
    Maddocks J.H. and Pego R.L. (1995). An unconstrained Hamiltonian formulation for incompressible fluid flow. Comm. Math. Phys. 170: 207–217 MATHCrossRefADSMathSciNetGoogle Scholar
  23. 23.
    Mielke A. (2002). On the energetic stability of solitary water waves. Phil. Trans. R. Soc. Lond. A 360: 2337–2358 MATHCrossRefADSMathSciNetGoogle Scholar
  24. 24.
    Milder D.M. (1977). A note regarding ‘On Hamilton’s principle for water waves’. J. Fluid Mech. 83: 159–161 MATHCrossRefADSGoogle Scholar
  25. 25.
    Miles J.W. (1977). On Hamilton’s principle for water waves. J. Fluid Mech. 83: 153–158 MATHCrossRefADSMathSciNetGoogle Scholar
  26. 26.
    Stoker J.J. (1957). Water Waves. The Mathematical Theory with Applications. Interscience, New York MATHGoogle Scholar
  27. 27.
    Swan C., Cummins I.P. and James R.L. (2001). An experimental study of two-dimensional surface water waves propagating on depth-varying currents. J. Fluid Mech. 428: 273–304 MATHCrossRefADSGoogle Scholar
  28. 28.
    Teles da Silva A.F. and Peregrine D.H. (1988). Steep, steady surface waves on water of finite depth with constant vorticity. J. Fluid Mech. 195: 281–302 CrossRefADSMathSciNetGoogle Scholar
  29. 29.
    Thomas, G., Klopman, G.: Wave-current interactions in the nearshore region, in Gravity waves in water of finite depth, pp. 215–319. Southhampton, United Kingdom, (1997)Google Scholar
  30. 30.
    Wahlén, E.: Steady periodic capillary waves with vorticity. Ark. Mat. (to appear)Google Scholar
  31. 31.
    Wahlén E. (2006). Steady periodic capillary-gravity waves with vorticity. SIAM J. Math. Anal. 38: 921–943 CrossRefMathSciNetGoogle Scholar
  32. 32.
    Zakharov V.E. (1968). Stability of periodic waves of finite amplitude on the surface of a deep fluid. Zh. Prikl. Mekh. Tekh. Fiz. 9: 86–94Google Scholar

Copyright information

© Springer 2007

Authors and Affiliations

  1. 1.Department of MathematicsLund UniversityLundSweden

Personalised recommendations