Stokes Waves in a Constant Vorticity Flow

  • Sergey A. Dyachenko
  • Vera Mikyoung HurEmail author
Part of the Tutorials, Schools, and Workshops in the Mathematical Sciences book series (TSWMS)


The Stokes wave problem in a constant vorticity flow is formulated via conformal mapping as a modified Babenko equation. The associated linearized operator is self-adjoint, whereby efficiently solved by the Newton-conjugate gradient method. For strong positive vorticity, a fold develops in the wave speed versus amplitude plane, and a gap as the vorticity strength increases, bounded by two touching waves, whose profile contacts with itself, enclosing a bubble of air. More folds and gaps follow as the vorticity strength increases further. Touching waves at the beginnings of the lowest gaps tend to the limiting Crapper wave as the vorticity strength increases indefinitely, while a fluid disk in rigid body rotation at the ends of the gaps. Touching waves at the boundaries of higher gaps contain more fluid disks.


Stokes wave Constant vorticity Conformal Numerical 

Mathematics Subject Classification (2000)

Primary 76B15; Secondary 76B07 30C30 65T50 



VMH is supported by the US National Science Foundation under the Faculty Early Career Development (CAREER) Award DMS-1352597, and SD is supported by the National Science Foundation under DMS-1716822. VMH is grateful to the Erwin Schrödinger International Institute for Mathematics and Physics for its hospitality during the workshop Nonlinear Water Waves. This work was completed with the support of our TE X-pert.


  1. 1.
    K.I. Babenko, Some remarks on the theory of surface waves of finite amplitude. Soviet Math. Doklady 35, 599–603 (1987) (See also loc. cit. 647–650)Google Scholar
  2. 2.
    B. Buffoni, J.F. Toland, Analytic Theory of Global Bifurcation: An Introduction. Princeton Series in Applied Mathematics (Princeton University Press, Princeton, 2003)Google Scholar
  3. 3.
    B. Buffoni, E.N. Dancer, J.F. Toland, The regularity and local bifurcation of steady periodic water waves. Arch. Ration. Mech. Anal. 152, 207–240 (2002)MathSciNetCrossRefGoogle Scholar
  4. 4.
    B. Buffoni, E.N. Dancer, J.F. Toland, The sub-harmonic bifurcation of Stokes waves. Arch. Ration. Mech. Anal. 152, 241–271 (2002)MathSciNetCrossRefGoogle Scholar
  5. 5.
    W. Choi, Nonlinear surface waves interacting with a linear shear current. Math. Comput. Simul. 80, 101–110 (2009)MathSciNetCrossRefGoogle Scholar
  6. 6.
    A. Constantin, W. Strauss, E. Varvaruca, Global bifurcation of steady gravity water waves with critical layers. Acta. Math. 217, 195–262 (2016)MathSciNetCrossRefGoogle Scholar
  7. 7.
    G.D. Crapper, An exact solution for progressive capillary waves of arbitrary amplitude. J. Fluid Mech. 2, 532–540 (1957)MathSciNetCrossRefGoogle Scholar
  8. 8.
    S.A. Dyachenko, V.M. Hur, Stokes waves with constant vorticity: I. Numerical computation. Stud. Appl. Math. 142, 162–189 (2019)MathSciNetzbMATHGoogle Scholar
  9. 9.
    S.A. Dyachenko, V.M. Hur, Stokes waves with constant vorticity: folds, gaps, and fluid bubbles. J. Fluid Mech. 878, 502–521 (2019)MathSciNetCrossRefGoogle Scholar
  10. 10.
    A.I. Dyachenko, E.A. Kuznetsov, M.D. Spector, V.E. Zakharov, Analytical description of the free surface dynamics of an ideal fluid (canonical formalism and conformal mapping). Phys. Lett. A 1, 73–79 (1996)CrossRefGoogle Scholar
  11. 11.
    S.A. Dyachenko, P.M. Lushnikov, A.O. Korotkevich, Branch cuts of Stokes waves on deep water. Part I: numerical solutions and Padé approximation. Stud. Appl. Math. 137, 419–472 (2016)zbMATHGoogle Scholar
  12. 12.
    V.H. Hur, Symmetry of steady periodic water waves with vorticity. Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 365, 2203–2214 (2007)MathSciNetCrossRefGoogle Scholar
  13. 13.
    M.S. Longuet-Higgins, Some new relations between Stokes’s coefficients in the theory of gravity waves. J. Inst. Math. Appl. 22, 261–273 (1978)MathSciNetCrossRefGoogle Scholar
  14. 14.
    M.S. Longuet-Higgins, M.J.H. Fox, Theory of the almost-highest wave. Part 2. Matching and analytic extension. J. Fluid Mech. 85, 769–786 (1978)MathSciNetCrossRefGoogle Scholar
  15. 15.
    P.M. Lushnikov, Structure and location of branch point singularities for Stokes waves on deep water. J. Fluid Mech. 800, 557–594 (2016)MathSciNetCrossRefGoogle Scholar
  16. 16.
    P.M. Lushnikov, S.A. Dyachenko, D.A. Silantyev, New conformal mapping for adaptive resolving of the complex singularities of Stokes wave. Proc. R. Soc. A 473, 20170198, 19 pp. (2017)MathSciNetCrossRefGoogle Scholar
  17. 17.
    P.I. Plotnikov, Nonuniqueness of solutions of the problem of solitary waves and bifurcation of critical points of smooth functionals. Math. USSR Izvetiya 38, 333–357 (1992)MathSciNetCrossRefGoogle Scholar
  18. 18.
    R. Ribeiro, P.A. Milewski, A. Nachbin, Flow structure beneath rotational water waves with stagnation points. J. Fluid Mech. 812, 792–814 (2017)MathSciNetCrossRefGoogle Scholar
  19. 19.
    J.A. Simmen, P.G. Saffman, Steady deep-water waves on a linear shear current. Stud. Appl. Math. 73, 35–57 (1985)MathSciNetCrossRefGoogle Scholar
  20. 20.
    G.G. Stokes, On the theory of oscillatory waves. Trans. Camb. Phil. Soc. 8, 441–445 (1847)Google Scholar
  21. 21.
    G.G. Stokes, Considerations relative to the greatest height of oscillatory irrotational waves which can be propagated without change of form, in Mathematical and Physical Papers, vol. I (Cambridge University Press, Cambridge, 1880), pp. 225–228Google Scholar
  22. 22.
    W.A. Strauss, Steady water waves. Bull. Amer. Math. Soc. (N.S.) 47, 671–694 (2010)MathSciNetCrossRefGoogle Scholar
  23. 23.
    A.F. Teles da Silva, D.H. Peregrine, Steep, steady surface waves on water of finite depth with constant vorticity. J. Fluid Mech. 195, 281–302 (1988)MathSciNetCrossRefGoogle Scholar
  24. 24.
    J.F. Toland, Stokes waves. Topol. Methods Nonlinear Anal. 7, 1–48 (1996)MathSciNetCrossRefGoogle Scholar
  25. 25.
    J.-M. Vanden-Broeck, Periodic waves with constant vorticity in water of infinite depth. IMA J. Appl. Math. 56, 207–217 (1996)CrossRefGoogle Scholar
  26. 26.
    A. Zygmund, Trigonometric Series I & II, corrected reprint (1968) of 2nd edn. (Cambridge University Press, Cambridge, 1959)Google Scholar

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Authors and Affiliations

  1. 1.Department of Applied MathematicsUniversity of WashingtonSeattleUSA
  2. 2.Department of MathematicsUniversity of Illinois at Urbana-ChampaignUrbanaUSA

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