Integrable Models of Internal Gravity Water Waves Beneath a Flat Surface

  • Alan C. Compelli
  • Rossen I. IvanovEmail author
  • Tony Lyons
Part of the Tutorials, Schools, and Workshops in the Mathematical Sciences book series (TSWMS)


A two-layer fluid system separated by a pycnocline in the form of an internal wave is considered. The lower layer is bounded below by a flat bottom and the upper layer is bounded above by a flat surface. The fluids are incompressible and inviscid and Coriolis forces as well as currents are taken into consideration. A Hamiltonian formulation is presented and appropriate scaling leads to a KdV approximation. Additionally, considering the lower layer to be infinitely deep leads to a Benjamin–Ono approximation.


Internal waves Currents Nonlinear waves Long waves Hamiltonian systems Solitons 

Mathematics Subject Classification (2000)

Primary: 35Q35 35Q51 35Q53; Secondary: 37K10 



The authors are grateful to the Erwin Schrödinger International Institute for Mathematics and Physics (ESI), Vienna (Austria) for the opportunity to participate in the workshop Nonlinear Water Waves—an Interdisciplinary Interface, 2017 where a significant part of this work has been accomplished. AC is also funded by SFI grant 13/CDA/2117.


  1. 1.
    T.B. Benjamin, Internal waves of permanent form in fluids of great depth. J. Fluid Mech. 29, 559–562 (1967)CrossRefGoogle Scholar
  2. 2.
    T.B. Benjamin, T.J. Bridges, Reappraisal of the Kelvin-Helmholtz problem. Part 1. Hamiltonian structure. J. Fluid Mech. 333, 301–325 (1997)Google Scholar
  3. 3.
    T.B. Benjamin, T.J. Bridges, Reappraisal of the Kelvin-Helmholtz problem. Part 2. Interaction of the Kelvin-Helmholtz, superharmonic and Benjamin-Feir instabilities. J. Fluid Mech. 333, 327–373 (1997)Google Scholar
  4. 4.
    T.B. Benjamin, P.J. Olver, Hamiltonian structure, symmetries and conservation laws for water waves. J. Fluid Mech. 125, 137–185 (1982)MathSciNetCrossRefGoogle Scholar
  5. 5.
    J.L. Bona, D. Lannes, J.-C. Saut, Asymptotic models for internal waves. J. Math. Pures Appl. 89, 538–566 (2008)MathSciNetCrossRefGoogle Scholar
  6. 6.
    A. Compelli, Hamiltonian formulation of 2 bounded immiscible media with constant non-zero vorticities and a common interface. Wave Motion 54, 115–124 (2015)MathSciNetCrossRefGoogle Scholar
  7. 7.
    A. Compelli, Hamiltonian approach to the modeling of internal geophysical waves with vorticity. Monatsh. Math. 179(4), 509–521 (2016)MathSciNetCrossRefGoogle Scholar
  8. 8.
    A. Compelli, R. Ivanov, The dynamics of flat surface internal geophysical waves with currents. J. Math. Fluid Mech. 19(2), 329–344 (2017)MathSciNetCrossRefGoogle Scholar
  9. 9.
    A. Compelli, R. Ivanov, Benjamin-Ono model of an equatorial pycnocline. Discrete Contin. Dynam. Syst. A 39(8), 4519–4532 (2019). CrossRefGoogle Scholar
  10. 10.
    A. Constantin, J. Escher, Symmetry of steady periodic surface water waves with vorticity. J. Fluid Mech. 498, 171–181 (2004)MathSciNetCrossRefGoogle Scholar
  11. 11.
    A. Constantin, J. Escher, Analyticity of periodic traveling free surface water waves with vorticity. Ann. Math. 173, 559–568 (2011)MathSciNetCrossRefGoogle Scholar
  12. 12.
    A. Constantin, R. Ivanov, A Hamiltonian approach to wave-current interactions in two-layer fluids. Phys. Fluids 27, 08660 (2015)CrossRefGoogle Scholar
  13. 13.
    A. Constantin, R.S. Johnson, The dynamics of waves interacting with the Equatorial Undercurrent. Geophys. Astrophys. Fluid Dyn. 109(4), 311–358 (2015)MathSciNetCrossRefGoogle Scholar
  14. 14.
    A. Constantin, W. Strauss, Exact steady periodic water waves with vorticity. Commun. Pure Appl. Math. 57, 481–527 (2004)MathSciNetCrossRefGoogle Scholar
  15. 15.
    A. Constantin, D. Sattinger, W. Strauss, Variational formulations for steady water waves with vorticity. J. Fluid Mech. 548, 151–163 (2006)MathSciNetCrossRefGoogle Scholar
  16. 16.
    A. Constantin, R. Ivanov, E. Prodanov, Nearly-Hamiltonian structure for water waves with constant vorticity. J. Math. Fluid Mech. 9, 1–14 (2007)MathSciNetCrossRefGoogle Scholar
  17. 17.
    W. Craig, M. Groves, Hamiltonian long-wave approximations to the water-wave problem. Wave Motion 19, 367–389 (1994)MathSciNetCrossRefGoogle Scholar
  18. 18.
    W. Craig, C. Sulem, Numerical simulation of gravity waves. J. Comput. Phys. 108, 73–83 (1993)MathSciNetCrossRefGoogle Scholar
  19. 19.
    W. Craig, P. Guyenne, H. Kalisch, Hamiltonian long-wave expansions for free surfaces and interfaces. Commun. Pure Appl. Math. 58, 1587–1641 (2005)MathSciNetCrossRefGoogle Scholar
  20. 20.
    W. Craig, P. Guyenne, C. Sulem, Coupling between internal and surface waves. Nat. Hazards 57(3), 617–642 (2011)CrossRefGoogle Scholar
  21. 21.
    S.A. Elder, J. Williams, Fluid Physics for Oceanographers and Physicists: An Introduction to Incompressible Flow (Pergamon Press, Oxford, 1989)Google Scholar
  22. 22.
    A.V. Fedorov, J.N. Brown, Equatorial waves, in Encyclopedia of Ocean Sciences ed. by J. Steele (Academic, San Diego, 2009), pp. 3679–3695Google Scholar
  23. 23.
    A.S. Fokas, M.J. Ablowitz, The inverse scattering transform for the Benjamin-Ono equation – a pivot to multidimensional problems. Stud. Appl. Math. 68, 1–10 (1983)MathSciNetCrossRefGoogle Scholar
  24. 24.
    J.K. Hunter, B. Nachtergaele, Applied Analysis (World Scientific, Singapore, 2001)CrossRefGoogle Scholar
  25. 25.
    R.S. Johnson, Mathematical Theory of Water Waves. Cambridge Texts in Applied Mathematics (Cambridge University Press, Cambridge, 1997)Google Scholar
  26. 26.
    R.S. Johnson, Application of the ideas and techniques of classical fluid mechanics to some problems in physical oceanography. Phil. Trans. R. Soc. A 376, 20170092 (2018)MathSciNetCrossRefGoogle Scholar
  27. 27.
    D. Milder, A note regarding “On Hamilton’s principle for water waves”. J. Fluid Mech. 83, 159–161 (1977)CrossRefGoogle Scholar
  28. 28.
    J. Miles, On Hamilton’s principle for water waves. J. Fluid Mech. 83(1), 153–158 (1977)MathSciNetCrossRefGoogle Scholar
  29. 29.
    H. Ono, Algebraic solitary waves in stratified fluids. J. Phys. Soc. Jpn. 39, 1082–1091 (1975)MathSciNetCrossRefGoogle Scholar
  30. 30.
    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
  31. 31.
    E. Wahlén, A Hamiltonian formulation of water waves with constant vorticity. Lett. Math. Phys. 79, 303–315 (2007)MathSciNetCrossRefGoogle Scholar
  32. 32.
    V. Zakharov, Stability of periodic waves of finite amplitude on the surface of a deep fluid (in Russian). Zh. Prikl. Mekh. Tekh. Fiz. 9, 86–94 (1968); J. Appl. Mech. Tech. Phys. 9, 190–194 (1968) (English translation)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Alan C. Compelli
    • 1
  • Rossen I. Ivanov
    • 2
    Email author
  • Tony Lyons
    • 3
  1. 1.School of Mathematical SciencesUniversity College CorkCorkIreland
  2. 2.School of Mathematical SciencesTechnological University DublinDublinIreland
  3. 3.Department of Computing and MathematicsWaterford Institute of TechnologyWaterfordIreland

Personalised recommendations