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Integrable Models of Internal Gravity Water Waves Beneath a Flat Surface

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Abstract

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.

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References

  1. T.B. Benjamin, Internal waves of permanent form in fluids of great depth. J. Fluid Mech. 29, 559–562 (1967)

    Article  Google Scholar 

  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. 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. T.B. Benjamin, P.J. Olver, Hamiltonian structure, symmetries and conservation laws for water waves. J. Fluid Mech. 125, 137–185 (1982)

    Article  MathSciNet  Google Scholar 

  5. J.L. Bona, D. Lannes, J.-C. Saut, Asymptotic models for internal waves. J. Math. Pures Appl. 89, 538–566 (2008)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  7. A. Compelli, Hamiltonian approach to the modeling of internal geophysical waves with vorticity. Monatsh. Math. 179(4), 509–521 (2016)

    Article  MathSciNet  Google Scholar 

  8. A. Compelli, R. Ivanov, The dynamics of flat surface internal geophysical waves with currents. J. Math. Fluid Mech. 19(2), 329–344 (2017)

    Article  MathSciNet  Google Scholar 

  9. A. Compelli, R. Ivanov, Benjamin-Ono model of an equatorial pycnocline. Discrete Contin. Dynam. Syst. A 39(8), 4519–4532 (2019). https://doi.org/10.3934/dcds.2019185

    Article  Google Scholar 

  10. A. Constantin, J. Escher, Symmetry of steady periodic surface water waves with vorticity. J. Fluid Mech. 498, 171–181 (2004)

    Article  MathSciNet  Google Scholar 

  11. A. Constantin, J. Escher, Analyticity of periodic traveling free surface water waves with vorticity. Ann. Math. 173, 559–568 (2011)

    Article  MathSciNet  Google Scholar 

  12. A. Constantin, R. Ivanov, A Hamiltonian approach to wave-current interactions in two-layer fluids. Phys. Fluids 27, 08660 (2015)

    Article  Google Scholar 

  13. A. Constantin, R.S. Johnson, The dynamics of waves interacting with the Equatorial Undercurrent. Geophys. Astrophys. Fluid Dyn. 109(4), 311–358 (2015)

    Article  MathSciNet  Google Scholar 

  14. A. Constantin, W. Strauss, Exact steady periodic water waves with vorticity. Commun. Pure Appl. Math. 57, 481–527 (2004)

    Article  MathSciNet  Google Scholar 

  15. A. Constantin, D. Sattinger, W. Strauss, Variational formulations for steady water waves with vorticity. J. Fluid Mech. 548, 151–163 (2006)

    Article  MathSciNet  Google Scholar 

  16. A. Constantin, R. Ivanov, E. Prodanov, Nearly-Hamiltonian structure for water waves with constant vorticity. J. Math. Fluid Mech. 9, 1–14 (2007)

    Article  MathSciNet  Google Scholar 

  17. W. Craig, M. Groves, Hamiltonian long-wave approximations to the water-wave problem. Wave Motion 19, 367–389 (1994)

    Article  MathSciNet  Google Scholar 

  18. W. Craig, C. Sulem, Numerical simulation of gravity waves. J. Comput. Phys. 108, 73–83 (1993)

    Article  MathSciNet  Google Scholar 

  19. W. Craig, P. Guyenne, H. Kalisch, Hamiltonian long-wave expansions for free surfaces and interfaces. Commun. Pure Appl. Math. 58, 1587–1641 (2005)

    Article  MathSciNet  Google Scholar 

  20. W. Craig, P. Guyenne, C. Sulem, Coupling between internal and surface waves. Nat. Hazards 57(3), 617–642 (2011)

    Article  Google Scholar 

  21. S.A. Elder, J. Williams, Fluid Physics for Oceanographers and Physicists: An Introduction to Incompressible Flow (Pergamon Press, Oxford, 1989)

    Google Scholar 

  22. A.V. Fedorov, J.N. Brown, Equatorial waves, in Encyclopedia of Ocean Sciences ed. by J. Steele (Academic, San Diego, 2009), pp. 3679–3695

    Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  24. J.K. Hunter, B. Nachtergaele, Applied Analysis (World Scientific, Singapore, 2001)

    Book  Google Scholar 

  25. R.S. Johnson, Mathematical Theory of Water Waves. Cambridge Texts in Applied Mathematics (Cambridge University Press, Cambridge, 1997)

    Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  27. D. Milder, A note regarding “On Hamilton’s principle for water waves”. J. Fluid Mech. 83, 159–161 (1977)

    Article  Google Scholar 

  28. J. Miles, On Hamilton’s principle for water waves. J. Fluid Mech. 83(1), 153–158 (1977)

    Article  MathSciNet  Google Scholar 

  29. H. Ono, Algebraic solitary waves in stratified fluids. J. Phys. Soc. Jpn. 39, 1082–1091 (1975)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  31. E. Wahlén, A Hamiltonian formulation of water waves with constant vorticity. Lett. Math. Phys. 79, 303–315 (2007)

    Article  MathSciNet  Google Scholar 

  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 

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Acknowledgements

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.

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, University College Cork, Cork, Ireland

    Alan C. Compelli

  2. School of Mathematical Sciences, Technological University Dublin, Dublin, Ireland

    Rossen I. Ivanov

  3. Department of Computing and Mathematics, Waterford Institute of Technology, Waterford, Ireland

    Tony Lyons

Authors
  1. Alan C. Compelli
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  2. Rossen I. Ivanov
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  3. Tony Lyons
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Corresponding author

Correspondence to Rossen I. Ivanov .

Editor information

Editors and Affiliations

  1. School of Mathematical Sciences, University College Cork, Cork, Ireland

    David Henry

  2. Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge, UK

    Konstantinos Kalimeris

  3. School of Mathematics, University of East Anglia, Norwich, UK

    Emilian I. Părău

  4. Department of Mathematics, University College London, London, UK

    Jean-Marc Vanden-Broeck

  5. Centre for Mathematical Sciences, Lund University, Lund, Sweden

    Erik Wahlén

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Compelli, A.C., Ivanov, R.I., Lyons, T. (2019). Integrable Models of Internal Gravity Water Waves Beneath a Flat Surface. In: Henry, D., Kalimeris, K., Părău, E., Vanden-Broeck, JM., Wahlén, E. (eds) Nonlinear Water Waves . Tutorials, Schools, and Workshops in the Mathematical Sciences . Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-33536-6_6

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