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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T.B. Benjamin, Internal waves of permanent form in fluids of great depth. J. Fluid Mech. 29, 559–562 (1967)
T.B. Benjamin, T.J. Bridges, Reappraisal of the Kelvin-Helmholtz problem. Part 1. Hamiltonian structure. J. Fluid Mech. 333, 301–325 (1997)
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)
T.B. Benjamin, P.J. Olver, Hamiltonian structure, symmetries and conservation laws for water waves. J. Fluid Mech. 125, 137–185 (1982)
J.L. Bona, D. Lannes, J.-C. Saut, Asymptotic models for internal waves. J. Math. Pures Appl. 89, 538–566 (2008)
A. Compelli, Hamiltonian formulation of 2 bounded immiscible media with constant non-zero vorticities and a common interface. Wave Motion 54, 115–124 (2015)
A. Compelli, Hamiltonian approach to the modeling of internal geophysical waves with vorticity. Monatsh. Math. 179(4), 509–521 (2016)
A. Compelli, R. Ivanov, The dynamics of flat surface internal geophysical waves with currents. J. Math. Fluid Mech. 19(2), 329–344 (2017)
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
A. Constantin, J. Escher, Symmetry of steady periodic surface water waves with vorticity. J. Fluid Mech. 498, 171–181 (2004)
A. Constantin, J. Escher, Analyticity of periodic traveling free surface water waves with vorticity. Ann. Math. 173, 559–568 (2011)
A. Constantin, R. Ivanov, A Hamiltonian approach to wave-current interactions in two-layer fluids. Phys. Fluids 27, 08660 (2015)
A. Constantin, R.S. Johnson, The dynamics of waves interacting with the Equatorial Undercurrent. Geophys. Astrophys. Fluid Dyn. 109(4), 311–358 (2015)
A. Constantin, W. Strauss, Exact steady periodic water waves with vorticity. Commun. Pure Appl. Math. 57, 481–527 (2004)
A. Constantin, D. Sattinger, W. Strauss, Variational formulations for steady water waves with vorticity. J. Fluid Mech. 548, 151–163 (2006)
A. Constantin, R. Ivanov, E. Prodanov, Nearly-Hamiltonian structure for water waves with constant vorticity. J. Math. Fluid Mech. 9, 1–14 (2007)
W. Craig, M. Groves, Hamiltonian long-wave approximations to the water-wave problem. Wave Motion 19, 367–389 (1994)
W. Craig, C. Sulem, Numerical simulation of gravity waves. J. Comput. Phys. 108, 73–83 (1993)
W. Craig, P. Guyenne, H. Kalisch, Hamiltonian long-wave expansions for free surfaces and interfaces. Commun. Pure Appl. Math. 58, 1587–1641 (2005)
W. Craig, P. Guyenne, C. Sulem, Coupling between internal and surface waves. Nat. Hazards 57(3), 617–642 (2011)
S.A. Elder, J. Williams, Fluid Physics for Oceanographers and Physicists: An Introduction to Incompressible Flow (Pergamon Press, Oxford, 1989)
A.V. Fedorov, J.N. Brown, Equatorial waves, in Encyclopedia of Ocean Sciences ed. by J. Steele (Academic, San Diego, 2009), pp. 3679–3695
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)
J.K. Hunter, B. Nachtergaele, Applied Analysis (World Scientific, Singapore, 2001)
R.S. Johnson, Mathematical Theory of Water Waves. Cambridge Texts in Applied Mathematics (Cambridge University Press, Cambridge, 1997)
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)
D. Milder, A note regarding “On Hamilton’s principle for water waves”. J. Fluid Mech. 83, 159–161 (1977)
J. Miles, On Hamilton’s principle for water waves. J. Fluid Mech. 83(1), 153–158 (1977)
H. Ono, Algebraic solitary waves in stratified fluids. J. Phys. Soc. Jpn. 39, 1082–1091 (1975)
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)
E. Wahlén, A Hamiltonian formulation of water waves with constant vorticity. Lett. Math. Phys. 79, 303–315 (2007)
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)
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.
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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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