Abstract
A two-dimensional water wave system is examined consisting of two discrete incompressible fluid domains separated by a free common interface. In a geophysical context this is a model of an internal wave, formed at a pycnocline or thermocline in the ocean. The system is considered as being bounded at the bottom and top by a flatbed and wave-free surface respectively. A current profile with depth-dependent currents in each domain is considered. The Hamiltonian of the system is determined and expressed in terms of canonical wave-related variables. Limiting behaviour is examined and compared to that of other known models. The linearised equations as well as long-wave approximations are presented.
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References
Benjamin T.B., Bridges T.J.: Reappraisal of the Kelvin-Helmholtz problem. Part 1. Hamiltonian structure. J. Fluid Mech. 333, 301–325 (1997)
Benjamin T.B., Bridges T.J.: 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)
Benjamin T.B., Olver P.J.: Hamiltonian structure, symmetries and conservation laws for water waves. J. Fluid Mech. 125, 137–185 (1982)
Compelli A.: Hamiltonian formulation of 2 bounded immiscible media with constant non-zero vorticities and a common interface. Wave Motion 54, 115–124 (2015). doi:10.1016/j.wavemoti.2014.11.015
Compelli A.: Hamiltonian approach to the modeling of internal geophysical waves with vorticity. Monatsh. Math. 179(4), 509–521 (2016). doi:10.1007/s00605-014-0724-1
Compelli, A., Ivanov, R.I.: On the dynamics of internal waves interacting with the Equatorial Undercurrent. J. Nonlinear Math. Phys. 22, 531–539 (2015). doi:10.1080/14029251.2015.1113052. arXiv:1510.04096 [math-ph]
Compelli, A., Ivanov, R.I.: Hamiltonian approach to internal wave-current interactions in a two-media fluid with a rigid lid. Pliska Stud. Math. Bulgar. 25, 7–18 (2015). arXiv:1607.01358 [physics.flu-dyn]
Constantin A.: On the deep water wave motion. J. Phys. A 34, 1405–1417 (2001). doi:10.1088/0305-4470/34/7/313
Constantin, A.: Nonlinear water waves with applications to wave-current interactions and tsunamis. In: CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 81. SIAM, Philadelphia (2011)
Constantin A., Escher J.: Symmetry of steady periodic surface water waves with vorticity. J. Fluid Mech. 498, 171–181 (2004). doi:10.1017/S0022112003006773
Constantin A., Escher J.: Analyticity of periodic traveling free surface water waves with vorticity. Ann. Math. 173, 559–568 (2011). doi:10.4007/annals.2011.173.1.12
Constantin A., Germain P.: Instability of some equatorially trapped waves. J. Geophys. Res. Oceans 118, 2802–2810 (2013). doi:10.1002/jgrc.20219
Constantin, A., Ivanov, R.I.: On an integrable two-component Camassa-Holm shallow water system. Phys. Lett. A. 372, 7129–7132 (2008). doi:10.1016/j.physleta.2008.10.050. arXiv:0806.0868 [nlin.SI]
Constantin A., Ivanov R.I.: A Hamiltonian approach to wave-current interactions in two-layer fluids. Phys. Fluids 27, 086603 (2015). doi:10.1063/1.4929457
Constantin, A., Ivanov, R.I., Martin, C.I.: Hamiltonian formulation for wave-current interactions in stratified rotational flows. Arch. Ration. Mech. Anal. 221(3), 1417–1447 (2016). doi:10.1007/s00205-016-0990-2
Constantin, A., Ivanov, R.I., Prodanov, E.M.: Nearly-Hamiltonian structure for water waves with constant vorticity. J. Math. Fluid Mech. 9, 1–14 (2007). doi:10.1007/s00021-006-0230-x. arXiv:math-ph/0610014
Constantin A., Johnson R.S.: The dynamics of waves interacting with the equatorial undercurrent. Geophys. Astrophys. Fluid Dyn. 109, 311–358 (2015). doi:10.1080/03091929.2015.1066785
Constantin A., Sattinger D., Strauss W.: Variational formulations for steady water waves with vorticity. J. Fluid Mech. 548, 151–163 (2006). doi:10.1017/S0022112005007469
Constantin A., Strauss W.: Exact steady periodic water waves with vorticity. Commun. Pure Appl. Math. 57, 481–527 (2004). doi:10.1002/cpa.3046
Cotter, C.J., Holm, D.D., Percival, J.R.: The square root depth wave equations. Proc. R. Soc. A. 466, 3621–3633 (2010). doi:10.1098/rspa.2010.0124
Craig W., Guyenne P., Kalisch H.: Hamiltonian long wave expansions for free surfaces and interfaces. Commun. Pure Appl. Math. 24, 1587–1641 (2005)
Craig W., Guyenne P., Sulem C.: Coupling between internal and surface waves. Nat. Hazards 57, 617–642 (2011). doi:10.1007/s11069-010-9535-4
Escher J., Henry D., Kolev B., Lyons T.: Two-component equations modelling water waves with constant vorticity. Annali di Matematica Pura ed Applicata 195, 249–271 (2016)
Fan L., Gao H., Liu Y.: On the rotation-two-component Camassa–Holm system modelling the equatorial water waves. Adv. Math. 291, 59–89 (2016). doi:10.1016/j.aim.2015.11.049
Genoud F., Henry D.: Instability of equatorial water waves with an underlying current. J. Math. Fluid Mech. 16, 661–667 (2014). doi:10.1007/s00021-014-0175-4
Henry D., Hsu H.-C.: Instability of equatorial water waves in the f-plane. Discrete Contin. Dyn. Syst. 35, 906–916 (2015). doi:10.3934/dcds.2015.35.909
Holm, D.D., Ivanov, R.I.: Two-component CH system: Inverse scattering, peakons and geometry. Inverse Probl. 27, 045013 (2011). doi:10.1088/0266-5611/27/4/045013. arXiv:1009.5374v1 [nlin.SI]
Ivanov, R.I.: Two component integrable systems modelling shallow water waves: the constant vorticity case. Wave Motion 46, 389–396 (2009). doi:10.1016/j.wavemoti.2009.06.012. arXiv:0906.0780
Ivanov, R.I. and Lyons, T.: Integrable models for shallow water with energy dependent spectral problems. Journal of Nonlinear Mathematical Physics 19, Suppl. 1, 1240008 (2012) (17 pages). doi:10.1142/S1402925112400086. arXiv:1211.5567 [nlin.SI]
Johnson R.S.: Camassa-Holm Korteweg-de Vries and related models for water waves. J. Fluid. Mech. 457, 63–82 (2002). doi:10.1017/S0022112001007224
Johnson R.S.: The Camassa-Holm equation for water waves moving over a shear flow. Fluid Dyn. Res. 33, 97–111 (2003). doi:10.1016/S0169-5983(03)00036-4
Johnson, R.S.: The classical problem of water waves: a reservoir of integrable and nearly-integrable equations. J. Nonlinear Math. Phys. 10(suppl. 1), 72–92 (2003)
Jonsson, I.G.: Wave-current interactions. In: The sea, pp, 65–120. Willey, New York (1990)
Kaup D.J.: A higher-order water-wave equation and the method for solving it. Progr. Theor. Phys. 54, 396–408 (1975)
Korteweg, D., de Vries, G.: On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves. Philos. Mag. 39, 422–443 (1895) [reprint: Philos. Mag. 91, 1007–1028 (2011). doi:10.1080/14786435.2010.547337]
Lamb K.: Internal wave breaking and dissipation mechanisms on the continental slope/shelf. Annu. Rev. Fluid Mech. 46, 231–254 (2014)
Martin C.I.: Dispersion relations for gravity water flows with two rotational layers. Eur. J. Mech. B/Fluids 50, 9–18 (2015). doi:10.1016/j.euromechflu.2014.10.005
Milder D.M.: A note regarding On Hamilton‘s principle for water waves. J. Fluid Mech. 83, 159–161 (1977)
Miles J.W.: Hamiltonian formulations for surface waves. Appl. Sci. Res. 37, 103–110 (1981)
Miles J.W.: On Hamilton’s principle for water waves. J. Fluid Mech. 83, 153–158 (1977)
Peregrine D.H.: Interaction of water waves and currents. Adv. Appl. Mech. 16, 9–117 (1976)
Sattinger D.H., Szmigielski J.: A Riemann-Hilbert problem for an energy dependent Schrödinger operator. Inverse Probl. 12, 1003–1025 (1996)
Teles da Silva A.F., Peregrine D.H.: Steep, steady surface waves on water of finite depth with constant vorticity. J. Fluid Mech. 195, 281–302 (1988)
Thomas R., Kharif C., Manna M.: A nonlinear Schrödinger equation for water waves on finite depth with constant vorticity. Phys. Fluids. 24, 127102 (2012). doi:10.1063/1.4768530
Thomas, G.P., Klopman, G.: Wave-current interactions in the near-shore region. In: Gravity waves in water of finite depth, edited by Hunt, J.N. (computational Mechanics, Southampton), pp. 215–319 (1997)
Wahlén E.: A Hamiltonian formulation of water waves with constant vorticity. Lett. Math. Phys. 79, 303–315 (2007). doi:10.1007/s11005-007-0143-5
Wahlén E.: Hamiltonian Long Wave Approximations of Water Waves with Constant Vorticity. Phys. Lett. A. 372, 2597–2602 (2008). doi:10.1016/j.physleta.2007.12.018
Zakharov, V.E.: Stability of periodic waves of finite amplitude on the surface of a deep fluid. Zh. Prikl. Mekh. Tekh. Fiz. 9, 86–94 (1968, in Russian) [J. Appl. Mech. Tech. Phys. 9, 190–194 (1968, English translation)]
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Compelli, A., Ivanov, R.I. The Dynamics of Flat Surface Internal Geophysical Waves with Currents. J. Math. Fluid Mech. 19, 329–344 (2017). https://doi.org/10.1007/s00021-016-0283-4
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DOI: https://doi.org/10.1007/s00021-016-0283-4