July  2022, 21(7): 2291-2307. doi: 10.3934/cpaa.2022029

Hamiltonian description of internal ocean waves with Coriolis force

School of Mathematical Sciences, Technological University Dublin, City Campus, Grangegorman Lower, Dublin D07 ADY7, Ireland

* Corresponding author

Received  October 2021 Revised  December 2021 Published  July 2022 Early access  January 2022

Fund Project: R.I. is partially supported by the Bulgarian National Science Fund, grant KΠ -06H42/2 from 27.11.2020

The interfacial internal waves are formed at the pycnocline or thermocline in the ocean and are influenced by the Coriolis force due to the Earth's rotation. A derivation of the model equations for the internal wave propagation taking into account the Coriolis effect is proposed. It is based on the Hamiltonian formulation of the internal wave dynamics in the irrotational case, appropriately extended to a nearly Hamiltonian formulation which incorporates the Coriolis forces. Two propagation regimes are examined, the long-wave and the intermediate long-wave propagation with a small amplitude approximation for certain geophysical scales of the physical variables. The obtained models are of the type of the well-known Ostrovsky equation and describe the wave propagation over the two spatial horizontal dimensions of the ocean surface.

Citation: Joseph D. Cullen, Rossen I. Ivanov. Hamiltonian description of internal ocean waves with Coriolis force. Communications on Pure and Applied Analysis, 2022, 21 (7) : 2291-2307. doi: 10.3934/cpaa.2022029
References:
[1]

T. B. Benjamin, Internal waves of finite amplitude and permanent form, J. Fluid Mech., 16 (1966), 241-270.  doi: 10.1017/S0022112066001630.

[2]

T. B. Benjamin, Internal waves of permanent form in fluids of great depth, J. Fluid Mech., 29 (1967), 559-592.  doi: 10.1017/S002211206700103X.

[3]

J. L. BonaD. Lannes and J.-C. Saut, Asymptotic models for internal waves, J. Math. Pures Appl., 89 (2008), 538-566.  doi: 10.1016/j.matpur.2008.02.003.

[4]

V. V. Bulatov and Y. V. Vladimirov, Fundamental problems of internal gravity waves dynamics in ocean, J. Basic Appl. Sci., 9 (2013), 69-81. 

[5]

H. H. Chen and Y. C. Lee, Internal wave solitons of fluids of finite depth, Phys. Rev. Lett., 43 (1979), 264-266.  doi: 10.1103/PhysRevLett.43.264.

[6]

W. Choi and R. Camassa, Fully nonlinear internal waves in a two-fluid system, J. Fluid Mech., 396 (1999), 1–36. doi: 10.1017/S0022112099005820.

[7]

R. ChoudhuryR. I. Ivanov and Y. Liu, Hamiltonian formulation, nonintegrability and local bifurcations for the Ostrovsky equation, Chaos, Solitons and Fractals, 34 (2007), 544-550.  doi: 10.1016/j.chaos.2006.03.057.

[8]

A. Compelli, Hamiltonian approach to the modeling of internal geophysical waves with vorticity, Monatsh. Math., 179 (2016), 509-521.  doi: 10.1007/s00605-014-0724-1.

[9]

A. Compelli and R. Ivanov, The dynamics of flat surface internal geophysical waves with currents, J. Math. Fluid Mech., 19 (2017), 329-344.  doi: 10.1007/s00021-016-0283-4.

[10]

A. Compelli and R. Ivanov, Benjamin-Ono model of an internal wave under a flat surface, Discrete Contin. Dyn. Syst., 39 (2019), 4519-4532.  doi: 10.3934/dcds.2019185.

[11]

A. Constantin and R. Ivanov, A Hamiltonian approach to wave-current interactions in two-layer fluids, Phys. Fluids, 27 (2015), 086603.  doi: 10.1063/1.4929457.

[12]

A. Constantin and R. Ivanov, Equatorial wave-current interactions, Commun. Math. Phys., 370 (2019), 1C-48. doi: 10.1007/s00220-019-03483-8.

[13]

A. ConstantinR. Ivanov and C. -I. Martin, Hamiltonian formulation for wave-current interactions in stratified rotational flows, Arch. Rational Mech. Anal., 221 (2016), 1417-1447.  doi: 10.1007/s00205-016-0990-2.

[14]

A. ConstantinR. Ivanov and E. Prodanov, Nearly-Hamiltonian structure for water waves with constant vorticity, J. Math. Fluid Mech., 9 (2007), 1-14.  doi: 10.1007/s00021-006-0230-x.

[15]

A. Constantin and R. S. Johnson, On the role of nonlinearity in geostrophic ocean flows on a sphere, in Nonlinear systems and their remarkable mathematical structures (ed. N. Euler), Vol. 1, pp. 500–519, CRC Press, Boca Raton, FL, 2019. doi: 10.1201/9780429470462-18.

[16]

A. Constantin and R. S. Johnson, On the modelling of large-scale atmospheric flow, J. Differ. EqU., 285 (2021), 751-798.  doi: 10.1016/j.jde.2021.03.019.

[17]

A. Constantin and R. S. Johnson, On the propagation of waves in the atmosphere, Proc. Roy. Soc. A, 477 (2021), 25 pp. doi: 10.1098/rspa.2020.0424.

[18]

A. Constantin and R. S. Johnson, An exact, steady, purely azimuthal flow as a model for the Antarctic Circumpolar Current, J. Phys. Oceanogr., 46 (2016), 3585-3594.  doi: 10.1175/JPO-D-16-0121.1.

[19]

W. CraigP. Guyenne and C. Sulem, Coupling between internal and surface waves, Nat. Hazards, 57 (2011), 617-642.  doi: 10.1007/s11069-010-9535-4.

[20]

W. CraigP. Guyenne and H. Kalisch, Hamiltonian long-wave expansions for free surfaces and interfaces, Commun. Pure Appl. Math., 58 (2005), 1587-1641.  doi: 10.1002/cpa.20098.

[21]

J. Cullen and R. Ivanov, On the intermediate long wave propagation for internal waves in the presence of currents, Euro. J. Mech.-B/Fluids, 84 (2020), 325-333.  doi: 10.1016/j.euromechflu.2020.07.001.

[22]

V. D. Djordjevic and L. G. Redekopp, The fission and disintegration of internal solitary waves moving over two-dimensional topography, J. Phys. Ocean., 8 (1978), 1016-1024.  doi: 10.1175/1520-0485(1978)008<1016:TFADOI>2.0.CO;2.

[23]

A. V. Fedorov and J. N. Brown, Equatorial waves, in Encyclopedia of Ocean Sciences (ed. J. Steele), Academic, San Diego, Calif., 2009, 3679–3695. doi: 10.1016/B978-012374473-9.00610-X.

[24]

A. S. Fokas and M. J. Ablowitz, The Inverse Scattering Transform for the Benjamin-Ono equation – a pivot to multidimensional problems, Stud. Appl. Math., 68 (1983), 1-10.  doi: 10.1002/sapm19836811.

[25]

G. G. Grahovski and R. I. Ivanov, Generalised Fourier transform and perturbations to soliton equations, Discrete Contin. Dyn. Syst. Ser. B, 12 (2009), 579-595.  doi: 10.3934/dcdsb.2009.12.579.

[26]

A. R. Giniyatullin, A. A. Kurkin, O. E. and Kurkina and Y. A. Stepanyants, Generalized Korteweg–de Vries equation for internal waves in two-layer fluid, Fundamental and Applied Hydrophysics, 7 (2014), 16–28, (in Russian).

[27]

R. Grimshaw, Evolution equations for weakly nonlinear, long internal waves in a rotating fluid, Stud. Appl. Math., 73 (1985) 1–33. doi: 10.1002/SAPM19857311.

[28]

R. GrimshawL. OstrovskyV. Shrira and Y. Stepanyants, Long nonlinear surface and internal waves in a rotating ocean, Surveys Geophys., 19 (1998), 289-338.  doi: 10.1023/A:1006587919935.

[29]

S. V. Haziot and K. Marynets, Applying the stereographic projection to modeling of the flow of the Antarctic Circumpolar Current, Oceanography, 31 (2018), 68-75.  doi: 10.5670/oceanog.2018.311.

[30]

K. R. Helfrich and W. K. Melville, Long nonlinear internal waves, Annu. Rev. Fluid Mech., 38 (2006), 395-425.  doi: 10.1146/annurev.fluid.38.050304.092129.

[31]

D. Henry, Energy considerations for nonlinear equatorial water waves, Commun. Pure Appl. Anal., to appear.

[32]

R. I. Ivanov, Hamiltonian model for coupled surface and internal waves in the presence of currents, Nonlinear Analysis: Real World Applications, 34 (2017) 316–334. doi: 10.1016/j.nonrwa.2016.09.010.

[33]

R. I. Ivanov, On the Coriolis effect for internal ocean waves, in: Floating Offshore Energy Devices, eds: C. Mc Goldrick et al., Materials Research Proceedings 20 (2022), 20–25. doi: 10.21741/9781644901731-3.

[34]

R. S. Johnson, Camassa-Holm, Korteweg-de Vries and related models for water waves, J. Fluid Mech., 457 (2002), 63-82.  doi: 10.1017/S0022112001007224.

[35] R. S. Johnson, A Modern Introduction to the Mathematical Theory of Water Waves, Cambridge University Press, Cambridge, 1997.  doi: 10.1017/CBO9780511624056.
[36]

R. I. Joseph, Solitary waves in a finite depth fluid, J. Phys. A: Math. Gen., 10 (1977), L225–L227. doi: 10.1088/0305-4470/10/12/002.

[37]

B. B. Kadomtsev and V. I. Petviashvili, On the stability of solitary waves in weakly dispersing media, Dokl. Akad. Nauk SSSR, 192 (1970), 753–756, (in Russian).

[38]

D. J. Kaup, Y. Matsuno, The Inverse Scattering Transform for the Benjamin-Ono Equation, Stud. Appl. Math., 101, (1998), 73–98. doi: 10.1111/1467-9590.00086.

[39]

D. Korteweg and G. de Vries, On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves, Philosophical Magazine, 39 (1895), 422–443; reprint: Philosophical Magazine, 39 (2011), 1007–1028. doi: 10.1080/14786435.2010.547337.

[40]

K. R. KhusnutdinovaY. A. Stepanyants and M. R. Tranter, Soliton solutions to the fifth-order Korteweg–de Vries equation and their applications to surface and internal water waves, Physics of Fluids, 30 (2018), 022104.  doi: 10.1063/1.5009965.

[41]

C. G. Koop and G. Butler, An investigation of internal solitary waves in a two-fluid system, J. Fluid Mech., 112 (1981), 225-251.  doi: 10.1017/S0022112081000372.

[42]

T. KubotaD. R. S. Ko and L. D. Dobbs, Weakly-nonlinear, long internal gravity waves in stratified fluids of finite depth, J. Hydronautics, 12 (1978), 157-165.  doi: 10.2514/3.63127.

[43]

A. I. Leonov, The effect of the Earth's rotation on the propagation of weak nonlinear surface and internal long oceanic waves, Ann. NY Acad. Sci., 373 (1981), 150-159.  doi: 10.1111/j.1749-6632.1981.tb51140.x.

[44]

S. P. Novikov, S. V. Manakov, L. P. Pitaevsky and V. E. Zakharov, Theory of Solitons: The Inverse Scattering Method, New York, Plenum, 1984.

[45]

H. Ono, Algebraic solitary waves in stratified fluids, J. Phys. Soc. Japan, 39 (1975), 1082-1091.  doi: 10.1143/JPSJ.39.1082.

[46]

A. R. Osborne and T. L. Burch, Internal solitons in the Andaman Sea, Science, 208 (1980), 451-460.  doi: 10.1126/science.208.4443.451.

[47]

L. A. Ostrovsky, Nonlinear internal waves in a rotating ocean, Okeanologia, 18 (1978), 181–191, (in Russian).

[48]

J. Pedlosky, Geophysical Fluid Dynamics, Springer-Verlag, New York, 1979. doi: 10.1007/978-1-4612-4650-3.

[49]

V. Varlamov and Y. Liu, Cauchy problem for the Ostrovsky equation, Discrete and Contin. Dyn. Sys., 10 (2004), 731-751.  doi: 10.3934/dcds.2004.10.731.

[50]

V. E. Zakharov, Stability of periodic waves of finite amplitude on the surface of a deep fluid, Zh. Prikl. Mekh. Tekh. Fiz., 9 (1968), 86–94 (in Russian); J. Appl. Mech. Tech. Phys., 9 (1968), 190–194 (English translation). doi: 10.1007/BF00913182.

show all references

References:
[1]

T. B. Benjamin, Internal waves of finite amplitude and permanent form, J. Fluid Mech., 16 (1966), 241-270.  doi: 10.1017/S0022112066001630.

[2]

T. B. Benjamin, Internal waves of permanent form in fluids of great depth, J. Fluid Mech., 29 (1967), 559-592.  doi: 10.1017/S002211206700103X.

[3]

J. L. BonaD. Lannes and J.-C. Saut, Asymptotic models for internal waves, J. Math. Pures Appl., 89 (2008), 538-566.  doi: 10.1016/j.matpur.2008.02.003.

[4]

V. V. Bulatov and Y. V. Vladimirov, Fundamental problems of internal gravity waves dynamics in ocean, J. Basic Appl. Sci., 9 (2013), 69-81. 

[5]

H. H. Chen and Y. C. Lee, Internal wave solitons of fluids of finite depth, Phys. Rev. Lett., 43 (1979), 264-266.  doi: 10.1103/PhysRevLett.43.264.

[6]

W. Choi and R. Camassa, Fully nonlinear internal waves in a two-fluid system, J. Fluid Mech., 396 (1999), 1–36. doi: 10.1017/S0022112099005820.

[7]

R. ChoudhuryR. I. Ivanov and Y. Liu, Hamiltonian formulation, nonintegrability and local bifurcations for the Ostrovsky equation, Chaos, Solitons and Fractals, 34 (2007), 544-550.  doi: 10.1016/j.chaos.2006.03.057.

[8]

A. Compelli, Hamiltonian approach to the modeling of internal geophysical waves with vorticity, Monatsh. Math., 179 (2016), 509-521.  doi: 10.1007/s00605-014-0724-1.

[9]

A. Compelli and R. Ivanov, The dynamics of flat surface internal geophysical waves with currents, J. Math. Fluid Mech., 19 (2017), 329-344.  doi: 10.1007/s00021-016-0283-4.

[10]

A. Compelli and R. Ivanov, Benjamin-Ono model of an internal wave under a flat surface, Discrete Contin. Dyn. Syst., 39 (2019), 4519-4532.  doi: 10.3934/dcds.2019185.

[11]

A. Constantin and R. Ivanov, A Hamiltonian approach to wave-current interactions in two-layer fluids, Phys. Fluids, 27 (2015), 086603.  doi: 10.1063/1.4929457.

[12]

A. Constantin and R. Ivanov, Equatorial wave-current interactions, Commun. Math. Phys., 370 (2019), 1C-48. doi: 10.1007/s00220-019-03483-8.

[13]

A. ConstantinR. Ivanov and C. -I. Martin, Hamiltonian formulation for wave-current interactions in stratified rotational flows, Arch. Rational Mech. Anal., 221 (2016), 1417-1447.  doi: 10.1007/s00205-016-0990-2.

[14]

A. ConstantinR. Ivanov and E. Prodanov, Nearly-Hamiltonian structure for water waves with constant vorticity, J. Math. Fluid Mech., 9 (2007), 1-14.  doi: 10.1007/s00021-006-0230-x.

[15]

A. Constantin and R. S. Johnson, On the role of nonlinearity in geostrophic ocean flows on a sphere, in Nonlinear systems and their remarkable mathematical structures (ed. N. Euler), Vol. 1, pp. 500–519, CRC Press, Boca Raton, FL, 2019. doi: 10.1201/9780429470462-18.

[16]

A. Constantin and R. S. Johnson, On the modelling of large-scale atmospheric flow, J. Differ. EqU., 285 (2021), 751-798.  doi: 10.1016/j.jde.2021.03.019.

[17]

A. Constantin and R. S. Johnson, On the propagation of waves in the atmosphere, Proc. Roy. Soc. A, 477 (2021), 25 pp. doi: 10.1098/rspa.2020.0424.

[18]

A. Constantin and R. S. Johnson, An exact, steady, purely azimuthal flow as a model for the Antarctic Circumpolar Current, J. Phys. Oceanogr., 46 (2016), 3585-3594.  doi: 10.1175/JPO-D-16-0121.1.

[19]

W. CraigP. Guyenne and C. Sulem, Coupling between internal and surface waves, Nat. Hazards, 57 (2011), 617-642.  doi: 10.1007/s11069-010-9535-4.

[20]

W. CraigP. Guyenne and H. Kalisch, Hamiltonian long-wave expansions for free surfaces and interfaces, Commun. Pure Appl. Math., 58 (2005), 1587-1641.  doi: 10.1002/cpa.20098.

[21]

J. Cullen and R. Ivanov, On the intermediate long wave propagation for internal waves in the presence of currents, Euro. J. Mech.-B/Fluids, 84 (2020), 325-333.  doi: 10.1016/j.euromechflu.2020.07.001.

[22]

V. D. Djordjevic and L. G. Redekopp, The fission and disintegration of internal solitary waves moving over two-dimensional topography, J. Phys. Ocean., 8 (1978), 1016-1024.  doi: 10.1175/1520-0485(1978)008<1016:TFADOI>2.0.CO;2.

[23]

A. V. Fedorov and J. N. Brown, Equatorial waves, in Encyclopedia of Ocean Sciences (ed. J. Steele), Academic, San Diego, Calif., 2009, 3679–3695. doi: 10.1016/B978-012374473-9.00610-X.

[24]

A. S. Fokas and M. J. Ablowitz, The Inverse Scattering Transform for the Benjamin-Ono equation – a pivot to multidimensional problems, Stud. Appl. Math., 68 (1983), 1-10.  doi: 10.1002/sapm19836811.

[25]

G. G. Grahovski and R. I. Ivanov, Generalised Fourier transform and perturbations to soliton equations, Discrete Contin. Dyn. Syst. Ser. B, 12 (2009), 579-595.  doi: 10.3934/dcdsb.2009.12.579.

[26]

A. R. Giniyatullin, A. A. Kurkin, O. E. and Kurkina and Y. A. Stepanyants, Generalized Korteweg–de Vries equation for internal waves in two-layer fluid, Fundamental and Applied Hydrophysics, 7 (2014), 16–28, (in Russian).

[27]

R. Grimshaw, Evolution equations for weakly nonlinear, long internal waves in a rotating fluid, Stud. Appl. Math., 73 (1985) 1–33. doi: 10.1002/SAPM19857311.

[28]

R. GrimshawL. OstrovskyV. Shrira and Y. Stepanyants, Long nonlinear surface and internal waves in a rotating ocean, Surveys Geophys., 19 (1998), 289-338.  doi: 10.1023/A:1006587919935.

[29]

S. V. Haziot and K. Marynets, Applying the stereographic projection to modeling of the flow of the Antarctic Circumpolar Current, Oceanography, 31 (2018), 68-75.  doi: 10.5670/oceanog.2018.311.

[30]

K. R. Helfrich and W. K. Melville, Long nonlinear internal waves, Annu. Rev. Fluid Mech., 38 (2006), 395-425.  doi: 10.1146/annurev.fluid.38.050304.092129.

[31]

D. Henry, Energy considerations for nonlinear equatorial water waves, Commun. Pure Appl. Anal., to appear.

[32]

R. I. Ivanov, Hamiltonian model for coupled surface and internal waves in the presence of currents, Nonlinear Analysis: Real World Applications, 34 (2017) 316–334. doi: 10.1016/j.nonrwa.2016.09.010.

[33]

R. I. Ivanov, On the Coriolis effect for internal ocean waves, in: Floating Offshore Energy Devices, eds: C. Mc Goldrick et al., Materials Research Proceedings 20 (2022), 20–25. doi: 10.21741/9781644901731-3.

[34]

R. S. Johnson, Camassa-Holm, Korteweg-de Vries and related models for water waves, J. Fluid Mech., 457 (2002), 63-82.  doi: 10.1017/S0022112001007224.

[35] R. S. Johnson, A Modern Introduction to the Mathematical Theory of Water Waves, Cambridge University Press, Cambridge, 1997.  doi: 10.1017/CBO9780511624056.
[36]

R. I. Joseph, Solitary waves in a finite depth fluid, J. Phys. A: Math. Gen., 10 (1977), L225–L227. doi: 10.1088/0305-4470/10/12/002.

[37]

B. B. Kadomtsev and V. I. Petviashvili, On the stability of solitary waves in weakly dispersing media, Dokl. Akad. Nauk SSSR, 192 (1970), 753–756, (in Russian).

[38]

D. J. Kaup, Y. Matsuno, The Inverse Scattering Transform for the Benjamin-Ono Equation, Stud. Appl. Math., 101, (1998), 73–98. doi: 10.1111/1467-9590.00086.

[39]

D. Korteweg and G. de Vries, On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves, Philosophical Magazine, 39 (1895), 422–443; reprint: Philosophical Magazine, 39 (2011), 1007–1028. doi: 10.1080/14786435.2010.547337.

[40]

K. R. KhusnutdinovaY. A. Stepanyants and M. R. Tranter, Soliton solutions to the fifth-order Korteweg–de Vries equation and their applications to surface and internal water waves, Physics of Fluids, 30 (2018), 022104.  doi: 10.1063/1.5009965.

[41]

C. G. Koop and G. Butler, An investigation of internal solitary waves in a two-fluid system, J. Fluid Mech., 112 (1981), 225-251.  doi: 10.1017/S0022112081000372.

[42]

T. KubotaD. R. S. Ko and L. D. Dobbs, Weakly-nonlinear, long internal gravity waves in stratified fluids of finite depth, J. Hydronautics, 12 (1978), 157-165.  doi: 10.2514/3.63127.

[43]

A. I. Leonov, The effect of the Earth's rotation on the propagation of weak nonlinear surface and internal long oceanic waves, Ann. NY Acad. Sci., 373 (1981), 150-159.  doi: 10.1111/j.1749-6632.1981.tb51140.x.

[44]

S. P. Novikov, S. V. Manakov, L. P. Pitaevsky and V. E. Zakharov, Theory of Solitons: The Inverse Scattering Method, New York, Plenum, 1984.

[45]

H. Ono, Algebraic solitary waves in stratified fluids, J. Phys. Soc. Japan, 39 (1975), 1082-1091.  doi: 10.1143/JPSJ.39.1082.

[46]

A. R. Osborne and T. L. Burch, Internal solitons in the Andaman Sea, Science, 208 (1980), 451-460.  doi: 10.1126/science.208.4443.451.

[47]

L. A. Ostrovsky, Nonlinear internal waves in a rotating ocean, Okeanologia, 18 (1978), 181–191, (in Russian).

[48]

J. Pedlosky, Geophysical Fluid Dynamics, Springer-Verlag, New York, 1979. doi: 10.1007/978-1-4612-4650-3.

[49]

V. Varlamov and Y. Liu, Cauchy problem for the Ostrovsky equation, Discrete and Contin. Dyn. Sys., 10 (2004), 731-751.  doi: 10.3934/dcds.2004.10.731.

[50]

V. E. Zakharov, Stability of periodic waves of finite amplitude on the surface of a deep fluid, Zh. Prikl. Mekh. Tekh. Fiz., 9 (1968), 86–94 (in Russian); J. Appl. Mech. Tech. Phys., 9 (1968), 190–194 (English translation). doi: 10.1007/BF00913182.

Figure 1.  System with an internal wave. The fluid domain $ \Omega $ is the fluid of higher density. The pycnocline/thermocline is the interface that separates the two fluid domains $ \Omega $ and $ \Omega_1 $. The function $ \eta(x,t) $ describes the elevation of the internal wave
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