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Benjamin-Ono model of an internal wave under a flat surface
Shallow water models for stratified equatorial flows
1. | Delft University of Technology, Delft Institute of Applied Mathematics, Faculty of EEMCS, Mekelweg 4, 2628 CD Delft, The Netherlands |
2. | KTH Royal Institute of Technology, Department of Mathematics, Lindstedtsvägen 25,100 44 Stockholm, Sweden |
Our aim is to study the effect of a continuous prescribed density variation on the propagation of ocean waves. More precisely, we derive KdV-type shallow water model equations for unidirectional flows along the Equator from the full governing equations by taking into account a prescribed but arbitrary depth-dependent density distribution. In contrast to the case of constant density, we obtain for each fixed water depth a different model equation for the horizontal component of the velocity field. We derive explicit formulas for traveling wave solutions of these model equations and perform a detailed analysis of the effect of a given density distribution on the depth-structure of the corresponding traveling waves.
References:
[1] |
T. B. Benjamin,
The stability of solitary waves, Proc. Roy. Soc. London A, 328 (1972), 153-183.
doi: 10.1098/rspa.1972.0074. |
[2] |
J. L. Bona, P. E. Souganidis and W. Strauss,
Stability and instability of solitary waves of korteweg-de vries type, Proc. Roy. Soc. London A, 411 (1987), 395-412.
doi: 10.1098/rspa.1987.0073. |
[3] |
A. Constantin, Nonlinear Water Waves with Applications to Wave-Current Interactions and Tsunamis, CBMS-NSF Regional Conference Series in Applied Mathematics, SIAM, Philadelphia, 2011.
doi: 10.1137/1.9781611971873. |
[4] |
A. Constantin, On the modelling of equatorial waves, Geophys. Res. Lett., 39 (2012), Art. No. L05602.
doi: 10.1029/2012GL051169. |
[5] |
A. Constantin, An exact solution for equatorially trapped waves, J. Geophys. Res. Oceans, 117 (2012), Art. No. C05029.
doi: 10.1029/2012JC007879. |
[6] |
A. Constantin,
Some three-dimensional nonlinear equatorial flows, J. Phys. Oceanogr, 43 (2013), 165-175.
doi: 10.1175/JPO-D-12-062.1. |
[7] |
A. Constantin,
Some nonlinear, equatorially trapped, nonhydrostatic internal geophysical waves, J. Phys. Oceanogr, 44 (2014), 781-789.
doi: 10.1175/JPO-D-13-0174.1. |
[8] |
A. Constantin and R. I. Ivanov, A hamiltonian approach to wave-current interactions in two-layer fluids, Phys. Fluids., 27 (2015), Art. No. 086603.
doi: 10.1063/1.4929457. |
[9] |
A. Constantin, R. I. Ivanov and C. I. Martin,
Hamiltonian formulation for wave-current interactions in stratified rotational flows, Arch. Ration. Mech. Anal., 221 (2016), 1417-1447.
doi: 10.1007/s00205-016-0990-2. |
[10] |
A. Constantin and R. S. Johnson,
On the non-dimensionalisation, scaling and resulting interpretation of the classical governing equations for water waves, J. Nonlinear Math. Phys., 15 (2008), 58-73.
doi: 10.2991/jnmp.2008.15.s2.5. |
[11] |
A. Constantin and R. S. Johnson,
The dynamics of waves interacting with the Equatorial Undercurrent, Geophys. Astrophys. Fluid Dyn., 109 (2015), 311-358.
doi: 10.1080/03091929.2015.1066785. |
[12] |
A. Constantin and R. S. Johnson,
An exact, steady, purely azimuthal equatorial flow with a free surface, J. Phys. Oceanogr, 46 (2016), 1935-1945.
doi: 10.1175/JPO-D-15-0205.1. |
[13] |
A. Constantin and R. S. Johnson,
Current and future prospects for the application of systematic theoretical methods to the study of problems in physical oceanography, Phys. Lett. A, 380 (2016), 3007-3012.
doi: 10.1016/j.physleta.2016.07.036. |
[14] |
A. Constantin and R. S. Johnson, A nonlinear, three-dimensional model for ocean flows, motivated by some observations of the Pacific Equatorial Undercurrent and thermocline, Phys. Fluids., 29 (2017), 056604. |
[15] |
B. Deconinck and T. Kapitula,
The orbital stability of the cnoidal waves of the Korteweg-de Vries equation, Phys. Lett. A, 374 (2010), 4018-4022.
doi: 10.1016/j.physleta.2010.08.007. |
[16] |
M. W. Dingemans, Water Wave Propagation Over Uneven Bottoms, World Scientific, 1997. |
[17] |
P. G. Drazin and R. S. Johnson, Solitons: An Introduction, Cambridge University Press, Cambridge, UK, 1989.
doi: 10.1017/CBO9781139172059. |
[18] |
L. D. Faddeev and V. E. Zakharov, Kortweg-de Vries equation: A completely integrable Hamiltonian system, (Russian) Funkcional. Anal. i Prilovien., 5 (1971), 18–27. |
[19] |
A. V. Fedorov and J. N. Brown, Equatorial waves, Encyclopedia of Ocean Sciences, edited by J. Steele, 3679–3695, Academic Press: New York (2009). |
[20] |
C. S. Gardner, J. M. Greene, M. D. Kruskal and R. M. Miura,
Method for solving the Korteweg-de Vries equation, Phys. Rev. Lett., 19 (1967), 1095-1097.
|
[21] |
A. Geyer and R. Quirchmayr, Shallow water equations for equatorial tsunami waves, Phil. Trans. R. Soc. A, 376 (2018), 20170100, 12pp.
doi: 10.1098/rsta.2017.0100. |
[22] |
D. Ionescu-Kruse and C. I. Martin,
Periodic equatorial water flows from a Hamiltonian perspective, J. Differ. Equ., 262 (2017), 4451-4474.
doi: 10.1016/j.jde.2017.01.001. |
[23] |
R. I. Ivanov,
Hamiltonian model for coupled surface and internal waves in the presence of currents, Nonlinear Anal. Real World Appl., 34 (2017), 316-334.
doi: 10.1016/j.nonrwa.2016.09.010. |
[24] |
R. S. Johnson, A Modern Introduction to the Mathematical Theory of Water Waves, Cambridge University Press, Cambridge, UK, 1997.
doi: 10.1017/CBO9780511624056.![]() ![]() ![]() |
[25] |
R. S. Johnson,
Camassa-Holm, Korteweg-de Vries and related models for water waves, J. Fluid Mech., 455 (2002), 63-82.
doi: 10.1017/S0022112001007224. |
[26] |
R. S. Johnson,
An ocean undercurrent, a thermocline, a free surface, with waves: a problem in classical fluid mechanics, J. Nonlinear Math. Phys., 22 (2015), 475-493.
doi: 10.1080/14029251.2015.1113042. |
[27] |
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 (2018), 20170092, 19pp.
doi: 10.1098/rsta.2017.0092. |
[28] |
T. Kappeler and J. Pöschel, KdV & KAM, Ergeb. der Math. und ihrer Grenzgeb., Springer, Berlin-Heidelberg-New York, 2003.
doi: 10.1007/978-3-662-08054-2. |
[29] |
D. J. 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, Phil. Mag., 39 (1895), 422-443.
doi: 10.1080/14786449508620739. |
[30] |
P. Lax,
Integrals of nonlinear equations of evolution and solitary waves, Comm. Pure Appl. Math., 21 (1968), 467-490.
doi: 10.1002/cpa.3160210503. |
[31] |
P. H. LeBlond and L. A. Mysak, Waves in the Ocean, Elsevier, Amsterdam, 1978. |
[32] |
C. I. Martin,
Dynamics of the thermocline in the equatorial region of the Pacific ocean, J. Nonlinear Math. Phys., 22 (2015), 516-522.
doi: 10.1080/14029251.2015.1113049. |
[33] |
S. Walsh,
Stratified steady periodic water waves, SIAM J. Math. Anal., 41 (2009), 1054-1105.
doi: 10.1137/080721583. |
show all references
References:
[1] |
T. B. Benjamin,
The stability of solitary waves, Proc. Roy. Soc. London A, 328 (1972), 153-183.
doi: 10.1098/rspa.1972.0074. |
[2] |
J. L. Bona, P. E. Souganidis and W. Strauss,
Stability and instability of solitary waves of korteweg-de vries type, Proc. Roy. Soc. London A, 411 (1987), 395-412.
doi: 10.1098/rspa.1987.0073. |
[3] |
A. Constantin, Nonlinear Water Waves with Applications to Wave-Current Interactions and Tsunamis, CBMS-NSF Regional Conference Series in Applied Mathematics, SIAM, Philadelphia, 2011.
doi: 10.1137/1.9781611971873. |
[4] |
A. Constantin, On the modelling of equatorial waves, Geophys. Res. Lett., 39 (2012), Art. No. L05602.
doi: 10.1029/2012GL051169. |
[5] |
A. Constantin, An exact solution for equatorially trapped waves, J. Geophys. Res. Oceans, 117 (2012), Art. No. C05029.
doi: 10.1029/2012JC007879. |
[6] |
A. Constantin,
Some three-dimensional nonlinear equatorial flows, J. Phys. Oceanogr, 43 (2013), 165-175.
doi: 10.1175/JPO-D-12-062.1. |
[7] |
A. Constantin,
Some nonlinear, equatorially trapped, nonhydrostatic internal geophysical waves, J. Phys. Oceanogr, 44 (2014), 781-789.
doi: 10.1175/JPO-D-13-0174.1. |
[8] |
A. Constantin and R. I. Ivanov, A hamiltonian approach to wave-current interactions in two-layer fluids, Phys. Fluids., 27 (2015), Art. No. 086603.
doi: 10.1063/1.4929457. |
[9] |
A. Constantin, R. I. Ivanov and C. I. Martin,
Hamiltonian formulation for wave-current interactions in stratified rotational flows, Arch. Ration. Mech. Anal., 221 (2016), 1417-1447.
doi: 10.1007/s00205-016-0990-2. |
[10] |
A. Constantin and R. S. Johnson,
On the non-dimensionalisation, scaling and resulting interpretation of the classical governing equations for water waves, J. Nonlinear Math. Phys., 15 (2008), 58-73.
doi: 10.2991/jnmp.2008.15.s2.5. |
[11] |
A. Constantin and R. S. Johnson,
The dynamics of waves interacting with the Equatorial Undercurrent, Geophys. Astrophys. Fluid Dyn., 109 (2015), 311-358.
doi: 10.1080/03091929.2015.1066785. |
[12] |
A. Constantin and R. S. Johnson,
An exact, steady, purely azimuthal equatorial flow with a free surface, J. Phys. Oceanogr, 46 (2016), 1935-1945.
doi: 10.1175/JPO-D-15-0205.1. |
[13] |
A. Constantin and R. S. Johnson,
Current and future prospects for the application of systematic theoretical methods to the study of problems in physical oceanography, Phys. Lett. A, 380 (2016), 3007-3012.
doi: 10.1016/j.physleta.2016.07.036. |
[14] |
A. Constantin and R. S. Johnson, A nonlinear, three-dimensional model for ocean flows, motivated by some observations of the Pacific Equatorial Undercurrent and thermocline, Phys. Fluids., 29 (2017), 056604. |
[15] |
B. Deconinck and T. Kapitula,
The orbital stability of the cnoidal waves of the Korteweg-de Vries equation, Phys. Lett. A, 374 (2010), 4018-4022.
doi: 10.1016/j.physleta.2010.08.007. |
[16] |
M. W. Dingemans, Water Wave Propagation Over Uneven Bottoms, World Scientific, 1997. |
[17] |
P. G. Drazin and R. S. Johnson, Solitons: An Introduction, Cambridge University Press, Cambridge, UK, 1989.
doi: 10.1017/CBO9781139172059. |
[18] |
L. D. Faddeev and V. E. Zakharov, Kortweg-de Vries equation: A completely integrable Hamiltonian system, (Russian) Funkcional. Anal. i Prilovien., 5 (1971), 18–27. |
[19] |
A. V. Fedorov and J. N. Brown, Equatorial waves, Encyclopedia of Ocean Sciences, edited by J. Steele, 3679–3695, Academic Press: New York (2009). |
[20] |
C. S. Gardner, J. M. Greene, M. D. Kruskal and R. M. Miura,
Method for solving the Korteweg-de Vries equation, Phys. Rev. Lett., 19 (1967), 1095-1097.
|
[21] |
A. Geyer and R. Quirchmayr, Shallow water equations for equatorial tsunami waves, Phil. Trans. R. Soc. A, 376 (2018), 20170100, 12pp.
doi: 10.1098/rsta.2017.0100. |
[22] |
D. Ionescu-Kruse and C. I. Martin,
Periodic equatorial water flows from a Hamiltonian perspective, J. Differ. Equ., 262 (2017), 4451-4474.
doi: 10.1016/j.jde.2017.01.001. |
[23] |
R. I. Ivanov,
Hamiltonian model for coupled surface and internal waves in the presence of currents, Nonlinear Anal. Real World Appl., 34 (2017), 316-334.
doi: 10.1016/j.nonrwa.2016.09.010. |
[24] |
R. S. Johnson, A Modern Introduction to the Mathematical Theory of Water Waves, Cambridge University Press, Cambridge, UK, 1997.
doi: 10.1017/CBO9780511624056.![]() ![]() ![]() |
[25] |
R. S. Johnson,
Camassa-Holm, Korteweg-de Vries and related models for water waves, J. Fluid Mech., 455 (2002), 63-82.
doi: 10.1017/S0022112001007224. |
[26] |
R. S. Johnson,
An ocean undercurrent, a thermocline, a free surface, with waves: a problem in classical fluid mechanics, J. Nonlinear Math. Phys., 22 (2015), 475-493.
doi: 10.1080/14029251.2015.1113042. |
[27] |
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 (2018), 20170092, 19pp.
doi: 10.1098/rsta.2017.0092. |
[28] |
T. Kappeler and J. Pöschel, KdV & KAM, Ergeb. der Math. und ihrer Grenzgeb., Springer, Berlin-Heidelberg-New York, 2003.
doi: 10.1007/978-3-662-08054-2. |
[29] |
D. J. 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, Phil. Mag., 39 (1895), 422-443.
doi: 10.1080/14786449508620739. |
[30] |
P. Lax,
Integrals of nonlinear equations of evolution and solitary waves, Comm. Pure Appl. Math., 21 (1968), 467-490.
doi: 10.1002/cpa.3160210503. |
[31] |
P. H. LeBlond and L. A. Mysak, Waves in the Ocean, Elsevier, Amsterdam, 1978. |
[32] |
C. I. Martin,
Dynamics of the thermocline in the equatorial region of the Pacific ocean, J. Nonlinear Math. Phys., 22 (2015), 516-522.
doi: 10.1080/14029251.2015.1113049. |
[33] |
S. Walsh,
Stratified steady periodic water waves, SIAM J. Math. Anal., 41 (2009), 1054-1105.
doi: 10.1137/080721583. |



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