Shallow water asymptotic models for the propagation of internal waves

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April 2014, 7(2): 239-269. doi: 10.3934/dcdss.2014.7.239

Shallow water asymptotic models for the propagation of internal waves

Vincent Duchêne ^1, , Samer Israwi ^2, and Raafat Talhouk ^3,

1.	IRMAR - UMR6625, Univ. Rennes 1, CNRS, Campus de Beaulieu, F-35042 Rennes cedex, France
2.	Laboratory of Mathematics-EDST and Faculty of Sciences I, Lebanese University, Center for Research in Applied Mathematics and Statistics, Arts Sciences and Technology University in Lebanon (AUL), 113-7504 Beirut, Lebanon
3.	Laboratory of Mathematics-EDST and Faculty of Sciences I, Lebanese University, Beirut, Lebanon

Received May 2013 Revised July 2013 Published September 2013

Abstract
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We are interested in asymptotic models for the propagation of internal waves at the interface between two shallow layers of immiscible fluid, under the rigid-lid assumption. We review and complete existing works in the literature, in order to offer a unified and comprehensive exposition. Anterior models such as the shallow water and Boussinesq systems, as well as unidirectional models of Camassa-Holm type, are shown to descend from a broad Green-Naghdi model, that we introduce and justify in the sense of consistency. Contrarily to earlier works, our Green-Naghdi model allows a non-flat topography, and horizontal dimension $d=2$. Its derivation follows directly from classical results concerning the one-layer case, and we believe such strategy may be used to construct interesting models in different regimes than the shallow-water/shallow-water studied in the present work.

Keywords: long waves, Internal gravity waves, shallow water, intermediate waves, asymptotic models..

Mathematics Subject Classification: 76B55, 35C20, 35Q3.

Citation: Vincent Duchêne, Samer Israwi, Raafat Talhouk. Shallow water asymptotic models for the propagation of internal waves. Discrete and Continuous Dynamical Systems - S, 2014, 7 (2) : 239-269. doi: 10.3934/dcdss.2014.7.239

References:

[1]	T. Alazard, N. Burq and C. Zuily, On the Cauchy problem for gravity water waves, arXiv:1212.0626, (2012).
[2]	T. Alazard and J.-M. Delort, Global solutions and asymptotic behavior for two dimensional gravity water waves, arXiv:1305.4090, (2013).
[3]	B. Alvarez-Samaniego and D. Lannes, Large time existence for 3D water-waves and asymptotics, Invent. Math., 171 (2008), 485-541. doi: 10.1007/s00222-007-0088-4.
[4]	C. T. Anh, Influence of surface tension and bottom topography on internal waves, Math. Models Methods Appl. Sci., 19 (2009), 2145-2175. doi: 10.1142/S0218202509004078.
[5]	J. L. Bona, M. Chen and J.-C. Saut, Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media. I. Derivation and linear theory, J. Nonlinear Sci., 12 (2002), 283-318. doi: 10.1007/s00332-002-0466-4.
[6]	J. L. Bona, M. Chen and J.-C. Saut, Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media. {II. The nonlinear theory,} Nonlinearity, 17 (2004), 925-952. doi: 10.1088/0951-7715/17/3/010.
[7]	J. L. Bona, T. Colin and D. Lannes, Long wave approximations for water waves, Arch. Ration. Mech. Anal., 178 (2005), 373-410. doi: 10.1007/s00205-005-0378-1.
[8]	J. L. Bona, D. Lannes and J.-C. Saut, Asymptotic models for internal waves, J. Math. Pures Appl. (9), 89 (2008), 538-566. doi: 10.1016/j.matpur.2008.02.003.
[9]	J. Boussinesq, Théorie de l'intumescence liquide appelée onde solitaire ou de translation se propageant dans un canal rectangulaire, C. R. Acad. Sci. Paris Sér. A-B, 72 (1871), 755-759.
[10]	J. Boussinesq, Théorie des ondes et des remous qui se propagent le long d'un canal rectangulaire horizontal, en communiquant au liquide contenu dans ce canal des vitesses sensiblement pareilles de la surface au fond, J. Math. Pures Appl., 17 (1872), 55-108.
[11]	F. Chazel, Influence of bottom topography on long water waves, M2AN Math. Model. Numer. Anal., 41 (2007), 771-799. doi: 10.1051/m2an:2007041.
[12]	F. Chazel, On the Korteweg-de Vries approximation for uneven bottoms, Eur. J. Mech. B Fluids, 28 (2009), 234-252. doi: 10.1016/j.euromechflu.2008.10.003.
[13]	W. Choi and R. Barros and T.-C. Jo, A regularized model for strongly nonlinear internal solitary waves, J. Fluid Mech., 629 (2009), 73-85. doi: 10.1017/S0022112009006594.
[14]	W. Choi and R. Camassa, Weakly nonlinear internal waves in a two-fluid system, J. Fluid Mech., 313 (1996), 83-103. doi: 10.1017/S0022112096002133.
[15]	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.
[16]	A. Constantin and D. Lannes, The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations, Arch. Ration. Mech. Anal., 192 (2009), 165-186. doi: 10.1007/s00205-008-0128-2.
[17]	C. J. Cotter and D. D. Holm and J. R. Percival, The square root depth wave equations, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 466 (2010), 3621-3633. doi: 10.1098/rspa.2010.0124.
[18]	W. Craig, An existence theory for water waves and the Boussinesq and Korteweg-de Vries scaling limits, Comm. Partial Differential Equations, 10 (1985), 787-1003. doi: 10.1080/03605308508820396.
[19]	W. Craig, P. Guyenne and H. Kalisch, Hamiltonian long-wave expansions for free surfaces and interfaces, Comm. Pure Appl. Math., 58 (2005), 1587-1641. doi: 10.1002/cpa.20098.
[20]	W. Craig and C. Sulem, Numerical simulation of gravity waves, J. Comput. Phys., 108 (1993), 73-83. doi: 10.1006/jcph.1993.1164.
[21]	V. Duchêne, Asymptotic shallow water models for internal waves in a two-fluid system with a free surface, SIAM J. Math. Anal., 42 (2010), 2229-2260. doi: 10.1137/090761100.
[22]	V. Duchêne, Boussinesq/Boussinesq systems for internal waves with a free surface, and the KdV approximation, M2AN Math. Model. Numer. Anal., 46 (2011), 145-185. doi: 10.1051/m2an/2011037.
[23]	V. Duchêne, Decoupled and unidirectional asymptotic models for the propagation of internal waves, M3AS: Math. Models Methods Appl. Sci, 2013.
[24]	V. Duchene, S. Israwi and R. Talhouk, A new Green-Naghdi model in the Camassa-Holm regime and full justification of asymptotic models for the propagation of internal waves, arXiv:1304.4554, (2013).
[25]	P. Germain, N. Masmoudi and J. Shatah, Global solutions for the gravity water waves equation in dimension 3, Ann. of Math. (2), 175 (2012), 691-754. doi: 10.4007/annals.2012.175.2.6.
[26]	A. E. Green and P. M. Naghdi, A derivation of equations for wave propagation in water of variable depth, J. Fluid Mech., 78 (1976), 237-246. doi: 10.1017/S0022112076002425.
[27]	P. Guyenne, D. Lannes and J.-C. Saut, Well-posedness of the Cauchy problem for models of large amplitude internal waves, Nonlinearity, 23 (2010), 237-275. doi: 10.1088/0951-7715/23/2/003.
[28]	K. R. Helfrich and W. K. Melville, Long nonlinear internal waves, in "Annual Review of Fluid Mechanics. Vol. 38," Annual Reviews, Palo Alto, CA, (2006), 395-425. doi: 10.1146/annurev.fluid.38.050304.092129.
[29]	A. D. Ionescu and F. Pusateri, Global solutions for the gravity water waves system in 2d, arXiv:1303.5357, (2013).
[30]	S. Israwi, Derivation and analysis of a new 2D Green-Naghdi system, Nonlinearity, 23 (2010), 2889-2904. doi: 10.1088/0951-7715/23/11/009.
[31]	S. Israwi, Variable depth KdV equations and generalizations to more nonlinear regimes, M2AN Math. Model. Numer. Anal., 44 (2010), 347-370. doi: 10.1051/m2an/2010005.
[32]	S. Israwi, Large time existence for 1d Green-Naghdi equations, Nonlinear Analysis: Theory, Methods & Applications, 74 (2011), 81-93. doi: 10.1016/j.na.2010.08.019.
[33]	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.
[34]	T. Kano and T. Nishida, A mathematical justification for Korteweg-de Vries equation and Boussinesq equation of water surface waves, Osaka J. Math., 23 (1986), 389-413.
[35]	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, Philos. Mag., 5 (1895), 422-443. doi: 10.1080/14786449508620739.
[36]	D. Lannes, Well-posedness of the water-waves equations, J. Amer. Math. Soc., 18 (2005), 605-654. doi: 10.1090/S0894-0347-05-00484-4.
[37]	D. Lannes, A stability criterion for two-fluid interfaces and applications, Arch. Ration. Mech. Anal., 208 (2013), 481-567. doi: 10.1007/s00205-012-0604-6.
[38]	D. Lannes, "The Water Waves Problem. Mathematical Analysis and Asymptotics," Mathematical Surveys and Monographs, 188, American Mathematical Society, Providence, RI, 2013.
[39]	R. Liska and L. Margolin and B. Wendroff, Nonhydrostatic two-layer models of incompressible flow, Comput. Math. Appl., 29 (1995), 25-37. doi: 10.1016/0898-1221(95)00035-W.
[40]	Y. Matsuno, A unified theory of nonlinear wave propagation in two-layer fluid systems, J. Phys. Soc. Japan, 62 (1993), 1902-1916. doi: 10.1143/JPSJ.62.1902.
[41]	V. I. Nalimov, The Cauchy-Poisson problem, Dinamika Splošn. Sredy, Vyp. 18 Dinamika Židkost. so Svobod. Granicami, (1974), 104-210, 254.
[42]	B. de Saint-Venant, Théorie du mouvement non-permanent des eaux, avec application aux crues des rivières et à l'introduction des marées dans leur lit, C. R. Acad. Sci. Paris, 73 (1871), 147-154.
[43]	J.-C. Saut, Lectures on asymptotic models for internal waves, in "Lectures on the Analysis of Nonlinear Partial Differential Equations," Vol. 2, MLM 2, (2012), 147-201.
[44]	J.-C. Saut and L. Xu, The Cauchy problem on large time for surface waves Boussinesq systems, J. Math. Pures Appl. (9), 97 (2012), 635-662. doi: 10.1016/j.matpur.2011.09.012.
[45]	G. Schneider and C. E. Wayne, The long-wave limit for the water wave problem. I. The case of zero surface tension, Comm. Pure Appl. Math., 53 (2000), 1475-1535.
[46]	F. Serre, Contribution à l'étude des écoulements permanents et variables dans les canaux, La Houille Blanche, 6 (1953), 830-872.
[47]	S. Wu, Well-posedness in Sobolev spaces of the full water wave problem in $2$-D, Invent. Math., 130 (1997), 39-72. doi: 10.1007/s002220050177.
[48]	S. Wu, Well-posedness in Sobolev spaces of the full water wave problem in 3-D, J. Amer. Math. Soc., 12 (1999), 445-495. doi: 10.1090/S0894-0347-99-00290-8.
[49]	S. Wu, Almost global wellposedness of the 2-D full water wave problem, Invent. Math., 177 (2009), 45-135. doi: 10.1007/s00222-009-0176-8.
[50]	S. Wu, Global wellposedness of the 3-D full water wave problem, Invent. Math., 184 (2011), 125-220. doi: 10.1007/s00222-010-0288-1.
[51]	H. Yosihara, Gravity waves on the free surface of an incompressible perfect fluid of finite depth, Publ. Res. Inst. Math. Sci., 18 (1982), 49-96. doi: 10.2977/prims/1195184016.
[52]	V. E. Zakharov, Stability of periodic waves of finite amplitude on the surface of a deep fluid, J. Appl. Mech. Tech. Phys., 9 (1968), 190-194. doi: 10.1007/BF00913182.

show all references

References:

[1]	T. Alazard, N. Burq and C. Zuily, On the Cauchy problem for gravity water waves, arXiv:1212.0626, (2012).
[2]	T. Alazard and J.-M. Delort, Global solutions and asymptotic behavior for two dimensional gravity water waves, arXiv:1305.4090, (2013).
[3]	B. Alvarez-Samaniego and D. Lannes, Large time existence for 3D water-waves and asymptotics, Invent. Math., 171 (2008), 485-541. doi: 10.1007/s00222-007-0088-4.
[4]	C. T. Anh, Influence of surface tension and bottom topography on internal waves, Math. Models Methods Appl. Sci., 19 (2009), 2145-2175. doi: 10.1142/S0218202509004078.
[5]	J. L. Bona, M. Chen and J.-C. Saut, Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media. I. Derivation and linear theory, J. Nonlinear Sci., 12 (2002), 283-318. doi: 10.1007/s00332-002-0466-4.
[6]	J. L. Bona, M. Chen and J.-C. Saut, Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media. {II. The nonlinear theory,} Nonlinearity, 17 (2004), 925-952. doi: 10.1088/0951-7715/17/3/010.
[7]	J. L. Bona, T. Colin and D. Lannes, Long wave approximations for water waves, Arch. Ration. Mech. Anal., 178 (2005), 373-410. doi: 10.1007/s00205-005-0378-1.
[8]	J. L. Bona, D. Lannes and J.-C. Saut, Asymptotic models for internal waves, J. Math. Pures Appl. (9), 89 (2008), 538-566. doi: 10.1016/j.matpur.2008.02.003.
[9]	J. Boussinesq, Théorie de l'intumescence liquide appelée onde solitaire ou de translation se propageant dans un canal rectangulaire, C. R. Acad. Sci. Paris Sér. A-B, 72 (1871), 755-759.
[10]	J. Boussinesq, Théorie des ondes et des remous qui se propagent le long d'un canal rectangulaire horizontal, en communiquant au liquide contenu dans ce canal des vitesses sensiblement pareilles de la surface au fond, J. Math. Pures Appl., 17 (1872), 55-108.
[11]	F. Chazel, Influence of bottom topography on long water waves, M2AN Math. Model. Numer. Anal., 41 (2007), 771-799. doi: 10.1051/m2an:2007041.
[12]	F. Chazel, On the Korteweg-de Vries approximation for uneven bottoms, Eur. J. Mech. B Fluids, 28 (2009), 234-252. doi: 10.1016/j.euromechflu.2008.10.003.
[13]	W. Choi and R. Barros and T.-C. Jo, A regularized model for strongly nonlinear internal solitary waves, J. Fluid Mech., 629 (2009), 73-85. doi: 10.1017/S0022112009006594.
[14]	W. Choi and R. Camassa, Weakly nonlinear internal waves in a two-fluid system, J. Fluid Mech., 313 (1996), 83-103. doi: 10.1017/S0022112096002133.
[15]	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.
[16]	A. Constantin and D. Lannes, The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations, Arch. Ration. Mech. Anal., 192 (2009), 165-186. doi: 10.1007/s00205-008-0128-2.
[17]	C. J. Cotter and D. D. Holm and J. R. Percival, The square root depth wave equations, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 466 (2010), 3621-3633. doi: 10.1098/rspa.2010.0124.
[18]	W. Craig, An existence theory for water waves and the Boussinesq and Korteweg-de Vries scaling limits, Comm. Partial Differential Equations, 10 (1985), 787-1003. doi: 10.1080/03605308508820396.
[19]	W. Craig, P. Guyenne and H. Kalisch, Hamiltonian long-wave expansions for free surfaces and interfaces, Comm. Pure Appl. Math., 58 (2005), 1587-1641. doi: 10.1002/cpa.20098.
[20]	W. Craig and C. Sulem, Numerical simulation of gravity waves, J. Comput. Phys., 108 (1993), 73-83. doi: 10.1006/jcph.1993.1164.
[21]	V. Duchêne, Asymptotic shallow water models for internal waves in a two-fluid system with a free surface, SIAM J. Math. Anal., 42 (2010), 2229-2260. doi: 10.1137/090761100.
[22]	V. Duchêne, Boussinesq/Boussinesq systems for internal waves with a free surface, and the KdV approximation, M2AN Math. Model. Numer. Anal., 46 (2011), 145-185. doi: 10.1051/m2an/2011037.
[23]	V. Duchêne, Decoupled and unidirectional asymptotic models for the propagation of internal waves, M3AS: Math. Models Methods Appl. Sci, 2013.
[24]	V. Duchene, S. Israwi and R. Talhouk, A new Green-Naghdi model in the Camassa-Holm regime and full justification of asymptotic models for the propagation of internal waves, arXiv:1304.4554, (2013).
[25]	P. Germain, N. Masmoudi and J. Shatah, Global solutions for the gravity water waves equation in dimension 3, Ann. of Math. (2), 175 (2012), 691-754. doi: 10.4007/annals.2012.175.2.6.
[26]	A. E. Green and P. M. Naghdi, A derivation of equations for wave propagation in water of variable depth, J. Fluid Mech., 78 (1976), 237-246. doi: 10.1017/S0022112076002425.
[27]	P. Guyenne, D. Lannes and J.-C. Saut, Well-posedness of the Cauchy problem for models of large amplitude internal waves, Nonlinearity, 23 (2010), 237-275. doi: 10.1088/0951-7715/23/2/003.
[28]	K. R. Helfrich and W. K. Melville, Long nonlinear internal waves, in "Annual Review of Fluid Mechanics. Vol. 38," Annual Reviews, Palo Alto, CA, (2006), 395-425. doi: 10.1146/annurev.fluid.38.050304.092129.
[29]	A. D. Ionescu and F. Pusateri, Global solutions for the gravity water waves system in 2d, arXiv:1303.5357, (2013).
[30]	S. Israwi, Derivation and analysis of a new 2D Green-Naghdi system, Nonlinearity, 23 (2010), 2889-2904. doi: 10.1088/0951-7715/23/11/009.
[31]	S. Israwi, Variable depth KdV equations and generalizations to more nonlinear regimes, M2AN Math. Model. Numer. Anal., 44 (2010), 347-370. doi: 10.1051/m2an/2010005.
[32]	S. Israwi, Large time existence for 1d Green-Naghdi equations, Nonlinear Analysis: Theory, Methods & Applications, 74 (2011), 81-93. doi: 10.1016/j.na.2010.08.019.
[33]	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.
[34]	T. Kano and T. Nishida, A mathematical justification for Korteweg-de Vries equation and Boussinesq equation of water surface waves, Osaka J. Math., 23 (1986), 389-413.
[35]	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, Philos. Mag., 5 (1895), 422-443. doi: 10.1080/14786449508620739.
[36]	D. Lannes, Well-posedness of the water-waves equations, J. Amer. Math. Soc., 18 (2005), 605-654. doi: 10.1090/S0894-0347-05-00484-4.
[37]	D. Lannes, A stability criterion for two-fluid interfaces and applications, Arch. Ration. Mech. Anal., 208 (2013), 481-567. doi: 10.1007/s00205-012-0604-6.
[38]	D. Lannes, "The Water Waves Problem. Mathematical Analysis and Asymptotics," Mathematical Surveys and Monographs, 188, American Mathematical Society, Providence, RI, 2013.
[39]	R. Liska and L. Margolin and B. Wendroff, Nonhydrostatic two-layer models of incompressible flow, Comput. Math. Appl., 29 (1995), 25-37. doi: 10.1016/0898-1221(95)00035-W.
[40]	Y. Matsuno, A unified theory of nonlinear wave propagation in two-layer fluid systems, J. Phys. Soc. Japan, 62 (1993), 1902-1916. doi: 10.1143/JPSJ.62.1902.
[41]	V. I. Nalimov, The Cauchy-Poisson problem, Dinamika Splošn. Sredy, Vyp. 18 Dinamika Židkost. so Svobod. Granicami, (1974), 104-210, 254.
[42]	B. de Saint-Venant, Théorie du mouvement non-permanent des eaux, avec application aux crues des rivières et à l'introduction des marées dans leur lit, C. R. Acad. Sci. Paris, 73 (1871), 147-154.
[43]	J.-C. Saut, Lectures on asymptotic models for internal waves, in "Lectures on the Analysis of Nonlinear Partial Differential Equations," Vol. 2, MLM 2, (2012), 147-201.
[44]	J.-C. Saut and L. Xu, The Cauchy problem on large time for surface waves Boussinesq systems, J. Math. Pures Appl. (9), 97 (2012), 635-662. doi: 10.1016/j.matpur.2011.09.012.
[45]	G. Schneider and C. E. Wayne, The long-wave limit for the water wave problem. I. The case of zero surface tension, Comm. Pure Appl. Math., 53 (2000), 1475-1535.
[46]	F. Serre, Contribution à l'étude des écoulements permanents et variables dans les canaux, La Houille Blanche, 6 (1953), 830-872.
[47]	S. Wu, Well-posedness in Sobolev spaces of the full water wave problem in $2$-D, Invent. Math., 130 (1997), 39-72. doi: 10.1007/s002220050177.
[48]	S. Wu, Well-posedness in Sobolev spaces of the full water wave problem in 3-D, J. Amer. Math. Soc., 12 (1999), 445-495. doi: 10.1090/S0894-0347-99-00290-8.
[49]	S. Wu, Almost global wellposedness of the 2-D full water wave problem, Invent. Math., 177 (2009), 45-135. doi: 10.1007/s00222-009-0176-8.
[50]	S. Wu, Global wellposedness of the 3-D full water wave problem, Invent. Math., 184 (2011), 125-220. doi: 10.1007/s00222-010-0288-1.
[51]	H. Yosihara, Gravity waves on the free surface of an incompressible perfect fluid of finite depth, Publ. Res. Inst. Math. Sci., 18 (1982), 49-96. doi: 10.2977/prims/1195184016.
[52]	V. E. Zakharov, Stability of periodic waves of finite amplitude on the surface of a deep fluid, J. Appl. Mech. Tech. Phys., 9 (1968), 190-194. doi: 10.1007/BF00913182.

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