# American Institute of Mathematical Sciences

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

## Shallow water asymptotic models for the propagation of internal waves

 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

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.
Citation: Vincent Duchêne, Samer Israwi, Raafat Talhouk. Shallow water asymptotic models for the propagation of internal waves. Discrete & 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,, , (2012). Google Scholar [2] T. Alazard and J.-M. Delort, Global solutions and asymptotic behavior for two dimensional gravity water waves,, , (2013). Google Scholar [3] B. Alvarez-Samaniego and D. Lannes, Large time existence for 3D water-waves and asymptotics,, Invent. Math., 171 (2008), 485. doi: 10.1007/s00222-007-0088-4. Google Scholar [4] C. T. Anh, Influence of surface tension and bottom topography on internal waves,, Math. Models Methods Appl. Sci., 19 (2009), 2145. doi: 10.1142/S0218202509004078. Google Scholar [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. doi: 10.1007/s00332-002-0466-4. Google Scholar [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. doi: 10.1088/0951-7715/17/3/010. Google Scholar [7] J. L. Bona, T. Colin and D. Lannes, Long wave approximations for water waves,, Arch. Ration. Mech. Anal., 178 (2005), 373. doi: 10.1007/s00205-005-0378-1. Google Scholar [8] J. L. Bona, D. Lannes and J.-C. Saut, Asymptotic models for internal waves,, J. Math. Pures Appl. (9), 89 (2008), 538. doi: 10.1016/j.matpur.2008.02.003. Google Scholar [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. Google Scholar [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. Google Scholar [11] F. Chazel, Influence of bottom topography on long water waves,, M2AN Math. Model. Numer. Anal., 41 (2007), 771. doi: 10.1051/m2an:2007041. Google Scholar [12] F. Chazel, On the Korteweg-de Vries approximation for uneven bottoms,, Eur. J. Mech. B Fluids, 28 (2009), 234. doi: 10.1016/j.euromechflu.2008.10.003. Google Scholar [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. doi: 10.1017/S0022112009006594. Google Scholar [14] W. Choi and R. Camassa, Weakly nonlinear internal waves in a two-fluid system,, J. Fluid Mech., 313 (1996), 83. doi: 10.1017/S0022112096002133. Google Scholar [15] W. Choi and R. Camassa, Fully nonlinear internal waves in a two-fluid system,, J. Fluid Mech., 396 (1999), 1. doi: 10.1017/S0022112099005820. Google Scholar [16] A. Constantin and D. Lannes, The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations,, Arch. Ration. Mech. Anal., 192 (2009), 165. doi: 10.1007/s00205-008-0128-2. Google Scholar [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. doi: 10.1098/rspa.2010.0124. Google Scholar [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. doi: 10.1080/03605308508820396. Google Scholar [19] W. Craig, P. Guyenne and H. Kalisch, Hamiltonian long-wave expansions for free surfaces and interfaces,, Comm. Pure Appl. Math., 58 (2005), 1587. doi: 10.1002/cpa.20098. Google Scholar [20] W. Craig and C. Sulem, Numerical simulation of gravity waves,, J. Comput. Phys., 108 (1993), 73. doi: 10.1006/jcph.1993.1164. Google Scholar [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. doi: 10.1137/090761100. Google Scholar [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. doi: 10.1051/m2an/2011037. Google Scholar [23] V. Duchêne, Decoupled and unidirectional asymptotic models for the propagation of internal waves,, M3AS: Math. Models Methods Appl. Sci, (2013). Google Scholar [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,, , (2013). Google Scholar [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. doi: 10.4007/annals.2012.175.2.6. Google Scholar [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. doi: 10.1017/S0022112076002425. Google Scholar [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. doi: 10.1088/0951-7715/23/2/003. Google Scholar [28] K. R. Helfrich and W. K. Melville, Long nonlinear internal waves,, in, (2006), 395. doi: 10.1146/annurev.fluid.38.050304.092129. Google Scholar [29] A. D. Ionescu and F. Pusateri, Global solutions for the gravity water waves system in 2d,, , (2013). Google Scholar [30] S. Israwi, Derivation and analysis of a new 2D Green-Naghdi system,, Nonlinearity, 23 (2010), 2889. doi: 10.1088/0951-7715/23/11/009. Google Scholar [31] S. Israwi, Variable depth KdV equations and generalizations to more nonlinear regimes,, M2AN Math. Model. Numer. Anal., 44 (2010), 347. doi: 10.1051/m2an/2010005. Google Scholar [32] S. Israwi, Large time existence for 1d Green-Naghdi equations,, Nonlinear Analysis: Theory, 74 (2011), 81. doi: 10.1016/j.na.2010.08.019. Google Scholar [33] R. S. Johnson, Camassa-Holm, Korteweg-de Vries and related models for water waves,, J. Fluid Mech., 455 (2002), 63. doi: 10.1017/S0022112001007224. Google Scholar [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. Google Scholar [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. doi: 10.1080/14786449508620739. Google Scholar [36] D. Lannes, Well-posedness of the water-waves equations,, J. Amer. Math. Soc., 18 (2005), 605. doi: 10.1090/S0894-0347-05-00484-4. Google Scholar [37] D. Lannes, A stability criterion for two-fluid interfaces and applications,, Arch. Ration. Mech. Anal., 208 (2013), 481. doi: 10.1007/s00205-012-0604-6. Google Scholar [38] D. Lannes, "The Water Waves Problem. Mathematical Analysis and Asymptotics,", Mathematical Surveys and Monographs, 188 (2013). Google Scholar [39] R. Liska and L. Margolin and B. Wendroff, Nonhydrostatic two-layer models of incompressible flow,, Comput. Math. Appl., 29 (1995), 25. doi: 10.1016/0898-1221(95)00035-W. Google Scholar [40] Y. Matsuno, A unified theory of nonlinear wave propagation in two-layer fluid systems,, J. Phys. Soc. Japan, 62 (1993), 1902. doi: 10.1143/JPSJ.62.1902. Google Scholar [41] V. I. Nalimov, The Cauchy-Poisson problem,, Dinamika Splošn. Sredy, (1974), 104. Google Scholar [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. Google Scholar [43] J.-C. Saut, Lectures on asymptotic models for internal waves,, in, (2012), 147. Google Scholar [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. doi: 10.1016/j.matpur.2011.09.012. Google Scholar [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. Google Scholar [46] F. Serre, Contribution à l'étude des écoulements permanents et variables dans les canaux,, La Houille Blanche, 6 (1953), 830. Google Scholar [47] S. Wu, Well-posedness in Sobolev spaces of the full water wave problem in $2$-D,, Invent. Math., 130 (1997), 39. doi: 10.1007/s002220050177. Google Scholar [48] S. Wu, Well-posedness in Sobolev spaces of the full water wave problem in 3-D,, J. Amer. Math. Soc., 12 (1999), 445. doi: 10.1090/S0894-0347-99-00290-8. Google Scholar [49] S. Wu, Almost global wellposedness of the 2-D full water wave problem,, Invent. Math., 177 (2009), 45. doi: 10.1007/s00222-009-0176-8. Google Scholar [50] S. Wu, Global wellposedness of the 3-D full water wave problem,, Invent. Math., 184 (2011), 125. doi: 10.1007/s00222-010-0288-1. Google Scholar [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. doi: 10.2977/prims/1195184016. Google Scholar [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. doi: 10.1007/BF00913182. Google Scholar

show all references

##### References:
 [1] T. Alazard, N. Burq and C. Zuily, On the Cauchy problem for gravity water waves,, , (2012). Google Scholar [2] T. Alazard and J.-M. Delort, Global solutions and asymptotic behavior for two dimensional gravity water waves,, , (2013). Google Scholar [3] B. Alvarez-Samaniego and D. Lannes, Large time existence for 3D water-waves and asymptotics,, Invent. Math., 171 (2008), 485. doi: 10.1007/s00222-007-0088-4. Google Scholar [4] C. T. Anh, Influence of surface tension and bottom topography on internal waves,, Math. Models Methods Appl. Sci., 19 (2009), 2145. doi: 10.1142/S0218202509004078. Google Scholar [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. doi: 10.1007/s00332-002-0466-4. Google Scholar [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. doi: 10.1088/0951-7715/17/3/010. Google Scholar [7] J. L. Bona, T. Colin and D. Lannes, Long wave approximations for water waves,, Arch. Ration. Mech. Anal., 178 (2005), 373. doi: 10.1007/s00205-005-0378-1. Google Scholar [8] J. L. Bona, D. Lannes and J.-C. Saut, Asymptotic models for internal waves,, J. Math. Pures Appl. (9), 89 (2008), 538. doi: 10.1016/j.matpur.2008.02.003. Google Scholar [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. Google Scholar [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. Google Scholar [11] F. Chazel, Influence of bottom topography on long water waves,, M2AN Math. Model. Numer. Anal., 41 (2007), 771. doi: 10.1051/m2an:2007041. Google Scholar [12] F. Chazel, On the Korteweg-de Vries approximation for uneven bottoms,, Eur. J. Mech. B Fluids, 28 (2009), 234. doi: 10.1016/j.euromechflu.2008.10.003. Google Scholar [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. doi: 10.1017/S0022112009006594. Google Scholar [14] W. Choi and R. Camassa, Weakly nonlinear internal waves in a two-fluid system,, J. Fluid Mech., 313 (1996), 83. doi: 10.1017/S0022112096002133. Google Scholar [15] W. Choi and R. Camassa, Fully nonlinear internal waves in a two-fluid system,, J. Fluid Mech., 396 (1999), 1. doi: 10.1017/S0022112099005820. Google Scholar [16] A. Constantin and D. Lannes, The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations,, Arch. Ration. Mech. Anal., 192 (2009), 165. doi: 10.1007/s00205-008-0128-2. Google Scholar [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. doi: 10.1098/rspa.2010.0124. Google Scholar [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. doi: 10.1080/03605308508820396. Google Scholar [19] W. Craig, P. Guyenne and H. Kalisch, Hamiltonian long-wave expansions for free surfaces and interfaces,, Comm. Pure Appl. Math., 58 (2005), 1587. doi: 10.1002/cpa.20098. Google Scholar [20] W. Craig and C. Sulem, Numerical simulation of gravity waves,, J. Comput. Phys., 108 (1993), 73. doi: 10.1006/jcph.1993.1164. Google Scholar [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. doi: 10.1137/090761100. Google Scholar [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. doi: 10.1051/m2an/2011037. Google Scholar [23] V. Duchêne, Decoupled and unidirectional asymptotic models for the propagation of internal waves,, M3AS: Math. Models Methods Appl. Sci, (2013). Google Scholar [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,, , (2013). Google Scholar [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. doi: 10.4007/annals.2012.175.2.6. Google Scholar [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. doi: 10.1017/S0022112076002425. Google Scholar [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. doi: 10.1088/0951-7715/23/2/003. Google Scholar [28] K. R. Helfrich and W. K. Melville, Long nonlinear internal waves,, in, (2006), 395. doi: 10.1146/annurev.fluid.38.050304.092129. Google Scholar [29] A. D. Ionescu and F. Pusateri, Global solutions for the gravity water waves system in 2d,, , (2013). Google Scholar [30] S. Israwi, Derivation and analysis of a new 2D Green-Naghdi system,, Nonlinearity, 23 (2010), 2889. doi: 10.1088/0951-7715/23/11/009. Google Scholar [31] S. Israwi, Variable depth KdV equations and generalizations to more nonlinear regimes,, M2AN Math. Model. Numer. Anal., 44 (2010), 347. doi: 10.1051/m2an/2010005. Google Scholar [32] S. Israwi, Large time existence for 1d Green-Naghdi equations,, Nonlinear Analysis: Theory, 74 (2011), 81. doi: 10.1016/j.na.2010.08.019. Google Scholar [33] R. S. Johnson, Camassa-Holm, Korteweg-de Vries and related models for water waves,, J. Fluid Mech., 455 (2002), 63. doi: 10.1017/S0022112001007224. Google Scholar [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. Google Scholar [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. doi: 10.1080/14786449508620739. Google Scholar [36] D. Lannes, Well-posedness of the water-waves equations,, J. Amer. Math. Soc., 18 (2005), 605. doi: 10.1090/S0894-0347-05-00484-4. Google Scholar [37] D. Lannes, A stability criterion for two-fluid interfaces and applications,, Arch. Ration. Mech. Anal., 208 (2013), 481. doi: 10.1007/s00205-012-0604-6. Google Scholar [38] D. Lannes, "The Water Waves Problem. Mathematical Analysis and Asymptotics,", Mathematical Surveys and Monographs, 188 (2013). Google Scholar [39] R. Liska and L. Margolin and B. Wendroff, Nonhydrostatic two-layer models of incompressible flow,, Comput. Math. Appl., 29 (1995), 25. doi: 10.1016/0898-1221(95)00035-W. Google Scholar [40] Y. Matsuno, A unified theory of nonlinear wave propagation in two-layer fluid systems,, J. Phys. Soc. Japan, 62 (1993), 1902. doi: 10.1143/JPSJ.62.1902. Google Scholar [41] V. I. Nalimov, The Cauchy-Poisson problem,, Dinamika Splošn. Sredy, (1974), 104. Google Scholar [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. Google Scholar [43] J.-C. Saut, Lectures on asymptotic models for internal waves,, in, (2012), 147. Google Scholar [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. doi: 10.1016/j.matpur.2011.09.012. Google Scholar [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. Google Scholar [46] F. Serre, Contribution à l'étude des écoulements permanents et variables dans les canaux,, La Houille Blanche, 6 (1953), 830. Google Scholar [47] S. Wu, Well-posedness in Sobolev spaces of the full water wave problem in $2$-D,, Invent. Math., 130 (1997), 39. doi: 10.1007/s002220050177. Google Scholar [48] S. Wu, Well-posedness in Sobolev spaces of the full water wave problem in 3-D,, J. Amer. Math. Soc., 12 (1999), 445. doi: 10.1090/S0894-0347-99-00290-8. Google Scholar [49] S. Wu, Almost global wellposedness of the 2-D full water wave problem,, Invent. Math., 177 (2009), 45. doi: 10.1007/s00222-009-0176-8. Google Scholar [50] S. Wu, Global wellposedness of the 3-D full water wave problem,, Invent. Math., 184 (2011), 125. doi: 10.1007/s00222-010-0288-1. Google Scholar [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. doi: 10.2977/prims/1195184016. Google Scholar [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. doi: 10.1007/BF00913182. Google Scholar
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