Boussinesq/Boussinesq systems for internal waves with a free surface, and the KdV approximation

Duchêne, Vincent

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 46 (2012) no. 1, pp. 145-185.

We study here some asymptotic models for the propagation of internal and surface waves in a two-fluid system. We focus on the so-called long wave regime for one-dimensional waves, and consider the case of a flat bottom. Following the method presented in [J.L. Bona, T. Colin and D. Lannes, Arch. Ration. Mech. Anal. 178 (2005) 373-410] for the one-layer case, we introduce a new family of symmetric hyperbolic models, that are equivalent to the classical Boussinesq/Boussinesq system displayed in [W. Choi and R. Camassa, J. Fluid Mech. 313 (1996) 83-103]. We study the well-posedness of such systems, and the asymptotic convergence of their solutions towards solutions of the full Euler system. Then, we provide a rigorous justification of the so-called KdV approximation, stating that any bounded solution of the full Euler system can be decomposed into four propagating waves, each of them being well approximated by the solutions of uncoupled Korteweg-de Vries equations. Our method also applies for models with the rigid lid assumption, using the Boussinesq/Boussinesq models introduced in [J.L. Bona, D. Lannes and J.-C. Saut, J. Math. Pures Appl. 89 (2008) 538-566]. Our explicit and simultaneous decomposition allows to study in details the behavior of the flow depending on the depth and density ratios, for both the rigid lid and free surface configurations. In particular, we consider the influence of the rigid lid assumption on the evolution of the interface, and specify its domain of validity. Finally, solutions of the Boussinesq/Boussinesq systems and the KdV approximation are numerically computed, using a Crank-Nicholson scheme with a predictive step inspired from [C. Besse, C. R. Acad. Sci. Paris Sér. I Math. 326 (1998) 1427-1432; C. Besse and C.H. Bruneau, Math. Mod. Methods Appl. Sci. 8 (1998) 1363-1386].

Classification : 76B55, 35Q35, 35L55, 35Q53, 35C07
Mots clés : internal waves, free surface, rigid lid configuration, long waves, Korteweg-de Vries approximation, Boussinesq models

[1] T.B. Benjamin, Internal waves of permanent form in fluids of great depth. J. Fluid Mech. 29 (1967) 559-592. | Zbl

[2] T.B. Benjamin, J.L. Bona and J.J. Mahony, Model equations for long waves in nonlinear dispersive systems. Philos. Trans. R. Soc. Lond. Ser. A 272 (1972) 47-78. | MR | Zbl

[3] C. Besse, Schéma de relaxation pour l'équation de Schrödinger non linéaire et les systèmes de Davey et Stewartson. C. R. Acad. Sci. Paris Sér. I Math. 326 (1998) 1427-1432. | MR | Zbl

[4] C. Besse and C.H. Bruneau, Numerical study of elliptic-hyperbolic Davey-Stewartson system: dromions simulation and blow-up. Math. Mod. Methods Appl. Sci. 8 (1998) 1363-1386. | MR | Zbl

[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. | MR | Zbl

[6] J.L. Bona, T. Colin and D. Lannes, Long wave approximations for water waves. Arch. Ration. Mech. Anal. 178 (2005) 373-410. | MR | Zbl

[7] J.L. Bona, D. Lannes and J.-C. Saut, Asymptotic models for internal waves. J. Math. Pures Appl. 89 (2008) 538-566. | MR | Zbl

[8] 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. | JFM

[9] 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. | JFM | Numdam

[10] F. Chazel, On the Korteweg-de Vries approximation for uneven bottoms. Eur. J. Mech. B Fluids 28 (2009) 234-252. | MR | Zbl

[11] W. Choi and R. Camassa, Weakly nonlinear internal waves in a two-fluid system. J. Fluid Mech. 313 (1996) 83-103. | MR | Zbl

[12] J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Sharp global well-posedness for KdV and modified KdV on $ℝ$ and $𝕋$. J. Amer. Math. Soc. 16 (2003) 705-749 (electronic). | MR | Zbl

[13] W. Craig, An existence theory for water waves and the Boussinesq and Korteweg-de Vries scaling limits. Commun. Partial Differ. Equ. 10 (1985) 787-1003. | MR | Zbl

[14] W. Craig, P. Guyenne and H. Kalisch, Hamiltonian long-wave expansions for free surfaces and interfaces. Comm. Pure Appl. Math. 58 (2005) 1587-1641. | MR | Zbl

[15] V.D. Djordjevic and L.G. Redekopp, The fission and disintegration of internal solitary waves moving over two-dimensional topography. J. Phys. Oceanogr. 8 (1978) 1016-1024.

[16] 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. | MR | Zbl

[17] M. Duruflé and S. Israwi, A numerical study of variable depth KdV equations and generalizations of Camassa-Holm-like equations. Preprint, available at http://hal.archives-ouvertes.fr/hal-00454495/en/. | MR | Zbl

[18] M. Funakoshi and M. Oikawa, Long internal waves of large amplitude in a two-layer fluid. J. Phys. Soc. Japan 55 (1986) 128-144.

[19] R. Grimshaw, E. Pelinovsky and T. Talipova, The modified korteweg-de vries equation in the theory of large-amplitude internal waves. Nonlin. Process. Geophys. 4 (1997) 237-250.

[20] P. Guyenne, Large-amplitude internal solitary waves in a two-fluid model. C. R. Mec. 334 (2006) 341-346. | Zbl

[21] K.R. Helfrich and W.K. Melville, Long nonlinear internal waves, in Annual review of fluid mechanics 38. Palo Alto, CA (2006) 395-425. | MR | Zbl

[22] T. Kakutani and N. Yamasaki, Solitary waves on a two-layer fluid. J. Phys. Soc. Japan 45 (1978) 674-679.

[23] T. Kato and G. Ponce, Commutator estimates and the Euler and Navier-Stokes equations. Comm. Pure Appl. Math. 41 (1988) 891-907. | MR | Zbl

[24] C.E. Kenig, G. Ponce and L. Vega, Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle. Commun. Pure Appl. Math. 46 (1993) 527-620. | MR | Zbl

[25] G.H. Keulegan, Characteristics of internal solitary waves. J. Res. Nat. Bur. Stand 51 (1953) 133-140. | Zbl

[26] C.G. Koop and G. Butler, An investigation of internal solitary waves in a two-fluid system. J. Fluid Mech. 112 (1981) 225-251. | MR | Zbl

[27] 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. | JFM

[28] D. Lannes, Secular growth estimates for hyperbolic systems. J. Diff. Equ. 190 (2003) 466-503. | MR | Zbl

[29] D. Lannes, A stability criterion for two-fluid interfaces and applications. preprint arXiv:1005.4565. | MR | Zbl

[30] C. Leone, H. Segur and J.L. Hammack, Viscous decay of long internal solitary waves. Phys. Fluids 25 (1982) 942-944. | Zbl

[31] R.R. Long, Long waves in a two-fluid system. J. Meteorol. 13 (1956) 70-74. | MR

[32] Y. Matsuno, A unified theory of nonlinear wave propagation in two-layer fluid systems. J. Phys. Soc. Japan 62 (1993) 1902-1916.

[33] H. Michallet and E. Barthélemy, Ultrasonic probes and data processing to study interfacial solitary waves. Exp. Fluids 22 (1997) 380-386.

[34] H. Michallet and E. Barthélemy, Experimental study of interfacial solitary waves. J. Fluid Mech. 366 (1998) 159-177. | Zbl

[35] H. Ono, Algebraic solitary waves in stratified fluids. J. Phys. Soc. Japan 39 (1975) 1082-1091. | MR

[36] L.A. Ostrovsky and Y.A. Stepanyants, Internal solitons in laboratory experiments: comparison with theoretical models. Chaos 15 (2005) 1-28. | MR | Zbl

[37] T. Sakai and L.G. Redekopp, Models for strongly-nonlinear evolution of long internal waves in a two-layer stratification. Nonlin. Process. Geophys. 14 (2007) 31-47.

[38] G. Schneider and C.E. Wayne, The long-wave limit for the water wave problem. I. The case of zero surface tension. Commun. Pure Appl. Math. 53 (2000) 1475-1535. | MR | Zbl

[39] H. Segur and J.L. Hammack, Soliton models of long internal waves. J. Fluid Mech. 118 (1982) 285-304. | MR | Zbl

[40] M.E. Taylor, Partial differential equations, III Nonlinear equations, Applied Mathematical Sciences 117. Springer-Verlag, New York (1997). | MR | Zbl

[41] L.R. Walker, Interfacial solitary waves in a two-fluid medium. Phys. Fluids 16 (1973) 1796-1804.

[42] 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.