Water waves over a rough bottom in the shallow water regime
Craig, Walter ; Lannes, David ; Sulem, Catherine
Annales de l'I.H.P. Analyse non linéaire, Tome 29 (2012), p. 233-259 / Harvested from Numdam
Access to full text
Full (PDF)
Full (DJVU)

This is a study of the Euler equations for free surface water waves in the case of varying bathymetry, considering the problem in the shallow water scaling regime. In the case of rapidly varying periodic bottom boundaries this is a problem of homogenization theory. In this setting we derive a new model system of equations, consisting of the classical shallow water equations coupled with nonlocal evolution equations for a periodic corrector term. We also exhibit a new resonance phenomenon between surface waves and a periodic bottom. This resonance, which gives rise to secular growth of surface wave patterns, can be viewed as a nonlinear generalization of the classical Bragg resonance. We justify the derivation of our model with a rigorous mathematical analysis of the scaling limit and the resulting error terms. The principal issue is that the shallow water limit and the homogenization process must be performed simultaneously. Our model equations and the error analysis are valid for both the two- and the three-dimensional physical problems.

Publié le : 2012-01-01
DOI : https://doi.org/10.1016/j.anihpc.2011.10.004
Classification:  76B15,  35Q35 
@article{AIHPC_2012__29_2_233_0,
     author = {Craig, Walter and Lannes, David and Sulem, Catherine},
     title = {Water waves over a rough bottom in the shallow water regime},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     volume = {29},
     year = {2012},
     pages = {233-259},
     doi = {10.1016/j.anihpc.2011.10.004},
     mrnumber = {2901196},
     zbl = {1329.76069},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIHPC_2012__29_2_233_0}
}

Craig, Walter; Lannes, David; Sulem, Catherine. Water waves over a rough bottom in the shallow water regime. Annales de l'I.H.P. Analyse non linéaire, Tome 29 (2012) pp. 233-259. doi : 10.1016/j.anihpc.2011.10.004. http://gdmltest.u-ga.fr/item/AIHPC_2012__29_2_233_0/

[1] G. Allaire, Homogenization and two-scale convergence, SIAM J. Math. Anal. 23 (1992), 1482-1518 | MR 1185639 | Zbl 0770.35005

[2] B. Alvarez-Samaniego, D. Lannes, Large time existence for 3D water-waves and asymptotics, Invent. Math. 171 (2008), 485-541 | MR 2372806 | Zbl 1131.76012

[3] A. Bensoussan, J.-L. Lions, G. Papanicolaou, Asymptotic Analysis for Periodic Structures, Studies in Mathematics and its Applications vol. 5, North-Holland Publishing Co., Amsterdam, New York (1978)

[4] J.L. Bona, T. Colin, D. Lannes, Long waves approximations for water waves, Arch. Ration. Mech. Anal. 178 (2005), 373-410 | MR 2196497 | Zbl 1108.76012

[5] F. Chazel, Influence of bottom topography on long water waves, ESAIM: M2AN 41 (2007), 771-799 | Numdam | MR 2362914 | Zbl 1144.76005

[6] J. Choi, P. Milewski, Long nonlinear waves in resonance with topography, Stud. Appl. Math. 110 (2003), 21-48 | MR 1946440 | Zbl 1141.76348

[7] L. Chupin, Roughness effect on the Neumann boundary condition, http://hal.archives-ouvertes.fr/hal-00551872 | Zbl 1248.35014

[8] W. Craig, An existence theory for water waves and the Boussinesq and Korteweg–de Vries scaling limits, Comm. Partial Differential Equations 10 no. 8 (1985), 787-1003 | MR 795808 | Zbl 0577.76030

[9] W. Craig, C. Sulem, Numerical simulation of gravity waves, J. Comput. Phys. 108 (1993), 73-83 | MR 1239970 | Zbl 0778.76072

[10] W. Craig, P. Guyenne, D. Nicholls, C. Sulem, Hamiltonian long-wave expansions for water waves over a rough bottom, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 461 (2005), 839-873 | MR 2121939 | Zbl 1145.76325

[11] W. Craig, U. Schanz, C. Sulem, The modulational regime of three-dimensional water waves and the Davey–Stewartson system, Ann. Inst. H. Poincaré Anal. Non Linéaire 14 no. 5 (1997), 615-667 | Numdam | MR 1470784 | Zbl 0892.76008

[12] J. Garnier, R. Kraenkel, A. Nachbin, Optimal Boussinesq model for shallow-water waves interacting with a microstructure, Phys. Rev. E 76 (2007), 046311 | MR 2365630

[13] J. Garnier, J.C. Muñoz Grajales, A. Nachbin, Effective behavior of solitary waves over random topography, Multiscale Model. Simul. 6 (2007), 995-1025 | MR 2368977 | Zbl 1151.76388

[14] G. Grataloup, C.C. Mei, Long waves in shallow water over a random seabed, Phys. Rev. E 68 (2003), 026314

[15] T. Iguchi, A long wave approximation for capillary-gravity waves and an effect of the bottom, Comm. Partial Differential Equations 32 (2007), 37-85 | MR 2304142 | Zbl 1136.35081

[16] F. John, Delayed singularity formation in solutions of nonlinear waves in higher dimensions, Comm. Pure Appl. Math. 29 (1976), 649-682 | MR 435612 | Zbl 0332.35044

[17] T. Kano, T. Nishida, Sur les ondes de surface de lʼeau avec une justification mathématique des équations des ondes en eau peu profonde, J. Math. Kyoto Univ. 19 (1979), 335-370 | MR 545714 | Zbl 0419.76013

[18] D. Lannes, Well-posedness of the water-waves equations, J. Amer. Math. Soc. 18 (2005), 605-654 | MR 2138139 | Zbl 1069.35056

[19] Y. Liu, D.K.P. Yue, On generalized Bragg scattering of surface waves by bottom ripples, J. Fluid Mech. 356 (1998), 297-326 | MR 1608008 | Zbl 0908.76014

[20] C.C. Mei, Resonant reflection of surface waves by bottom ripples, J. Fluid Mech. 152 (1985), 315-335 | Zbl 0588.76022

[21] J. Miles, On Hamiltonʼs principle for surface waves, J. Fluid Mech. 83 (1977), 153-158 | MR 489348 | Zbl 0377.76014

[22] A. Nachbin, K. Sølna, Apparent diffusion due to topographic microstructure in shallow waters, Phys. Fluids 15 (2002), 66-77 | MR 1971793 | Zbl 1185.76271

[23] O. Nakoulima, N. Zahibo, E. Pelinovsky, T. Talipova, A. Kurkin, Solitary wave dynamics in shallow water over periodic topography, Chaos 15 (2005), 037107 | MR 2184883 | Zbl 1144.37391

[24] L.V. Ovsjannikov, To the shallow water theory foundation, Arch. Math. Stos. 26 (1974), 407-422 | MR 347218 | Zbl 0283.76012

[25] L.V. Ovsjannikov, Cauchy problem in a scale of Banach spaces and its application to the shallow water theory justification, Applications of Methods of Functional Analysis to Problems in Mechanics, Lecture Notes in Math. vol. 503, Springer, Berlin (1976), 426-437 | MR 670760

[26] J.H. Pihl, C.C. Mei, M. Hancock, Surface gravity waves over a two-dimensional random seabed, Phys. Rev. E 66 (2002), 016611 | MR 1919755

[27] R. Rosales, G. Papanicolaou, Gravity waves in a channel with a rough bottom, Stud. Appl. Math. 68 (1983), 89-102 | MR 693716 | Zbl 0516.76018

[28] G. Schneider, 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 | MR 1780702 | Zbl 1034.76011

[29] M. Teng, Th.Y.-T. Wu, Nonlinear water waves in channels of arbitrary shape, J. Fluid Mech. 242 (1994), 211-233 | MR 1184522

[30] V.E. Zakharov, Stability of periodic waves of finite amplitude on the surface of a deep fluid, J. Appl. Mech. Tech. Phys. 2 (1968), 190-194