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) no. 2, pp. 233-259.

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.

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

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

[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 | Zbl

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

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

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

[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 | Zbl

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

[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 | Zbl

[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 | EuDML | Numdam | MR | Zbl

[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

[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 | Zbl

[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 | Zbl

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

[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 | Zbl

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

[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 | Zbl

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

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

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

[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 | Zbl

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

[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

[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

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

[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 | Zbl

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

[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