Influence of bottom topography on long water waves

Florent Chazel

ESAIM: Mathematical Modelling and Numerical Analysis (2007)

  • Volume: 41, Issue: 4, page 771-799
  • ISSN: 0764-583X

Abstract

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We focus here on the water waves problem for uneven bottoms in the long-wave regime, on an unbounded two or three-dimensional domain. In order to derive asymptotic models for this problem, we consider two different regimes of bottom topography, one for small variations in amplitude, and one for strong variations. Starting from the Zakharov formulation of this problem, we rigorously compute the asymptotic expansion of the involved Dirichlet-Neumann operator. Then, following the global strategy introduced by Bona et al. [Arch. Rational Mech. Anal.178 (2005) 373–410], we derive new symetric asymptotic models for each regime. The solutions of these systems are proved to give good approximations of solutions of the water waves problem. These results hold for solutions that evanesce at infinity as well as for spatially periodic ones.

How to cite

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Chazel, Florent. "Influence of bottom topography on long water waves." ESAIM: Mathematical Modelling and Numerical Analysis 41.4 (2007): 771-799. <http://eudml.org/doc/250058>.

@article{Chazel2007,
abstract = { We focus here on the water waves problem for uneven bottoms in the long-wave regime, on an unbounded two or three-dimensional domain. In order to derive asymptotic models for this problem, we consider two different regimes of bottom topography, one for small variations in amplitude, and one for strong variations. Starting from the Zakharov formulation of this problem, we rigorously compute the asymptotic expansion of the involved Dirichlet-Neumann operator. Then, following the global strategy introduced by Bona et al. [Arch. Rational Mech. Anal.178 (2005) 373–410], we derive new symetric asymptotic models for each regime. The solutions of these systems are proved to give good approximations of solutions of the water waves problem. These results hold for solutions that evanesce at infinity as well as for spatially periodic ones. },
author = {Chazel, Florent},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Water waves; uneven bottoms; bottom topography; long-wave approximation; asymptotic expansion; hyperbolic systems; Dirichlet-Neumann operator.; Zakharov formulation; Dirichlet-Neumann operator},
language = {eng},
month = {10},
number = {4},
pages = {771-799},
publisher = {EDP Sciences},
title = {Influence of bottom topography on long water waves},
url = {http://eudml.org/doc/250058},
volume = {41},
year = {2007},
}

TY - JOUR
AU - Chazel, Florent
TI - Influence of bottom topography on long water waves
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2007/10//
PB - EDP Sciences
VL - 41
IS - 4
SP - 771
EP - 799
AB - We focus here on the water waves problem for uneven bottoms in the long-wave regime, on an unbounded two or three-dimensional domain. In order to derive asymptotic models for this problem, we consider two different regimes of bottom topography, one for small variations in amplitude, and one for strong variations. Starting from the Zakharov formulation of this problem, we rigorously compute the asymptotic expansion of the involved Dirichlet-Neumann operator. Then, following the global strategy introduced by Bona et al. [Arch. Rational Mech. Anal.178 (2005) 373–410], we derive new symetric asymptotic models for each regime. The solutions of these systems are proved to give good approximations of solutions of the water waves problem. These results hold for solutions that evanesce at infinity as well as for spatially periodic ones.
LA - eng
KW - Water waves; uneven bottoms; bottom topography; long-wave approximation; asymptotic expansion; hyperbolic systems; Dirichlet-Neumann operator.; Zakharov formulation; Dirichlet-Neumann operator
UR - http://eudml.org/doc/250058
ER -

References

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  1. S. Alinhac and P. Gérard, Opérateurs pseudo-différentiels et théorème de Nash-Moser. Savoirs Actuels, InterEditions, Paris, Editions du Centre National de la Recherche Scientifique (CNRS), Meudon (1991) p. 190.  
  2. B. Alvarez-Samaniego and D. Lannes, Large time existence for 3D water-waves and asymptotics. Technical report (), Université Bordeaux I, IMB (2007).  URIhttp://fr.arxiv.org/abs/math/0702015v1
  3. B. Alvarez-Samaniego and D. Lannes, A Nash-Moser theorem for singular evolution equations. Application to the Serre and Green-Naghdi equations. Preprint (), Indiana University Mathematical Journal (2007) (to appear).  URIhttp://arxiv.org/abs/math.AP/0701681v1
  4. T.B. Benjamin, J.L. Bona and J.J. Mahony, Model equations for long waves in nonlinear dispersive systems. Philos. Trans. Roy. Soc. London Ser. A272 (1972) 47–78.  
  5. J.L. Bona and M. Chen, A Boussinesq system for two-way propagation of nonlinear dispersive waves. Physica D116 (2004) 191–224.  
  6. 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.  
  7. J.L. Bona, M. Chen and J.-C. Saut, Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media. II: Nonlinear theory. Nonlinearity17 (2004) 925–952.  
  8. J.L. Bona, T. Colin and D. Lannes, Long waves approximations for water waves. Arch. Rational Mech. Anal.178 (2005) 373–410.  
  9. M.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-B72 (1871) 755–759.  
  10. M. Chen, Equations for bi-directional waves over an uneven bottom. Math. Comput. Simulation62 (2003) 3–9.  
  11. W. Craig, An existence theory for water waves and the Boussinesq and Korteweg-de Vries scaling limits. Comm. Partial Differential Equations10 (1985) 787–1003.  
  12. M.W. Dingemans, Water Wave Propagation over uneven bottoms. Part I: Linear Wave Propagation. Adanced Series on Ocean Engineering13. World Scientific (1997).  
  13. M.W. Dingemans, Water Wave Propagation over uneven bottoms. Part II: Non-linear Wave Propagation. Adanced Series on Ocean Engineering13. World Scientific (1997).  
  14. 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.  
  15. T. Iguchi, A long wave approximation for capillary-gravity waves and an effect of the bottom. Preprint (2005).  
  16. T. Iguchi, A mathematical justification of the forced Korteweg-de Vries equation for capillary-gravity waves. Kyushu J. Math.60 (2006) 267–303.  
  17. J.T. Kirby, Gravity Waves in Water of Finite Depth, Advances in Fluid Mechanics 10, in J.N. Hunt Ed., Computational Mechanics Publications (1997) 55–125.  
  18. D. Lannes, Sur le caractère bien posé des équations d'Euler avec surface libre. Séminaire EDP de l'École Polytechnique (2004), Exposé no. XIV.  
  19. D. Lannes, Well-posedness of the water-waves equations. J. Amer. Math. Soc.18 (2005) 605–654.  
  20. D. Lannes and J.C. Saut, Weakly transverse Boussinesq systems and the Kadomtsev–Petviashvili approximation. Nonlinearity19 (2006) 2853–2875.  
  21. P.A. Madsen, R. Murray and O.R. Sorensen, A new form of the Boussinesq equations with improved linear dispersion characteristics (Part 1). Coastal Eng.15 (1991) 371–388.  
  22. G. Métivier, Small Viscosity and Boundary Layer Methods: Theory, Stability Analysis, and Applications. Modeling and Simulation in Science, Engineering and Technology, Birkhäuser, Boston-Basel-Berlin (2004).  
  23. D.P. Nicholls and F. Reitich, A new approach to analyticity of Dirichlet-Neumann operators. Proc. Royal Soc. Edinburgh Sect. A131 (2001) 1411–1433.  
  24. V.I. Nalimov, The Cauchy-Poisson problem. (Russian) Dinamika Splošn. Sredy Vyp. 18, Dinamika Zidkost. so Svobod. Granicami 254 (1974) 104–210.  
  25. O. Nwogu, Alternative form of Boussinesq equations for nearshore wave propagation. J. Waterw. Port Coastal Eng. ASCE119 (1993) 618–638.  
  26. D.H. Peregrine, Long waves on a beach. J. Fluid Mech.27 (1967) 815–827.  
  27. 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.162 (2002) 247–285.  
  28. G. Wei and J.T. Kirby, A time-dependent numerical code for extended Boussinesq equations. J. Waterw. Port Coastal Ocean Engineering120 (1995) 251–261.  
  29. G. Wei, J.T. Kirby, S.T. Grilli and R. Subramanya, A fully nonlinear Boussinesq model for surface waves. I. Highly nonlinear, unsteady waves. J. Fluid Mechanics294 (1995) 71–92.  
  30. S. Wu, Well-posedness in Sobolev spaces of the full water wave problem in 2-D. Invent. Math.130 (1997) 39–72.  
  31. S. Wu, Well-posedness in Sobolev spaces of the full water wave problem in 3-D. J. Amer. Math. Soc.12 (1999) 445–495.  
  32. 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.  

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