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Water waves over a variable bottom: a non-local formulation and conformal mappings

Published online by Cambridge University Press:  24 February 2012

A. S. Fokas  and
A. Nachbin
A. S. Fokas
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge CB3 0WA, UK Research Center of Mathematics of the Academy of Athens, Athens, 11527, Greece
A. Nachbin*
Affiliation:
Instituto Nacional de Matemática Pura e Aplicada, Estrada D. Castorina, 110, Rio de Janeiro, RJ, CEP 22460-320, Brazil
*
†Email address for correspondence: [email protected]
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Abstract

In Ablowitz, Fokas & Musslimani (J. Fluid Mech., vol. 562, 2006, pp. 313–343) a novel formulation was proposed for water waves in three space dimensions. In the flat-bottom case, this formulation consists of the Bernoulli equation, as well as of a non-local equation. The variable-bottom case, which now involves two non-local equations, was outlined but not explored in the above paper. Here, the variable-bottom formulation is addressed in more detail. First, it is shown that in the weakly nonlinear, weakly dispersive regime, the above system of three equations can be reduced to a system of two equations. Second, by combining the novel non-local formulation of the above authors with conformal mappings, it is shown that in the two-dimensional case, it is possible to obtain a system of two equations without any asymptotic approximations. Furthermore, for the weakly nonlinear, weakly dispersive regime, the nonlinear equations are simpler than the equations obtained without conformal mappings, since they contain lower order derivatives for the terms involving the bottom variable.

JFM classification

Waves/Free-surface Flows: Surface gravity waves Geophysical and Geological Flows: Topographic effects
Type
Papers
Information
Journal of Fluid Mechanics , Volume 695 , 25 March 2012 , pp. 288 - 309
Copyright
Copyright © Cambridge University Press 2012

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References

1. Ablowitz, M. J., Fokas, A. S. & Musslimani, Z. H. 2006 On a new non-local formulation of water waves. J. Fluid Mech. 562, 313–343.CrossRefGoogle Scholar
2. Artiles, W. & Nachbin, A. 2004 Nonlinear evolution of surface gravity waves over highly variable depth. Phys. Rev. Lett. 93, 234501-1.CrossRefGoogle ScholarPubMed
3. Ashton, A. C. L. & Fokas, A. S. 2011 A non-local formulation of rotational water waves. J. Fluid Mech. 689, 129–148.CrossRefGoogle Scholar
4. Deconinck, D. & Oliveras, K. 2011 The instability of periodic surface gravity waves. J. Fluid Mech. 675, 141–167.CrossRefGoogle Scholar
5. Driscoll, T. A. & Trefethen, L. 2002 Schwarz–Christoffel Mapping. Cambridge University Press.CrossRefGoogle Scholar
6. Floryan, J. M. 1985 Conformal-mapping based coordinate generation method for channel flows. J. Comput. Phys. 58, 229–245.CrossRefGoogle Scholar
7. Fokas, A. S. 2000 On the integrability of linear and nonlinear partial differential equations. J. Math. Phys. 41 (6), 4188–4237.CrossRefGoogle Scholar
8. Fokas, A. S. & Sung, L. Y. 2005 Generalised Fourier transforms, their nonlinearization and the imaging of the brain. Not. AMS 52, 1176–1190.Google Scholar
9. Garnier, J., Kraenkel, R. A. & Nachbin, A. 2007 Optimal Boussinesq model for shallow-water waves interacting with a microstructure. Phys. Rev. E 76, 046311.CrossRefGoogle ScholarPubMed
10. Haut, T. S. & Ablowitz, M. J. 2009 A reformulation and applications of interfacial fluids with a free surface. J. Fluid Mech. 631, 375–396.CrossRefGoogle Scholar
11. Keller, J. 2003 Shallow-water theory for arbitrary slopes of the bottom. J. Fluid Mech. 489, 345–348.CrossRefGoogle Scholar
12. Mei, C. C., Stiassnie, M. & Yue, D. K.-P. 2005 Theory and Applications of Ocean Surface Waves, Part 2: Nonlinear Aspects. World Scientific.Google Scholar
13. Muñoz, J. C. & Nachbin, A. 2005 Stiff microscale forcing and solitary wave refocusing. SIAM Multiscale Model. Simul. 3 (3), 680–705.Google Scholar
14. Muñoz, J. C. & Nachbin, A. 2006 Improved Boussinesq-type equations for highly-variable depths. IMA J. Appl. Maths 71, 600–633.Google Scholar
15. Nachbin, A. 2003 A terrain-following Boussinesq system. SIAM J. Appl. Maths 63 (3), 905–922.CrossRefGoogle Scholar
16. Nwogu, O. 1993 Alternative form of Boussinesq equations for nearshore wave propagation. J. Waterway Port Coast. Ocean Engng 119 (6), 618–638.CrossRefGoogle Scholar
17. Peregrine, D. H. 1967 Long waves on a beach. J. Fluid Mech. 27, 815–827.CrossRefGoogle Scholar
18. Ruiz de Zárate, A. R., Alfaro-Vigo, D. G., Nachbin, A. & Choi, W. 2009 A higher-order internal wave model accounting for large bathymetric variations. Stud. Appl. Maths 122, 275–294.CrossRefGoogle Scholar
19. Xu, L. & Guyenne, P. 2009 Numerical simulation of three-dimensional nonlinear water waves. J. Comput. Phys. 228, 8446–8466.CrossRefGoogle Scholar