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A three-dimensional Dirichlet-to-Neumann operator for water waves over topography

Published online by Cambridge University Press:  25 April 2018

D. Andrade Open the ORCID record for D. Andrade [Opens in a new window]  and
A. Nachbin
D. Andrade*
Affiliation:
Instituto Nacional de Matemática Pura e Aplicada, Estrada D. Castorina 110, Rio de Janeiro, RJ, CEP 22460-320, Brazil
A. Nachbin*
Affiliation:
Instituto Nacional de Matemática Pura e Aplicada, Estrada D. Castorina 110, Rio de Janeiro, RJ, CEP 22460-320, Brazil
*
†Email addresses for correspondence: [email protected], [email protected]
†Email addresses for correspondence: [email protected], [email protected]
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Abstract

Surface water waves are considered propagating over highly variable non-smooth topographies. For this three-dimensional problem a Dirichlet-to-Neumann (DtN) operator is constructed reducing the numerical modelling and evolution to the two-dimensional free surface. The corresponding discrete Fourier integral operator is defined through a matrix decomposition. The topographic component of the decomposition requires special care, and a Galerkin method is provided accordingly. One-dimensional numerical simulations, along the free surface, validate the DtN formulation in the presence of a large-amplitude rapidly varying topography. An alternative conformal-mapping-based method is used for benchmarking. A two-dimensional simulation in the presence of a Luneburg lens (a particular submerged mound) illustrates the accurate performance of the three-dimensional DtN operator.

JFM classification

Geophysical and Geological Flows: Geophysical and Geological Flows Waves/Free-surface Flows: Surface gravity waves Geophysical and Geological Flows: Topographic effects
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 845 , 25 June 2018 , pp. 321 - 345
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Copyright
© 2018 Cambridge University Press 

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References

Ablowitz, M. J., Fokas, A. S. & Musslimani, Z. H. 2006 On a new non-local formulation of water waves. J. Fluid Mech. 592, 313–343.CrossRefGoogle Scholar
Ambrose, D. M., Bona, J. L. & Nicholls, D. P. 2013 On ill-posedness of truncated series models for water waves. Proc. R. Soc. Lond. A 470, 20130849.Google Scholar
Arcar, D. & Segur, H. 2012 Seismically generated tsunamis. Phil. Trans. R. Soc. Lond. A 370, 1505–1542.Google Scholar
Artiles, W. & Nachbin, A. 2004 Nonlinear evolution of surface gravity waves over highly variable depth. Phys. Rev. Lett. 93, 234501.CrossRefGoogle ScholarPubMed
Ashton, A. C. L. & Fokas, A. S. 2011 A non-local formulation of rotational water waves. J. Fluid Mech. 689, 129–148.CrossRefGoogle Scholar
Cathala, M. 2016 Asymptotic shallow water models with non smooth topographies. Monatshefte für Mathematik 179, 325–353.CrossRefGoogle Scholar
Choi, J. & Milewski, P. A. 2003 Long nonlinear waves in resonance with topography. Stud. Appl. Maths 110, 21–47.CrossRefGoogle Scholar
Constantin, A. 2011 Nonlinear Water Waves with Applications to Wave–Current Interactions and Tsunamis. SIAM.CrossRefGoogle Scholar
Craig, W., Guyenne, P., Nicholls, D. P. & Sulem, C. 2005 Hamiltonian long-wave expansions for water waves over a rough bottom. Proc. R. Soc. Lond. A 461, 839–873.Google Scholar
Craig, W. & Sulem, C. 1993 Numerical simulation of gravity waves. J. Comput. Phys. 108, 73–83.CrossRefGoogle Scholar
Dean, R. G. & Dalrymple, R. A. 1991 Water Wave Mechanics for Engineers and Scientists. World Scientific.CrossRefGoogle Scholar
Deconinck, B. & Oliveras, K. 2011 The instability of periodic surface gravity waves. J. Fluid Mech. 675, 141–167.CrossRefGoogle Scholar
Dingemans, M. W. 1997 Water Wave Mechanics for Engineers and Scientists. World Scientific.Google Scholar
Driscoll, T. A. & Trefethen, L. N. 2002 Schwarz–Christoffel Mapping. Cambridge University Press.CrossRefGoogle Scholar
Fokas, A. S. & Nachbin, A. 2012 Water waves over a variable bottom: a non-local formulation and conformal mappings. J. Fluid Mech. 695, 288–309.CrossRefGoogle Scholar
Guyenne, P. & Nicholls, D. P. 2005 Numerical simulation of solitary waves on plane slopes. Maths Comput. Simul. 69, 269–281.CrossRefGoogle Scholar
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
Johnson, R. S. 1997 A Modern Introduction to the Mathematical Theory of Water Waves. Cambridge University Press.CrossRefGoogle Scholar
Lannes, D. 2005 Well-posedness of the water-waves equations. J. Amer. Math. Soc. 18, 605–654.CrossRefGoogle Scholar
Mei, C. C. 1985 Resonant reflection of surface water waves by periodic sandbars. J. Fluid Mech. 152, 315–335.CrossRefGoogle Scholar
Milewski, P. A. 1998 A formulation for water waves over topography. Stud. Appl. Maths 100, 95–106.CrossRefGoogle Scholar
Nachbin, A. 1993 Modelling of Water Waves in Shallow Channels. Computational Mechanics.Google Scholar
Nachbin, A. 2003 A terrain-following Boussinesq system. SIAM J. Appl. Maths 63, 905–922.CrossRefGoogle Scholar
Rosales, R. R. & Papanicolaou, G. C. 1983 Gravity waves in a channel with a rough bottom. Stud. Appl. Maths 68, 89–102.CrossRefGoogle Scholar
Vargas-Magaña, R. M. & Panayotaros, P. 2016 A Whitham–Boussinesq long-wave model for variable topography. Wave Motion 65, 156–174.CrossRefGoogle Scholar
Vasan, V. & Deconinck, B. 2013 The inverse water wave problem of bathymetry detection. J. Fluid Mech. 714, 562–590.CrossRefGoogle Scholar
Whitham, G. B. 1999 Linear and Nonlinear Waves. John Wiley.CrossRefGoogle Scholar
Wilkening, J. & Vasan, V. 2015 Comparison of five methods of computing the Dirichlet–Neumann operator for the water wave problem. Contemp. Maths 635, 175–210.CrossRefGoogle Scholar
Zauderer, E. 1989 Partial Differential Equations of Applied Mathematics. John Wiley.Google Scholar

Andrade and Nachbin supplementary movie

Focusing by the Luneburg lens: evolution and refraction of the velocity potential by the submerged circular mound.

Download Andrade and Nachbin supplementary movie(Video)
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