Abstract.
This paper reviews the authors' recent work on water waves in heterogeneous media, namely nonlinear reduced models for water waves propagating over a region of highly variable depth. Both surface and internal waves are considered. Through different strategies for the mathematical modeling, at the level of equations, we show how diferent types of water wave models (namely systems of partial differential equations) can arise: weakly or fully dispersive, weakly or strongly nonlinear. The applications for these types of long waves are usually from Geophysics: in particular oceanography and meteorology. Here the emphasis is on coastal waves related to Physical Oceanography. An overview of the mathematical formulation is presented together with different asymptotic simplifications at the level of the partial differential equations (PDEs). References for recent work on the asymptotic analysis of solutions is also provided. These describe the regime where long pulse shaped waves propagate over rapidly varying topographic heterogeneities, which are modeled through rapidly varying coefficients in the PDEs.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
R. Burridge, G. Papanicolaou, B. White, SIAM J. Appl. Math. 47, 146 (1987)
C.C. Mei, The Applied Dynamics of Ocean Surface Waves (John Wiley, 1983)
P.G. Baines, Topographic Effects in Stratified Flows (Cambridge University Press, 1995)
European Centre for Medium-Range Weather Forecasts (ECMWF), Orography (1998)
D.H. Peregrine, J. Fluid Mech. 27, 815 (1967)
A.B. Kennedy, J.T. Kirby, Q. Chen, R.A. Dalrymple, Wave Motion 33, 225 (2001)
P.A. Madsen, R. Murray, O.R. Sørensen, Coastal Engg. 15, 371 (1991)
P.A. Madsen, O.R. Sørensen, Coastal Engg. 18, 183 (1992)
O. Nwogu, Coastal Ocean Engg. 119, 618 (1993)
Y. Matsuno, Phys. Rev. Lett. 69, 609 (1992)
H.A. Schäffer, P.A. Madsen, Coastal Engg. 26, 1 (1995)
A. Nachbin, SIAM Appl. Math. 63, 905 (2003)
J.C. Muñoz Grajales, A. Nachbin, SIAM J. Appl. Math. 64, 977 (2004)
J.C. Muñoz Grajales, A. Nachbin, SIAM Multiscale Model. Simul. 3, 680 (2005)
J.P. Fouque, J. Garnier, A. Nachbin, SIAM J. Appl. Math. 64, 1810 (2004)
J.P. Fouque, J. Garnier, J.C. Muñoz, A. Nachbin, Phys. Rev. Lett. 92, 094502-1 (2004)
J.R. Quintero, J.C. Muñoz Grajales, Meth. Appl. Anal. 11, 15 (2004)
J.C. Muñoz Grajales, A. Nachbin, IMA J. Appl. Math. 71, 600 (2006)
W. Artiles, A. Nachbin, Phys. Rev. Lett. 93, 234501 (2004)
W. Artiles, A. Nachbin, Meth. Appl. Anal. 11, 1 (2004)
C. Pires, M.A. Miranda, J. Geophys. Res. 106, 19733 (2001)
G.B. Whitham, Linear and Nonlinear Waves (John Wiley, 1974)
R.R. Rosales, G.C. Papanicolaou, Stud. Appl. Math. 68, 89 (1983)
T. Driscoll, http://www.math.udel.edu/ driscoll/software
T. Driscoll, L.N. Trefethen, Schwarz-Christoffel Mapping (Cambridge University Press, 2002)
M. Asch, W. Kohler, G. Papanicolaou, M. Postel, B. White, SIAM Rev. 33, 519 (1991)
J.P. Fouque, A. Nachbin, SIAM Multiscale Model. Simul. 1, 609 (2003)
J.P. Fouque, J. Garnier, A. Nachbin, Physica D 195, 324 (2004)
J.P. Fouque, J. Garnier, G.C. Papanicolaou, K. Sølna, Wave Propagation and Time Reversal in Randomly Layered Media (Springer-Verlag, 2007) (to appear)
J. Garnier, A. Nachbin, Phys. Rev. Lett. 93, 154501 (2004)
A. Nachbin, Modelling of Water Waves in Shallow Channels (Comp. Mech. Publ., Southampton, UK, 1993)
A. Nachbin, J. Fluid Mech. 296, 353 (1995)
A. Nachbin, G.C. Papanicolaou, J. Fluid Mech. 241, 311 (1992)
S.B. Yoon, P.L.F. Liu, J. Fluid Mech. 205, 397 (1989)
J. Hamilton, J. Fluid Mech. 83, 289 (1977)
J.G.B. Byatt-Smith, J. Fluid Mech. 49, 625 (1971)
W. Craig, C. Sulem, J. Comput. Phys. 108, 73 (1993)
Y. Matsuno, J. Fluid Mech. 249, 121 (1993)
V.E. Zakharov, A.I. Dyachenko, O.A. Vasilyev, Euro. J. Mech. B/Fluids 21, 283 (2002)
J.B. Keller, J. Fluid Mech. 489, 345 (2003)
P. Guidotti, J. Comput. Phys. 190, 325 (2003)
M.E. Taylor, Partial Differential Equations I: Basic Theory (Springer-Verlag, 1996)
K.M. Berger, P.A. Milewski, SIAM J. Appl. Math. 63, 1121 (2003)
W. Choi, R. Camassa, J. Fluid Mech. 313, 83 (1996)
H. Lamb, Hydrodynamics (Dover, 1932)
T.B. Benjamin, J. Fluid Mech. 25, 241 (1966)
R.I. Joseph, J. Phys. A: Math. Gen. 10, L225 (1977)
K. Tung, T.F. Chan, T. Kubota, Stud. Appl. Math. 66, 1 (1982)
T.B. Benjamin, J. Fluid Mech. 29, 559 (1967)
H. Ono, J. Phys. Soc. Jpn. 39, 1082 (1975)
Y. Matsuno, J. Phys. Soc. Jpn. 62, 1902 (1993)
W. Choi, R. Camassa, J. Fluid Mech. 396, 1 (1999)
W. Choi, R. Camassa, Phys. Rev. Lett. 77, 1759 (1996)
M. Miyata, Proc. IUTAM Symp. on Nonlinear Water Waves, edited by H. Horikawa, H. Maruo, 399 (1988)
R. Camassa, W. Choi, H. Michallet, P.-O. Rusås, J.K. Sveen, J. Fluid Mech. 549, 1 (2006)
C.H. Su, C.S. Gardner, J. Math. Phys. 10, 536 (1969)
A.E. Green, P.M. Naghdi, J. Fluid Mech. 78, 237 (1976)
Y.A. Li, J.M. Hyman, W. Choi, Stud. Appl. Math. 113, 303 (2004)
T. Jo, W. Choi, Stud. Appl. Math. 109, 205 (2002)
J. Grue, H.A. Friis, E. Palm, P.O. Rusas, J. Fluid Mech. 351, 223 (1997)
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Nachbin, A., Choi, W. Nonlinear waves over highly variable topography. Eur. Phys. J. Spec. Top. 147, 113–132 (2007). https://doi.org/10.1140/epjst/e2007-00205-9
Issue Date:
DOI: https://doi.org/10.1140/epjst/e2007-00205-9