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Asymptotics of surface waves over random bathymetry

Author(s): Walter Craig; Catherine Sulem
Journal: Quart. Appl. Math.
MSC (2000): Primary 76B15
Posted: October 21, 2009
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Abstract: This paper addresses the propagation of free surface water waves over a variable seabed in the long wavelength scaling regime. We consider the situation in which the bathymetry is given by a stationary random process which has a correlation length substantially shorter than the wavelength of the principal surface wave components. An asymptotic description shows that the water waves problem is modeled by an effective system of equations that is related to the KdV, however in a reference frame given in terms of random characteristic coordinates, and in addition with a random amplitude modulation and random scattered component. The resulting random processes are strongly correlated and have canonical limits due to the Donsker invariance principle. Our analysis is based on the Hamiltonian description of water waves and long wave perturbation theory and a new criterion for asymptotic expansions of partial differential equations with rapidly varying coefficients. In this paper we give a detailed analysis of the transformation to random characteristic coordinates and the asymptotic form of the resulting transformed partial differential equations. A companion paper (de Bouard, A., Craig, W., Dıaz-Espinosa, O., Guyenne, P., Sulem, C., Long wave expansions for water waves over random topography, Nonlinearity 21 (2008), 2143-2178) analyses in detail the asymptotic behavior of the resulting expression for solutions, and their consistency with the derivation of the effective model equations.

References:

[1]
Alvarez-Samaniego, B., Lannes, D., Large time existence for $ 3D$ water waves and asymptotics, Invent. Math. 171 (2008), 485-541. MR 2372806 (2009b:35324)

[2]
Billingsley, P., Convergence of probability measures, John Wiley & Sons, Inc., New York-London-Sydney (1968). MR 0233396 (38:1718)

[3]
de Bouard, A., Craig, W., Dıaz-Espinosa, O., Guyenne, P., Sulem, C., Long wave expansions for water waves over random topography, Nonlinearity 21 (2008), 2143-2178. MR 2430666

[4]
Chazel, F., Influence of bottom topography on long water waves, M2AN Math. Model. Numer. Anal. 41 (2007), no. 4, 771-799. MR 2362914 (2008k:76017)

[5]
Craig, W., Guyenne, P. and Kalisch, H., Hamiltonian long-wave expansions for free surfaces and interfaces. Commun. Pure Appl. Math. 58 (2005), 1587-1641. MR 2177163 (2006i:76012)

[6]
Craig, W., Guyenne, P., Nicholls, D. and Sulem, C., Hamiltonian long-wave expansions for water waves over a rough bottom. Proc. Roy. Soc. Lond. - A 461 (2005), no. 2055, 839-87. MR 2121939 (2006e:76019)

[7]
Craig, W., Guyenne, P. and Sulem, C., Water waves over a random bottom. J. Fluid Mech. (2008), submitted.

[8]
Doukhan, P., Mixing. Properties and Examples, Lecture Notes in Statistics 85, Springer-Verlag, 1994. MR 1312160 (96b:60090)

[9]
Garnier, J., Kraenkel, R. A. and Nachbin, A., Optimal Boussinesq model for shallow-water waves interacting with a microstructure, Phys. Rev. E (3) 76 (2007), 046311. MR 2365630 (2008i:76022)

[10]
Garnier, J., Muñoz Grajales, J. C. and Nachbin, A., Effective behavior of solitary waves over random topography, Multiscale Model. Simul. 6 (2007), 995-1025. MR 2368977 (2009a:76018)

[11]
Mei, C.C. and Hancock, M., Weakly nonlinear surface waves over a random seabed, J. Fluid Mech. 475 (2003), 247-268. MR 2012498 (2004h:76026)

[12]
Mei, C.C. and Li, Y., Evolution of solitons over a randomly rough seabed, Phys. Rev E 70 (2004), 016302. MR 2125713 (2005j:76016)

[13]
Rosales, R. and Papanicolaou, G., Gravity waves in a channel with a rough bottom. Stud. Appl. Math. 68 (1983), no. 2, 89-102. MR 693716 (84e:76019)

[14]
Zakharov, V. E., Stability of periodic waves of finite amplitude on the surface of a deep fluid. J. Appl. Mech. Tech. Phys. 9 (1968), 1990-1994.

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Additional Information:

Walter Craig
Affiliation: Department of Mathematics, McMaster University, Hamilton, Ontario L8S 4K1, Canada
Email: craig@math.mcmaster.ca

Catherine Sulem
Affiliation: Department of Mathematics, University of Toronto, Toronto, Ontario M5S 3G3, Canada
Email: sulem@math.toronto.edu

PII: S0033-569X-09-01177-9
Received by editor(s): December 31, 2008
Posted: October 21, 2009
Additional Notes: The first author was partially supported by the Canada Research Chairs Program and NSERC grant \#238452-06. The second author was partially supported by NSERC grant \#46179-05
Dedicated: In honor of Walter Strauss on the occasion of his seventieth birthday
Copyright of article: Copyright 2009, Brown University
The copyright for this article reverts to public domain after 28 years from publication.


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