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A high-order spectral method for the study of nonlinear gravity waves. (English) Zbl 0638.76016

We develop a robust numerical method for modelling nonlinear gravity waves which is based on the Zakharov equation/mode-coupling idea but is generalized to include interactions up to an arbitrary order M in wave steepness. A large number \((N=0(1000))\) of free wave modes are typically used whose amplitude evolutions are determined through a pseudospectral treatment of the nonlinear free-surface conditions. The computational effort is directly proportional to N and M, and the convergence with N and M is exponentially fast for waves up to approximately 80 % of Stokes limiting steepness (ka\(\sim 0.35)\). The efficiency and accuracy of the method is demonstrated by comparisons to fully nonlinear semi-Lagrangian computations; calculations of long-time evolution of wavetrains using the modified fourth-order Zakharov equations; and experimental measurements of a travelling wave packet. As a final example of the usefulness of the method, we consider the nonlinear interactions between two colliding wave envelopes of different carrier frequencies.

MSC:

76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
76M99 Basic methods in fluid mechanics
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[1] DOI: 10.1017/S002211208200158X · Zbl 0485.76017 · doi:10.1017/S002211208200158X
[2] DOI: 10.1063/1.863708 · doi:10.1063/1.863708
[3] DOI: 10.1017/S0022112085000180 · Zbl 0603.76014 · doi:10.1017/S0022112085000180
[4] Fornberg, Phil. Trans. R. Soc. Lond. 289 pp 373– (1978)
[5] Dysthe, Proc. R. Soc. Lond. 369 pp 105– (1979)
[6] DOI: 10.1017/S0022112081003169 · Zbl 0459.76011 · doi:10.1017/S0022112081003169
[7] DOI: 10.1063/1.861481 · Zbl 0341.76005 · doi:10.1063/1.861481
[8] Bryant, J. Austral. Math. Soc. 25 pp 2– (1983)
[9] DOI: 10.1017/S0022112062001469 · Zbl 0117.43605 · doi:10.1017/S0022112062001469
[10] Stiassnie, J. Fluid Mech. 174 pp 299– (1987)
[11] DOI: 10.1017/S0022112084001257 · Zbl 0551.76016 · doi:10.1017/S0022112084001257
[12] DOI: 10.1017/S0022112074000802 · Zbl 0286.76016 · doi:10.1017/S0022112074000802
[13] DOI: 10.1017/S0022112081002851 · Zbl 0494.76019 · doi:10.1017/S0022112081002851
[14] DOI: 10.1017/S0022112060001043 · Zbl 0094.41101 · doi:10.1017/S0022112060001043
[15] DOI: 10.1017/S0022112071001940 · Zbl 0229.76029 · doi:10.1017/S0022112071001940
[16] DOI: 10.1143/JPSJ.37.486 · doi:10.1143/JPSJ.37.486
[17] DOI: 10.1063/1.863075 · doi:10.1063/1.863075
[18] DOI: 10.1017/S0022112062000245 · Zbl 0105.20303 · doi:10.1017/S0022112062000245
[19] Longuet-Higgins, Proc. R. Soc. Lond. 350 pp 1– (1976)
[20] Zakharov, Sov. Phys., J. Exp. Theor. Phys. 34 pp 62– (1972)
[21] DOI: 10.1007/BF00913182 · doi:10.1007/BF00913182
[22] Yuen, Adv. Appl. Mech. 22 pp 67– (1982)
[23] DOI: 10.1063/1.862122 · doi:10.1063/1.862122
[24] DOI: 10.1063/1.1694844 · Zbl 0294.76011 · doi:10.1063/1.1694844
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