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Instability of edge waves along a sloping beach. (English) Zbl 1295.35062

The stability of three-dimensional edge waves along a sloping beach described in the Lagrangian framework is investigated by the theory of short-wavelength perturbations. The author proves that the edge waves with the steepness parameter higher than \(\frac{7}{18}\sin{\alpha}\), \(\alpha\) being the sloping angle of the beach, are unstable. One mentions also the potential applicability of the short-wavelength instability method to other explicit Lagrangian solutions pointed out by the author.

MSC:

35B35 Stability in context of PDEs
76E99 Hydrodynamic stability
35Q35 PDEs in connection with fluid mechanics
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[1] Bayly, B. J., Three-dimensional instabilities in quasi-two dimensional inviscid flows, (Miksad, R. W.; etal., Nonlinear Wave Interactions in Fluids, (1987), ASME New York), 71-77
[2] Bayly, B. J., Three-dimensional centrifugal-type instabilities in inviscid two-dimensional flows, Phys. Fluids, 31, 56-64, (1988) · Zbl 0651.76018
[3] Bayly, B. J.; Holm, D. D.; Lifschitz, A., Three-dimensional stability of elliptical vortex columns in external strain flows, Philos. Trans. R. Soc. Lond. Ser. A, 354, 895-926, (1996) · Zbl 0872.76041
[4] Constantin, A., On the deep water wave motion, J. Phys. A, 34, 1405-1417, (2001) · Zbl 0982.76015
[5] Constantin, A., Edge waves along a sloping beach, J. Phys. A, 34, 9723-9731, (2001) · Zbl 1005.76009
[6] Constantin, A., The trajectories of particles in Stokes waves, Invent. Math., 166, 523-535, (2006) · Zbl 1108.76013
[7] Constantin, A., Nonlinear water waves with applications to wave-current interactions and tsunamis, CBMS-NSF Regional Conf. Ser. in Appl. Math., vol. 81, (2011), SIAM Philadelphia · Zbl 1266.76002
[8] Constantin, A., An exact solution for equatorially trapped waves, J. Geophys. Res., 117, C05029, (2012)
[9] Constantin, A., Some three-dimensional nonlinear equatorial flows, J. Phys. Oceanogr., 43, 165-175, (2013)
[10] Constantin, A.; Germain, P., Instability of some equatorially trapped waves, J. Geophys. Res., Oceans, 118, 2802-2810, (2013)
[11] Constantin, A.; Strauss, W., Pressure beneath a Stokes wave, Comm. Pure Appl. Math., 53, 533-557, (2010) · Zbl 1423.76061
[12] Ehrenmark, U., Oblique wave incidence on a plane beach: the classical problem revisited, J. Fluid Mech., 368, 291-319, (1998) · Zbl 0939.76012
[13] Friedlander, S.; Lipton-Lifschitz, A., Localized instabilities in fluids, (Friedlander, S.; Serre, D., Handbook of Mathematical Fluid Dynamics, vol. 2, (2003), North-Holland), 289-354 · Zbl 1187.76663
[14] Friedlander, S.; Vishik, M. M., Instability criteria for the flow of an inviscid incompressible fluid, Phys. Rev. Lett., 66, 2204-2206, (1991) · Zbl 0968.76543
[15] Friedlander, S.; Vishik, M. M., Instability criteria for steady flows of a perfect fluid, Chaos, 2, 455-460, (1992) · Zbl 1055.76518
[16] Friedlander, S.; Yudovich, V., Instabilities in fluid motion, Notices Amer. Math. Soc., 46, 1358-1367, (1999) · Zbl 0948.76003
[17] Gerstner, F., Theorie der wellen samt einer daraus abgeleiteten theorie der deichprofile, Ann. Phys., 2, 412-445, (1809)
[18] Henry, D., The trajectories of particles in deep-water Stokes waves, Int. Math. Res. Not. IMRN, (2006), Art. ID 23405, 13 pp · Zbl 1157.35449
[19] Henry, D., On Gerstner’s water wave, J. Nonlinear Math. Phys., 15, 87-95, (2008) · Zbl 1362.76009
[20] Henry, D., An exact solution for equatorial geophysical water waves with an underlying current, Eur. J. Mech. B Fluids, 38, 18-21, (2013) · Zbl 1297.86002
[21] Henry, D.; Mustafa, O., Existence of solutions for a class of edge wave equations, Discrete Contin. Dyn. Syst. Ser. B, 6, 1113-1119, (2006) · Zbl 1115.76015
[22] Howd, P. A.; Bowen, A. J.; Holman, R. A., Edge waves in the presence of strong longshore currents, J. Geophys. Res., 97, 11357-11371, (1992)
[23] Johnson, R. S., A modern introduction to the mathematical theory of water waves, (1997), Cambridge University Press · Zbl 0892.76001
[24] Johnson, R. S., Edge waves: theories past and present, Philos. Trans. R. Soc. Lond. Ser. A, 365, 2359-2376, (2007) · Zbl 1152.76301
[25] Lamb, H., Hydrodynamics, (1895), Cambridge University Press Cambridge, UK · JFM 26.0868.02
[26] Leblanc, S., Local stability of Gerstner’s waves, J. Fluid Mech., 506, 245-254, (2004) · Zbl 1062.76019
[27] Lifschitz, A., Short wavelength instabilities of incompressible three-dimensional flows and generation of vorticity, Phys. Lett. A, 157, 481-487, (1991)
[28] Lifschitz, A., On the instability of three-dimensional flows or an ideal incompressible fluid, Phys. Lett. A, 167, 465-474, (1992)
[29] Lifschitz, A., On the instability of certain motions of an ideal incompressible fluid, Adv. in Appl. Math., 15, 404-436, (1994) · Zbl 0813.76030
[30] Lifschitz, A.; Hameiri, E., Local stability conditions in fluid dynamics, Phys. Fluids, 3, 2644-2651, (1991) · Zbl 0746.76050
[31] Liftschitz, A.; Hameiri, E., Localized instabilities of vortex rings with swirl, Comm. Pure Appl. Math., 46, 1379-1408, (1993) · Zbl 0796.76043
[32] Matioc, A.-V., An exact solution for geophysical equatorial edge waves over a sloping beach, J. Phys. A, 45, 365501, (2012), 10 pp · Zbl 1339.86001
[33] Mollo-Christensen, E., Allowable discontinuities in a gerstner wave, Phys. Fluids, 25, 586-587, (1982)
[34] Stokes, G. G., Report on recent researches in hydrodynamics, (Rep. 16th Brit. Assoc. Adv. Sci., (1846)), Papers, vol. 1, 157-187, (1880), Cambridge University Press, see also
[35] Stuhlmeier, R., On edge waves in stratified water along a sloping beach, J. Nonlinear Math. Phys., 18, 127-137, (2011) · Zbl 1394.76030
[36] Ursell, F., Edge waves on a sloping beach, Proc. R. Soc. Lond. Ser. A, 214, 79-97, (1952) · Zbl 0047.43803
[37] Whitham, G. B., Nonlinear effects in edge waves, J. Fluid Mech., 74, 353-368, (1976) · Zbl 0352.76015
[38] Whitham, G. B., Lecture on wave propagation, (1979), Springer New York, (published for Tata Institute of Fundamental Research) · Zbl 0454.76015
[39] Yih, C. S., Note on edge waves in a stratified fluid, J. Fluid Mech., 24, 765-767, (1966) · Zbl 0142.45602
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