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Instability of internal equatorial water waves. (English) Zbl 1446.76081

Summary: In the following paper we present criteria for the hydrodynamical instability of internal equatorial water waves. We show, by way of the short-wavelength perturbation approach, that certain geophysical waves propagating above the equatorial thermocline are linearly unstable when the wave steepness exceeds a given threshold.

MSC:

76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
76B55 Internal waves for incompressible inviscid fluids
76E20 Stability and instability of geophysical and astrophysical flows
86A05 Hydrology, hydrography, oceanography
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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] Bennett, A., Lagrangian fluid dynamics, (2006), Cambridge University Press Cambridge · Zbl 1105.76002
[3] Constantin, A., On the deep water wave motion, J. Phys. A, 34, 1405-1417, (2001) · Zbl 0982.76015
[4] Constantin, A., Edge waves along a sloping beach, J. Phys. A, 34, 9723-9731, (2001) · Zbl 1005.76009
[5] Constantin, A., The trajectories of particles in Stokes waves, Invent. Math., 166, 523-535, (2006) · Zbl 1108.76013
[6] 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
[7] Constantin, A., An exact solution for equatorially trapped waves, J. Geophys. Res., 117, C05029, (2012)
[8] Constantin, A., On the modelling of equatorial waves, Geophys. Res. Lett., 39, L05602, (2012)
[9] Constantin, A., Some three-dimensional nonlinear equatorial flows, J. Phys. Oceanogr., 43, 165-175, (2013)
[10] Constantin, A., On equatorial wind waves, Differential Integral Equations, 26, 237-252, (2013) · Zbl 1289.86002
[11] Constantin, A., Some nonlinear, equatorially trapped, nonhydrostatic internal geophysical waves, J. Phys. Oceanogr., 44, 781-789, (2014)
[12] Constantin, A.; Germain, P., Instability of some equatorially trapped waves, J. Geophys. Res. Oceans, 118, 2802-2810, (2013)
[13] Constantin, A.; Strauss, W., Pressure beneath a Stokes wave, Comm. Pure Appl. Math., 63, 533-557, (2010) · Zbl 1423.76061
[14] Cushman-Roisin, B.; Beckers, J.-M., Introduction to geophysical fluid dynamics: physical and numerical aspects, (2011), Academic Press Waltham, MA · Zbl 1319.86001
[15] Drazin, P. G., Introduction to hydrodynamic stability, (2002), Cambridge University Press Cambridge · Zbl 0997.76001
[16] Drazin, P. G.; Reid, W. H., Hydrodynamic stability, (2004), Cambridge University Press Cambridge · Zbl 1055.76001
[17] Fedorov, A. V.; Brown, J. N., Equatorial waves, (Steele, J., Encyclopedia of Ocean Sciences, (2009), Academic Press San Diego, CA), 3679-3695
[18] 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
[19] Friedlander, S.; Yudovich, V., Instabilities in fluid motion, Notices Amer. Math. Soc., 46, 1358-1367, (1999) · Zbl 0948.76003
[20] Friedlander, S., Lectures on stability and instability of an ideal fluid, (Hyperbolic Equations and Frequency Interactions, IAS/Park City Math. Ser., vol. 5, (1999), Amer. Math. Soc. Providence, RI), 227-304 · Zbl 0921.76068
[21] Genoud, F.; Henry, D., Instability of equatorial water waves with an underlying current, J. Math. Fluid Mech., 16, 661-667, (2014) · Zbl 1308.76035
[22] Gerstner, F., Theorie der wellen samt einer daraus abgeleiteten theorie der deichprofile, Ann. Phys., 2, 412-445, (1809)
[23] Henry, D., On the deep-water Stokes flow, Int. Math. Res. Not. IMRN, 2008, (2008), 7 pp · Zbl 1245.76008
[24] Henry, D., On Gerstner’s water wave, J. Nonlinear Math. Phys., 15, 87-95, (2008) · Zbl 1362.76009
[25] 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
[26] Henry, D.; Hsu, H.-C., Instability of equatorial water waves in the f-plane, Discrete Contin. Dyn. Syst., 35, 909-916, (2015) · Zbl 1304.35698
[27] Henry, D.; Matioc, A., On the existence of equatorial wind waves, Nonlinear Anal., 101, 113-123, (2014) · Zbl 1286.86010
[28] Hsu, H.-C., An exact solution for nonlinear internal equatorial waves in the f-plane approximation, J. Math. Fluid Mech., 16, 463-471, (2014) · Zbl 1308.76054
[29] Hsu, H.-C., Some nonlinear internal equatorial waves with a strong underlying current, Appl. Math. Lett., 34, 1-6, (2014) · Zbl 1314.35193
[30] Hsu, H.-C., Some nonlinear internal equatorial flows, Nonlinear Anal. Real World Appl., 18, 69-74, (2014) · Zbl 1367.35182
[31] Hsu, H.-C., An exact solution of equatorial waves, Monatsh. Math., (2014), in press
[32] Ionescu-Kruse, D., Instability of edge waves along a sloping beach, J. Differential Equations, 256, 3999-4012, (2014) · Zbl 1295.35062
[33] Izumo, T., The equatorial current, meridional overturning circulation, and their roles in mass and heat exchanges during the el niño events in the tropical Pacific Ocean, Ocean Dyn., 55, 110-123, (2005)
[34] Leblanc, S., Local stability of Gerstner’s waves, J. Fluid Mech., 506, 245-254, (2004) · Zbl 1062.76019
[35] Lifschitz, A.; Hameiri, E., Local stability conditions in fluid dynamics, Phys. Fluids, 3, 2644-2651, (1991) · Zbl 0746.76050
[36] Matioc, A. V., An exact solution for geophysical equatorial edge waves over a sloping beach, J. Phys. A, 45, 365501, (2012) · Zbl 1339.86001
[37] Mollo-Christensen, E., Gravitational and geostrophic billows: some exact solutions, J. Atmospheric Sci., 35, 1395-1398, (1978)
[38] Stuhlmeier, R., On edge waves in stratified water along a sloping beach, J. Nonlinear Math. Phys., 18, 127-137, (2011) · Zbl 1394.76030
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