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Exact solution and instability for geophysical waves in modified equatorial \(\beta\)-plane approximation with and without centripetal forces. (English) Zbl 1474.86006

Summary: In the paper, we first present an exact solution for geophysical internal waves with centripetal forces and geophysical edge waves without centripetal forces in modified equatorial \(\beta\)-plane approximation. Then we apply the short-wavelength perturbation method to show an instability threshold for geophysical internal waves propagating eastward with centripetal forces and geophysical edge waves without centripetal forces in the modified equatorial \(\beta\)-plane approximation.

MSC:

86A05 Hydrology, hydrography, oceanography
76U60 Geophysical flows
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[1] Henry, D., Equatorially trapped nonlinear water waves in a \(\beta \)-plane approximation with centripetal forces, J. Fluid Mech., 804, R1 (2016) · Zbl 1454.76024
[2] Su, D.; Gao, H. J., An exact solution for geophysical internal waves with underlying current in modified equatorial \(\beta \)-plane approximation, J. Nonlinear Math. Phys., 26, 579-603 (2019) · Zbl 1418.86006
[3] Constantin, A., Some three-dimensional nonlinear equatorial flows, J. Phys. Oceanogr., 43, 165-175 (2013)
[4] Constantin, A., Some nonlinear, equatorially trapped, nonhydrostatic internal geophysical waves, J. Phys. Oceanogr., 44, 781-789 (2014)
[5] Fedorov, A. V.; Brown, J. N., Equatorial waves, (Steele, J., J. Encyclopedia of Ocean Sciences (2009), Academic Press: Academic Press San Diego), 3679-3695
[6] Izumo, T., The equatorial current, meridional overturning circulation, and their roles in mass and heat exchanges during the El \(Ni \widetilde{n}\) o events in the tropical Pacific ocean, Ocean Dyn., 55, 110-123 (2005)
[7] Matioc, A. V., An exact solution for geophysical equatorial edge waves over a sloping beach, J. Phys. A, 45, Article 365501 pp. (2012) · Zbl 1339.86001
[8] Johnson, R. S., Edge waves: Theories past and present, Phil. Trans. R. Soc. A, 365, 2359-2376 (2007) · Zbl 1152.76301
[9] Constantin, A., Edge wave along a sloping beach, J. Phys. A: Math. Gen., 45, 9723-9731 (2001) · Zbl 1005.76009
[10] 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
[11] Constantin, A., An exact solution for equatorially trapped waves, J. Geophys. Res. Oceans, 117, Article C05029 pp. (2012)
[12] Constantin, A., On the modelling of equatorial waves, Geophys. Res. Lett., 39, Article L05602 pp. (2012)
[13] Constantin, A., On equatorial wind waves, Differential Integral Equations, 26, 237-252 (2013) · Zbl 1289.86002
[14] Henry, D.; Matioc, A., On the existence of equatorial wind waves, Nonlinear Anal., 101, 113-123 (2014) · Zbl 1286.86010
[15] Cushman-Roisin, B.; Beckers, J.-M., Introduction to Geophysical Fluid Dynamics: Physical and Numerical Aspects (2011), Academic Press: Academic Press Waltham, MA · Zbl 1319.86001
[16] Constantin, A.; Germain, P., Instability of some equatorially trapped waves, J. Geophys. Res. Oceans, 118, 2802-2810 (2013)
[17] Constantin, A.; Johnson, R. S., The dynamics of waves interacting with the equatorial undercurrent, Geophys. Astrophys. Fluid Dyn., 109, 311-358 (2015) · Zbl 07658791
[18] Constantin, A.; Johnson, R. S., An exact, steady, purely azimuthal equatorial flow with a free surface, J. Phys. Oceanogr., 46, 1935-1945 (2016)
[19] Constantin, A.; Johnson, R. S., An exact, steady, purely azimuthal flow as a model for the antarctic circumpolar current, J. Phys. Oceanogr., 46, 3585-3594 (2016)
[20] Constantin, A.; Johnson, R. S., A nonlinear, three-dimensional model for ocean flows, motivated by some observations of the pacific equatorial undercurrent and thermocline, Phys. Fluids, 29, Article 056604 pp. (2017)
[21] Ionescu-Kruse, D., A three-dimensional autonomous nonlinear dynamical system modelling equatorial ocean flows, J. Differ. Equ., 264, 4650-4668 (2018) · Zbl 1386.35403
[22] Johnson, R. S., Application of the ideas and techniques of classical fluid mechanics to some problems in physical oceanography, Phil. Trans. R. Soc. A, 376, 2111, Article 20170092 pp. (2018) · Zbl 1404.86018
[23] Henry, D., On Gerstner’s water wave, J. Nonlinear Math. Phys., 15, 87-95 (2008) · Zbl 1362.76009
[24] 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
[25] Mollo-Christensen, E., Gravitational and geostrophic billows: some exact solutions, J. Atmos. Sci., 35, 1395-1398 (1978)
[26] Bayly, B. J., Three dimensional instabilities in quasi-two dimensional inviscid flows, (Miksad, R. W.; etal., Nonlinear Wave Interactions in Fluids (1987), ASME: ASME New York), 71-77
[27] 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
[28] Lifschitz, A.; Hameiri, E., Local stability condition in fluid dynamics, Phys. Fluids, 3, 2644-2651 (1991) · Zbl 0746.76050
[29] Ionescu-Kruse, D., Instability of edge waves along a sloping beach, J. Differential Equations, 256, 12, 3999-4012 (2014) · Zbl 1295.35062
[30] Leblanc, S., Local stability of Gerstner’s waves, J. Fluid Mech., 506, 245-254 (2004) · Zbl 1062.76019
[31] Henry, D.; Genoud, F., Instability of equatorial water waves with an underlying current, J. Math. Fluid Mech., 16, 661-667 (2014) · Zbl 1308.76035
[32] Henry, D.; Hsu, H.-C., Instability of internal equatorial water waves, J. Differential Equations, 258, 1015-1024 (2015) · Zbl 1446.76081
[33] Henry, D.; Hsu, H.-C., Instability of internal equatorial water waves in the f-plane, Discrete Contin. Dyn. Syst., 35, 909-916 (2015) · Zbl 1304.35698
[34] Ionescu-Kruse, D., Instability of equatorially trapped waves in stratified water, Ann. Mat. Pura Appl., 195, 589-599 (2016) · Zbl 1352.35191
[35] Fan, L. L.; Gao, H. J., Instability of equatorial edge waves in the back flow, Proc. Amer. Math. Soc., 145, 765-778 (2017) · Zbl 1355.35182
[36] Chu, J. F.; Ionescu-Kruse, D.; Yang, Y. J., Exact solution and instability for geophysical waves with cenrtipetal force and at arbitrary latitude, J. Math. Fluid Mech., 19, 1422-6928 (2019)
[37] Friedlander, S.; Yudovich, V., Instabilities in fluid motion, Notices Amer. Math. Soc., 46, 1358-1367 (1999) · Zbl 0948.76003
[38] Friedlander, S., Lectures on stability and instability of an ideal fluid, (Hyperbolic Equations and Frequency Interactions. Hyperbolic Equations and Frequency Interactions, IAS/Park City Mathematics Series, vol. 5 (1999), Notices Amer. Math. Soc.: Notices Amer. Math. Soc. Providence, RI), 227-304 · Zbl 0921.76068
[39] Ionescu-Kruse, D., Short-wavelength instability of edge waves in stratified water, Discrete Contin. Dyn. Syst., 35, 2053-2066 (2014) · Zbl 1302.76070
[40] Bennett, A., Lagranian Fluid Dynamics (2006), Cambridge University Press: Cambridge University Press Cambridge · Zbl 1105.76002
[41] Henry, D., A modified equatorial \(\beta \)-plane approximation modelling nonlinear wave-current interactions, J. Differential Equations, 263, 2554-2566 (2017) · Zbl 1365.76020
[42] Gallagher, I.; Saint-Raymond, L., On the influence of the Earth’s rotation on geophysical flows, (Handbook of Mathematical Fluid Mechanics, Vol. 4 (2007)), 201-329
[43] Kluczek, M., Exact and explicit internal equatorial water waves with underlying currents, J. Math. Fluid Mech., 19, 305-314 (2017) · Zbl 1365.76021
[44] Ionescu-Kruse, D., An exact solution for geophysical edge waves in the \(f\)-plane approximation, Nonlinear Anal., 24, 190-195 (2015) · Zbl 1330.35458
[45] Ionescu-Kruse, D., An exact solution for geophysical edge waves in the \(\beta \)-plane approximation, J. Math. Fluid Mech., 17, 699-706 (2015) · Zbl 1329.35311
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