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Instability of nonlinear wave-current interactions in a modified equatorial \(\beta\)-plane approximation. (English) Zbl 1411.76015

Summary: We present an instability analysis of some exact and explicit solutions to the geophysical equatorial \(\beta\)-plane equations incorporating a gravitational-correction term. A criterion for the instability is given by means of the short-wavelength perturbation method. Thresholds for both, a solution with a zonal current under constant density and a solution admitting stratification, are derived and expressed in terms of the steepness of the waves.

MSC:

76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
76E20 Stability and instability of geophysical and astrophysical flows
37H15 Random dynamical systems aspects of multiplicative ergodic theory, Lyapunov exponents
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[1] Bayly, B.J.: Three-dimensional instabilities in quasi-two dimensional inviscid flows. In: American Society of Mechanical Engineers, Applied Mechanics Division, AMD, pp. 71-77. ASME (1987)
[2] Boyd, J.P.: Nonlinear wavepackets and nonlinear schroedinger equation. In: Dynamics of the Equatorial Ocean, pp. 405-464. Springer, Berlin (2018)
[3] Constantin, A., The trajectories of particles in Stokes waves, Invent. Math., 166, 523-535, (2006) · Zbl 1108.76013
[4] Constantin, A.: Nonlinear water waves with applications to wave-current interactions and tsunamis. In: CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 81. SIAM, Philadelphia, PA (2011) · Zbl 1266.76002
[5] Constantin, A., An exact solution for equatorially trapped waves, J. Geophys. Res. Oceans, 117, c05029, (2012)
[6] Constantin, A., Some three-dimensional nonlinear equatorial flows, J. Phys. Oceanogr., 43, 165-175, (2013)
[7] Constantin, A., Some nonlinear, equatorially trapped, nonhydrostatic internal geophysical waves, J. Phys. Oceanogr., 44, 781-789, (2014)
[8] Constantin, A.; Germain, P., Instability of some equatorially trapped waves, J. Geophys. Res. Oceans, 118, 2802-2810, (2013)
[9] Constantin, A.; Ivanov, RI; Martin, C-I, Hamiltonian formulation for wave-current interactions in stratified rotational flows, Arch. Ration. Mech. Anal., 221, 1417-1447, (2016) · Zbl 1344.35084
[10] Constantin, A.; Ivanov, R., A hamiltonian approach to wave-current interactions in two-layer fluids, Phys. Fluids, 27, 086603, (2015) · Zbl 1326.76021
[11] Constantin, A.; Johnson, RS, The dynamics of waves interacting with the Equatorial Undercurrent, Geophys. Astrophys. Fluid Dyn., 109, 311-358, (2015)
[12] Constantin, A.; Johnson, R., An exact, steady, purely azimuthal equatorial flow with a free surface, J. Phys. Oceanogr., 46, 1935-1945, (2016)
[13] Constantin, A.; Johnson, R., Large gyres as a shallow-water asymptotic solution of euler’s equation in spherical coordinates, Proc. R. Soc. A, 473, 20170063, (2017) · Zbl 1404.86015
[14] Constantin, A.; Johnson, R., A nonlinear, three-dimensional model for ocean flows, motivated by some observations of the pacific equatorial undercurrent and thermocline, Phys. Fluids, 29, 056604, (2017)
[15] Constantin, A.; Johnson, R., Steady large-scale ocean flows in spherical coordinates, Oceanography, 31, 42-50, (2018)
[16] Constantin, A.; Monismith, S., Gerstner waves in the presence of mean currents and rotation, J. Fluid Mech., 820, 511-528, (2017) · Zbl 1387.86009
[17] Cushman-Roisin, B., Beckers, J.M.: Chapter 21—Equatorial Dynamics, International Geophysics, vol. 101. Academic Press, Cambridge (2011) · Zbl 1319.86001
[18] Dellar, PJ, Variations on a beta-plane: derivation of non-traditional beta-plane equations from hamilton’s principle on a sphere, J. Fluid Mech., 674, 174-195, (2011) · Zbl 1241.76428
[19] Drazin, P.G., Reid, W.H.: Hydrodynamic stability. Technical report. Cambridge university press (1981) · Zbl 0449.76027
[20] Friedlander, S.; Vishik, MM, Instability criteria for the flow of an inviscid incompressible fluid, Phys. Rev. Lett., 66, 2204, (1991) · Zbl 0968.76543
[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 three-dimensional gerstner-like equatorial water waves, Philos. Trans. R. Soc. A Math. Phys. Eng. Sci., 376, 20170088, (2017) · Zbl 1404.76040
[24] Henry, D.: On the deep-water Stokes wave flow. Int. Math. Res. Not. IMRN (2008). Art. ID rnn 071, 7 · Zbl 1245.76008
[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., A modified equatorial \(\beta \)-plane approximation modelling nonlinear wave-current interactions, J. Differ. Equ., 263, 2554-2566, (2017) · Zbl 1365.76020
[27] Henry, D.; Hsu, H-C, Instability of internal equatorial water waves, J. Differ. Equ., 258, 1015-1024, (2015) · Zbl 1446.76081
[28] Henry, D.; Martin, C-I, Exact, purely azimuthal stratified equatorial flows in cylindrical coordinates, Dyn. Partial Differ. Equ., 15, 337-349, (2018) · Zbl 1406.35251
[29] Henry, D., Martin, C.-I.: Free-surface, purely azimuthal equatorial flows in spherical coordinates with stratification. J. Differ. Equ. (2018). https://doi.org/10.1016/j.jde.2018.11.017 · Zbl 1412.35241
[30] Hsu, H-C, Some nonlinear internal equatorial flows, Nonlinear Anal. Real World Appl., 18, 69-74, (2014) · Zbl 1367.35182
[31] Ionescu-Kruse, D., Short-wavelength instabilities of edge waves in stratified water, Discrete Contin. Dyn. Syst. A, 35, 2053-2066, (2015) · Zbl 1302.76070
[32] Ionescu-Kruse, D., Instability of equatorially trapped waves in stratified water, Annali di Matematica Pura ed Applicata (1923-), 195, 585-599, (2016) · Zbl 1352.35191
[33] Ionescu-Kruse, D., Instability of Pollard’s exact solution for geophysical ocean flows, Phys. Fluids, 28, 086601, (2016)
[34] Ionescu-Kruse, D., On the short-wavelength stabilities of some geophysical flows, Philos. Trans. R. Soc. A Math. Phys. Eng. Sci., 376, 20170090, (2017) · Zbl 1404.76290
[35] Ionescu-Kruse, D.; Martin, C-I, Local stability for an exact steady purely azimuthal equatorial flow, J. Math. Fluid Mech., 20, 27-34, (2018) · Zbl 1394.76020
[36] Kluczek, M., Equatorial water waves with underlying currents in the f-plane approximation, Appl. Anal., 97, 1867-1880, (2018) · Zbl 1433.76024
[37] Leblanc, S., Local stability of gerstner’s waves, J. Fluid Mech., 506, 245-254, (2004) · Zbl 1062.76019
[38] Lifschitz, A.; Hameiri, E., Local stability conditions in fluid dynamics, Phys. Fluids A Fluid Dyn., 3, 2644-2651, (1991) · Zbl 0746.76050
[39] Martin, CI, On the vorticity of mesoscale ocean currents, Oceanography, 31, 28-35, (2018)
[40] Matioc, A-V, Exact geophysical waves in stratified fluids, Appl. Anal., 92, 2254-2261, (2013) · Zbl 1292.76018
[41] Rodriguez-Sanjurjo, A., Global diffeomorphism of the lagrangian flow-map for equatorially-trapped internal water waves, Nonlinear Anal. Theory Methods Appl., 149, 156-164, (2017) · Zbl 1354.35161
[42] Sastre-Gomez, S., Global diffeomorphism of the lagrangian flow-map defining equatorially trapped water waves, Nonlinear Anal., 125, 725-731, (2015) · Zbl 1330.35341
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