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Free-surface, purely azimuthal equatorial flows in spherical coordinates with stratification. (English) Zbl 1412.35241

Summary: In this paper we derive an exact solution to the governing equations for geophysical fluid dynamics in spherical coordinates which incorporates fluid stratification. This solution represents a steady, purely-azimuthal equatorial flow with an associated free-surface. Following the derivation of the solution we demonstrate that there is a well-defined relationship between the imposed pressure at the free-surface and the resulting distortion of the surface’s shape. Finally, the solution for stratified fluid flow is subjected to a short-wavelength stability analysis.

MSC:

35Q31 Euler equations
35Q35 PDEs in connection with fluid mechanics
35Q86 PDEs in connection with geophysics
35R35 Free boundary problems for PDEs
76E20 Stability and instability of geophysical and astrophysical flows
76U05 General theory of rotating fluids
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: ASME New York), 71-77
[2] Berger, M. S., Nonlinearity and Functional Analysis, (1977), Academic Press: Academic Press New York
[3] Constantin, A., An exact solution for equatorially trapped waves, J. Geophys. Res., Oceans, 117, (2012)
[4] Constantin, A., Some three-dimensional nonlinear equatorial flows, J. Phys. Oceanogr., 43, 165-175, (2013)
[5] Constantin, A., Some nonlinear, equatorially trapped, nonhydrostatic internal geophysical waves, J. Phys. Oceanogr., 44, 2, 781-789, (2014)
[6] Constantin, A.; Germain, P., Instability of some equatorially trapped waves, J. Geophys. Res., Oceans, 118, 2802-2810, (2013)
[7] Constantin, A.; Johnson, R. S., The dynamics of waves interacting with the equatorial undercurrent, Geophys. Astrophys. Fluid Dyn., 109, 4, 311-358, (2015)
[8] Constantin, A.; Johnson, R. S., An exact, steady, purely azimuthal equatorial flow with a free surface, J. Phys. Oceanogr., 46, 6, 1935-1945, (2016)
[9] Constantin, A.; Johnson, R. S., An exact, steady, purely azimuthal flow as a model for the antarctic circumpolar current, J. Phys. Oceanogr., 46, 12, 3585-3594, (2016)
[10] 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, 5, (2017), doi
[11] Constantin, A.; Johnson, R. S., Large gyres as a shallow-water asymptotic solution of Euler’s equation in spherical coordinates, Proc. R. Soc. Lond. A, 473, (2017) · Zbl 1404.86015
[12] Constantin, A.; Monismith, S., Gerstner waves in the presence of mean currents and rotation, J. Fluid Mech., 820, 511-528, (2017) · Zbl 1387.86009
[13] Cushman-Roisin, B.; Beckers, J.-M., Introduction to Geophysical Fluid Dynamics: Physical and Numerical Aspects, (2011), Academic: Academic Waltham, Mass · Zbl 1319.86001
[14] Dijkstra, H. A., Nonlinear Physical Oceanography, (2000), Kluwer Acad. Publ.: Kluwer Acad. Publ. Dordrecht · Zbl 0964.86003
[15] Fedorov, A. V.; Brown, J. N., Equatorial waves, (Steele, J., Encyclopedia of Ocean Sciences, (2009), Academic Press: Academic Press New York), 3679-3695
[16] 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
[17] Genoud, F.; Henry, D., Instability of equatorial water waves with an underlying current, J. Math. Fluid Mech., 16, 661-667, (2014) · Zbl 1308.76035
[18] Gill, A., Atmosphere-Ocean Dynamics, (1982), Academic Press: Academic Press New York
[19] 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
[20] Henry, D., Equatorially trapped nonlinear water waves in a β-plane approximation with centripetal forces, J. Fluid Mech., 804, (2016) · Zbl 1454.76024
[21] Henry, D., On three-dimensional Gerstner-like equatorial water waves, Philos. Trans. R. Soc. A, 376, (2018) · Zbl 1404.76040
[22] Henry, D.; Hsu, H.-C., Instability of internal equatorial water waves, J. Differential Equations, 258, 4, 1015-1024, (2015) · Zbl 1446.76081
[23] D. Henry, C.I. Martin, Exact, purely azimuthal stratified equatorial flows in cylindrical coordinates, accepted in Dyn. Partial Differ. Equ. · Zbl 1406.35251
[24] Hsu, H.-C., An exact solution for equatorial waves, Monatsh. Math., 176, 1, 143-152, (2015) · Zbl 1304.76011
[25] Hsu, H.-C.; Martin, C. I., Free-surface capillary-gravity azimuthal equatorial flows, Nonlinear Anal., 144, 1-9, (2016) · Zbl 1346.35153
[26] Ionescu-Kruse, D., Instability of equatorially trapped waves in stratified water, Ann. Mat. Pura Appl., 195, 585-599, (2016) · Zbl 1352.35191
[27] Ionescu-Kruse, D., Instability of Pollard’s exact solution for geophysical ocean flows, Phys. Fluids, 28, (2016)
[28] Ionescu-Kruse, D., On the short-wavelength stabilities of some geophysical flows, Philos. Trans. R. Soc. A, 376, (2018) · Zbl 1404.76290
[29] Ionescu-Kruse, D., Local stability for an exact steady purely azimuthal flow which models the antarctic circumpolar current, J. Math Fluid Mech. · Zbl 1458.76045
[30] Ionescu-Kruse, D.; Martin, C. I., Local stability for an exact steady purely azimuthal equatorial flow, J. Math. Fluid Mech. · Zbl 1394.76020
[31] 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)
[32] Johnson, G. C.; McPhaden, M. J.; Firing, E., Equatorial Pacific ocean horizontal velocity, divergence, and upwelling, J. Phys. Oceanogr., 31, 839-849, (2001)
[33] Johnson, R. S., Application of the ideas and techniques of classical fluid mechanics to some problems in physical oceanography, Philos. Trans. R. Soc. A, 376, (2018) · Zbl 1404.86018
[34] Kessler, W. S.; McPhaden, M. J., Oceanic equatorial waves and the 1991-93 El Niño, J. Climate, 8, 1757-1774, (1995)
[35] Lifschitz, A.; Hameiri, E., Local stability conditions in fluid dynamics, Phys. Fluids, 3, 2644-2651, (1991) · Zbl 0746.76050
[36] Martin, C. I., On the existence of free-surface azimuthal equatorial flows, Appl. Anal., 96, 7, 1207-1214, (2017) · Zbl 1368.35215
[37] Matioc, A.-V., An explicit solution for deep water waves with Coriolis effects, J. Nonlinear Math. Phys., 19, (2012) · Zbl 1362.76015
[38] McCreary, J. P., Modeling equatorial ocean circulation, Annu. Rev. Fluid Mech., 17, 359-409, (1985) · Zbl 0596.76115
[39] Philander, S., Equatorial waves in the presence of the equatorial undercurrent, J. Phys. Oceanogr., 9, 254-262, (1979)
[40] Sirven, J., The equatorial undercurrent in a two layer shallow water model, J. Mar. Syst., 9, 171-186, (1996)
[41] Vallis, G. K., Atmospheric and Oceanic Fluid Dynamics, (2006), Cambridge University Press
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