Oceanography The Official Magazine of
The Oceanography Society
Volume 31 Issue 03

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Volume 31, No. 3
Pages 22 - 27

Nonlinear Features of Equatorial Ocean Flows

David Henry
Article Abstract

We examine whether certain oceanographic features of equatorial flows can be modeled using a linear theoretical framework. In particular, we show, using elementary mathematical considerations, that linearization fails to capture the emergence and persistence of large coherent structures that are representative of upwelling and downwelling processes.


Henry, D. 2018. Nonlinear features of equatorial ocean flows. Oceanography 31(3):22–27, https://doi.org/10.5670/oceanog.2018.305.


Boyd, J.P. 2018. Dynamics of the Equatorial Ocean. Springer, Berlin.

Constantin, A. 2011. Nonlinear Water Waves with Applications to Wave-Current Interactions and Tsunamis. Vol. 81 in CBMS-NSF Conference Series in Applied Mathematics, Society for Industrial and Applied Mathematics, https://doi.org/​10.1137/1.9781611971873.

Constantin, A. 2012. An exact solution for equatorially trapped waves. Journal of Geophysical Research 117, C05029, https://doi.org/10.1029/2012JC007879.

Constantin, A. 2013. Some three-dimensional nonlinear equatorial flows. Journal of Physical Oceanography 43:165–175, https://doi.org/10.1175/JPO-D-12-062.1.

Constantin, A. 2014. Some nonlinear, equatorially trapped, nonhydrostatic internal geophysical waves. Journal of Physical Oceanography 44:781–789, https://doi.org/​10.1175/JPO-D-13-0174.1.

Constantin, A., and R.S. Johnson. 2015. The dynamics of waves interacting with the Equatorial Undercurrent. Geophysical & Astrophysical Fluid Dynamics 109:311–358, https://doi.org/10.1080/03091929.2015.1066785.

Constantin, A., and R.S. Johnson. 2016a. An exact, steady, purely azimuthal equatorial flow with a free surface. Journal of Physical Oceanography 46:1,935–1,945, https://doi.org/10.1175/JPO-D-15-0205.1.

Constantin, A., and R.S. Johnson. 2016b. An exact, steady, purely azimuthal flow as a model for the Antarctic Circumpolar Current. Journal of Physical Oceanography 46:3,585–3,594, https://doi.org/10.1175/JPO-D-16-0121.1.

Constantin, A., and R.S. Johnson. 2016c. Current and future prospects for the application of systematic theoretical methods to the study of problems in physical oceanography. Physics Letters A 380:3,007–3,012, https://doi.org/10.1016/​j.physleta.2016.07.036.

Constantin, A., and R.S. Johnson. 2017. A nonlinear, three-dimensional model for ocean flows, motivated by some observations of the Pacific equatorial undercurrent and thermocline. Physics of Fluids 29, 056604, https://doi.org/​10.1063/1.4984001.

Cushman-Roisin, B., and J.-M. Beckers. 2011. Introduction to Geophysical Fluid Dynamics: Physical and Numerical Aspects. Academic Press, Waltham, MA, 875 pp.

Fedorov, A.V., and J.N. Brown. 2009. Equatorial waves. Pp. 3,679 –3,695 in Encyclopedia of Ocean Sciences, J. Steele, ed., Academic Press, San Diego.

Gerkema, T., J.T.F. Zimmerman, L.R.M. Maas, and H. van Haren. 2008. Geophysical and astrophysical fluid dynamics beyond the traditional approximation. Reviews of Geophysics 46, RG2004, https://doi.org/10.1029/2006RG000220.

Gill, A. 1982. Atmosphere-Ocean Dynamics. New York: Academic Press, 662 pp.

Henry, D. 2016. Equatorially trapped nonlinear water waves in a β-plane approximation with centripetal forces. Journal of Fluid Mechanics 804, R1, https://doi.org/​10.1017/jfm.2016.544.

Henry, D. 2017. A modified equatorial β-plane approximation modelling nonlinear wave-current interactions. Journal of Differential Equations 263:2,554–2,566, https://doi.org/10.1016/j.jde.2017.04.007.

Henry, D. 2018. On nonlinearity in three-dimensional equatorial flows. Journal of Nonlinear Mathematical Physics 25:351–357, https://doi.org/10.1080/14029251.​2018.1494780.

Izumo, T. 2005. 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 Dynamics 55:110–123, https://doi.org/10.1007/s10236-005-0115-1.

Johnson, G.C., M.J. McPhaden, and E. Firing. 2001. Equatorial Pacific Ocean horizontal velocity, divergence, and upwelling. Journal of Physical Oceanography 31(3):839–849, https://doi.org/10.1175/1520-0485(2001)031​<0839:EPOHVD>2.0.CO;2.

Johnson, R.S. 2018. Application of the ideas and techniques of classical fluid mechanics to some problems in physical oceanography. Philosophical Transactions of the Royal Society A 276, 20170092, https://doi.org/10.1098/rsta.2017.0092.

Lighthill, J. 1978. Waves in Fluids. Cambridge University Press, Cambridge and New York.

Majda, A.J., and A.L. Bertozzi. 2002. Vorticity and Incompressible Flow. Cambridge University Press, Cambridge.

Marshall, J., and R.A. Plumb. 2016. Atmosphere, Ocean and Climate Dynamics: An Introductory Text. Academic Press, London.

Maslowe, S.A. 1986. Critical layers in shear flows. Annual Review of Fluid Mechanics 18:405–432, https://doi.org/10.1146/annurev.fl.18.010186.002201.

McCreary, J.P. 1985. Modeling equatorial ocean circulation. Annual Review of Fluid Mechanics 17(1):359–409, https://doi.org/10.1146/annurev.fl.17.010185.002043.

Meiss, J.D. 2007. Differential Dynamical Systems. Society for Industrial and Applied Mathematics, Philadelphia, https://doi.org/10.1137/1.9780898718232.

Philander, S.G.H. 1979. Equatorial waves in the presence of the equatorial undercurrent. Journal of Physical Oceanography 9:254–262, https://doi.org/​10.1175/​1520-0485(1979)009<0254:EWITPO>2.0.CO;2.

Sirven, J. 1996. The equatorial undercurrent in a two layer shallow water model. Journal of Marine Systems 9:171–186, https://doi.org/10.1016/S0924-7963(96)00041-3.

Stewart, A.L., and P.J. Dellar. 2010. Multilayer shallow water equations with complete Coriolis force: Part I. Derivation on a non-traditional beta-plane. Journal of Fluid Mechanics 651:387–413, https://doi.org/10.1017/S0022112009993922.

Vallis, G.K. 2017. Atmospheric and Oceanic Fluid Dynamics, 2nd ed. Cambridge University Press.

Walton, D.W.H. 2013. Antarctica: Global Science from a Frozen Continent. Cambridge University Press, Cambridge, 352 pp.

Wunsch, C. 2015. Modern Observational Physical Oceanography. Princeton University Press, New Jersey, 512 pp.