• Berger, M. S., 1977: Nonlinearity and Functional Analysis. Academic Press, 417 pp.

  • Constantin, A., 2011: Nonlinear Water Waves with Applications to Wave-Current Interactions and Tsunamis. SIAM, 321 pp.

  • Constantin, A., 2012: An exact solution for equatorially trapped waves. J. Geophys. Res., 117, C05029, doi:10.1029/2012JC007879.

  • Constantin, A., 2014: Some nonlinear, equatorially trapped, nonhydrostatic internal geophysical waves. J. Phys. Oceanogr., 44, 781789, doi:10.1175/JPO-D-13-0174.1.

    • Search Google Scholar
    • Export Citation
  • Constantin, A., , and R. S. Johnson, 2015: The dynamics of waves interacting with the Equatorial Undercurrent. Geophys. Astrophys. Fluid Dyn., 109, 311358, doi:10.1080/03091929.2015.1066785.

    • Search Google Scholar
    • Export Citation
  • Deser, C., 1994: Daily surface wind variations over the equatorial Pacific Ocean. J. Geophys. Res., 99, 23 07123 078, doi:10.1029/94JD02155.

    • Search Google Scholar
    • Export Citation
  • Dijkstra, H. A., 2000: Nonlinear Physical Oceanography: A Dynamical Systems Approach to the Large Scale Ocean Circulation and El Niño. Kluwer, 456 pp.

  • Drazin, P., , and N. Riley, 2007: The Navier-Stokes Equations: A Classification of Flows and Exact Solutions. Cambridge University Press, 196 pp.

  • Gerstner, F., 1809: Theorie der Wellen. Ann. Phys., 32, 412445, doi:10.1002/andp.18090320808.

  • Henry, D., 2013: An exact solution for equatorial geophysical water waves with an underlying current. Eur. J. Mech. B/Fluids, 38, 1821, doi:10.1016/j.euromechflu.2012.10.001.

    • Search Google Scholar
    • Export Citation
  • Hsu, H.-C., 2014: Some nonlinear internal equatorial flows. Nonlinear Anal. Real World Appl., 18, 6974, doi:10.1016/j.nonrwa.2013.12.011.

    • Search Google Scholar
    • Export Citation
  • Johnson, G. C., , and M. J. McPhaden, 2001: Equatorial Pacific ocean horizontal velocity, divergence, and upwelling. J. Phys. Oceanogr., 31, 839849, doi:10.1175/1520-0485(2001)031<0839:EPOHVD>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Johnson, R. S., 1997: A Modern Introduction to the Mathematical Theory of Water Waves. Cambridge University Press, 445 pp.

  • Kessler, W. S., , and M. J. McPhaden, 1995: Oceanic equatorial waves and the 1991–93 El Niño. J. Climate, 8, 17571774, doi:10.1175/1520-0442(1995)008<1757:OEWATE>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • McCreary, J. P., 1985: Modeling equatorial ocean circulation. Annu. Rev. Fluid Mech., 17, 359409, doi:10.1146/annurev.fl.17.010185.002043.

    • Search Google Scholar
    • Export Citation
  • McPhaden, M. J., , J. A. Proehl, , and L. M. Rothstein, 1986: The interaction of equatorial Kelvin waves with realistically sheared zonal currents. J. Phys. Oceanogr., 16, 14991515, doi:10.1175/1520-0485(1986)016<1499:TIOEKW>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Segar, D. A., 2012: Introduction to Ocean Science. 3rd ed. Segar, 565 pp. [Available online at https://reefimages.com/oceans/oceans.html.]

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An Exact, Steady, Purely Azimuthal Equatorial Flow with a Free Surface

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  • 1 Faculty of Mathematics, University of Vienna, Vienna, Austria
  • | 2 School of Mathematics and Statistics, Newcastle University, Newcastle upon Tyne, United Kingdom
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Abstract

The general problem of an ocean on a rotating sphere is considered. The governing equations for an inviscid, incompressible fluid, written in spherical coordinates that are fixed at a point on the rotating Earth, together with the free surface and rigid bottom boundary conditions, are introduced. An exact solution of this system is presented; this describes a steady flow that is moving only in the azimuthal direction, with no variation in this direction. However, this azimuthal velocity component has an arbitrary variation with depth (i.e., radius), and so, for example, an Equatorial Undercurrent (EUC) can be accommodated. The pressure boundary condition at the free surface relates this pressure to the shape of the surface via a Bernoulli relation; this provides the constraint on the existence of a solution, although the restrictions are somewhat involved in spherical coordinates. To examine this constraint in more detail, the corresponding problems in model cylindrical coordinates (with the equator “straightened” to become a generator of the cylinder), and then in the tangent-plane version (with the β-plane approximation incorporated), are also written down. Both these possess similar exact solutions, with a Bernoulli condition that is more readily interpreted in terms of the choices available. Some simple examples of the surface pressure, and associated surface distortion, are presented. The relevance of these exact solutions to more complicated, and physically realistic, flow structures is briefly mentioned.

Denotes Open Access content.

Corresponding author address: A. Constantin, Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria. E-mail: adrian.constantin@univie.ac.at

Abstract

The general problem of an ocean on a rotating sphere is considered. The governing equations for an inviscid, incompressible fluid, written in spherical coordinates that are fixed at a point on the rotating Earth, together with the free surface and rigid bottom boundary conditions, are introduced. An exact solution of this system is presented; this describes a steady flow that is moving only in the azimuthal direction, with no variation in this direction. However, this azimuthal velocity component has an arbitrary variation with depth (i.e., radius), and so, for example, an Equatorial Undercurrent (EUC) can be accommodated. The pressure boundary condition at the free surface relates this pressure to the shape of the surface via a Bernoulli relation; this provides the constraint on the existence of a solution, although the restrictions are somewhat involved in spherical coordinates. To examine this constraint in more detail, the corresponding problems in model cylindrical coordinates (with the equator “straightened” to become a generator of the cylinder), and then in the tangent-plane version (with the β-plane approximation incorporated), are also written down. Both these possess similar exact solutions, with a Bernoulli condition that is more readily interpreted in terms of the choices available. Some simple examples of the surface pressure, and associated surface distortion, are presented. The relevance of these exact solutions to more complicated, and physically realistic, flow structures is briefly mentioned.

Denotes Open Access content.

Corresponding author address: A. Constantin, Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria. E-mail: adrian.constantin@univie.ac.at
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