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

View Issue TOC
Volume 31, No. 3
Pages 42 - 50

Steady Large-Scale Ocean Flows in Spherical Coordinates

Adrian Constantin Robin Stanley Johnson
Article Abstract

We show that rotating spherical coordinates can and should be used for the study of steady, large-scale ocean flows, and describe the role of the stream function in simplifying the governing equations and for generating solutions.

Citation

Constantin, A., and R.S. Johnson. 2018. Steady large-scale ocean flows in spherical coordinates. Oceanography 31(3):42–50, https://doi.org/10.5670/oceanog.2018.308.

References

Amitai, Y., Y. Lehahn, A. Lazar, and E. Heifetz. 2010. Surface circulation of the eastern Mediterranean Levantine basin: Insights from analyzing 14 years of satellite altimetry data. Journal of Geophysical Research 115(C10), https://doi.org/​10.1029/​2010JC006147.

Andreev, V.K., O.V. Kaptsov, V.V. Pukhnachev, and A.A. Rodionov. 1998. Applications of Group-Theoretical Methods in Hydrodynamics. Kluwer Academic Publishers, Dordrecht, 396 pp., https://doi.org/10.1007/978-94-017-0745-9.

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

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. 2016. 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. 2017a. 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.

Constantin, A., and R.S. Johnson. 2017b. Large gyres as a shallow-water asymptotic solution of Euler’s equation in spherical coordinates. Proceedings of the Royal Society A 473(2200), https://doi.org/10.1098/rspa.2017.0063.

Dellar, P.J. 2011. Variations on a beta-plane: Derivation of non-traditional beta-plane equations from Hamilton’s principle on a sphere. Journal of Fluid Mechanics 674:174–195, https://doi.org/10.1017/S0022112010006464.

Ekman, V.W. 1905. On the influence of the Earth’s rotation on ocean currents. Arkiv för matematik, astronomi och fysik 2:1–52.

Fofonoff, N.P. 1954. Steady flow in a frictionless homogeneous ocean. Journal of Marine Research 13:254–262.

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.

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.

McWilliams, J.C., and E. Huckle. 2006. Ekman layer rectification. Journal of Physical Oceanography 36:1,646–1,659, https://doi.org/10.1175/JPO2912.1.

Newns, W.F. 1967. Functional dependence. The American Mathematical Monthly 74:911–920, https://doi.org/10.2307/2315264.

O’Neill, M.E., K.E. Emanuel, and G.R. Flierl. 2015. Polar vortex formation in giant-planet atmospheres due to moist convection. Nature Geoscience 8:523–527, https://doi.org/10.1038/ngeo2459.

Özkömen, T.M., and E.C. Chassignet. 1998. Emergence of inertial gyres in a two-layer quasigeostrophic ocean model. Journal of Physical Oceanography 28:461–484, https://doi.org/10.1175/1520-0485(1998)028​<0461:EOIGIA>2.0.CO;2.

Poulain, P.-M., M. Menna, and E. Mauri. 2012. Surface geostrophic circulation of the Mediterranean Sea derived from drifter and satellite altimeter data. Journal of Physical Oceanography 42:973–990, https://doi.org/10.1175/JPO-D-11-0159.1.

Rudnik, D.L. 2003. Observations of momentum transfer in the upper ocean: Did Ekman get it right? Pp. 163–170 in Near-Boundary Processes and Their Parametrization: Proceedings of the 13th‘Aha Huliko‘a Winter Workshop. University of Hawaii at Manoa. Honolulu, HI.

Talley, L.D., G.L. Pickard, W.J. Emery, and J.H. Swift. 2011. Descriptive Physical Oceanography: An Introduction. Academic Press, 560 pp.

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

Zikanov, O., D.N. Slinn, and M.R. Dhanak. 2003. Large eddy simulations of the wind-induced turbulent Ekman layer. Journal of Fluid Mechanics 495:343–368, https://doi.org/10.1017/S0022112003006244.