Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Numerical studies of transient planetary circulations in a wind-driven ocean

  • 23 Accesses

  • 3 Citations


The time-dependent primitive equations for a shallow homogeneous ocean with a free surface are solved for a bounded basin on the sphere, driven by a steady zonal wind stress and subject to lateral viscous dissipation. These are the vertically integrated equations for a free-surface model, and are integrated to 60 days from an initial state of rest by an explicit centered-difference method with zero-slip lateral boundary conditions. In a series of comparative numerical solutions it is shown that at least a 2-deg resolution is needed to resolve the western boundary currents adequately and to avoid undue distortion of the transient (Rossby waves. The β-plane formulation is shown to be an adequate approximation for the mean circulation in the lower and middle latitudes, but noticeably intensifies the transports poleward of about 50 deg and both slows and distorts the transients in the central basin. The influence of the (southern) zonal boundary on the transport solutions is confined to the southernmost gyre, except in the region of the western boundary currents where its influence spreads to the northern edge of the basin by 30 days. The total boundary current transport is shown to be approximately proportional to the zonal width of the basin and independent of the basin's (uniform) depth, while the elevation of the free water surface is inversely proportional to the basin depth, in accordance with linear theory. The introduction of bottom friction has a marked damping effect on the transient Rossby waves, and also reduces the maximum boundary-current transport. The solutions throughout are approximately geostrophic and are only slightly nonlinear.

The root-mean-square (rms) transport variability during the period 30 to 60 days is concentrated in the southwest portion of the basin through the reflection of the transient Rossby waves from the western shore and has a maximum corresponding to an rms current variability of about 3 cm sec−1. The transport variabilities are about 10 percent of the mean zonal transport and more than 100 percent of the mean meridional transport over a considerable region of the western basin (outside the western boundary current regime). Some 99 percent of the total kinetic energy is associated with the zonal mean and standing zonal waves, which are also responsible for the bulk of the meridional transport of zonal angular momentum. Although the transient Rossby waves systematically produce a momentum flux convergence at the latitude of the maximum eastward current, much in the manner of their atmospheric counterparts, this is only a relatively small contribution to the zonal oceanic momentum balance; the bulk of the mean zonal stress is here balanced by a nearly stationary net pressure torque exerted against the meridional boundaries by the wind-raised water. In an ocean without such boundaries the role of the transient circulations may be somewhat more important.

This is a preview of subscription content, log in to check access.


  1. [1]

    A. Arakawa,Computational design for long-term numerical integration of the equations of fluid motion: two-dimensional incompressible flow, part 1, J. Comput. Phys.1 (1966), 119–143.

  2. [2]

    K. Bryan,A numerical investigation of a nonlinear model of a wind-driven ocean, J. Atmos. Sci.20 (1963), 594–606.

  3. [3]

    K. Bryan andM. D. Cox,A numerical investigation of the oceanic general circulation, Tellus19, (1967), 54–80.

  4. [4]

    K. Bryan andM. D. Cox,A nonlinear solution of an ocean driven by wind and differential heating: Parts I and II, J. Atmos. Sci.25 (1968), 945–967, 968–978.

  5. [5]

    W. P. Crowley,A numerical model for viscous, nondivergent, barotropic, wind-driven ocean circulations, J. Comput. Phys.6 (1970), 183–199.

  6. [6]

    W. L. Gates,A numerical study of transient Rossby waves in a wind-driven homogeneous ocean, J. Atmos. Sci.25 (1968), 3–22.

  7. [7]

    W. L. Gates,The effects of coastal orientation on Rossby wave reflection and the resulting largescale oceanic circulation, J. Geophys. Res.75 (1970), 4105–4120.

  8. [8]

    W. L. Gates,A note on the Reynolds stress and lateral eddy viscosity due to transient oceanic Rossby waves, Pure and Applied Geophys.96 (1972), 217–227.

  9. [9]

    W. R. Holland,On the wind-driven circulation in an ocean with bottom topography, Tellus19 (1967), 582–600.

  10. [10]

    E. O. Holopainen,On the mean meridional circulation and the flux of angular momentum over the northern hemisphere, Tellus19 (1967), 1–13.

  11. [11]

    V. M. Kamenkovich,On the coefficients of eddy diffusion and eddy viscosity in large-scale oceanic and atmospheric motions, Izv. Acad. Sci. USSR, Atmos. Oc. Phys.3 (1967), 777–781.

  12. [12]

    M. S. Longuet-Higgins,Planetary waves on a rotating sphere, Proc. Roy. Soc. London, Ser. A279, (1964), 446–473.

  13. [13]

    M. S. Longuet-Higgins,Planetary waves on a rotating sphere, Proc. Roy. Soc. London, Ser. A284 (1965), 40–68.

  14. [14]

    W. M. Munk,On the wind-driven ocean circulation, J. Meteor.7 (1950), 79–93.

  15. [15]

    J. Pedlosky,A study of the time dependent ocean circulation., J. Atmos. Sci.22 (1965), 267–272.

  16. [16]

    J. Pedlosky,Fluctuating winds and ocean circulation, Tellus19 (1967), 250–267.

  17. [17]

    N. A. Phillips,Large-scale eddy motion in the western Atlantic, J. Geophys. Res.71 (1966), 3883–3891.

  18. [18]

    N. A. Phillips,The equations of motion for a shallow rotating atmosphere and the ‘traditional approximation’. J. Atmos. Sci.23 (1966), 626–628.

  19. [19]

    V. P. Starr,Physics of Negative Viscosity Phenomena (McGray-Hill, New York, 1968), 256 pp.

  20. [20]

    H. Stommel,The westward intensification of ocean currents, Trans. Amer. Geophys. Un.29 (1948), 202–206.

  21. [21]

    H. U. Sverdrup,Wind-driven currents in a baroclinic ocean: with application to the equatorial currents of the eastern Pacific, Proc. Nat. Acad. Sci.33 (1947), 318–326.

  22. [22]

    K. Takano,Effet de la sphéricité de la terre sur la circulation générale dans un ocean, J. Oc. Soc. Japan22 (1966), 255–263.

  23. [23]

    G. Veronis,On the approximations involved in transforming the equations of motion from a spherical surface to the β-plane, I. Barotropic systems, J. Mar. Res.21 (1963), 110–124.

  24. [24]

    G. Veronis,Generation of mean ocean circulation by fluctuating winds, Tellus18 (1966), 67–76.

  25. [25]

    G. Veronis,Wind-driven ocean circulation-part I. Linear theory and perturbation analysis, Deep-Sea Res.13 (1966), 17–29.

  26. [26]

    G. Veronis,Wind-driven ocean circulation-part II. Numerical solutions of the nonlinear problem, Deep-Sea Res.13 (1966), 31–55.

  27. [27]

    F. Webster,The effect of meanders on the kinetic energy balance of the Gulf Stream, Tellus13 (1961), 392–401.

Download references

Author information

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Gates, W.L. Numerical studies of transient planetary circulations in a wind-driven ocean. PAGEOPH 99, 169–200 (1972). https://doi.org/10.1007/BF00875275

Download citation


  • Rossby Wave
  • Western Boundary Current
  • Free Water Surface
  • Zonal Wind Stress
  • Meridional Transport