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Homogenization of potential vorticity in planetary gyres

Published online by Cambridge University Press:  20 April 2006

Peter B. Rhines  and
William R. Young
Peter B. Rhines
Affiliation:
Woods Hole Oceanographic Institution, Woods Hole, Massachusetts 02543
William R. Young
Affiliation:
Woods Hole Oceanographic Institution, Woods Hole, Massachusetts 02543
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Abstract

The mean circulation of planetary fluids tends to develop uniform potential vorticity q in regions where closed time-mean streamlines or closed isolines of mean potential vorticity exist. This state is established in statistically steady flows by geostrophic turbulence or by wave-induced potential-vorticity flux. At the outer edge of the closed contours the expelled gradients of q are concentrated. Beyond this transition lies motionless fluid, or a different flow regime in which the planetary gradient of q may be dominant. The homogenized regions occur where direct forcing by external stress or heating within the closed isoline is negligible, upon the potential-density surface under consideration. In the stably stratified ocean such regions are found at depths greater than those of direct wind-induced stress or penetrative cooling. In ‘channel’ models of the atmosphere we again find constant q when mesoscale eddies cause the dominant potential-vorticity flux. In the real atmosphere the results here can apply only where internal heating is negligible. The derivations given here build upon the Prandtl–Batchelor theorem, which applies to non-rotating, steady two-dimensional flow. Supporting evidence is found in numerical circulation models and oceanic observations.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 122 , September 1982 , pp. 347 - 367
Copyright
© 1982 Cambridge University Press

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References

Anderson, D. & Gill, A. E. 1975 Spin-up of a stratified ocean with application to upwelling. Deep-Sea Res. 22, 583–596.Google Scholar
Batchelor, G. K. 1956 Steady laminar flow with closed streamlines at large Reynolds numbers. J. Fluid Mech. 1, 177–190.Google Scholar
Behringer, D. & Stommel, H. 1980 The β-spiral in the North Atlantic subtropical gyre. Deep-Sea Res. 27, 225–238.Google Scholar
Benney, D. & Bergeron, R. 1969 A new class of nonlinear waves in parallel flows. Stud. Appl. Math. 48, 181–204.Google Scholar
Davis, R. E. 1969 On the high Reynolds number flow over a wavy boundary. J. Fluid Mech. 36, 337–351.Google Scholar
Grimshaw, R. 1969 On steady recirculating flows. J. Fluid Mech. 39, 695–703.Google Scholar
Holland, W. R. & Rhines, P. B. 1980 An example of eddy-induced circulation. J. Phys. Oceanogr. 10, 1010–1031.Google Scholar
Ingersoll, A. P. 1969 Inertial Taylor columns and Jupiter's great red spot. J. Atmos. Sci. 26, 744–752.Google Scholar
Leetma, A., Niiler, P. P. & Stommel, H. 1977 Does the Sverdrup relation account for the Mid-Atlantic circulation?. J. Mar. Res. 35, 1–10.Google Scholar
Mcdowell, S., Rhines, P. B. & Keffer, T. 1982 North Atlantic potential vorticity and its relation to the general circulation. J. Phys. Oceanogr. (in the press).Google Scholar
Mcwilliams, J. C. & Chow, J. H. S. 1981 Equilibrium geostrophic turbulence. I. A reference solution in a β-plane channel. J. Phys. Oceanogr. 11, 921–949.Google Scholar
Mcwilliams, J., Holland, W. R. & Chow, J. H. S. 1978 A description of numerical Antarctic circumpolar currents. Dyn. Atmos. Oceans 2, 213–291.Google Scholar
Moffatt, H. K. 1978 Magnetic Field Generation in Electrically Conducting Fluids. Cambridge University Press.
Montgomery, R. B. 1938 Circulation in upper layers of southern North Atlantic deduced with use of isentropic analysis. Pap. Phys. Oceanogr. Meteorol. W.H.O.I./M.I.T.
Proctor, M. R. E. 1975 Nonlinear mean-field dynamo models and related topics. Ph.D. thesis, Cambridge University.
Redekopp, L. 1980 Solitary Rossby waves with critical layers. Geophysical Fluid Dynamics Lectures, Woods Hole Oceanographic Institution, pp. 55–72.
Rhines, P. B. 1977 The dynamics of unsteady currents. In The Sea, vol. VI (ed. E. Goldberg), pp. 189–318. Wiley.
Rhines, P. B. & Holland, W. R. 1979 A theoretical discussion of eddy-induced circulation. Dyn. Atmos. Oceans 3, 285–325.Google Scholar
Rhines, P. B. & Young, W. R. 1982 A theory of wind-driven ocean circulation. J. Mar. Res. (in the press).Google Scholar
Rooth, C., Stommel, H. M. & Veronis, G. 1978 On motion in steady layered geostrophic models. J. Oceanog. Soc. Japan 34, 265–267.Google Scholar
Rossby, C. G. 1947 On the distribution of angular velocity in gaseous envelopes under the influence of large-scale horizontal mixing processes. Bull. Am. Met. Soc. 28, 53–68.Google Scholar
Stommel, H. & Schott, F. 1977 The β-spiral and the determination of the absolute velocity field from hydrographic station data. Deep-Sea Res. 24, 325–329.Google Scholar
Weiss, N. O. 1966 The expulsion of magnetic fluxing eddies. Proc. R. Soc. Lond. A 293, 310–328.Google Scholar
Welander, P. 1969 Effects of planetary topography on deep-sea circulation. Deep-Sea Res. Suppl. 16, 369–391.Google Scholar
Yamagata, T. 1981 On a steady mean flow in a generalized closed geostrophic contour – a generalization of Prandtl–Batchelor theorem. J. Met. Soc. Japan (in the press).Google Scholar