• Andrews, D. G., and M. E. McIntyre, 1976: Planetary waves in horizontal and vertical shear: The generalized Eliassen–Palm relation and the zonal mean acceleration. J. Atmos. Sci, 33 , 20312048.

    • Search Google Scholar
    • Export Citation
  • Andrews, D. G., and M. E. McIntyre, 1978: Generalized Eliassen–Palm and Charney–Drazin theorems for waves on axisymmetric mean flows in compressible atmosphere. J. Atmos. Sci, 35 , 175185.

    • Search Google Scholar
    • Export Citation
  • Andrews, D. G., J. R. Holton, and C. B. Leovy, 1987: Middle Atmosphere Dynamics. Academic Press, 489 pp.

  • de Szoeke, R. A., and A. F. Bennett, 1993: Microstructure fluxes across density surfaces. J. Phys. Oceanogr, 23 , 22542264.

  • Gille, S. T., and R. E. Davis, 1999: The influence of mesoscale eddies on coarsely resolved density: An examination of subgrid-scale parameterization. J. Phys. Oceanogr, 29 , 11091123.

    • Search Google Scholar
    • Export Citation
  • Held, I. M., and T. Schneider, 1999: The surface branch of the zonally averaged mass transport circulation in the troposphere. J. Atmos. Sci, 56 , 16881697.

    • Search Google Scholar
    • Export Citation
  • Killworth, P. D., 2001: Boundary conditions on quasi–Stokes velocities in parameterization. J. Phys. Oceanogr, 31 , 11321155.

  • Kushner, P. J., and I. M. Held, 1999: Potential vorticity fluxes and wave–mean flow interactions. J. Atmos. Sci, 56 , 948958.

  • McDougall, T. J., 1998: Three-dimensional residual-mean theory. Ocean Modelling and Parameterization, E. P. Chassignet and J. Verron, Eds., NATO Science Series, Vol. 516, Kluwer, 269– 302.

    • Search Google Scholar
    • Export Citation
  • McDougall, T. J., and P. C. McIntosh, 2001: The temporal-residual-mean velocity. Part II: Isopycnal interpretation and the tracer and momentum equations. J. Phys. Oceanogr, 31 , 12221246.

    • Search Google Scholar
    • Export Citation
  • McIntosh, P. C., and T. J. McDougall, 1996: Isopycnal averaging and the residual mean circulation. J. Phys. Oceanogr, 26 , 16551660.

  • Nurser, A. J. G., and M-M. Lee, 2004: Isopycnal averaging at constant height. Part I: The formulation and a case study. J. Phys. Oceanogr, 34 , 27212739.

    • Search Google Scholar
    • Export Citation
All Time Past Year Past 30 Days
Abstract Views 0 0 0
Full Text Views 77 27 0
PDF Downloads 50 20 0

Isopycnal Averaging at Constant Height. Part II: Relating to the Residual Streamfunction in Eulerian Space

View More View Less
  • 1 Southampton Oceanography Centre, Empress Dock, Southampton, United Kingdom
Restricted access

Abstract

In , the “vertical” transport streamfunction was defined as resulting from isopycnic averaging at constant height in the same way that the meridional streamfunction results from averaging at constant latitude. Part II here discusses the relationship between these two isopycnic streamfunctions and the Eulerian residual streamfunction that arises from the transformed Eulerian mean (TEM). It is known that the meridional isopycnic streamfunction can be approximated by a Taylor expansion to give an Eulerian residual streamfunction involving the horizontal eddy flux. This Taylor expansion approximation works well in the interior, removing the spurious mixing associated with the simple Eulerian-averaged streamfunction. However, it fails near the surface where isopycnals outcrop to the surface. It can be shown in a similar way that the vertical isopycnic streamfunction can formally be approximated by a residual streamfunction involving the vertical eddy flux. However, if horizontal isopycnal displacements are large, this approximation fails even in the ocean interior. Inspired by the two different residual streamfunctions, a more general form of TEM formulation is explored. It is shown that the different TEM residual streamfunctions arise from decomposing the eddy flux into a component along isopycnals, which leads to advective flow, and a remaining diffusive component, which is oriented either vertically or horizontally. In theory the diffusive flux can be oriented in any direction, although in practice the orientation should be such that neither the advective flow nor the diffusive flux cross any boundary (surface, sidewalls, and bottom). However, it is not clear how to merge the continuously changing orientation in a physically meaningful way. A variety of approaches are discussed.

Corresponding author address: A. J. George Nurser, James Rennell Division for Ocean Circulation and Climate, Southampton Oceanography Centre, Empress Dock, Southampton SO14 3ZH, United Kingdom. Email: g.nurser@soc.soton.ac.uk

Abstract

In , the “vertical” transport streamfunction was defined as resulting from isopycnic averaging at constant height in the same way that the meridional streamfunction results from averaging at constant latitude. Part II here discusses the relationship between these two isopycnic streamfunctions and the Eulerian residual streamfunction that arises from the transformed Eulerian mean (TEM). It is known that the meridional isopycnic streamfunction can be approximated by a Taylor expansion to give an Eulerian residual streamfunction involving the horizontal eddy flux. This Taylor expansion approximation works well in the interior, removing the spurious mixing associated with the simple Eulerian-averaged streamfunction. However, it fails near the surface where isopycnals outcrop to the surface. It can be shown in a similar way that the vertical isopycnic streamfunction can formally be approximated by a residual streamfunction involving the vertical eddy flux. However, if horizontal isopycnal displacements are large, this approximation fails even in the ocean interior. Inspired by the two different residual streamfunctions, a more general form of TEM formulation is explored. It is shown that the different TEM residual streamfunctions arise from decomposing the eddy flux into a component along isopycnals, which leads to advective flow, and a remaining diffusive component, which is oriented either vertically or horizontally. In theory the diffusive flux can be oriented in any direction, although in practice the orientation should be such that neither the advective flow nor the diffusive flux cross any boundary (surface, sidewalls, and bottom). However, it is not clear how to merge the continuously changing orientation in a physically meaningful way. A variety of approaches are discussed.

Corresponding author address: A. J. George Nurser, James Rennell Division for Ocean Circulation and Climate, Southampton Oceanography Centre, Empress Dock, Southampton SO14 3ZH, United Kingdom. Email: g.nurser@soc.soton.ac.uk

Save