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The Eliassen–Palm flux tensor

Published online by Cambridge University Press:  16 July 2013

J. R. Maddison  and
D. P. Marshall
J. R. Maddison*
Affiliation:
Atmospheric, Oceanic and Planetary Physics, Department of Physics, University of Oxford, Oxford OX1 3PU, UK
D. P. Marshall
Affiliation:
Atmospheric, Oceanic and Planetary Physics, Department of Physics, University of Oxford, Oxford OX1 3PU, UK
*
†Email address for correspondence: maddison@atm.ox.ac.uk
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Abstract

The aim of this paper it to derive general coordinate-invariant forms of the Eliassen–Palm flux tensor and thereby characterize the true geometric nature of the eddy–mean-flow interaction in hydrostatic Boussinesq rotating fluids. In the quasi-geostrophic limit previous forms of the Eliassen–Palm flux tensor are shown to be related to each other via a gauge transformation; a general form is stated and its geometric properties are discussed. Similar methodology is applied to the hydrostatic Boussinesq Navier–Stokes equations to re-derive the residual-mean equations in a coordinate-invariant form. Thickness-weighted averaging in buoyancy coordinates is carefully described, via the definition of a volume-form-weighted average, constructed so as to commute with the covariant divergence of a vector. The procedures leading to the thickness-weight averaged equation are discussed, and forms of the Eliassen–Palm flux tensor which arise are identified.

JFM classification

Geophysical and Geological Flows: Geostrophic turbulence Geophysical and Geological Flows: Ocean processes Geophysical and Geological Flows: Quasi-geostrophic flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 729 , 25 August 2013 , pp. 69 - 102
Copyright
©2013 Cambridge University Press 

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References

Andrews, D. G. 1983 A finite-amplitude Eliassen–Palm theorem in isentropic coordinates. J. Atmos. Sci. 40, 1877–1883.Google Scholar
Andrews, D. G., Holton, J. R. & Conway, C. B. 1987 Middle Atmosphere Dynamics, International Geophysics Series, vol. 40. Academic.Google Scholar
Andrews, D. G. & McIntyre, M. E. 1976 Planetary waves in a horizontal and vertical shear: the generalized Eliassen–Palm relation and the mean zonal acceleration. J. Atmos. Sci. 33, 2031–2048.Google Scholar
Andrews, D. G. & McIntyre, M. E. 1978 Generalized Eliassen–Palm and Charney–Drazin theorems for waves on axisymmetric mean flows in compressible atmospheres. J. Atmos. Sci. 35 (2), 175–185.Google Scholar
Bretherton, F. P. 1966 Critical layer instability in baroclinic flows. Q. J. R. Meteorol. Soc. 92 (393), 325–334.Google Scholar
Charney, J. G. & Drazin, P. G. 1961 Propagation of planetary-scale disturbances from the lower into the upper troposphere. J. Geophys. Res. 66 (1), 83–109.Google Scholar
Cronin, M. 1996 Eddy-mean flow interaction in the Gulf Stream at $6{8}^{\circ } \mathrm{W} $ . Part II: eddy forcing on the time-mean flow. J. Phys. Oceanogr. 26 (10), 2132–2151.Google Scholar
De, U. C., Shaikh, A. A. & Sengupta, J. 2005 Tensor Calculus. Alpha Science.Google Scholar
Eden, C., Greatbatch, R. J. & Olbers, D. 2007 Interpreting eddy fluxes. J. Phys. Oceanogr. 37 (5), 1282–1296.Google Scholar
Eliassen, A. & Palm, E. 1961 On the transfer of energy in stationary mountain waves. Geofysicke Publikasjoner 22 (3), 1–23.Google Scholar
Ferreira, D. & Marshall, J. 2006 Formulation and implementation of a ‘residual-mean’ ocean circulation model. Ocean Model. 13 (1), 86–107.Google Scholar
Gent, P. R. & McWilliams, J. C. 1990 Isopycnal mixing in ocean circulation models. J. Phys. Oceanogr. 20 (1), 150–155.Google Scholar
Gent, P. R. & McWilliams, J. C. 1996 Eliassen–Palm fluxes and the momentum equation in non-eddy-resolving ocean circulation models. J. Phys. Oceanogr. 26 (11), 2539–2546.Google Scholar
Gent, P. R., Willebrand, J., McDougall, T. J. & McWilliams, J. C. 1995 Parameterizing eddy-induced tracer transports in ocean circulation models. J. Phys. Oceanogr. 25 (4), 463–474.Google Scholar
Greatbatch, R. J. & Lamb, K. G. 1990 On parameterizing vertical mixing of momentum in non-eddy resolving ocean models. J. Phys. Oceanogr. 20 (10), 1634–1637.Google Scholar
Green, J. S. A. 1970 Transfer properties of the large-scale eddies and the general circulation of the atmosphere. Q. J. R. Meteorol. Soc. 96 (408), 157–185.Google Scholar
Haynes, P. H. & McIntyre, M. E. 1990 On the conservation and impermeability theorems for potential vorticity. J. Atmos. Sci. 47 (16), 2021–2031.Google Scholar
Held, I. M. & Schneider, T. 1999 The surface branch of the zonally averaged mass transport circulation in the troposphere. J. Atmos. Sci. 56 (11), 1688–1697.Google Scholar
Holton, J. R. 1981 An advective model for two-dimensional transport of stratospheric trace species. J. Geophys. Res. 86 (C12), 11989–11994.Google Scholar
Hoskins, B. J., James, I. N. & White, G. H. 1983 The shape, propagation and mean-flow interaction of large-scale weather systems. J. Atmos. Sci. 40 (7), 1595–1612.Google Scholar
Jackett, D. R. & McDougall, T. J. 1997 A neutral density variable for the world’s oceans. J. Phys. Oceanogr. 27 (2), 237–263.Google Scholar
Lee, M.-M. & Leach, H. 1996 Eliassen–Palm flux and eddy potential vorticity flux for a nonquasigeostrophic time-mean flow. J. Phys. Oceanogr. 26 (7), 1304–1319.Google Scholar
Marshall, D. P., Maddison, J. R. & Berloff, P. S. 2012 A framework for parameterizing eddy potential vorticity fluxes. J. Phys. Oceanogr. 42 (4), 539–557.Google Scholar
Marshall, J. C. 1981 On the parameterization of geostrophic eddies in the ocean. J. Phys. Oceanogr. 11 (2), 257–271.Google Scholar
McDougall, T. J. & McIntosh, P. C. 1996 The temporal-residual-mean velocity. Part I: derivation and the scalar conservation equations. J. Phys. Oceanogr. 26 (12), 2653–2665.Google Scholar
McDougall, T. J. & McIntosh, P. C. 2001 The temporal-residual-mean velocity. Part II: isopycnal interpretation and the tracer and momentum equations. J. Phys. Oceanogr. 31 (5), 1222–1246.2.0.CO;2>CrossRefGoogle Scholar
Medvedev, A. S. & Greatbatch, R. J. 2004 On advection and diffusion in the mesosphere and lower thermosphere: the role of rotational fluxes. J. Geophys. Res. 109, D07104.Google Scholar
Miyahara, S. 2006 A three-dimensional wave activity flux applicable to inertio-gravity waves. Sci. Online Lett. Atmos. 2, 108–111.Google Scholar
Muraki, D. J., Snyder, C. & Rotunno, R. 1999 The next-order corrections to quasigeostrophic theory. J. Atmos. Sci. 56 (11), 1547–1560.Google Scholar
Nurser, A. J. G. & Lee, M.-M. 2004 Isopycnal averaging at constant height. Part II: relating to the residual streamfunction in Eulerian space. J. Phys. Oceanogr. 34 (12), 2740–2755.Google Scholar
Pedlosky, J. 1987 Geophysical Fluid Dynamics, 2nd edn. Springer.Google Scholar
Plumb, R. A. 1985 On the three-dimensional propagation of stationary waves. J. Atmos. Sci. 42 (3), 217–229.Google Scholar
Plumb, R. A. 1986 Three-dimensional propagation of transient quasi-geostrophic eddies and its relationship with the eddy forcing of the time-mean flow. J. Atmos. Sci. 43 (16), 1657–1678.2.0.CO;2>CrossRefGoogle Scholar
Plumb, R. A. & Ferrari, R. 2005 Transformed Eulerian-mean theory. Part I: nonquasigeostrophic theory for eddies on a zonal-mean flow. J. Phys. Oceanogr. 35 (2), 165–174.Google Scholar
Prandtl, L. 1942 Bemerkungen zur Theorie der freien Turbulenz. Z. Angew. Math. Mech. 22 (5), 241–243.Google Scholar
Rhines, P. B. & Young, W. R. 1982 Homogenization of potential vorticity in planetary gyres. J. Fluid Mech. 122, 347–367.Google Scholar
Schneider, T., Held, I. M. & Garner, S. T. 2003 Boundary effects in potential vorticity dynamics. J. Atmos. Sci. 60 (8), 1024–1040.Google Scholar
Smith, R. D. 1999 The primitive equations in the stochastic theory of adiabatic stratified turbulence. J. Phys. Oceanogr. 29 (8), 1865–1880.Google Scholar
Stewart, R. W. & Thomson, R. E. 1977 Re-examination of vorticity transfer theory. Proc. R. Soc. Lond. A 354 (1676), 1–8.Google Scholar
de Szoeke, R. A. & Bennett, A. F. 1993 Microstructure fluxes across density surfaces. J. Phys. Oceanogr. 23 (10), 2254–2264.Google Scholar
Taylor, G. I. 1915 Eddy motion in the atmosphere. Phil. Trans. R. Soc. Lond. A 215, 1–26.Google Scholar
Taylor, G. I. 1932 The transport of vorticity and heat through fluids in turbulent motion. Phil. Trans. R. Soc. Lond. A 135 (828), 685–702.Google Scholar
Tung, K. T. 1986 Nongeostrophic theory of zonally averaged circulation. Part I: formulation. J. Atmos. Sci. 43 (22), 2600–2617.Google Scholar
Welander, P. 1973 Lateral friction in the oceans as an effect of potential vorticity mixing. Geophys. Fluid Dyn. 5 (1), 173–189.Google Scholar
Young, W. R. 2012 An exact thickness-weighted average formulation of the Boussinesq equations. J. Phys. Oceanogr. 42 (5), 692–707.Google Scholar
Zhao, R. & Vallis, G. 2008 Parameterizing mesoscale eddies with residual and Eulerian schemes, and a comparison with eddy-permitting models. Ocean Model. 23 (1–2), 1–12.Google Scholar