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The Eliassen–Palm flux tensor

Published online by Cambridge University Press:  16 July 2013

J. R. Maddison*
Atmospheric, Oceanic and Planetary Physics, Department of Physics, University of Oxford, Oxford OX1 3PU, UK
D. P. Marshall
Atmospheric, Oceanic and Planetary Physics, Department of Physics, University of Oxford, Oxford OX1 3PU, UK
Email address for correspondence:


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.

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