A Nonacceleration Theorem for Transient Quasi-geostrophic Eddies on a Three-Dimensional Time-Mean Flow

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  • 1 CSIRO Division of Atmospheric Research Aspendale, Australia
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Abstract

A nonacceleration theorem is derived for small-amplitude, transient quasi-geostrophic eddies on a three-dimensional time-mean flow. This theorem states that the divergence of the eddy potential vorticity flux—and hence the forcing by the eddies of the mean geostrophic flow—vanishes to leading order under conditions that (i) the eddies and mean flow are conservative, (ii) the eddy enstrophy density and the quantity Fu/|ū| (where Fu is the component of the eddy potential vorticity flux in the direction of the time-mean flow ū) are constant along time-mean streamlines, and (iii) the boundary conditions on the mean geostrophic flow are independent of the eddies.

The requirement of downstream–constant eddy amplitudes parallels that of steadiness of eddy amplitudes in the equivalent theorem for eddies on a zonal-mean flow. In general, when this condition is not met, the divergence of the transient eddy flux of potential vorticity is nonzero. Thus, unlike in the zonal-mean problem, small-amplitude, conservative, transient eddies propagating on a steady, three-dimensional mean flow will, in some if not in most cases, influence the mean flow in a nontrivial way, even though their amplitudes are steady in time. There are, however, some constraints on the nature of this interaction; conservative eddies do not impact on the global time-mean enstrophy budget, while small-amplitude conservative eddies on a conservative mean flow make no explicit contribution to the global budget of time-mean energy.

Abstract

A nonacceleration theorem is derived for small-amplitude, transient quasi-geostrophic eddies on a three-dimensional time-mean flow. This theorem states that the divergence of the eddy potential vorticity flux—and hence the forcing by the eddies of the mean geostrophic flow—vanishes to leading order under conditions that (i) the eddies and mean flow are conservative, (ii) the eddy enstrophy density and the quantity Fu/|ū| (where Fu is the component of the eddy potential vorticity flux in the direction of the time-mean flow ū) are constant along time-mean streamlines, and (iii) the boundary conditions on the mean geostrophic flow are independent of the eddies.

The requirement of downstream–constant eddy amplitudes parallels that of steadiness of eddy amplitudes in the equivalent theorem for eddies on a zonal-mean flow. In general, when this condition is not met, the divergence of the transient eddy flux of potential vorticity is nonzero. Thus, unlike in the zonal-mean problem, small-amplitude, conservative, transient eddies propagating on a steady, three-dimensional mean flow will, in some if not in most cases, influence the mean flow in a nontrivial way, even though their amplitudes are steady in time. There are, however, some constraints on the nature of this interaction; conservative eddies do not impact on the global time-mean enstrophy budget, while small-amplitude conservative eddies on a conservative mean flow make no explicit contribution to the global budget of time-mean energy.

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