A new formulation of an approximate conservation relation is proposed, which is applicable for either stationary or migratory quasi-geostrophic (QG) eddies on a zonally-varying basic flow.
Generally, a wave-activity flux consists of quadratic terms of velocity, pressure, and/or temperature fluctuations associated with the eddies. Hence the unaveraged wave-activity flux inherently includes an oscillatory component on a scale of one-half wave length. This component can be removed by taking some sort of average so as to represent statistics averaged over the wave phase. For transient, migratory eddies, time averaging is in general appropriate as in Plumb (1986), but not for a ``snapshot'' analysis. For stationary waves, of cource, time averaging is inappropriate since it is not equivalent to phase averaging. Zonal averaging as used in the conventional Eliassen-Palm flux cannot represent zonal propagation. Hence, for a stationary eddies or a ``snapshot'' analysis of migratory eddies, a conservation relation meant to represent three-dimensional propagation should be derived without any averaging. Plumb (1985) was the first to derive such a conservation law for stationary eddies on a zonally-uniform basic flow, but we adopt a somewhat different approach to obtain a more generalized relation.
We utilize that a quantity A proportional to wave enstrophy and another quantity E proportional to wave energy are both related to the wave activity pseudomomentum. It is shown for QG eddies on a slowly varying, unforced non-zonal flow that a particular linear combination of A and E, namely, M = (A + E)/2, is independent of the wave phase, even if unaveraged, in the limit of a small-amplitude plane wave. In the same limit, the flux of M is also free from an oscillatory component on a scale of one-half wavelength even without any averaging. It is shown that M is conserved under steady, unforced and nondissipative conditions and the flux of M is parallel to the local three-dimensional group velocity in the WKB limit. Our conservation relation based on a straightforward derivation is a generalization of that for stationary Rossby waves as derived by Plumb and others.
Applying our wave-activity flux to the simulated and observed atmospheric data, we can verify their phase-independency and validity to depict the wave-packet propagation of both stationary and migratory QG eddies. In particular, the flux is the first diagnostic tool capable of illustrating an instantaneous three-dimensional propagation of migratory wave packet in a phase-independent manner. The flux may not be particularly suited for evaluating the exact local budget of M because of several assumptions imposed in our derivation. Nevertheless, in our assessment based on the observed and simulated data, it is also verified that these assumptions seem qualitatively valid. We claim that our wave-activity flux is a useful diagnostic tool for representing a ``snapshot'' of a packet of stationary or migratory QG wave disturbances, and thereby for inferring where the packet is emitted and absorbed.
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12th Conference on Atmospheric and Oceanic Fluid Dynamics