A Formulation of a Phase-Independent Wave-Activity Flux for Stationary and Migratory Quasigeostrophic Eddies on a Zonally Varying Basic Flow

A new formulation of an approximate conservation relation of wave-activity pseudomomentum is derived, which is applicable for either stationary or migratory quasigeostrophic (QG) eddies on a zonally varying basic flow. The authors utilize a combination of a quantity A that is proportional to wave enstrophy and another quantity E that is proportional to wave energy. Both A and E are approximately related to the wave-activity pseudomomentum. It is shown for QG eddies on a slowly varying, unforced nonzonal 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, a 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. The authors’ conservation relation based on a straightforward derivation is a generalization of that for stationary Rossby waves on a zonally uniform basic flow as derived by Plumb and others. A dynamical interpretation is presented for each term of such a phase-independent flux of the authors or Plumb. Terms that consist of eddy heat and momentum fluxes are shown to represent systematic upstream transport of the mean-flow westerly momentum by a propagating wave packet, whereas other terms proportional to eddy streamfunction anomalies are shown to represent an ageostrophic flux of geopotential in the direction of the local group velocity. In such a flux, these two dynamical processes acting most strongly on the node lines and ridge/trough lines of the eddy streamfunction field, respectively, are appropriately combined to eliminate its phase dependency. The authors also derive generalized three-dimensional transformed Eulerian-mean equations with the residual circulation and eddy forcing both expressed in phase-independent forms. The flux may not be particularly suited for evaluating the exact local budget of M, because of several assumptions imposed in the derivation. Nevertheless, these assumptions seem qualitatively valid in the assessment based on observed and simulated data. The wave-activity flux is a useful diagnostic tool for illustrating a“snapshot” of a propagating packet of stationary or migratory QG wave disturbances and thereby for inferring where the packet is emitted and absorbed, as verified in several applications to the data. It may also be useful for routine climate diagnoses in an operational center.