Finite-amplitude wave activity and mean-flow adjustment in the atmosphere

The traditional flux-based theory for eddy-mean flow interaction quantifies how and how fast the mean flow is being modified by the eddies, but it does not necessarily quantify how much the mean flow has been modified already. To address this last point, we introduce finite-amplitude Rossby wave activity as a hybrid Lagrangian-Eulerian mean quantity and use it to construct a non-flux-based transformed Eulerian mean (TEM) theory in both quasigeostrophic and isentropic frameworks. Wave activity is defined as the net displacement of ‘potential vorticity substance' (Haynes and McIntyre 1990) from the line of equivalent latitude. This definition permits particularly simple (finite-amplitude) forms of Eliassen-Palm relation and nonacceleration theorem. In the isentropic formalism, the nonacceleration theorem derives from Kelvin's circulation theorem and in this sense it is closely related to its counterpart in the generalized Lagrangian mean (GLM) formalism of Andrews and McIntyre (1978) -- the two differ in the way Kelvin's circulation is partitioned into ‘mean flow' and ‘pseudomomentum/wave activity.' Then Kelvin's circulation (in the quasigeostrophic case potential vorticity gradient with respect to equivalent latitude) is used to define a quasi-conservative, eddy-free reference state. The instantaneous departure of the zonal-mean flow from this reference state is interpreted as ‘mean flow adjustment,' and it is shown to be related to wave activity and Stokes correction in the form of time-integrated TEM theory. The Eliassen-Palm flux and its divergence are implicit in all this and not diagnosed at all. The formalism is readily applicable to meteorological data and represents a practical alternative to the flux-based TEM and/or GLM formalisms.