Kushner, P. J., and I. M. Held, 1999: Potential vorticity thickness fluxes and wave-mean flow interaction. Journal of the Atmospheric Sciences, 56(7), 948-958.

Abstract: The use of eddy flux of thickness between density surfaces has become a familiar starting point in oceanographic studies of adiabatic eddy effects on the mean density distribution. In this study, a dynamical analogy with the density thickness flux approach is explored to reexamine the theory of nonzonal wave-mean flow interaction in two-dimensional horizontal flows. By analogy with the density thickness flux, the flux of thickness between potential vorticity (PV) surfaces is used as a starting point for a residual circulation formulation for nonzonal mean flows. Mean equations for barotropic PV dynamics are derived in which a modified mean velocity with an eddy-induced component advects a modified mean PV that also has an eddy-induced component. For small-amplitude eddies, the results are analogous to recent results of McDougall and McIntosh derived for stratified flow.

The dynamical implications of this approach are then examined. The modified mean PV equation provides a decomposition of the eddy forcing of the mean flow into contributions from wave transience, wave dissipation, and wave-induced mass redistribution between PV contours. If the mean flow is along the mean PV contours, the contribution from wave-induced mass redistribution is "workless" in Plumb's sense that it is equivalent to an eddy-induced stress that is perpendicular to the mean flow. This contribution is also associated with the convergence along the mean streamlines of a modified PV flux that is equal to the difference between the PV flux and the rotational PV flux term identified by Illari and Marshall. The cross-stream component of the modified PV flux is related to wave transience and dissipation.