On the interpretation of downward control - a new perspective on the interplay between potential vorticity, wave forcing and the residual circulation

A general theoretical perspective on the interplay between Rossby-Ertel potential vorticity (PV), the residual circulation and the transformed Eulerian mean (TEM) wave forcings is derived. We show, in the context of a zonal mean flow with meridional and vertical shear, that the TEM equations describe the evolution of the second order zonal mean PV associated with the mean flow, advected by the residual circulation and forced by the TEM wave forcings. We then recover the nonacceleration theorem in a unified and more intuitive way. The waves provide the sole internal forcing on the mean PV associated with the zonal mean flow when the nonacceleration conditions are violated. Hence, pedagogically, the TEM wave forcings ``control'' the flow in the same sense as ``a forced PV controls the flow.'' We suggest a similar interpretation in the QG limit: the ``downward control'' principle is fundamentally a statement on how the QG-PV controls the flow. We also show that a finite amplitude analysis provides essentially the same PV interpretation as the perturbation expansion up to second order: the residual velocities are those that advect the finite amplitude mean PV, forced by the finite amplitude TEM wave forcings. Overall, this PV perspective provides a new interpretation for the residual circulation in the primitive equations.