Transformed Eulerian-mean theory. Part 1: Nonquasigeostrophic theory for eddies on a zonal-mean flow

A theoretical formalism for nongeostrophic eddy transport in zonal-mean flows, using a transformed Eulerian-mean (TEM) approach in z coordinates, is discussed. By using Andrews and McIntyre's coordinate-independent definition of the quasi-Stokes streamfunction, it is argued that the surface boundary condition can be dealt with more readily than when the widely used quasigeostrophic definition is adopted. Along with the residual mean circulation, the concept of residual eddy flux arises naturally within the TEM framework, and it is argued that it is this residual eddy flux, and not the raw eddy flux, that might reasonably be expected to be downgradient. This distinction is shown to be especially important for Ertel potential vorticity (PV). The authors show how a closed set of transformed mean equations can be generated, and how the eddy forcing appears in the TEM momentum equations. Under adiabatic conditions, the eddy drag is just proportional to the residual eddy flux of PV along the mean isopycnals; in the diabatic layer close to the surface, it is more complicated, but becomes very simple for small Rossby number (without any assumption of small isopycnal slope).