A formulation of the wave-mean-flow interactions and Lagrangian-mean meridional circulation is derived in terms of mass-weighted isentropic zonal mean parameters. This expresses the nonacceleration theorem in a nongeostrophic and finite-amplitude sense and handles the lower boundary conditions exactly. For further convenience, the pressure-isentrope hybrid vertical (p+) coordinates are adopted, which is the isentropic zonal-mean of pressure . It leads us to traditional representations of the diagnostic form of continuity equation and the advective form of thermodynamic equation without any eddy terms. The non-divergent mass stream functions are fully equivalent for the so-called 'diabatic circulation'.

2. The p+ analysis of Lagrangian-mean circulation & Eliassen-Palm flux

The above formulations can reasonably be applied to the actual atmosphere. Using the NCAR/NCEP reanalysis data, we diagnose the Lagrangian-mean circulation and the Eliassen-Palm flux and compare them to those of the transformed-Eulerian mean (TEM) analysis. In the extratropical lower troposphere, the results of the p+ analysis are very different from the TEM. It suggests a practical advantage of handling the lower boundaries. Intersections of isentrope with the boundary may cause the mean equatorial flows and the EP flux divergence in the lower troposphere, which are consistent with the structure of baroclinic instability waves. Even in the extratropical upper troposphere and the stratosphere, there appear some considerable differences between the two analyses. Nonlinear and/or nongeostrophic effects may not be negligible.

3. Energy cycle between zonal means and eddies

Energy cycles between zonal means and eddies based on the mass-weighted zonal means looks very different from those by Lorenz (1955). Particularly in baroclinic instability waves, the cross conversion term from zonal mean kinetic energy into eddy available potential energy (KZ --> PE) plays an essential role.