The residual mean flow is expressed with the sum of the Eulerian-mean flow and the Stokes drift in the 2D TEM equations. Thus, a formulation is made for the 3D Stokes drift on the primitive equation (PRSD) from its original definition using a small amplitude theory for a slowly-varying mean flow. The PRSD is equivalent to the 3D Stokes drift derived by Kinoshita et al. for gravity waves under the constant Coriolis parameter assumption and to the 3D QG Stokes drift which is also derived in this study for the small Rossby number limit. The 3D wave activity flux (3D-flux-M), whose divergence corresponds to the wave force, is derived by using PRSD.
The 3D wave activity flux (3D-flux-W) describing wave propagation is formulated by modifying 3D-flux-M, as 3D-flux-M does not describe it. For this, a unified dispersion relation for gravity waves and Rossby waves is derived. Using this relation, modified wave activity density is introduced so as to relate the 3D-flux-W to the group velocity. It is shown that 3D-flux-W and modified wave activity density accord with those for gravity waves for the constant Coriolis parameter (Miyahara, 2006), and those for Rossby waves for the small Rossby number limit (Plumb, 1986).
A case study is made for dominant gravity waves around the Southern Andes by applying the PRSD and 3D-flux-M to outputs from the simulation by a gravity-wave resolving GCM (Watanabe et al. 2008). It is shown that the 3D residual mean flow driven by gravity waves in the stratosphere is poleward (equatorward) in the west (east) of the Southern Andes. This is likely due to such a horizontal structure of the variance in the zonal component of the mountain waves, which do not change much while they propagate upward.