A formulation of three-dimensional Transformed Eulerian-Mean equations applicable to both Rossby waves and gravity waves

The Transformed Eulerian-Mean (TEM) equations formulated by Andrews and McIntyre (1976, 1978) have been widely used to examine wave-mean flow interaction in the meridional cross section. Although a lot of efforts have been made to generalize the TEM equations to three dimensions so far, formulae derived by previous studies are applicable to particular kinds of waves, mainly Rossby waves on the quasi-geostrophic (QG) equations or gravity waves on the primitive equations. This study has formulated three-dimensional (3D) TEM equations which are applicable to both Rossby waves and gravity waves in order to examine the 3D material transport driven by these waves. The material transport driven by gravity waves around the Southern Andes is investigated by applying the derived formulae to a high-resolution general circulation model (GCM).

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.