A Formulation of Three-Dimensional Residual Mean Flow Applicable Both to Inertia-Gravity Waves and to Rossby Waves

The three-dimensional (3D) residual mean flow is expressed as the sum of the Eulerian-mean flow and the Stokes drift. The present study derives formulas that are approximately equal to the 3D Stokes drift for the primitive equation (PRSD) and for the quasigeostrophic equation (QGSD) using small-amplitude theory for a slowly varying time-mean flow. The PRSD has a broad utility that is applicable to both Rossby waves and inertia-gravity waves. The 3D wave activity flux whose divergence corresponds to the wave forcing is also derived using PRSD. The PRSD agrees with QGSD under the small-Rossby-number assumption, and it agrees with the 3D Stokes drift derived by S. Miyahara and by T. Kinoshita et al. for inertia-gravity waves under the constant-Coriolis-parameter assumption. Moreover, a phase-independent 3D Stokes drift is derived under the QG approximation. The 3D residual mean flow in the upper troposphere in April is investigated by applying the new formulas to the European Centre for Medium-Range Weather Forecasts (ECMWF) Interim Re-Analysis (ERA-Interim) data. It is observed that the PRSD is strongly poleward (weakly equatorward) upstream (downstream) of the storm track. A case study was also made for dominant gravity waves around the southern Andes in the simulation by a gravity wave-resolving general circulation model. The 3D residual mean flow associated with the gravity waves is poleward (equatorward) in the western (eastern) region of the southern Andes. This flow is due to the horizontal structure of the variance in the zonal component of the mountain waves, which do not change much while they propagate upward.