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The transformed Eulerian-mean (TEM) equations are useful in examining how the generation and/or dissipation of atmospheric waves drives the mean meridional circulation. However, the TEM equations do not provide a three-dimensional view of the transport. Several previous studies extended the TEM equation system to three dimensions but usually under the quasi-geostrophic assumption, which excludes small-scale phenomena such as gravity waves. Miyahara recently derived three-dimensional wave activity flux and the corresponding residual circulation applicable to gravity waves. However, his formulation has two flaws. First, the three-dimensional residual mean circulation does not satisfy the continuity equation. Second, the Eulerian-mean flow appears in the advection terms and the residual circulation appears in the Coriolis force term of the zonal momentum equation, unlike in the TEM one. The present study developed theoretical formulae of a three-dimensional residual mean circulation and wave activity flux on the basis of primitive equations that overcome these flaws. It is confirmed that the three-dimensional residual mean circulation accords with the sum of the Eulerian time-mean flow and the Stokes drift and that the three-dimensional wave activity flux accords with the mean tangential forces across material surfaces corrugated by the waves under an assumption similar to the TEM equations. A simple physical meaning is given for the terms including the shear of time-mean flow in the three-dimensional wave activity flux. Moreover, the time mean tracer transport equation is derived using the three-dimensional residual mean circulation. A simple case study using the new formulae was made on the three-dimensional transport of stratospheric ozone in the Southern Hemisphere. It is shown that the product of the Coriolis parameter and the strong poleward/equatorward Stokes drifts also balances the divergence/convergence of the three-dimensional wave-activity flux.

On the Three-Dimensional Residual Mean Circulation and Wave Activity Flux of the Primitive Equations