On the ThreeDimensional Residual Mean Circulation and Wave Activity Flux of the Primitive Equations
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Abstract
The transformed Eulerianmean (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 threedimensional view of the transport. Several previous studies extended the TEM equation system to three dimensions but usually under the quasigeostrophic assumption, which excludes smallscale phenomena such as gravity waves. Miyahara recently derived threedimensional wave activity flux and the corresponding residual circulation applicable to gravity waves. However, his formulation has two flaws. First, the threedimensional residual mean circulation does not satisfy the continuity equation. Second, the Eulerianmean 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 threedimensional residual mean circulation and wave activity flux on the basis of primitive equations that overcome these flaws. It is confirmed that the threedimensional residual mean circulation accords with the sum of the Eulerian timemean flow and the Stokes drift and that the threedimensional 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 timemean flow in the threedimensional wave activity flux. Moreover, the time mean tracer transport equation is derived using the threedimensional residual mean circulation. A simple case study using the new formulae was made on the threedimensional 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 threedimensional waveactivity flux.
Journal

 Journal of the Meteorological Society of Japan. Ser. II

Journal of the Meteorological Society of Japan. Ser. II 88(3), 373394, 2010
Meteorological Society of Japan