Balances on the planetary scale in the atmosphere

Single scale asymptotic analysis reveal that planetary scale atmospheric motions (length scales of the order of the earth's radius) are governed by the so-called planetary geostrophic equations (PGEs) and an evolution equation for the barotropic component of the flow. If a two scale asymptotic approach is applied, resolving both the planetary and the synoptic scales, the PGEs are again the leading order model for the planetary scale motion and the synoptic fluxes appear only in the evolution equation for the barotropic component. We test the asymptotic results with a primitive equation model by studying the balances in the vorticity transport. We consider the temporal variations of the spectral coefficients of different terms in the vorticity equation. For modes with a total wavenumber less than three we find that: first the horizontal divergence of the relative vorticity flux nearly vanish and second the advection of the planetary vorticity "f" and the horizontal divergence of the wind multiplied with "f" nearly balance. Further we observe that higher modes obey the quasi-geostrophic vorticity equation. We show that the results are consistent with the asymptotic analysis.