A Multiscale Model for the Planetary and Synoptic Motions in the Atmosphere

A reduced asymptotic model valid for the planetary and synoptic scales in the atmosphere is presented. The model is derived by applying a systematic multiple-scales asymptotic method to the full compressible-flow equations in spherical geometry. The synoptic-scale dynamics in the model is governed by modified quasigeostrophic equations, which take into account planetary-scale variations of the background stratification and of the Coriolis parameter. The planetary-scale background is described by the planetary geostrophic equations and a new closure condition in the form of a two-scale evolution equation for the barotropic component of the background flow. This closure equation provides a model revealing an interaction mechanism from the synoptic scale to the planetary scale. To obtain a quantitative assessment of the validity of the asymptotics, the balances on the planetary and synoptic scales are studied by utilizing a primitive equations model. For that purpose, spatial and temporal variations of different terms in the vorticity equation are analyzed. It is found that, for planetary-scale modes, the horizontal fluxes of relative and planetary vorticity are nearly divergence free. It is shown that the results are consistent with the asymptotic model.