Model equations for planetary and synoptic scale atmospheric motions associated with different background stratification

We consider planetary scale atmospheric motions (spatial scales comparable with the earth's radius and a corresponding advective time scale of the order of ten days) and the synoptic eddies (characteristic length and time scales: $1,000, {rm km}$ and $1, {rm day}$, respectively). We present reduced equations that aim to capture the relevant dynamics on the planetary scale and account for multiscale planetary-synoptic interactions. The derivation of the equations is based on an unified multiple-scales asymptotic approach. We examine two different flow regimes associated with different background stratification