We use for that the Eady model within the Boussinesq semi-geostrophic approximation. The forcing is split up into two independent parts, under the hypothesis that the scale of the mean orography is significantly different from the scale of the individual peaks. The first part is due to the gravity waves that interact with the large-scale flow at turning critical levels, where they produce a dipolar potential vorticity anomaly advected and steered by the shear in the mid-troposphere. The second part is due to the mean orography, which produces a vertical velocity at the ground but no potential vorticity.
First, we study the model response in the absence of any upper boundary. We show that, under a geometrical configuration such that many gravity waves encounter a critical level, the potential vorticity they produce can force boundary Eady modes as much substantial than those forced by the mean orography. Furthermore, we find that the breaking gravity waves can reinforce (i) the anticyclonic circulation and (ii) the downslope low which are produced by the mean orography. We show that the warm front configuration is inherently more efficient than the cold front configuration in this process.
In the presence of a rigid lid, baroclinic instability is allowed and the latter results are still valid. But in addition, the potential vorticity advected in the far field sustains very efficiently the developpment of baroclinic unstable Eady modes.
Although very theoretical and academic, these calculations illustrates in a well closed system, the significance of breaking gravity waves and turning critical levels for the synoptic circulation. They may help to appreciate the needs for the parametrisation of turning critical levels in GCMs, and in particular the need to parametrise mountain gravity waves in the spectral space. They also give some hints of the benefits to be expected from such parametrisations.