 Solution

In a simple northern-hemisphere layer with a fixed upper boundary, climbing a mountain compresses the column. Potential vorticity conservation, $(\zeta +f)/h=\mathrm{constant}$, then reduces absolute vorticity: at nearly fixed latitude the flow acquires anticyclonic relative vorticity. Descending the lee side stretches the column and restores or increases cyclonic vorticity. For an eastward streamline the idealized turning tendency is clockwise in the compressed region and counterclockwise on re-expansion; the resulting disturbance can launch a stationary downstream Rossby wave pattern.

This column argument is an idealization. Stratified air can move around the barrier, isentropic thickness changes need not equal the mountain height, and wave breaking or friction can change PV. Northern-hemisphere signs reverse if $f$ changes sign.