= Solution
In an idealized crossing flow, take a layer between a mountain and an approximately fixed upper boundary. If its upstream depth is H and the terrain height is $h_m$, climbing reduces its depth to $H-h_m$. Conservation of <shallow-water potential vorticity> gives
$$
\boxed{\frac{f+\zeta}{H-h_m}=\frac{f_0+\zeta_0}{H},\qquad
\zeta=\frac{H-h_m}{H}(f_0+\zeta_0)-f.}
$$
At nearly fixed northern latitude, with negligible upstream relative vorticity, $\zeta\simeq-fh_m/H<0$: compression produces an \b[anticyclonic tendency]. For an eastward flow this corresponds to a rightward/equatorward turning tendency. On the descending side, re-expansion restores cyclonic vorticity relative to that compressed state. Meridional displacement also changes f and participates in the balance.
The corresponding <mountain-crossing isentropic PV balance> uses $\sigma_\theta$ rather than geometric depth: reduced isentropic mass thickness requires reduced absolute vorticity, and increased thickness the reverse. The geometric cartoon assumes an upper boundary that does not simply rise with the terrain. A real mountain can displace isentropes, block low-level flow or generate a downstream Rossby-wave pattern; friction and latent heating can change PV. Thus the local compression sign is a model tendency, not a universal map of every orographic circulation.
\Image[/past-exam-of-the-atmospheric-physics-course-of-the-university-of-cambridge/2012/paper-1-ap2012-mountain-pv.png]
{title=Mountain PV compression}
{source=paper-1-ap2012-mountain-pv.py}
Back to article page
