Mountain-crossing isentropic PV balance ($f+{\zeta }_{\theta }=q_{\theta }{\sigma }_{\theta }$)

For an adiabatic, inviscid parcel, reduced isentropic mass thickness requires reduced absolute vertical vorticity; increased mass thickness requires the reverse. A rigid-upper-boundary geometric cartoon with layer depth $H-h_m$ gives an anticyclonic tendency while crossing a mountain in the Northern Hemisphere. Actual isentropes need not behave like a rigid lid. Blocking, meridional displacement, heating and friction must be assessed before applying that cartoon to real terrain.

A mountain does not by itself determine whether an atmospheric layer becomes thinner. In the fixed-lid model of mountain-crossing isentropic PV balance, raising the lower boundary by $h_m$ reduces layer depth from $H$ to $H-h_m$. Conserving potential vorticity then requires reduced absolute vertical vorticity; in the Northern Hemisphere this gives an anticyclonic relative tendency if other changes are neglected.

For an isentropic atmospheric layer, the relevant thickness is its isentropic mass density, not simply its geometric depth. Its upper potential temperature surface can rise, deform or exchange pressure loading as flow crosses terrain. Both boundaries and the thermodynamic structure must therefore be specified before inferring compression. The relation $f+{\zeta }_{\theta }=q_{\theta }{\sigma }_{\theta }$ gives the vorticity change once that mass-thickness change is known; it cannot determine the upper-boundary response on its own.