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.