Vortex stretching in layered quasi-geostrophic flow

The stretching contribution to upper-layer potential vorticity is $-F({\psi }_1-{\psi }_2)$, with the opposite sign below. An interface displacement thickens one layer and thins the other. In a parcel's potential-vorticity equation, its material change balances changes of relative and planetary vorticity and any forcing. The two layer derivatives cannot in general be replaced by one common advection operator.