 Ertel's theorem

An inviscid fluid with conservative body force, a materially conserved stratifying scalar $\alpha$, and an equation of state making $\nabla \rho ,\nabla p,\nabla \alpha$ coplanar conserves its Ertel potential vorticity. The stretching of absolute vorticity per unit density cancels the deformation of the scalar gradient; the remaining baroclinic triple product vanishes. In an incompressible density-stratified ideal fluid, choosing the advected density as $\alpha$ gives that cancellation directly.