Past exam of the atmospheric physics course of the University of Cambridge / 2012 / Paper 1 / 3 / a

For a hydrostatic, stably stratified atmosphere, define isentropic mass density ${\sigma }_{\theta }=-(1/g)\partial p/\partial \theta >0$ and relative vertical vorticity evaluated on an isentrope, ${\zeta }_{\theta }$. Then isentropic potential vorticity isIt is the hydrostatic isentropic form of dry Ertel potential vorticity, $q_E={\mathbf{\omega }}_a\cdot \nabla \theta /\rho$. For inviscid, adiabatic motion with fixed composition and conservative body forces, $D{q}_E/Dt=0$, so parcels carry their PV along isentropes. Stable stratification makes the coordinate transformation well defined.

Heating and friction spoil simple material conservation. More explicitly, with $\dot{\theta }=D\theta /Dt$ and nonconservative force per mass $\mathbf{F}$,Thus the distribution is especially useful away from strongly diabatic clouds and frictional boundary layers. Explaining a velocity field by potential-vorticity inversion additionally requires an appropriate balanced-flow approximation and boundary information; a PV value alone does not specify the entire flow. ^[ref]