 Isentropic potential vorticity

On potential-temperature surfaces, $q=({\zeta }_{\theta }+f)/{\sigma }_{\theta }$, where ${\sigma }_{\theta }=-(1/g)\partial p/\partial \theta$ is isentropic mass density. In adiabatic frictionless flow it is materially conserved. Diagnosing a velocity field also requires balance and boundary data.