Potential-vorticity generation by heating and friction ($D{q}_E/Dt=[{\mathbf{\omega }}_a\cdot \nabla \dot{\theta }+(\nabla \times \mathbf{F})\cdot \nabla \theta ]/\rho$)

For a composition-fixed equation of state, let $\dot{\theta }=D\theta /Dt$ and let $\mathbf{F}$ be nonconservative force per unit mass. The vorticity and scalar-gradient evolution equations give the source terms in the title. Spatially varying heating projected along absolute vorticity and the curl of friction can generate or destroy Ertel potential vorticity. Heating that changes potential temperature does not automatically change PV everywhere: its spatial gradient and orientation matter. Unresolved irreversible mixing can also alter the parcelwise conservation picture.