Dry atmospheric Ertel potential vorticity ($q_E={\rho }^{-1}(\nabla \times \mathbf{u}+2\mathbf{\Omega })\cdot \nabla \theta$)

For dry atmospheric thermodynamics, choose potential temperature as the materially conserved scalar in Ertel's theorem. The displayed density-normalized scalar has units $\mathrm{K}{\mathrm{m}}^2\mathrm{k}{\mathrm{g}}^{-1}{\mathrm{s}}^{-1}$. Its conservation in adiabatic inviscid motion depends on conservative body forces and a composition-fixed equation of state. It differs in normalization from the Boussinesq approximation convention using buoyancy without the variable density factor. Large-scale balanced weather can be organized by its distribution, with potential-vorticity inversion additionally requiring balance and boundary information.

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.