Ertel potential vorticity ($Q={\mathbf{\omega }}_a\cdot \nabla b$)

Ertel potential vorticity is the scalar product of absolute vorticity with the gradient of a materially conserved stratifying scalar. In Boussinesq flow with buoyancy $b$,It is materially conserved by inviscid, adiabatic motion.

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.

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.

For an adiabatic, inviscid parcel, reduced isentropic mass thickness requires reduced absolute vertical vorticity; increased mass thickness requires the reverse. A rigid-upper-boundary geometric cartoon with layer depth $H-h_m$ gives an anticyclonic tendency while crossing a mountain in the Northern Hemisphere. Actual isentropes need not behave like a rigid lid. Blocking, meridional displacement, heating and friction must be assessed before applying that cartoon to real terrain.

A mountain does not by itself determine whether an atmospheric layer becomes thinner. In the fixed-lid model of mountain-crossing isentropic PV balance, raising the lower boundary by $h_m$ reduces layer depth from $H$ to $H-h_m$. Conserving potential vorticity then requires reduced absolute vertical vorticity; in the Northern Hemisphere this gives an anticyclonic relative tendency if other changes are neglected.

For an isentropic atmospheric layer, the relevant thickness is its isentropic mass density, not simply its geometric depth. Its upper potential temperature surface can rise, deform or exchange pressure loading as flow crosses terrain. Both boundaries and the thermodynamic structure must therefore be specified before inferring compression. The relation $f+{\zeta }_{\theta }=q_{\theta }{\sigma }_{\theta }$ gives the vorticity change once that mass-thickness change is known; it cannot determine the upper-boundary response on its own.

${\sigma }_{\theta }=-(1/g)\partial p/\partial \theta$ measures mass per horizontal area per unit potential-temperature interval under hydrostatic balance. It is positive in a statically stable atmosphere.

Heating over land often favours a deeper cyclonic thermal low, while cooling favours a shallower anticyclonic thermal high. Diabatic heating and surface friction violate the assumptions of simple parcel PV conservation; a vertical profile is needed for a quantitative statement.

Spatially varying drag has a vorticity tendency beyond simple damping of existing relative vorticity. The cross-gradient term depends on wind direction and the gradient of surface resistance. Increasing roughness on moving from ocean to land does not alone specify a universal cyclonic or anticyclonic sign. Seasonal heating also changes stratification and isentropic mass thickness, so it cannot generally be replaced by adiabatic PV conservation.

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.