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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]

The isentropic potential vorticity is Ertel potential vorticity expressed on constant potential temperature surfaces:Here ${\zeta }_{\theta }$ is the vertical curl of horizontal velocity differentiated on an isentropic surface, and ${\sigma }_{\theta }$ is mass per unit area per unit potential-temperature interval. For smooth inviscid adiabatic motion, $Dq/Dt=0$. Conserved anomalies are advected and deformed, explaining much balanced large-scale weather variability. Diagnosing velocity from PV additionally needs balance, stratification and boundary information. Strong latent/radiative heating, friction and irreversible mixing generate or modify PV, limiting a purely conservative explanation.^[ref]

For an approximately conserved isentropic potential vorticity, ${\zeta }_{\theta }+f=q{\sigma }_{\theta }$: increased isentropic layer mass implies more cyclonic vorticity, and decreased layer mass more anticyclonic vorticity at fixed latitude. Equivalently, in the thin-layer analogy, stretching increases and compression decreases absolute vorticity.

The usual summer land-heating picture has a deeper, less stably stratified lower atmosphere and favours a cyclonic thermal low; winter land cooling gives a shallow stable layer and favours an anticyclonic thermal high. These are conditional lower-tropospheric tendencies, not a season-only conservation law. Heating is diabatic and the rougher land surface introduces friction and convergence, so parcel PV need not remain fixed. The sign at every altitude cannot be deduced from “ocean to land” alone without the heating/stability profile and background wind.