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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.

Mountain blocking can prevent low-level upstream air from crossing the barrier while air from higher levels passes over and descends. In stable stratification that elevated source air has higher potential temperature; adiabatic descent can bring it into a lee valley warmer than the blocked low-level air. This source-air change permits a dry foehn without requiring condensation and rainout along the arriving parcel's path.

A foehn wind is relatively warm, dry downslope air on a mountain's lee side. In the classical condensation pathway, ascending windward air expands and cools at the dry adiabatic lapse rate until the lifting condensation level. Condensation then releases latent heat, reducing the cooling rate to the moist adiabatic lapse rate. Atmospheric precipitation removes condensate, and the descending parcel warms by compression at approximately the dry rate. Its relative humidity falls.

For equal starting and finishing heights, complete condensate removal and a constant moist lapse rate above condensation height $z_c$, the precipitation-driven foehn warming isHere $H>z_c$ is the parcel's maximum height and ${\Gamma }_d>{\Gamma }_m$. The warming is adiabatic compression during descent, not heating by friction. Without condensation or a difference in source air, a reversible dry parcel returning to its original height would regain its original temperature.

Precipitation is not essential to every foehn event. Isentropic foehn drawdown can bring air of higher potential temperature from aloft down the lee slope; turbulent mixing and extra solar heating beneath clear skies can also contribute. The sketch shows the classical rainout pathway rather than asserting a unique mechanism.^[ref]

For dry atmospheric Ertel potential vorticity, use absolute vorticity ${\mathbf{\omega }}_a=\nabla \times \mathbf{u}+2\mathbf{\Omega }$ and potential temperature $\theta =T(p_0/p{)}^{R_d/c_p}$:The conservation statement is Ertel's theorem for smooth, inviscid, adiabatic flow with conservative body forces and a thermodynamic equation of state $p=p(\rho ,\theta )$. Stretching of ${\mathbf{\omega }}_a/\rho$ cancels deformation of $\nabla \theta$ in their material derivative; the remaining baroclinic triple product vanishes because $\nabla p$ lies in the span of $\nabla \rho ,\nabla \theta$. The atmospheric form includes $1/\rho$; the commonly used Boussinesq approximation version has a different normalization.

In a predominantly vertical stratification, $q_E\simeq (f+\zeta ){\partial }_z\theta /\rho$. Thus rotation and stratification jointly label air masses. Material conservation explains how parcels deform and stretch while preserving a useful scalar, supports Rossby waves, and helps diagnose sharp fronts, jets and stratospheric intrusions.

Under an appropriate balance model, potential-vorticity inversion recovers associated wind and temperature fields from a PV distribution together with stratification and boundary data. PV alone does not specify every component of an arbitrary atmospheric flow. Potential-vorticity generation by heating and friction limits conservative parcel reasoning near strong heating, surface stress or mixing.^[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]