Ertel's theorem on potential vorticity

Builds on Vorticity Material derivative Inviscid flow

Statement

Consider a smooth flow in a frame rotating with constant angular velocity $\Omega \in \R^3$ (possibly $\Omega = 0$ ), with velocity $u$ , density $\rho > 0$ and pressure $p$ . Assume the flow is inviscid, so that $\frac{Du}{Dt} + 2 \Omega \times u = -\frac{1}{\rho} \nabla p - \nabla \Phi, \qquad \frac{D\rho}{Dt} + \rho \, \nabla \cdot u = 0,$ for a body force potential $\Phi$ , where $D/Dt$ is the material derivative. Let $\psi$ be a smooth scalar field that is conserved by each parcel, $D\psi/Dt = 0$ . Assume that $\nabla \psi \cdot (\nabla \rho \times \nabla p) = 0$ . This holds for instance when $\psi$ is a function of $\rho$ and $p$ only, or when the fluid is barotropic. Let $\omega_a = \nabla \times u + 2\Omega$ be the absolute vorticity. Then the potential vorticity $q = \frac{\omega_a \cdot \nabla \psi}{\rho}$ is conserved by each parcel: $Dq/Dt = 0$ .

Remarks

The proof combines the vorticity equation for $\omega_a / \rho$ , which reads $D(\omega_a/\rho)/Dt = ((\omega_a/\rho) \cdot \nabla) u + \nabla \rho \times \nabla p / \rho^3$ , with the identity $D(\partial_i \psi)/Dt = \partial_i (D\psi/Dt) - \sum_j \partial_i u_j \, \partial_j \psi$ . The stretching terms cancel in the dot product, and the baroclinic term vanishes by the hypothesis on $\psi$ .

Example: in the atmosphere one takes $\psi$ to be the potential temperature, which is conserved in adiabatic motion and is a function of $p$ and $\rho$ . A column of air between two surfaces of constant potential temperature that is stretched vertically must then spin faster. This helps explain why cyclones tend to form on the lee side of mountain ranges.

Without the hypothesis on $\psi$ , the baroclinic term $\nabla \psi \cdot (\nabla \rho \times \nabla p)/\rho^3$ in general makes $q$ change.

Going deeper

For a student

The theorem is a local consequence of Kelvin's circulation theorem applied to small loops on surfaces of constant $\psi$ . In meteorology and oceanography, potential vorticity is the main conserved quantity of large-scale flows. Together with an inversion principle, which recovers the flow from $q$ , it reduces the dynamics of rotating stratified fluids to the transport of one scalar.

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Inviscid flow concept
Material derivative concept
Vorticity concept
Ertel's theorem on potential vorticity statement

Tag	`EH6K8VP1C3`
URI	`m2://EH6K8VP1C3`
Kind	statement, theorem
Writing	W0 stub
MSC	76U05, 86A10, 76B47
Status	This copy of the site has no build data. Run `m2 status EH6K8VP1C3`.
About	Vorticity
Uses	Vorticity, Material derivative, Inviscid flow
Changed	3 November 2026