Potential vorticity ($q$)

Potential vorticity combines vorticity with vortex stretching and density stratification. It also applies to nonrotating stratified flow through Ertel potential vorticity. In the inviscid shallow water equations,where $\zeta$ is relative vorticity and $h$ is layer thickness.

The spatial gradient of potential vorticity controls the anomaly created when a parcel is displaced. The linear potential-vorticity equation contains ${\mathbf{u}}^{\prime }\cdot \nabla \bar{q}$. A gradient of the Coriolis parameter contributes planetary vorticity, but gradients of relative vorticity and layer stretching can be equally important.

A potential vorticity budget has the formThe material derivative follows the fluid motion and $S_q$ collects forcing or dissipation. With $S_q=0$ it becomes potential-vorticity conservation. For a quasi-geostrophic streamfunction, horizontal advection is $J(\psi ,q)={\psi }_x{q}_y-{\psi }_y{q}_x$.

In unforced adiabatic inviscid motion, potential vorticity is conserved following a parcel:The linearization about a basic state is $({\partial }_t+\bar{\mathbf{u}}\cdot \nabla )q^{\prime }+{\mathbf{u}}^{\prime }\cdot \nabla \bar{q}=0$. This separates advection of a disturbance from its generation by displacement across a basic potential-vorticity gradient.

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.

Shallow-water potential vorticity is materially conserved by inviscid, unforced shallow water equations. Linearizing about rest with depth $H_0$ gives an anomaly proportional to

For a constant-depth resting layer on an f-plane, the depth-scaled linear potential vorticity anomaly isIt satisfies $q_t=0$ under the linearized shallow water equations. The physical perturbation of $(f+\zeta )/(H+\eta )$ is $q/H$. The signed scalar $q$, not its absolute value, is used in potential-vorticity inversion and balanced energy pairings.

For small Rossby number and small free-surface displacement, shallow-water potential vorticity reduces, up to a constant factor and additive constant, to relative vorticity plus background PV gradients and the stretching term $-\psi /L_D^2$.

The barotropic deformation radius is the horizontal scale at which free-surface gravity and rotation balance. It contributes $L_D^{-2}$ to the denominator of the one-layer Rossby-wave dispersion relation.