Potential-vorticity evolution equation ($Dq/Dt=S_q$)

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.

If quasi-geostrophic potential vorticity initially has gradient $\beta$ across a channel of width $W$ and is homogenized, a height-independent change of zonal velocity obeys $\delta Q=-\delta U_y$. Requiring $\delta U(0)=\delta U(W)=0$ gives uniform final potential vorticity equal to its original channel mean, $\delta U=\beta y(y-W)/2$, and channel-mean momentum change $-\beta W^2/12$. The negative change for $\beta >0$ is wave drag associated with mixing, not a loss of total momentum in an isolated unforced system.

In zonally averaged barotropic quasi-geostrophic approximation flow with zero mean meridional transport, the Taylor identity for quasi-geostrophic flux relates the meridional potential vorticity flux to the convergence of Reynolds stress. A local downgradient closure $\overline{v^{\prime }{q}^{\prime }}=-\kappa {\overline{Q}}_y$, with $\kappa \ge 0$, therefore gives westward mean acceleration where the background gradient is positive. Rossby wave breaking can stir PV into fine filaments; coarse-graining or weak diffusion produces the inferred mixing. Globally the momentum-flux convergence integrates to boundary stresses, so localized westward changes need compensating transport or momentum changes elsewhere in an isolated system.