= Momentum loss from uniform potential-vorticity mixing
{title2=$\langle\delta U\rangle=-\beta W^2/12$}
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.
Back to article page
