 Momentum loss from uniform potential-vorticity mixing

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.