Stationary Rossby waves in a sinusoidal zonal jet ($k^2+l^2=(\pi /L{)}^2$)

For a sinusoidal mean jet on an $f$ plane, its QG potential-vorticity gradient is $({\pi }^2/L^2+k_d^2)U$. Ground-stationary modes satisfy the displayed circle and can solve the full stationary shear equation even though their scale does not meet a local short-wave assumption. The local stationary group vector is $2Uk(k,l)/(k^2+l^2+k_d^2)$, reducing to a zonal scalar formula only for $l=0$.