Rossby-wave equation for a sheared zonal current ($({\partial }_t+U{\partial }_x)({\nabla }_h^2-R_D^{-2})\varphi +Q_y{\varphi }_x=0$)

Linearizing shallow-water quasi-geostrophic potential vorticity about a zonal current $U(y)$ gives $Q_y=\beta -U^{\prime \prime }+U/R_D^2$. A normal mode $\hat{\varphi }(y)e^{i(kx-\omega t)}$ obeys $(U-\omega /k)[{\hat{\varphi }}^{\prime \prime }-(k^2+R_D^{-2})\hat{\varphi }]+Q_y\hat{\varphi }=0$. A global transverse plane wave is generally unavailable for an arbitrary nonuniform jet; boundary conditions or a local short-wavelength approximation are required.

When a zonal current varies slowly compared with the wavelength, freeze its velocity and background potential vorticity gradient locally in the Rossby-wave equation for a sheared zonal current. The resulting local dispersion relation includes curvature $U^{\prime \prime }$ and, for a free surface, the basic surface-slope contribution $U/R_D^2$. In the nondivergent limit it becomes $\omega =kU-k(\beta -U^{\prime \prime })/(k^2+l^2)$.

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 $l=0$.