 Rossby-wave equation for a sheared zonal current

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.