Local Rossby-wave dispersion relation in a zonal jet ($\omega =kU-\frac{k(\beta -U^{\prime \prime }+U/R_D^2)}{k^2+l^2+R_D^{-2}}$)

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 $\omega =kU-k(\beta -U^{\prime \prime })/(k^2+l^2)$.