Rossby Wave Green's Functions in an Azimuthal Wind

Green's functions for Rossby waves in an azimuthal wind are obtained, in which the stream-function $\psi$ depends on $r$, $\phi$ and $t$, where $r$ is cylindrical radius and $\phi$ is the azimuthal angle in the $\beta$-plane relative to the easterly direction, in which the $x$-axis points east and the $y$-axis points north. The Rossby wave Green's function with no wind is obtained using Fourier transform methods, and is related to the previously known Green's function obtained for this case, which has a different but equivalent form to the Green's function obtained in the present paper. We emphasize the role of the wave eikonal solution, which plays an important role in the form of the solution. The corresponding Green's function for a rotating wind with azimuthal wind velocity ${\bf u}=\Omega r{\bf e}_\phi$ ($\Omega=$const.) is also obtained by Fourier methods, in which the advective rotation operator in position space is transformed to a rotation operator in ${\bf k}$ transform space. The finite Rossby deformation radius is included in the analysis. The physical characteristics of the Green's functions are delineated and applications are discussed. In the limit as $\Omega\to 0$, the rotating wind Green's function reduces to the Rossby wave Green function with no wind.