The group velocity and radiation pattern of Rossby waves

Abstract

The propagation properties of mid-latitude Rossby waves are well known and are usually revealed in terms of the dispersion equation either in its diagnostic or wave normal forms, both of which show that phase propagation is entirely westward and is permitted only if the wave frequency is less than a certain critical frequency. Here we show that the group velocity diagram is an ellipse whose focus lies at the origin. This simple result supplements the Longuet-Higgins (1964) interpretation in which the wave normal curve is an offset circle, in elucidating the propagation properties of Rossby waves. In the case of a general β, which describes both topographic zonal variations as well as the latitudinal and spatial variations of the Coriolis effect, we show that these results hold through a rotation to a new set of coordinates. The stationary phase method shows that the radiation pattern generated by a time harmonic spatially compact source consists of two sets of hyperbolae exhibiting westward pointing “Mach-Froude” like lines, in a manner analogous to the generation of capillary-gravity waves by a moving object on the surface of deep water. These results are confirmed by the Green’s function for the system which consists of a westward propagating wave superimposed on the Hankel function of zero order, appropriate to the 2-D Helmholtz equation.