Using the shallow water equations for a rotating layer of fluid, the wave and dispersion equations for Rossby wavesare developed for the cases of both the standard β-plane approximation for the latitudinal variation of theCoriolis parameter and a zonal variation of the shallow water speed. It is well known that the wave normal diagram for thestandard (mid-latitude) Rossby wave on a β-plane is a circle in wave number (,) space, whosecentre is displaced −β/2 ω units along the negative axis, and whose radius is less than this displacement, which means that phasepropagation is entirely westward. This form of anisotropy (arising from thelatitudinal variation of ), combined with the highly dispersive nature of the wave, gives rise to a groupvelocity diagram which permits eastward as well as westward propagation. It is shown that the group velocitydiagram is an ellipse, whose centre is displaced westward, and whose major and minor axes give themaximum westward, eastward and northward (southward) group speeds as functions of the frequency and a parameter which measures the ratio of the low frequency-long wavelength Rossby wave speed to the shallow water speed. Webelieve these properties of group velocity diagram have not been elucidated in this way before. We present asimilar derivation of the wave normal diagram and its associated group velocity curve for the case of a zonal() variation of the shallow water speed, which may arise when the depth of an ocean varies zonally from acontinental shelf.
展开▼
