Rossby waves in an azimuthal wind

Rossby waves in an azimuthal wind are analyzed using an eigen-function expansion. Solutions of the wave equation for the stream-function [GRAPHICS] for Rossby waves are obtained in which [GRAPHICS] depends on [GRAPHICS] where r is the cylindrical radius, [GRAPHICS] is the azimuthal angle measured in the [GRAPHICS] plane relative to the Easterly direction, (the [GRAPHICS] -plane is locally horizontal to the Earth's surface in which the x-axis points East, and the y-axis points North). The radial eigenfunctions in the [GRAPHICS] -plane are Bessel functions of order [GRAPHICS] and argument kr, where k is a characteristic wave number and have the form [GRAPHICS] in which the [GRAPHICS] satisfy recurrence relations involving [GRAPHICS] , [GRAPHICS] , and [GRAPHICS] . The recurrence relations for the [GRAPHICS] have solutions in terms of Bessel functions of order [GRAPHICS] where [GRAPHICS] is the frequency of the wave and [GRAPHICS] is the angular velocity of the wind and argument [GRAPHICS] . By summing the Bessel function series, the complete solution for [GRAPHICS] reduces to a single Bessel function of the first kind of order [GRAPHICS] . The argument of the Bessel function is a complicated expression depending on r, [GRAPHICS] , a, and kr. These solutions of the Rossby wave equation can be interpreted as being due to wave-wave interactions in a locally rotating wind about the local vertical direction. The physical characteristics of the rotating wind Rossby waves are investigated in the long and short wavelength limits; in the limit as the azimuthal wind velocity [GRAPHICS] ; and in the zero frequency limit [GRAPHICS] in which one obtains a stationary spatial pattern for the waves. The vorticity structure of the waves are investigated. Time dependent solutions with [GRAPHICS] are also investigated.