Rossby waves in an azimuthal wind

Abstract

Rossby waves in an azimuthal wind are analyzed using an eigen-function expansion. Solutions of the wave equation for the stream-function ψ for Rossby waves are obtained in which ψ depends on (r,φ,t) where r is the cylindrical radius, φ is the azimuthal angle measured in the β plane relative to the Easterly direction, (the β-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 β-plane are Bessel functions of order n and argument kr,where k is a characteristic wave number and have the form anJn(kr) in which the an satisfy recurrence relations involving an+1, an,andan−1. The recurrence relations for the an have solutions in terms of Bessel functions of order n − ω/Ω where ω is the frequency of the wave and Ω is the angular velocity of the wind and argument a = β/(kΩ). By summing the Bessel function series, the complete solution for ψ reduces to a single Bessel function of the first kind of order ω/Ω. The argument of the Bessel function is a complicated expression depending on r, φ, 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 Vw → 0; and in the zero frequency limit ω → 0 in which one obtains a stationary spatial pattern for the waves. The vorticity structure of the waves are investigated. Time dependent solutions with ω = 0 are also investigated.