Rossby waves in an azimuthal wind

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