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ON THE EXACT AND APPROXIMATE LINEAR THEORY OF VERTICALLY PROPAGATING PLANETARY ROSSBY WAVES FORCED AT A SPHERICAL LOWER BOUNDARY

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  • 1 Massachusetts Institute of Technology, Cambridge, Mass.
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Abstract

An approximate linearized model for the analysis of low frequency transient and stationary planetary scale atmospheric waves is derived. The problem of stationary waves forced at the lower boundary is solved exactly for an atmosphere in constant rotation by making use of the recent tidal theory analysis of Longuet-Higgins, and this solution is then compared with the solution of the approximate model equation. The approximate model is found to describe the Rossby wave modes with little error away from the Tropics. The eigenvalues of the approximate model are in good agreement with the eigenvalues of the exact model with the exception of eigenvalues of the lowest latitudinal modes. It is concluded that the model will be suitable for the purpose of linear theoretical analysis of the vertical propagation of planetary Rossby waves in the presence of zonal wind shears. Assuming an atmosphere in constant angular rotation and assuming westerly zonal wind velocities of the magnitude of the maximum winds observed in the midwinter stratospheric jet, there will always be two or more planetary wave modes that can propagate vertically. The constant angular wind velocity model is used together with the amplitude of observed stationary planetary waves in the winter lower stratosphere to predict the magnitude of planetary waves at the meteor wind level. Because the amplitude of the eddy winds so predicted exceeds observed values by at least an order of magnitude, we infer that horizontal wind shears and possibly also diabatic damping need be considered for the description of the propagation of planetary waves from the troposphere to the lower thermosphere.

Abstract

An approximate linearized model for the analysis of low frequency transient and stationary planetary scale atmospheric waves is derived. The problem of stationary waves forced at the lower boundary is solved exactly for an atmosphere in constant rotation by making use of the recent tidal theory analysis of Longuet-Higgins, and this solution is then compared with the solution of the approximate model equation. The approximate model is found to describe the Rossby wave modes with little error away from the Tropics. The eigenvalues of the approximate model are in good agreement with the eigenvalues of the exact model with the exception of eigenvalues of the lowest latitudinal modes. It is concluded that the model will be suitable for the purpose of linear theoretical analysis of the vertical propagation of planetary Rossby waves in the presence of zonal wind shears. Assuming an atmosphere in constant angular rotation and assuming westerly zonal wind velocities of the magnitude of the maximum winds observed in the midwinter stratospheric jet, there will always be two or more planetary wave modes that can propagate vertically. The constant angular wind velocity model is used together with the amplitude of observed stationary planetary waves in the winter lower stratosphere to predict the magnitude of planetary waves at the meteor wind level. Because the amplitude of the eddy winds so predicted exceeds observed values by at least an order of magnitude, we infer that horizontal wind shears and possibly also diabatic damping need be considered for the description of the propagation of planetary waves from the troposphere to the lower thermosphere.

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