A Calculation of the Structure of Stationary Planetary Waves in Winter

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Abstract
A propagation equation valid for determining the vertical and latitudinal structure of stationary planetary waves in winter is derived. This equation reduces to that used by Matsuno (1970) for an isothermal atmosphere where Rayleigh friction and Newtonian cooling are taken to be equal and constant. Quasi-analytical solutions to the propagation equation are obtained for the idealized case of an isothermal atmosphere in constant rotation to illustrate some of its important properties. Solutions are obtained numerically for altitudes between the forcing level at 100 mb and the 100 km level, where a radiation boundary condition is assumed. For various realistic models of the mean zonal wind field, radiative damping rates and mean temperatures profiles, we find that the structure of our computed stationary planetary-scale waves is more sensitive to the former two than to the latter. The structural behavior of the numerical solutions is interpreted using modal decomposition. Two principal modes compose most of the structure of wavenumber 1, while wavenumber 2 is dominated by a single mode. The trapping of these modes at different heights appears to explain the variability of the wave amplitude and phase with change in the mean wind structure. Overall, the numerical results give reasonable agreement with observations. We also discuss the associated energy fluxes and conversion terms due to vertically propagating planetary waves.

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