Non-linear saturation of vertically propagating Rossby waves

Abstract

Linear quasi-geostrophic theory predicts an exponential amplitude increase with height for Rossby waves propagating vertically through a stratified atmosphere, as a result of wave activity density conservation. At the same time layer-wise conservation of potential enstrophy constrains wave amplitudes, given the limited amount of potential enstrophy available in the initial mean flow. A break down of linear theory is thus expected above a certain critical wave amplitude, raising the question of how the non-linear flow reacts to limit the vertical penetration of waves. Keeping in mind the potential importance for the dynamics of the winter stratosphere, where strong wave penetration and amplitude growth are often observed, the issue of wave saturation in a non-linear flow is examined in a generally abstract context, through a variety of simple model studies. We thus consider the cases of a topographically forced barotropic beta plane channel model, of vertical propagation through a three-dimensional beta plane channel model, and of a polar coordinate model with realistic basic state and geometry. In the barotropic model transient wave growth is forced through the use of bottom topography and the deviations of the non-linear flow evolution from the predictions of both a linear and a quasi-linear analytical solution are examined for strong topographic anomalies. The growth of the forced wave is found to decelerate the zonal mean flow which in turn reduces the topographic forcing. Wave-mean flow interactions are thus found to be sufficient in leading to saturation of the eddy amplitudes. Interestingly it is the formation of zonal mean easterlies, rather than the depletion of mean available potential enstrophy, that is found to be the crucial factor in the saturation dynamics. Similar results are obtained for the case of vertical propagation through a three dimensional beta plane channel. The vertical penetration of the forced wave is shown to cause a reduction of the zonal mean winds and mean potential vorticity gradients in the center of the channel, eventually leading to the formation of either a critical line or a refractive index turning surface. In both cases the penetration of the wave to high altitudes is prohibited, thus constraining wave amplitudes. While signs of non-linear behaviour are clear in synoptic maps of potential vorticity, wave-wave interactions are found to play a secondary role in the saturation process. The results of the three-dimensional beta plane channel model are then extended to a more realistic set-up, using a polar coordinate model with a basic state based on the observed winter stratosphere climatology. The basic conclusions of the idealized study are shown to remain unchanged.