Saturation of vertically propagating Rossby waves

The interaction between vertical Rossby wave propagation and wave breaking is studied in the idealized context of a beta plane channel model. Considering the problem of linear propagation through a uniform zonal flow, which predicts exponential wave growth with height, we ask the question of how wave growth is limited in the non-linear case. Using a numerical model we examine behaviour of the non-linear flow, as the bottom forcing increases through values bound to lead to a breakdown of the linear solution. We focus on the final "equilibrated" state, as a function of the bottom forcing, and address the issue of the mechanisms involved in limiting wave growth and of the importance of non-linearities in affecting the dynamics.

Wave-mean flow interactions are found to be dominant even for strong bottom forcing values. Ultimately it is the deformation of the mean flow that is found to limit the vertical penetration of the forced wave, through either increased damping or reflection. Linear propagation theory is found to capture the wave structure surprisingly well, even when the total flow is highly deformed. It is found that the presence of even slight damping strongly counteracts non-linear advection, leading to results very different from the traditional wave-breaking description. Overall the numerical results seem to suggest that non-linearities do not enter violently the final "equilibrated" state, even at high forcing values. Wave-mean flow interaction limits wave growth sufficiently so that an additional enstrophy sink, through downscale cascade, does not become necessary.