Rigidity of Stratospheric Travelling Waves

We present some rigidity results for travelling waves that propagate zonally in bands of latitude in the stratosphere of the outer planets. Such waves arise as perturbations of a background zonally shear flow. The f-plane and $\beta$ -plane approximations are adequate for the small and moderately wide bands on Jupiter and Saturn, while for the broad zonal jets on Uranus and Neptune one must use spherical coordinates. We show that within the f-plane setting the wave speeds must lie in the range of the flow's zonal velocity, while in the $\beta$ -plane setting they cannot exceed the maximum of the zonal velocity but may be slightly less than its minimum. In spherical coordinates we obtain a rigidity result for travelling waves near a zonal flow. By means of some examples, we also show that such zonally travelling waves do not have to be zonally symmetric, and symmetry breaking may occur due to the rotation.