Computing large-amplitude progressive Rossby waves on a sphere

We consider the flow of a thin layer of incompressible fluid on a rotating sphere, bounded internally by the surface of the sphere and externally by a free surface. Progressive-wave solutions are sought for this problem, without tangent plane simplifications. A linearized theory is derived for small amplitude perturbations about a base westerly flow field, allowing calculation of the linearized progressive wavespeed. This result is then extended to the numerical solution of the full model, to obtain highly non-linear large-amplitude progressive-wave solutions in the form of Fourier series. A detailed picture is developed of how the progressive wavespeed depends on wave amplitude. This approach reveals the presence of non-linear resonance behaviour, with different disjointed solution branches existing at different values of the amplitude. Additionally, we show that the formation of localized low pressure systems cut off from the main flow field is an inherent feature of the non-linear dynamics, once the amplitude forcing reaches a certain critical level.