Linear and nonlinear progressive Rossby waves on a rotating sphere

We present an analysis of incompressible and compressible flow of a thin layer of fluid with a free-surface on a rotating sphere. Our general aim is to investigate the nature of progressive Rossby wave structures that are possible in this rotating system, with the goal of expanding previous research by conducting an in-depth analysis of wavespeed/amplitude relationships. A linearized theory for the incompressible dynamics, closely related to the theory developed by B. Haurwitz, is constructed, with good agreement observed between the two separate models. This result is then extended to the numerical solution of the full model, to obtain highly nonlinear large-amplitude progressive-wave solutions in the form of Fourier series. A detailed picture is developed of how the progressive wavespeed depends on the wave amplitude. This approach reveals the presence of nonlinear resonance behaviour, with different disjoint solution branches existing at different values of the amplitude. Additionally, we show that the formation of localised low pressure systems is an inherent feature of the nonlinear dynamics, once the forcing amplitude reaches a certain critical level. We then derive a new free-surface model for compressible fluid dynamics and repeat the above analysis by first constructing a linearized solution and then using this to guide the computation of nonlinear solutions via a bootstrapping process. It is shown that if the value of the pressure on the free-surface is assumed to be zero, which is consistent with the concept of the atmosphere terminating, then the model almost reduces to the incompressible dynamics with the only difference being a slightly modified conservation of mass equation. By forcing wave amplitude in the model we show that the resonant behaviour observed in the incompressible dynamics is again encountered in the compressible model. The effect of the compressibility is observed to become apparent through damped resonance behaviour in general, so that in some instances two neighbouring disjoint solution branches from the incompressible dynamics are seen to merge into one continuous solution branch when compressible dynamics are incorporated. In closing, some conjectures are made as to how these results might help explain certain observed atmospheric phenomena. In particular it is conjectured that the process of atmospheric blocking is a direct result of critically forced stationary Rossby waves.