Vortices and Rossby-wave radiation on the beta-plane
The Earth’s atmosphere and oceans contain strongly swirling coherent structures. The sphericity of the Earth’s surface, which may be modelled by the beta-effect, is responsible for the motion of these vortices, and also for the existence of Rossby waves. This dissertation examines the evolution of distributed vortices on the beta-plane when the vortices are much stronger than the background beta-effect. There is a small nondimensional parameter ε , and a solution to the problem may be sought as an asymptotic expansion in ε. The near-field equation is then, to first order, a forced radial Rayleigh equation. For the nondivergent case, with no vortex stretching, a different dynamical balance holds in the far field. Chapter 2 derives an exact mode-one solution for the radial Rayleigh operator and completely solves the classical inviscid instability initial-value problem for mode-one disturbances to a circular basic state. Disturbances tend to a steady-state solution if the basic state circulation is nonzero, and grow algebraically without bound otherwise. Chapter 3 examines properties of the causal Green’s function for the far-field Rossby wave equation. Chapter 4 calculates the first-order solution to the global problem in the nondivergent case by the method of Matched Asymptotic Expansions. For zero circulation, there is no far field. The resulting trajectory of vortices is computed; several ways of identifying the centre of the vortices are presented. Trajectories are calculated for the Rankine and Gaussian vortices. Chapter 5 calculates the asymptotic behaviour of the second-order solution, and examines the nonuniformity of the asymptotic expansion. In the case of nonzero circulation, the expansion loses validity for times t = 0( ε -2/3 ) and spatial scales r = 0( ε -1/3 ), rather than t = 0( ε -1) and r = 0(1) respectively. Chapter 6 solves the divergent problem numerically to first order, and also analytically for asymptotically large Rossby radii of deformation. The order of breakdown is then t = 0( ε -1). Chapter 7 presents conclusions and suggestions for further research.