Hamiltonian methods for some geophysical vortex dynamics models

We study integrable point-vortex dynamics, with particular emphasis on the two-layer geostrophic model. We derive the equations of relative motion for three or more point vortices in the two-layer model. We formulate the full geostrophic two-layer equations in an infinite-dimensional Hamiltonian form and present the analogous calculations for the point vortex equations. We present the Poisson bracket formalism and demonstrate new four-vortex integrable systems. Our study contains results on the location and characterization of both fixed and relative equilibria of three-vortex two-layer systems. This study also includes a trilinear plane analysis of the relative equilibria. We provide results on finite-time collision of three-vortices in the two-layer model as well as explicit configurations of vortex alignment for vortices in different layers. A numerical study of integrable two- and four-vortex systems is presented and includes an investigation of streamline patterns for rotating two-vortex configurations as well as various regimes of motion for integrable four-vortex systems. Aspects of integrable vortex motion on the sphere were also investigated and results on self-similar collapse were obtained, as new solutions including non-trivial four-vortex equilibria and explicit integration of three-vortex configurations. We also obtain new integrable coaxial four-vortex systems. Some negative results on vortex collapse in (planar) circular domains are given.