A general treatment of the dynamics and statistical mechanics of point vortices in bounded domains is introduced in Chapter 1. Chapter 2 then considers high positive energy statistical mechanics of 2D Euler vortices. In this case, the most-probable equilibrium dynamics are given by solutions of the sinh-Poisson equation and a particular heart-shaped domain is found in which below a critical energy the solution has a dipolar structure and above it a monopolar structure. Sinh-Poisson predictions are compared to long-time averages of dynamical simulations of the $N$ vortex system in the same domain. Chapter 3 introduces a new algorithm (VOR-MFS) for the solution of generalised point vortex dynamics in an arbitrary domain. The algorithm only requires knowledge of the free-space Green's function and utilises the exponentially convergent method of fundamental solutions to obtain an approximation to the vortex Hamiltonian by solution of an appropriate boundary value problem. A number of test cases are presented, including quasi-geostrophic shallow water (QGSW) point vortex motion (governed by a Bessel function). Chapter 4 concerns low energy (positive and negative) statistical mechanics of QGSW vortices in `Neumann oval' domains. In this case, the `vorticity fluctuation equation' -- analogous to the sinh-Poisson equation -- is derived and solved to give expressions for key thermodynamic quantities. These theoretical expressions are compared with results from direct sampling of the microcanonical ensemble, using VOR-MFS to calculate the energy of the QGSW system. Chapter 5 considers the distribution of 2D Euler vortices in a Neumann oval. At high energies, vortices of one sign cluster in one lobe of the domain and vortices of the other sign cluster in the other lobe. For long-time simulations, these clusters are found to switch lobes. This behaviour is verified using results from the microcanonical ensemble.
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:626583 |
| Date | January 2014 |
| Creators | Ashbee, T. L. |
| Publisher | University College London (University of London) |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
| Source | http://discovery.ucl.ac.uk/1427632/ |
Page generated in 0.0101 seconds