In this thesis we study the coherent vortices of a two-dimensional incompressible ideal fluid (the Euler equations) which is important to many physical systems, including the atmosphere of outer planets, two-dimensional turbulence, and pure electron plasma experiments. Using the statistical equilibrium theory derived recently which respects all the infinite conservation laws of the ideal fluid, we solve the coherent vortex solutions in a disk and an annulus. In addition to finding the solutions, we develop the formulation and numerical scheme for a bifurcation and a thermodynamic stability analysis. Numerical simulations of the Euler equations are also performed to study dynamical relaxation from an initial flow to final steady states.
In these studies we pay attention to the problem of the lack of ergodicity which results from incomplete flow mixing. Ergodicity is assumed in the statistical theory but has not been justified. Our general conclusion concerning ergodicity is as follows: Mixing processes during the dynamics have strong effects on the final steady states. Mixing may not be complete as required for ergodicity, but can happen in particular regions or periods, or even in some special flow modes. When strong mixing does occur, the flow structure follows very closely the prediction of statistical mechanics.
Specifically our statistical calculations are on the following questions: (a) single coherent vortices and their bifurcation behavior in a disk, and comparison with the final states in an electron plasma experiment, (b) the stable state of two identical vortices and the prediction, which agrees well with many experimental and simulation results, of the critical separation for merging, (c) the proposal of "vorticity localization" which is used to explain the recently observed states with regular multiple-vortex patterns, (d) work towards explaining the formation of stable vortices on the surfaces of outer planets under the influence of deeper shear flows. Finally using numerical simulations we study vorticity mixing during the relaxation of a vortex ring and its effect on the final states.
