| The field-reversed configuration (FRC) is a toroidal-shaped magnetic-field geometry used for confining plasmas for the purpose of the controlled, safe, steady-state production of fusion energy. As a result of angular invariance of the Solov'ev equilibrium, used for analytic and numerical study of the FRC, the full three dimensional Hamiltonian system can be expressed as two coupled highly nonlinear oscillators. Due to high nonlinearity of the equations of motion, the behavior of the system is highly complex, showing regimes of both chaotic and integrable motion, depending on the constants of motion and geometry of the FRC. Using analytic techniques from nonlinear dynamics and Poincare surface-of-section plots, the structure of phase space is investigated and shown to be highly sensitive to the parameters of the system. In the limit of a highly elongated geometry, there is a separation of time scales between the axial and radial motion of the ion, leading to adiabatic chaos. Integrability criteria are derived which distinguish between near-integrable and chaotic trajectories, based on the crossing of the phase-space separatrix. An averaged one-dimensional potential for near-integrable motion is derived. It is found that orbits with high radial energies are more integrable and confined closer to the midplane, suggesting that high temperature plasmas may be more stable and have lower resistivity. The affect of a small odd-parity rotating magnetic field (RMF) on ion heating inside the elongated FRC is investigated. The addition of RMF breaks the angular invariance leading to a more chaotic system. It is found that cyclotron orbits tend to interact more regularly with RMF than figure-8 orbits. Stochastic heating for cyclotron orbits occurs in a series of random steps in the regions of field-reversal, indicating that magnetic nulls are important to ion heating. The maximum energy gains are large, particularly for cyclotron orbits, confirming the affectiveness off odd-parity RMF in heating the ions. The interaction of figure-8 orbits with RMF in the midplane is investigated analytically and displays a set of resonances that increase and overlap closer to the phase-space separatrix. |
