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One of the most problematic instabilities in tokamak plasmas is tearing modes; they are driven by current and pressure gradients, and involve a reconfiguration of the magnetic and velocity fields localized into a narrow region located at a resonant magnetic surface. While the equilibrium magnetic field lines are located on concentric nested toroidal flux surfaces, the instability creates magnetic islands in which field lines connect flux tubes together, allowing for a high radial heat transport, and, thus, resulting in a loss of confinement, and, potentially, disruptions. In order for the magnetic field lines to break and reconnect, we need to take into account the resistivity of the plasma and solve the resistive magnetohydrodynamics (MHD) equations. The analytical solution consists of a boundary layer analysis (asymptotic matching) and takes advantage of the small radial width of the region where the perturbations vary significantly. Indeed, ideal magnetohydrodynamics can be used everywhere except in that narrow region where the full resistive problem must be solved. This dissertation addresses two related problems in the study of resistive tearing modes, and their interactions with externally induced resonant magnetic perturbations (error-fields). First, an in-depth investigation of the bifurcated states of a rotating, quasi-cylindrical, tokamak plasma in the presence of a resonant error-field is performed, within the context of constant-[greek letter psi] resistive MHD theory. The response of the rotating plasma is studied in both the linear, and the nonlinear regime. In general, there is a "forbidden band" of tearing mode rotation frequencies that separates a branch of high-frequency solutions from a branch of low-frequency solutions. When a high-frequency solution crosses the upper boundary of the forbidden band there is a bifurcation to a low-frequency solution, and vice versa. Second, the analysis is extended to include the study of braking and locking of tearing mode rotation by the interaction of the mode with an error-field. It is found that this interaction can brake the plasma rotation, suppress magnetic island evolution and drive locked modes.
This comprehensive research and reference text aims to outline a realistic, comprehensive, self-consistent, analytic theory of tearing mode dynamics in tokamak plasmas.
The tokamak is the principal tool in controlled fusion research. This book acts as an introduction to the subject and a basic reference for theory, definitions, equations, and experimental results. The fourth edition has been completely revised, describing their development of tokamaks to the point of producing significant fusion power.
In cylindrical geometry, we analyze the hypothesis that the major disruption in tokamaks is due to the interaction of tearing modes of different pitch. For a flat safety factor profile, we find that the m = 3/n = 2 tearing mode, which is linearly unstable, is strongly destabilized nonlinearly by the m = 2/n = 1 mode. Other modes are nonlinearly destabilized, particularly the m = 1/n = 1 and m = 5/n = 3 modes. Due to the development of islands of many different pitches, the toroidal current density profile is severely deformed. The corresponding region of island activity can extend essentially from the plasma center to the limiter. Presumably, this deformation of the field lines can lead to rapid transport of heat and particles from the plasma center to the limiter. The destabilization of the m = 3/n = 2 mode and other odd m modes is significant because it may correspond to the observed vertical asymmetry of some of the major disruptions in PLT.