Shaping Effects on Magnetohydrodynamic Instabilities in a Tokamak Plasma Surrounded by a Resistive Wall

The primary achievement of this study is the development of a new approach for optimizing the plasma shape in a tokamak fusion energy reactor. In the interest of producing the largest possible fusion power output, the shape is optimized to allow for the highest possible beta - the ratio of the fluid to magnetic pressure - that can be sustained without the onset of magnetohydrodynamic (MHD) instabilities. To this end, the study explores the beta-domain that is stabilizable by bulk plasma rotation, with rotation timescales comparable to the resistive dissipation time of the plasma tearing surfaces or of the surrounding vacuum chamber. Modern feedback control systems are able to apply external magnetic fields which are phased to emulate the effect of plasma rotation, making the technique applicable even to large tokamaks with inadequate plasma rotation. In order to explore how the rotationally stabilizable beta-domain is affected by plasma shaping, a new semi-analytic MHD model of a tokamak has been developed. In addition to shaped toroidal tokamak geometry, the model contains dissipative effects resulting from resistivity in both the plasma and in the vacuum-chamber wall. The inclusion of plasma and wall resistivity introduces a lower beta-limit, associated with the onset of an unstable MHD mode, which can become dominated by either resistive-plasma (tearing) or resistive-wall effects in different parts of the parameter space. The computation time for analyzing the mode stability is greatly reduced by approximating the plasma current to reside in a thin layer, a form known as a sharp-boundary model. With fast calculations that focus on the key physics of these MHD instabilities, the model is able to explore qualitative trends of rotational stabilizability over a broad range of plasma shapes. Results of this study predict that varying the elongation or triangularity of the plasma cross-section can lead to qualitatively different beta-limits for the rotationally stabilizable domain. As the shape is varied, the upper bound in beta for rotational stabilization is found to switch from resistive-wall dominated behavior to resistive-plasma dominated behavior. The optimal plasma shape, associated with the highest beta-limit achievable with plasma rotation, is shown to be at the crossing point between the two domains. This discovery provides a basis for understanding existing experimental results and lays the groundwork for more quantitative studies with larger codes.