This paper describes progress achieved since 2007 in understanding disruptions in tokamaks, when the effect of plasma current sharing with the wall was introduced into theory. As a result, the toroidal asymmetry of the plasma current measurements during vertical disruption event (VDE) on the Joint European Torus was explained. A new kind of plasma equilibria and mode coupling was introduced into theory, which can explain the duration of the external kink 1/1 mode during VDE. The paper presents first results of numerical simulations using a free boundary plasma model, relevant to disruptions.

Results from an array of theoretical and computational tools developed to treat the instabilities of most interest for high performance tokamak discharges are described. The theory and experimental diagnostic capabilities have now been developed to the point where detailed predictions can be productively tested so that competing effects can be isolated and either eliminated or confirmed. The predictions using high quality discharge equilibrium reconstructions are tested against the observations for the principal limiting phenomena in DIII-D: L-mode negative central shear (NCS) disruptions, H-mode NCS edge instabilities, and tearing and resistive wall modes (RWMs) in long pulse discharges. In the case of predominantly ideal MHD instabilities, agreement between the code predictions and experimentally observed stability limits and thresholds can now be obtained to within several percent, and the predicted fluctuations and growth rates to within the estimated experimental errors. Edge instabilities can be explained by a new model for edge localized modes as predominantly ideal low to intermediate n modes. Accurate ideal calculations are critical to demonstrating RWM stabilization by plasma rotation and the ideal eigenfunctions provide a good representation of the RWM structure when the rotation slows. Ideal eigenfunctions can then be used to predict stabilization using active feedback. For non-ideal modes, the agreement is approaching levels similar to that for the ideal comparisons; ∆' calculations, for example, indicate that some discharges are linearly unstable to classical tearing modes, consistent with the observed growth of islands in those discharges.

The stability of resistive modes is examined using reconstructions of experimental equilibria in the DIII-D tokamak [J. L. Luxon and L. G. Davis, Fusion Technol. 8, 441 (1985)], revealing the important physics in mode onset as discharges evolve to instability. Experimental attempts to access the highest β in tokamak discharges, including “hybrid” discharges, are typically terminated by the growth of a large 2∕1 tearing mode. Model equilibria, based on experimental reconstructions from one of these discharges with steady state axial q0≈1, are generated varying q0 and pressure. For each equilibrium, the PEST-III code [A. Pletzer, A. Bondeson, and R. L. Dewar, J. Comput. Phys. 115, 530 (1994)] is used to determine the ideal magnetohydrodynamic solution including both tearing and interchange parities. This outer region solution must be matched to the resistive inner layer solutions at the rational surface to determine resistive mode stability. From this analysis it is found that the approach to q=1 simultaneously causes the 2∕1 mode to become unstable and the nonresonant 1∕1 displacement to become large, as the ideal β limit rapidly decreases toward the experimental value. However, the 2∕2 harmonic on axis, which is also large and is coupled to the saturated steady state 3∕2 mode, is thought to contribute to the current drive sustaining q0 above 1 in these hybrid discharges. Thus, the approach to the q=1 resonance is self-limiting in this context. This work suggests that sustaining q0 slightly above 1 will avoid the 2∕1 instability and will allow access to significantly higher β values in these discharges.

Linear magnetohydrodynamic (MHD) and equilibrium evolution approaches describe linear and nonlinear axisymmetric displacement dynamics of free boundary plasma equilibrium configurations surrounded by conductors in an external magnetic field. A comparison of the two different approaches was made using DIII-D-like free boundary equilibria. Good agreement was found for up-down symmetric configurations. However, a considerable difference in growth rates is found for up-down asymmetric equilibria. The difference can be explained by taking into account surface current perturbations in the MHD model. Common and specific features of the two approaches are discussed.

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TAE Technologies, Inc. (TAE) is pursuing an alternative approach to magnetically confined fusion, which relies on field-reversed configuration (FRC) plasmas composed of mostly energetic and well-confined particles by means of a state-of-the-art tunable energy neutral-beam (NB) injector system. TAE’s current experimental device, C-2W (also called ‘Norman’), is the world’s largest compact-toroid device and has made significant progress in FRC performance, producing record breaking, high temperature (electron temperature, T e > 500 eV; total electron and ion temperature, T tot > 3 keV) advanced beam-driven FRC plasmas, dominated by injected fast particles and sustained in steady-state for up to 30 ms, which is limited by NB pulse duration. C-2W produces significantly better FRC performance than the preceding C-2U experiment, in part due to Google’s machine-learning framework for experimental optimization, which has contributed to the discovery of a new operational regime where novel settings for the formation section and the confinement region yield consistently reproducible, hot, and stable plasmas. An active plasma control system has been developed and utilized in C-2W to produce consistent FRC performance as well as for reliable machine operations using magnets, electrodes, gas injection, and tunable NBs. The active control system has demonstrated stabilization of FRC axial instability. Overall FRC performance is well correlated with NBs and edge-biasing system, where higher total plasma energy is obtained by increasing both NB injection power and applied-voltage on biasing electrodes. C-2W divertors have demonstrated a good electron heat confinement on open-field-lines using strong magnetic mirror fields as well as expanding the magnetic field in the divertors (expansion ratio > 30); the energy lost per electron ion pair, η e ∼ 6–8, is achieved, which is close to the ideal theoretical minimum.

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