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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