The maximum normalized beta achieved in long-pulse tokamak discharges at low collisionality falls significantly below both that observed in short pulse discharges and that predicted by the ideal MHD theory. Recent long-pulse experiments, in particular those simulating the International Thermonuclear Experimental Reactor ͑ITER͒ ͓M. Rosenbluth et al., Plasma Physics and Controlled Nuclear Fusion ͑International Atomic Energy Agency, Vienna, 1995͒, Vol. 2, p. 517͔ scenarios with low collisionality e * , are often limited by low-m/n nonideal magnetohydrodynamic ͑MHD͒ modes. The effect of saturated MHD modes is a reduction of the confinement time by 10%-20%, depending on the island size and location, and can lead to a disruption. Recent theories on neoclassical destabilization of tearing modes, including the effects of a perturbed helical bootstrap current, are successful in explaining the qualitative behavior of the resistive modes and recent results are consistent with the size of the saturated islands. Also, a strong correlation is observed between the onset of these low-m/n modes with sawteeth, edge localized modes ͑ELM͒, or fishbone events, consistent with the seed island required by the theory. We will focus on a quantitative comparison between both the conventional resistive and neoclassical theories, and the experimental results of several machines, which have all observed these low-m/n nonideal modes. This enables us to single out the key issues in projecting the long-pulse beta limits of ITER-size tokamaks and also to discuss possible plasma control methods that can increase the soft  limit, decrease the seed perturbations, and/or diminish the effects on confinement.
A self-consistent kinetic model is developed to study the stability of the resistive wall mode in toroidal plasmas. This model is compared with other models based on perturbative approaches. The degree of the kinetic modification to the stability of the mode depends on the plasma configurations. Both stabilizing and destabilizing kinetic effects are observed. The nonperturbative approach, with a self-consistent inclusion of the eigenfunctions and the eigenvalues of the resistive wall mode, normally finds less stabilization than the perturbative approach.
Understanding the physics of the edge pedestal and edge localized modes (ELMs) is of great importance for ITER and the optimization of the tokamak concept. The peeling-ballooning model has quantitatively explained many observations, including ELM onset and pedestal constraints, in the standard H-mode regime. The ELITE code has been developed to efficiently evaluate peeling-ballooning stability for comparison to observation and predictions for future devices. We briefly present recent progress in the peelingballooning model, including studies of the apparent power dependence of the pedestal, and studies of the impact of sheared toroidal flow. Nonlinear 3D simulations of ELM dynamics using the BOUT code are also described, leading to an emerging understanding of the physics of the onset and dynamics of ELMs in the standard intermediate to high collisionality regime. Recently, highly promising low collisionality regimes without ELMs have been discovered, including the quiescent H-mode (QH) and resonant magnetic perturbation (RMP) regimes. We present recent observations of the density, shape and rotation dependence of QH discharges, and studies of the peeling-ballooning stability in this regime. We propose a model of the QH-mode in which the observed edge harmonic oscillation (EHO) is a saturated kink/peeling mode which is destabilized by current and rotation, and drives significant transport, allowing a near steady-state edge plasma. The model quantitatively predicts the observed density dependence, and qualitatively predicts observed mode structure, rotation dependence, and outer gap dependence. Low density RMP discharges are found to operate in a similar regime, but with the EHO replaced by an applied magnetic perturbation.
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