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.

The theory of perturbed magnetohydrodynamic equilibria is presented for different formulations of the tokamak equilibrium problem. For numerical codes, it gives an explicit Newton scheme for solving the Grad–Shafranov equation subject to different constraints. The problem of stability of axisymmetric modes is shown to be a particular case of the equilibrium perturbation theory.

The stability of a test equilibrium relevant to the International Thermonuclear Experimental Reactor [Fusion Eng. Des. 36, 9 (1997)] has been studied within the framework of the neoclassical island theory. The most unstable modes, with the most positive matching index Δ′, have been found by solving the toroidal stability equation to have 3/2, 2/1, and 4/3 helicities. Quasilinear effects resulting from the flattening of the current profile as the island develops are important and stabilizing. Large saturated islands are predicted to arise from a combination of strong bootstrap current drive and weakly negative Δ′. The island size can be significantly reduced by applying a continuous current drive at the unstable rational surfaces. The so-obtained reduction of island width is approximately proportional to the current drive and inversely proportional to the square of the current channel. This stabilization relies on the removal of the free energy in the outer region and can thus be regarded as a Δ′ effect.

Fast Wave (FW) studies of mode conversion (MC) processes at the ion-ion hybrid layer in toroidal plasmas must capture the disparate scales of the FW and mode converted ion Bernstein (IBW) and ion cyclotron waves (ICW). Correct modeling of the MC layer requires resolving wavelengths on the order of k ⊥ ρi ∼ 1 which leads to a scaling of the maximum poloidal mode number, Mmax, proportional to 1/ρ * (ρ * ≡ ρi/L). The computational resources needed a scale with the number of radial (Nr), poloidal (N θ ), and toroidal (N φ ) elements as Nr * N φ * N

Non-uniform voltage distribution across the electrode area results in inhomogeneous thin-film RF plasma deposition in large-area reactors. In this work, a two-dimensional analytic model for the calculation of the voltage distribution across the electrode area is presented. The results of this model are in good agreement with measurements performed without plasma at 13.56 MHz and 70 MHz in a large-area reactor. The principal voltage inhomogeneities are caused by logarithmic singularities in the vicinity of RF connections and not by standing waves. These singularities are only described by a two-dimensional model and cannot be intuitively predicted by analogy to a one-dimensional case. Plasma light emission measurements and thickness homogeneity studies of a-Si:H deposited films show that the plasma reproduces these voltage inhomogeneities. Improvement of the voltage uniformity is investigated by changing the number and position of the RF connections.

For high β, highly shaped plasmas in the DIII-D tokamak [J. L. Luxon and L. G. Davis, Fusion Technol. 8, 441 (1985)], the value of the tearing stability index Δ′ calculated at a rational surface can be especially sensitive to the pressure and current profiles. Near marginal stability for a global ideal mode, a pole in Δ′ exists in equilibrium parameter space, as predicted by analytic theory. The proximity of an equilibrium reconstruction to this pole in parameter space strongly decreases the accuracy of the Δ′ calculations. Tearing stability calculations on kinetic equilibrium reconstructions of a series of times in three DIII-D discharges are presented, which indicate that the tearing modes in these discharges are classically unstable at the time of onset. The onset mechanism of two of these discharges (which are in H-mode) is related to the approach of ideal stability boundaries and the occurrence of poles in Δ′. Several ideal modes (sawteeth, edge localized modes, and resistive wall modes) are thought to seed neoclassical tearing modes (NTMs) through forced reconnection, after the ideal mode is unstable. However, tearing modes often appear suddenly and grow quickly without an obvious ideal mode causing a seed island through forced reconnection, which could be explained by this mechanism. This is proposed as an alternative mechanism for the onset of NTMs in tokamaks, which is not incompatible with forced reconnection.

Within the framework of studies of the stability of magneto-plasmas to non-ideal modes, such as resistive modes, the problem of determining the asymptotic matching data arising from the outer (ideal) region is considered. Modes possessing both tearing and interchange (ballooning) parity are considered in finite-pressure plasmas. The matching data, which form a matrix whose elements represent the small solution response to forcing by a big solution, are shown to derive from a variational (energy) principle. The variational principle, as presented, applies to both cylindrical and two-dimensional (toroidal) geometries. Allowing for the presence of multiple rational surfaces, a reciprocity relation between off-diagonal elements of the matching data matrix is obtained. The variational principle is suitable for numerical approximation, and, in particular, for the finite-element method, for which convergence rates are estimated. By packing nodes near the rational surface, maximum convergence, proportional to the inverse square of the number of mesh nodes for tent functions, is achieved.

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