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 detailed comparison is made between the tearing-type modes observed in TFTR supershot plasmas and the nonlinear, neoclassical pressure-gradient -driven tearing mode theory. Good agreement is found on the nonlinear evolution of single helicity magnetic islands (m/n = 3/2, 4/3, or 5/4, where m and n are the poloidal and toroidal mode numbers, respectively). The saturation of these neoclassical tearingtype modes requires 6' ( 0 (where 5' is the well-known parameter for classical current-driven tearing instability), which is also consistent with the numerical calculation using the experimental data.PACS numbers: 52.55. Fa, 52.30.Jb, 52.35.Py, Understanding the tearing-type MHD (magnetohydrodynamic) instabilities observed in TFTR neutral-beam (NB) heated supershot [1] plasmas has long been a challenge for plasma theory. These modes typically have low frequency ( f (50kHz) and low mode numbers (m/n = 3/2, 4/3, and 5/4). The m/n = 2/1 modes are not usually seen in the high-performance supershot plasmas. The important effects of these MHD modes on plasma performance have been discussed in Ref. [2]. It is found that when these modes are large they can cause a strong deterioration in plasma performance [2] as measured by the DD or DT neutron rate, plasma-stored energy, energy confinement time, etc. Considerable effort has been expended on the theoretical interpretation and numerical simulation of these modes. These works have been mostly based upon the classical current-driven tearing mode theory [3,4]. However, the results have been unsatisfactory [5,6]. In this Letter, we compare the experimental results with a relatively new theory, the neoclassical pressure-gradient-(7'p) driven tearing mode theory [7,8]. The results are found to be very encouraging.The evolution of two typical tearing-type modes in the high power NB heated, high P (plasma pressure/magnetic field pressure) supershot plasmas is shown in Fig. 1.Discharge A developed an m/n = 3/2 mode. Discharge B developed an m/n = 4/3 mode. Detailed analyses of the MHD modes and their deterioration effect on plasma transport [see Fig.

The moment equation approach to neoclassical transport theory has been generalized to nonaxisymmetric toroidal systems under the assumption of the existence of magnetic surfaces. In particular, the parallel plasma flows and bootstrap current are calculated in both the Pfirsch–Schlüter and banana regimes. It is found that both parallel plasma flows and the bootstrap current can be reduced as the toroidal bumpiness increases in an otherwise axisymmetric system.

The effectiveness of using localized current drive or heating to suppress the formation and growth of neoclassical magnetohydrodynamic ͑MHD͒ tearing modes is addressed. The most efficient way to use an auxiliary current source is to cause current to flow in the same direction as the equilibrium bootstrap current and phase the current relative to the magnetic island such that the current is deposited on the O-point of the island. Theoretical estimates for the amount of required current to suppress the formation of a large magnetic island is of order a few percent of the equilibrium current. If the suppression is successful, the magnetic island will saturate at a width of order the radial localization width of the current source. Localized heating at the O-point of the magnetic island can also produce stabilizing effects relative to magnetic island growth. The effects of the driven current or heating can be illustrated by using a phase diagram of the island growth. © 1997 American Institute of Physics.

The nonlinear dynamics of rotating low m (poloidal mode number) tearing modes in a tokamak with external resonant magnetic perturbations is examined. Nonlinear evolution equations for the island width and the toroidal rotation frequency are derived within the two-fluid magnetohydrodynamic model, taking into account the plasma rotation and neoclassical parallel viscosity. The nonlinear stability of magnetic islands interacting with a static external magnetic perturbation is considered, and the critical magnetic field for the appearance of a locked mode is determined. It is shown that the coupling of the perpendicular and longitudinal plasma flow due to the neoclassical plasma viscosity enhances the amplitude of the critical magnetic field compared to the value obtained in a slab approximation. The perpendicular plasma viscosity causes a finite phase shift between the applied external field and the magnetic island, and further increases the value of the critical magnetic field required to induce a magnetic island. 0 1995 American Institute of Ph.ysics.