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.

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The theory of tearing mode stabilization in toroidal plasmas by RF driven currents that are modulated in phase with the island rotation is investigated. A timescale analysis of the phenomena involved indicates that transient effects, such as finite time response of the driven currents, island rotation during the power pulses and the inductive response of the plasma, are intrinsically important. A dynamical model of such effects is developed, based on a 3-D Fokker-Planck code coupled to both the electric field diffusion equation and the island evolution equation. Extensive applications to both ECCD and LHCD in ITER are presented.

NIMROD is a code development project designed to study long-wavelength, lowfrequency, nonlinear phenomena in toroidal plasmas with realistic geometry and dynamics. The numerical challenges of solving the fluid equations for a fusion plasma are discussed and our discretization scheme is presented. Simulations of a resistive tearing mode show that time steps much greater than the Alfvén time are possible without loss of accuracy. Validation tests of a resistive interchange mode in a shaped equilibrium, a ballooning mode and nonlinear activity in reversed-field pinches are described.

Nonlinear numerical studies of macroscopic modes in a variety of magnetic fusion experiments are made possible by the flexible high-order accurate spatial representation and semi-implicit time advance in the NIMROD simulation code ͓A. H. Glasser et al., Plasma Phys. Controlled Fusion 41, A747 ͑1999͔͒. Simulation of a resistive magnetohydrodynamics mode in a shaped toroidal tokamak equilibrium demonstrates computation with disparate time scales, simulations of discharge 87009 in the DIII-D tokamak ͓J. L. Luxon et al., Plasma Physics and Controlled Nuclear Fusion Research 1986 ͑International Atomic Energy Agency, Vienna, 1987͒, Vol. I, p. 159͔ confirm an analytic scaling for the temporal evolution of an ideal mode subject to plasma-␤ increasing beyond marginality, and a spherical torus simulation demonstrates nonlinear free-boundary capabilities. A comparison of numerical results on magnetic relaxation finds the nϭ1 mode and flux amplification in spheromaks to be very closely related to the mϭ1 dynamo modes and magnetic reversal in reversed-field pinch configurations. Advances in local and nonlocal closure relations developed for modeling kinetic effects in fluid simulation are also described.

Heuristic closures are presented for use in simulations of neoclassical modifications to magnetohydrodynamic phenomenon in tokamaks. The closures capture the dominant physics expected from linear and quasilinear neoclassical instability theory and are computationally easy to implement. Numerical results from the NIMROD ͓A. H. Glasser, C. R. Sovinec, R. A. Nebel et al., Plasma Phys. Controlled Fusion 41, A747 ͑1999͔͒ code are shown which demonstrate poloidal flow damping, growth rate reduction due to the neoclassical enhancement of the polarization current, and generation of perturbed bootstrap currents and subsequent generation of a neoclassical tearing mode.

Numerical studies of the nonlinear evolution of magnetohydrodynamic-type tearing modes in three-dimensional toroidal geometry with neoclassical effects are presented. The inclusion of neoclassical physics introduces an additional free-energy source for the nonlinear formation of magnetic islands through the effects of a bootstrap current in Ohm's law. The neoclassical tearing mode is demonstrated to be destabilized in plasmas which are otherwise ⌬Ј stable, albeit once an island width threshold is exceeded. The plasma pressure dynamics and neoclassical tearing growth is shown to be sensitive to the choice of the ratio of the parallel to perpendicular diffusivity ( ʈ / Ќ ). The study is completed with a demonstration and theoretical comparison of the threshold for single helicity neoclassical magnetohydrodynamic tearing modes, which is described based on parameter scans of the local pressure gradient, the ratio of perpendicular to parallel pressure diffusivities Ќ / ʈ , and the magnitude of an initial seed magnetic perturbation.

The application of fluid models in studies of transport and macroscopic stability of magnetized, nearly collisionless plasmas requires closure relations that are inherently nonlocal. Such closures address the fact that particles are capable of carrying information over macroscopic parallel scale lengths. In this work, generalized closures that embody Landau, collisional and particle-trapping physics are derived and discussed. A gyro/bounce-averaged drift kinetic equation is solved via an expansion in eigenfunctions of the pitch-angle scattering operator and the resulting system of algebraic equations is solved by integrating along characteristics. The desired closure moments take the form of integral equations involving perturbations in the flow and temperature along magnetic field lines. Implementation of the closures in massively parallel plasma fluid simulation codes is also discussed. This implementation includes the use of a semi-implicit time advance of the fluid equations to stabilize the dominant closure terms which are introduced explicitly. Application of the nonlocal, parallel heat flow closure, q ʈ , in studies of temperature flattening across helical magnetic islands in toroidal geometry reveal a scaling of temperature versus critical island width for flattening of Tϳw d Ϫ1.5 . This result predicts more robust flattening at small island widths when compared to the diffusive scaling, Tϳw d Ϫ1.7 , which assumes a Braginskii-type parallel heat conductivity. Preliminary application of q ʈ to tokamak disruption simulations shows qualitative agreement of wall heat loads with experimental observations, smooth distribution in toroidal angle, and striation in the poloidal direction along the wall.