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The prevention of neoclassical tearing modes (NTMs) in tokamak plasmas is a major challenge for fusion. Ideal modes can seed NTMs through forced reconnection, yet in sawtoothing discharges it is not well understood why a particular sawtooth crash seeds a NTM after several preceding sawteeth did not. Also, tearing modes sometimes appear and grow without an obvious ideal mode causing a seed island. Based on theoretical and experimental results a new mechanism for tearing mode onset is proposed and tested which explains these puzzling observations. Tearing stability calculations based on experimental equilibria from the DIII-D tokamak [J. L. Luxon and L. G. Davis, Fusion Technol. 8, 441 (1985)] indicate that tearing modes can be driven unstable by a rapid increase in the linear tearing stability index Δ′ just before onset. Near the ideal kink limit in βN≡β/(I/aBT), Δ′ becomes large and positive as it approaches a pole discontinuity. The relative time scales of the change in Δ′ vs the effects of finite island width will determine the evolution of the island, and the eventual nonlinear state. Theoretical predictions of the onset point, the βN at a specified small island width, and the early evolution characteristics are compared with results from new DIII-D experiments designed specifically to test this hypothesis. Time integration of the island evolution equation shows that these predictions are consistent with the experimental observations. The nonlinear 3D resistive magnetohydrodynamic code NIMROD [A. H. Glasser et al., Plasma Phys. Control. Fusion 41, A747 (1999)] is used to evolve equilibria reconstructed from these experiments in time to provide a more comprehensive prediction of the island evolution during the early nonlinear phase. The simulations are also consistent with the proposed mechanism.

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.

The influence of toroidal sheared equilibrium flows on the nonlinear evolution of classical and neoclassical tearing modes is studied through numerical solutions of a set of reduced generalized MHD equations that include viscous force effects based on neoclassical closures. In general, differential flow is found to have a strong stabilizing influence leading to lower saturated island widths for the classical tearing mode and reduced growth rates for the neoclassical mode. Velocity shear, on the other hand, is seen to make a destabilizing contribution.

A new derivation of reduced magnetohydrodynamic ͑MHD͒ equations is presented. A multiple-time-scale expansion is employed. It has the advantage of clearly separating the three time scales of the problem associated with ͑1͒ MHD equilibrium, ͑2͒ fluctuations whose wave vector is aligned perpendicular to the magnetic field, and ͑3͒ those aligned parallel to the magnetic field. The derivation is carried out without relying on a large aspect ratio assumption; therefore this model can be applied to any general toroidal configuration. By accounting for the MHD equilibrium and constraints to eliminate the fast perpendicular waves, equations are derived to evolve scalar potential quantities on a time scale associated with the parallel wave vector ͑shear-Alfven wave time scale͒, which is the time scale of interest for MHD instability studies. Careful attention is given in the derivation to satisfy energy conservation and to have manifestly divergence-free magnetic fields to all orders in the expansion parameter. Additionally, neoclassical closures and equilibrium shear flow effects are easily accounted for in this model. Equations for the inner resistive layer are derived which reproduce the linear ideal and resistive stability criterion of Glasser, Greene, and Johnson ͓Phys. Fluids 18, 875 ͑1975͔͒.