The neoclassical tearing mode (NTM) has emerged as a limiting factor for magnetic pressure in plasma confinement systems. Because the instability grows on slow resistive times (0.2 s in DIII-D), it is expected to be more constraining in long pulse machines such as ITER. Observations show that there is a critical initial island width below which the mode is stable and the island decays. Finite islands grow as a result of the local modification of the plasma pressure profile around the island and associated bootstrap currents (driven by the drag between the trapped and untrapped particles in a toroidal plasma). Its growth is triggered by sudden MHD events, most commonly sawtooth crashes. Presumably these events drive a finite sized island and thereby initiate the instability. Unlike other large-scale reconnection phenomena, the NTM is driven by the thermal rather than the magnetic free energy of the system (the tearing mode stability parameter Delta prime is typically negative). It is restricted to long azimuthal wavelengths by the stabilizing effect of line bending. The critical questions for fusion are: when do these islands grow and what is the critical island size for instability onset? In order to converge on the answers to these questions we propose to conduct a campaign of simulations, theory and experiments aimed at isolating and understanding key physical effects. At the same time we will push forward with the development of the novel computational techniques that will be required to solve the complete problem and give quantitative answers to the critical questions.

The NTM growth is, like the sawtooth crash, at the intersection of MHD and turbulent transport modeling and this has hampered progress in understanding the phenomenon. The stability of the NTM is thought to depend primarily on the modification of the temperature, density and velocity (potential) profiles in the vicinity of the island. These profiles are determined by transport processes. For example, the temperature profile depends on the competition between the parallel conduction that tends to flatten the temperature within the island, and the perpendicular conduction that maintains a radial gradient. The cross-field transport in tokamaks is anomalous (turbulent) and is driven by gradients in temperature, density or flow velocity. One must solve for the turbulence in the vicinity of the island to determine the profiles self-consistently -- ambient turbulence-driven transport rates do not apply in the steep gradients that develop near the magnetic island. This represents a formidable challenge. In earlier models of island growth empirical transport coefficients have been used to represent the turbulent transport. However the choice of how these turbulent diffusivities vary with gradients and phase of the island is unclear. Furthermore islands are initiated with widths comparable to the turbulent eddy sizes and therefore the representation of transport as a diffusion process is itself questionable. The use of a local model for the bootstrap current in the vicinity of these same locally varying pressure gradients is also questionable.

Turbulence also determines the local viscosity and thereby the rotation of the island through the plasma. The importance of the rotation frequency lies in the fact that the polarization current has been shown to be stabilizing for islands propagating at frequencies lying between the ion and the electron diamagnetic frequency. When propagating in the electron diamagnetic direction the island emits drift waves and the resulting loss of momentum results in a slowing down torque. The reverse process, where the island accelerates by absorbing drift waves, is also possible. This behavior is analogous to the zonal flow excitation. Thus, the interaction between the magnetic island and turbulence can have a significant impact on the dynamics of islands and must be included in any credible theoretical model.

The key to unravelling the physics of the NTM lies in exploiting scale separation in both time and space. The critical island size (approximately 1 cm), the current carrying layer of the tearing mode (approximately 1 mm) and the turbulent radial correlation length (approximately 1 cm) are all very small compared to the typical size of the plasma (> 1 m). In the region outside the island (and strictly speaking outside the ion sound distance from the rational surface) the perturbed magnetic flux of the resonant mode can be calculated from the MHD model. Thus, the computational domain can be reduced by solving for the island evolution in a thin annulus around the rational surface and matching to an external solution based either on a linearized Delta prime or a nonlinear MHD model. Further simplification can be achieved by exploiting the timescale separation between the long island growth time (0.2 s), and the short turbulent eddy turnover time (10**-5 s) and profile relaxation time over the island (10**-4 s). Thus, on the turbulence timescale the island is stationary and on the island growth timescale the turbulence sets up a steady constant transport flux solution across the annulus. To solve for the complete evolution of the island we will use a multiscale (projective integration-type) algorithm. First the annular solutions with an island present are computed using a fine scale (kinetic) model for a few turbulent relaxation times and the island growth rate over this short time is evaluated. Then the island is advanced a long timestep using this computed growth rate. A new computation is then initiated with the new island size and a new growth rate evaluated. These steps are repeated until the island stops growing or disappears.