The sawtooth cycle remains perhaps the most studied of MHD phenomena in the fusion science program yet this central and important problem has remain unsolved. The basics of the cycle as proposed by Kadomtsev were that the peaking of the temperature in tokamaks would cause the current increase in the core until the safety factor q dropped below unity. At this point the m=1,n=1 tearing mode would drive reconnection across the central plasma, raising q above unity and expelling the high temperature core. Simulations reproduced the essence of Kadomtsev's theory and subsequent ideas on the onset of the sawtooth crash based on a balance between diamagnetic stabilization and the reconnection drive were consistent with observations. The time between sawtooth is approximately consistent with the time needed for the q profile to adjust from just above unity to an unstable profile -- although the behavior of the q profile is not clear, see below. A major surprise was that the sawtooth crash times were actually shorter in the largest and highest temperature machines, which was inconsistent with predictions of the Sweet-Parker theory with classical resistivity. The short time scale of the crash is not inconsistent with the recent kinetic model of reconnection. The development of turbulence-induced anomalous resistivity may also explain the fast crash of the sawtooth. There is in fact evidence for turbulence in the vicinity of the X-line during the crash. However the role of anomalous resistivity in the crash has not yet been convincingly established.

The clearest discrepancy between existing theoretical models and the observations concerns the observations that q does not rise above unity after the crash, indicating that reconnection stops short of complete reconnection of the core plasma. Remarkably, however, complete expulsion of the high temperature core plasma occurs. Why reconnection stops and how the core plasma can be expelled without complete reconnection has remained a mystery. Thus far, resistive MHD models have failed to reproduce the essential observations. A possible scenario that has been proposed is that the very strong gradient in the plasma pressure that develops at the magnetic X-line as the high temperature core plasma inside of the q=1 surface meets the lower temperature plasma outside of this surface will drive secondary ballooning instabilities or possibly resistive interchanges. Support for this idea comes from the observation that ballooning modes triggered by internal kinks are observed to cause very rapid disruptions. As the island grows the pressure gradient rises passing rapidly through marginal stability to the secondary modes. However, there is presently no consensus on what secondary modes dominate. Given that kinetic models suggest that the local pressure gradient near the X-line can approach the ion sound Larmor radius, the turbulence is likely to be kinetic rather than MHD-like. Finally, we note that the secondary instabilities effects must propagate into the magnetic axis to explain the collapse of the central electron temperature.

From this discussion it is evident that the sawtooth crash problem lies at the intersection of the MHD treatment of large scale instabilities and the kinetic treatment of turbulence and transport. Even a Hall MHD model is probably not adequate to capture the correct dynamics of secondary instabilities. Moving towards the resolution of this problem therefore clearly requires the development of new computational techniques and the close iteration of experiment and theory/computation. In spite of the difficulties outlined, we propose a concrete series of steps that will on the one hand make progress on the physics description of the problem and on the other hand lead to the development of novel computational approaches for addressing such challenging multiscale problems.

On the theory/computational side it is essential to begin to explore the range of instabilities that will dominate the region around the X-line during the sawtooth crash. We will accomplish this through an analysis of the linear instabilities in locally large pressure gradients. Computationally we will use a toroidal AMR MHD code {\tt BATS-R-US} developed by the group at the University of Michigan to explore the early stage of reconnection and the development of the large pressure gradient at the q=1 surface. The structure of the flux surfaces and pressure gradient in an annulus around this surface will then be used as input into a gyrokinetic model. The development of the turbulent spectrum of instabilities will then be explored. Of particular interest will be whether the turbulence can eat into the core plasma (interior to the q=1 surface), which will ultimately be required to explain the expulsion of the core energy. Our goal is develop the capability of coupling the gyrokinetic layer description to an MHD external description. In the initial phase of the project the external region will be treated as a linear perturbed MHD equilibrium using Resistive DCON (in collaboration with Alan Glasser) and with the external solution simply being represented by a linearized Delta-prime description. In later studies a nonlinear external region will be calculated with BATS-R-US. We note that these simulations will not include anomalous resistivity. Since the relevant instabilities are too high frequency for gyrokinetics, this requires an interface with the particle code, p3d.

In parallel with this theoretical effort scattering experiments will be carried out on the DIII-D and NSTX experiments to measure the fluctuations that develop near the X-line during the crash phase of the sawtooth.