Ideal MHD stability calculations for the wall stabilized DIII-D discharges #107607 and #107611 confirmed that the normalized β (β_{N}) limit with no wall is still close to 2.5*l*_{i} for times at the end of the high beta phase as well as at the initial heating phase for these low triangularity, low *l*_{i}, single null (SN) discharges. These discharges were similar to the discharge that reached twice the no-wall β limit, except that β was ramped downwards toward the end in order to avoid the disruption from the ideal with-wall kink instability. This confirms that the operational definition for the β_{N} limit of 2.5*l*_{i} in these discharges remains valid during the entire discharge evolution. This limit is also consistent with experiments in similar discharges in which the RWM onset also occurs close to 2.5*l*_{i} when the plasma rotation is slowed. The experiments utilized these SN low *l*_{i} targets in order to obtain a low β_{N} limit. For high triangularity, double null (DN) Advanced Tokamak targets, the no-wall limit has been found previously to be 4*l*_{i}.

MHD stability studies of Alcator C-Mod equilibria were carried out in collaboration with D. Mossessian of MIT as part of the C-Mod and DIII-D edge similarity experiments. Intermediate-n ideal MHD stability analysis with the ELITE code found that low-power EDA mode discharges in C-Mod are stable, whereas the high power shots in which small ELMs appear are unstable to peeling-ballooning modes. These peeling-ballooning modes have a mode structure that extends across the pedestal region, and growth rates that depend strongly on the edge current. Low q ELM-free shots have also been studied and found to be unstable only to very weak, strongly localized MHD modes, even with large edge pressure gradients. The contrast in stability between the ELMing and ELM-free discharges is consistent with our model of ELMs as intermediate n peeling-ballooning modes.

In a collaboration with PPPL, it has been shown that, by carefully treating the energetic-particle banana orbits in computing the torque-density source term for the angular momentum diffusive transport equation, ion cyclotron absorption of fast Alfven waves by energetic minority ions can lead to toroidal torque density distributions in the bulk plasma. Previous theoretical work, which neglected wave angular input ( valid when n=k_{||}R =0), showed that finite banana orbits of ICRF generated energetic ions created regions of positive and negative torque density with zero integrated torque. Integration of the transport equation produced a rotational shear layer, even though the fast waves deposited no net torque in the plasma. An extension of this work to travelling Alfven waves with n = 10 to 20, which do deposit a net torque, found only modest changes in the rotational profiles, in agreement with simple dimensional analysis. The magnitude and scaling of the rotation rate is characteristic of Alcator C-Mod and JET observations but differences remain concerning the sense of rotation and the boundary condition value. Estimates of the rotational shear find it sufficient to affect microinstability turbulence within a shear layer of about 10% of the minor radius in extent. This leads to the possibility of ICRF-controlled transport barriers, as has been observed on C-Mod.

A method to discretize the pitch-angle scattering operator on highly irregular grids has been developed for use in GYRO. For 5-point stencils, we make use of a topological constraint on point selection to ensure numerical stability of the time-advance and a direct sparse matrix package (UMFPACK) for factorizations and solves. With the added effect of electron collisions, we can now study turbulent transport even near the highly collisional edge using a fully gyrokinetic model for both ions and electrons.

In collaboration with Dr. Guo at RFX-Padova, the non-linear dynamics of a tearing mode interacting with an externally applied rotating field and a resistive wall was studied. A pair of coupled equations describing the evolution of the amplitude and frequency of a single tearing mode under the combined effect of the external field and the resistive wall was derived and solved numerically. Bifurcated stationary states of the tearing mode were found and the transition between these stationary states was studied. This investigation shows that the dynamics of the tearing mode is substantially affected by the external applied field; a tearing mode locked to the resistive wall can be unlocked to rotate with the external field.

**Disclaimer**

These highlights are reports of research work in progress and are accordingly subject to change or modification