### Theory Weekly Highlights for September 2001

##### September 28, 2001

A new class of “doublet FRC” equilibria that contain double magnetic axes and two teardrop shaped private flux regions contained within a figure-eight shaped internal separatrix has been found. In contrast, the internal flux surfaces of the usual FRC equilibria have a single magnetic axis with a separatrix envelope having the shape of an ellipse or a racetrack (a long straight central section terminated by high-curvature end regions) in the r-z plane. In the new solution, a common flux region lies between the internal separatrix and the outer separatrix, which for the most part is rectangular in shape with an indentation near the midplane. Analytical solutions of the Grad-Shafranov equation in cylinder coordinates were found by specializing to the pressure profile p ~ ψ2 that peaks at the two magnetic axes, and is zero at the outer separatrix. Numerical solutions have also been found for other pressure profiles using the GA equilibrium code EQCT. Doublet FRCs may possess more favorable stability properties: the common flux region near the internal separatrix has favorable average curvature, while finite pressure on the internal separatrix provides interchange stability in the private flux regions.

##### September 21, 2001

The GKS gyro-kinetic stability code has been implemented in the XPTOR transport code. The linear growth rates for both s-alpha and real geometry are in agreement with those from the Cray version of GKS that is currently in use. Using the MPI library, the code is parallel over the radial grid allowing the GKS eigenvalues to be found for an entire profile when executed on a Linux cluster. This facilitates the analysis of experimental data and paves the way for the future developments of the GLF23 transport model. In principle, the GKS growth rates can also be used directly to compute the turbulent fluxes in a time-dependent transport simulation.

##### September 14, 2001

The initial electromagnetic version of the GA gyrokinetic turbulence simulation code GYRO has been completed. Linear eigenvalues are in unequivocal agreement with those from GKS, a gyrokinetic stability code, and the nonlinear performance in flux-tube mode appears to be excellent. The only other code that can treat electromagnetic turbulence in toroidal geometry is GS2 (an Eulerian solver like GYRO) - but GS2 is limited to flux tubes and thus cannot treat profile variation. Global particle (rather than Eulerian) codes address profile variation, but are limited to electrostatic fluctuations because of noise-related numerical difficulties. GYRO, in contrast, is both global and electromagnetic. We are very excited at initial indications that GYRO is also substantially faster than competing codes for long-time nonlinear simulations, but more work is required to fully support this claim.

##### September 07, 2001

Stability analysis of a QDB discharge shows that it is stable to ideal n=1,2 modes with a conducting wall at 1.5a, but marginally stable to high n ideal ballooning modes in a small region near r / a ~ 0.45. The results are consistent with the GLF23 and GKS modeling results and with FIR and reflectometer data, indicating that core turbulence is not completely eliminated in QDB plasmas. The trajectory of the QDB discharge in the pressure peakedness and normalized beta stability space, from a time sequence of kinetic EFIT reconstruction, lies between the trajectories of weak central shear H-mode discharges with broad pressure profiles and negative central shear L-mode discharges with peaked pressure profiles. This is consistent with the presence of both an ITB and an H-mode edge in the QDB discharges.

**Disclaimer**

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