Page 7 - Compact Toroid Injection Fueling in a Large Field-Reversed Configuration

082512-7 Lau et al.
Phys. Plasmas 24, 082512 (2017)
with even faster growth. The resonant ions are at lower energy while the resonant electrons are, as in the g 1⁄4 1 case, the high energy locally barely trapped electrons, as seen in the bottom panel of Fig. 7. From the comparison of the three cases, it is clear that the motion of the locally barely trapped electrons is the resonance that drives the collisionless SOL instability.
D. Collisional effects
Using the Fokker-Planck model, the SOL instability for g 1⁄4 1 was simulated with pitch-angle scattering through electron-ion collisions over a range of collisionality, from
1⁄4 1.5 using equilibrium densities and temperatures detailed in Table I. With collisions, both frequency and growth-rate decrease, but as the collisionality is lowered, the collisionless frequency and growth-rate are recovered. While collisions are strongly stabilizing in the long wavelength case, there is negligible effect to the mode structure, which is strongly
dominated by the m 1⁄4 0 and m 1⁄4 1 harmonics.
The effect of collisions on this instability can be under-
stood from the locally trapped electron resonance. Pitch- angle scattering frequently moves electrons in and out of that particular energy-pitch position, essentially removing the drive of this instability. However, as seen in Figs. 2 and 3, there is still a significant instability which can exist at shorter wavelengths even when the collisionless mode is suppressed by collisions.
V. DISCUSSION
Local gyrokinetic simulations have been used to investi- gate electrostatic pressure gradient-driven drift-waves in the FRC. While the FRC core was expected to be less unstable1 due to the large ion gyro-radius, simulations found drift- waves in the core to be completely stable with C-2-like parameters for pressure gradient drives up to qi=LP Oð1Þ with equal temperature and density gradients (g 1⁄4 1) for ion- to-electron scale wavelengths (kfqe < 0.3). Our studies of limiting cases in the FRC strongly suggest that this stability is due to the short electron connection length with further stabilizing contribution from FLR8 and rB effects. Only when the FRC geometry is artificially elongated from the typical C-2-like FRC toward the field-reversed theta pinch (by Zlim/Z0 exceeding 5–7) does the known drift-wave insta- bility appear, consistent with the importance of electron par- allel dynamics. It should also be noted that the electron connection length along the field-lines in the FRC core is much shorter than those such as in tokamaks.
In the SOL, a pressure gradient driven mode with wave- lengths ranging from ion-scale to electron-scale has been found. This collisionless instability is driven by magnetically trapped electrons. The unstable mode peaks are correlated with the regions with weakest magnetic fields, strongest cur- vature, and local minima of rB. Collisions suppress this instability but allow a different lower frequency collisional instability at shorter wavelengths.
In experiments conducted by Schmitz et al.,35 density fluctuations measured in the core display a “depressed”
wavenumber spectrum in which fluctuation amplitudes are low at ion-scale wavelengths but peak at electron-scale wavelengths (in the range around kfqe 0.15–0.45). The fluctuations measured in the SOL have a more typical wave- number spectrum with higher amplitudes at ion-scale and exponentially decreasing amplitudes toward shorter wave- lengths. The stability exhibited in the simulations of the core is consistent with the experimental core fluctuation spec- trum; however, simulations using Vlasov ions are necessary to explore the possibility of higher frequency instabilities. The ion-to-electron-scale nature of the instability of the SOL displayed in simulations is consistent with the experimen- tally measured SOL fluctuation spectrum. In addition, exper- imental data show the existence of fluctuation thresholds35 at normalized drive strengths of jexp 3.9. In simulations, col- lisionless linear thresholds are found at jsim 3–5 in the SOL for the longer wavelength modes.
Based on the fastest growth-rates of the collisionless SOL instability driven by the largest and smallest simulated drive strengths j 8.1–1.3, the characteristic growth times are c 1.75 0.25Cs/R0 ! sSOL 1⁄4 1.4–10 ls. The fastest growing collisional SOL mode, driven by j 1⁄4 6.7, has a comparable growth time of c 0:65Cs=R0 ! sSOL 1⁄4 3:9 ls. In the C-2 experiments, the FRC plasma lifetimes are on the order of milliseconds, and so both the collisionless and colli- sional SOL modes have enough time to grow to a substantial amplitude to explain the fluctuation spectrum observed.
In experimental measurements of density fluctuations,35 the FRC core and SOL show distinct behaviors. The SOL dis- plays strong density fluctuations which follow an exponential scaling while the quiescent core density fluctuations are lower in amplitude by an order of magnitude. In agreement with these experimental results, our simulations find drift-waves to be robustly stable in the core and unstable in the SOL.
The surprising stability of electrostatic drift-waves in the core requires further studies to isolate the origin of the fluctuations observed in experiments. Higher frequency instabilities may exist in the core, but the origin of the fluctu- ations may also lie in the interaction between the SOL and core. Future work will focus on the physics of cross- separatrix interactions and the possible propagation of fluctu- ations from SOL to core.
ACKNOWLEDGMENTS
The authors would like to thank Sean Dettrick and the TAE team at Tri Alpha Energy, Inc., for equilibrium data as well as the ongoing insights and collaboration in the development of these simulations. This work was carried out at University of California, Irvine, with the support of the Norman Rostoker Fellowship (Grant No. TAE-200441) and by the DOE SciDAC GSEP center. Simulations used the resources of the Oak Ridge Leadership Computing Facility at the Oak Ridge National Laboratory (DOE Contract No. DE-AC05-00OR22725) and the National Energy Research Scientific Computing Center (DOE Contract No. DE-AC02-05CH11231).
1N. Rostoker, F. Wessel, H. Rahman, B. C. Maglich, B. Spivey, and A. Fisher, “Magnetic fusion with high energy self-colliding ion beams,” Phys. Rev. Lett.. 70, 1818–1821 (1993).
collisionless to the collision frequency defined for Z
eff