Page 10 - Tomographic imaging system for measuring impurity line emission in a field-reversed configuration

022506-10 D. C. Barnes and L. C. Steinhauer
Phys. Plasmas 21, 022506 (2014)
TABLE III. Summary of ‘ 1⁄4 2 stability results.
‘ 1⁄4 2 Non-reversed
Non-reversed Harned RR C-2 RR
C-2 RR
Thin SOL Thin SOL
Ion flow shear Ion flow shear
Figures
3
4 Tables I and II 9
10
13
14
...
...
a
0.8 þ1.6 0.8 0.3 þ0.8 0.83 þ1.19 þ0.89 0.35
c/x* &xR/x* (k1⁄40) 1.12, 1.41
0.51, 1.17 0.58, 1.26 0.20, 0.49 0.46, 0.92 0.75, 0.84 1.2, 1.26 1.29, 0.64 0.07, 0.52
kstab /k0
0.8 0.21 ? 0.28 0.76 1.47 2.0 3.23 0.40
a stab range @k1⁄40 0:4 < a < 1:2
... None None ... None ... None ...
The numerical model is applied to a range of equilibria with dependence of all quantities only on the cylindrical ra- dius. The modes of most interest have toroidal (azimuthal) wavenumbers of ‘ 1⁄4 1 and ‘ 1⁄4 2 and are investigated in each case. We focus on the fundamental radial mode throughout.
Examining non-reversed rigid rotor cases, we reproduce the previous results of Freidberg and Pearlstein.7 We next examine the rigid rotor FRC, first using Harned’s profiles and reproduce his results12 for axial wavenumber k 1⁄4 0.
Important new physics insights are provided by exten- sion of these previous results. Using Harned’s profiles, we studytheeffectsofk61⁄40onthe‘1⁄42modeandfindittobe strongly stabilizing when k is comparable to k0 1⁄4 p=L where L is the FRC length. A study of ‘ 1⁄4 2 modes for an extended set of profiles confirms this picture qualitatively.
We have studied rigid rotor profiles more representative of modern FRC experiments, added a realistically thin SOL to these, and studied the effects of sheared ion rotation. In all cases, the ‘ 1⁄4 2 mode is most unstable at k 1⁄4 0 and is stabi- lized for k near k0. This strongly suggests that the correct physics picture for the ‘ 1⁄4 2 mode is as follows. The mode is little affected by ion kinetics but is strongly affected by fi- nite axial variations. Profiles with steeper (thinner) SOL’s and/or rotational shear are slightly more unstable. Table III summarizes our ‘ 1⁄4 2 results.
Experimentally, there is a threshold near a 1⁄4 1 (X x ) associated with a typical FRC elongation of 4–7. Profiles with steeper SOL’s and/or sheared ion rotation are slightly more unstable and require shorter configurations for stability. The correct interpretation of HPF18 stabilization of this mode is likely that the rotation of the main column is reduced by the counter rotation of the SOL in these experiments.
TABLE IV. Summary of ‘ 1⁄4 1 wobble mode results.
The picture for the ‘ 1⁄4 1 wobble mode is complex. As studied in earlier work,7 this mode is close to marginal sta- bility and many effects compete. We find that for some FRC profiles, previous results with marginal stability at k 1⁄4 0 and instability over some finite k range is qualitatively repro- duced. For other profiles, the most unstable ‘ 1⁄4 1 mode is at k 1⁄4 0 and finite k is stabilizing. Finally, no ‘ 1⁄4 1 modes were found for our cases of strong rotational shear at the separa- trix, even though these modes are often observed experimen- tally even in the HPF regime. These results are summarized in Table IV.
A complete picture of ‘ 1⁄4 1 wobble stability will need to incorporate physics beyond our present scope. An impor- tant issue is whether line tying at the ends of the machine is complete or whether sheath, resistive and other non-ideal effects reduce the effectiveness. There is also the question of communication of end line tying along the considerable and potentially non-ideal length of the column. Additionally, recent positional stability work19 has shown that many FRC equilibria are transversely unstable to rigid displacements even without rotation, but that these modes are usually wall stabilized. Thus, it may be important to incorporate trans- verse destabilizing forces into the ‘ 1⁄4 1 analysis. One possi- bility for this within the present model would be to take k2 < 0 in our line tying formulation, but we do not investi- gate this further here.
The present work extends the physics content of previ- ous rotational mode studies to include important profile effects and to consider the FRC with equilibrium B passing through zero within a physically reasonable model. Observations about rotational modes presented here will likely be useful for interpretation and design of future experiments.
‘ 1⁄4 1 wobble Non-reversed
C-2 RR
C-2 RR
Thin SOL
Ion flow shear
Figs. a
1 0.8 2 þ1.6 7 0.8 8 þ1.4 12 0.83
... ...
c/x* & xR/x*
0.16,0.55 k/ko 1⁄4 0.3 0.125, 0.33 k/ko 1⁄4 0.15 0.47,0.52 k 1⁄4 0 0.18, 0.33k1⁄40 0.16,0.56 k/ko 1⁄4 0.33 No unstable modes
kstab /k0
0.58 0.26 0.30 0.061 0.60 ...
a stab range @k1⁄40 0 < a < 1
0:2 < a < 1:2 a > 0
...