Page 6 - Tomographic imaging system for measuring impurity line emission in a field-reversed configuration

022506-6 D. C. Barnes and L. C. Steinhauer
Phys. Plasmas 21, 022506 (2014)
The exact threshold depends on whether one includes the “tail” of the growth rate curves. This tail phenomena is very typical of our FRC results and reflects the most unstable mode changing localization (radially) as k increases. We find that higher k modes are more localized to the outer radius regions of the FRC, those near the separatrix, while lower k modes are more global in character. Depending on additional physics details, modes in the tail region may not be observed as troublesome ‘ 1⁄4 2 instabilities experimentally.
In any interpretation, we find that the 0 < a < 1 stability region of Ref. 7 vanishes because of the reduction of GV effects associated with a large fraction of unmagnetized ions in the FRC case, and that the stabilizing mechanism for ‘ 1⁄4 2 instability is likely the finite length of the FRC, giving a threshold of a slightly less than unity, as often observed. To see the much reduced effects of GV, we have compared the GV and MHD growth rates in Table I for k 1⁄4 0 modes. As can be seen, there is little effect on these modes using the GV model of Ref. 14. We discuss this point further in Sec. VI.
B. Additional rigid-rotor FRC results
We have extended these results to examine ‘ 1⁄4 1 modes and to additional equilibrium profiles. A rigid-rotor number density profile which is more representative of recent C-2 ex- perimental results18 (as compared with Harned’s profiles) is shown as the upper curve in Fig. 5.
FIG. 7. Growth rate (a) and real frequency (b) in MKS units for ‘ 1⁄4 1 mode with a 1⁄4 0.3, 0.4, 0.6, 0.8 (bottom to top curves). Normalization of frequencies to ð1 sÞx 1⁄41:02 105 s 1 and k0 1⁄4ð1 sÞx =VA 1⁄40:46m 1 may be used to compare to previous normalized results.
We have examined the ‘ 1⁄4 1 and ‘ 1⁄4 2 stability of this profile as a function of a and k. Results for ‘ 1⁄4 1 are shown in Figs. 7 and 8. No ‘ 1⁄4 1 instabilities were found for 0 < a < 1:4. Notice that these results differ qualitatively from the non-reversed results of Sec. III (Figs. 1 and 2) in that the most unstable modes occur for k 1⁄4 0. Thus, the FRC wobble is expected to be a nearly rigid shift of the column with little z dependence, unlike the theta pinch cases consid- ered previously. Additionally, we observe that the wobble is stabilized for very small axial wave number (corresponding to a length of 20–30m in this example) and will thus be strongly affected (stabilized) by end conditions which ap- proximate line tying. We discuss these results further subsequently.
The ‘ 1⁄4 2 results are shown in Figs. 9 and 10. These results are very similar to those derived with Harned’s pro- files (Fig. 7) and show stabilization at experimentally realis- tic lengths, as discussed previously. Notice that all plots with a “tail” of growth show a transition in the real frequency at the same point, as the mode localization shifts, with the global (or internal) mode becoming less unstable than one which is localized at radii near the separatrix.
C. Rigid ion rotation FRC with thin SOL
Additional results have also been obtained with an equi- librium generated by the methods of Ref. 16. Figure 11 shows a case with a realistically thin SOL. In this case, the separatrix radius is 0.36 m and the density scale length at the inflection point of the profile is 0.06 m, about 1/2 of that of
FIG. 8. Growth rate (a) and real frequency (b) in MKS units for ‘ 1⁄4 1 mode with a 1⁄4 1.4, 1.6, 1.8 (bottom to top a curves, top to bottom b curves). Again normalizing frequency is 1.02 105 and normalizing wavenumber is 0.46.