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012502-5 Tuszewski et al.
TABLE III. Simulated and standard FRC parameters.
Phys. Plasmas 24, 012502 (2017)
FIG. 7. Oxygen ion temperatures for C-2U FRCs with 2, 4, and 6 neutral beams.
Oxygen and deuterium ion temperatures are expected to be comparable because energy equipartition times are relatively short (<0.1 ms). The measured oxygen ion tem- peratures (red) in Fig. 7 are close to the standard ion tem- peratures T* Te (yellow curve with Te 1⁄4 100 eV), but tend to lower at late times, especially for larger injected neutral beam powers (NBs 1⁄4 4 and 6). This suggests higher fast ion pressures with higher neutral beam powers. More accurate ion temperatures TR Te (purple curves with Te 1⁄4 100 eV) offer a better comparison with the oxygen data. The values of TR are calculated from Eq. (1), with neR estimated from multi-chord interferometry.19
The fast ion pressure builds up on slowing down time scales (1–2ms for C-2 and C-2U FRCs). Initially, fast ion pressure is negligible and standard FRC analysis should be adequate. Later, the separatrix magnetic field is gradually reduced as the fast ion pressure increases, and rDU increases above rs if resistive decay of the FRC magnetic flux is negli- gible. Some rDU rises observed in C-2U discharges are shown in Fig. 8.
The fast ion pressure build-up obscures the FRC con- finement times. The evolution of the FRC magnetic flux U cannot be inferred any more from excluded flux measure- ments because of the competing effects of rDU rise, resistive magnetic flux decay, and possible fast ion current drive. The magnetic field must be measured to calculate U. Future non- perturbing internal magnetic field measurements from either motional stark effect or multi-chord polarimetry may be
Parameter
Simulation Standard (*)
TR (keV)
0.55 0.75
nR (1019 m 3) 1.6
2.3
Et/L (KJ/m)
0.7 1.4
for a typical C-2 FRC are compared in Table III to their cor- responding standard values.
The parameters from the Q1D simulation have lower than standard values. The difference is due to fast ion pres- sure, since the Q1D calculation does not include ion impuri- ties, toroidal magnetic field, plasma rotation, and magnetic field curvature. The calculated thermal plasma pressure nRkTR is lower than n*kT* by a factor of 2, suggesting com- parable thermal and fast ion pressures inside the FRC. The Q1D thermal plasma energy (per unit length) is lower than the standard value by a factor of 2. The Q1D simulation also yields lower values of the separatrix radius and of the FRC magnetic flux (rs 1⁄4 0.29 m and U 1⁄4 1.6 mWb) compared to standard values (r* 1⁄4 0.35 m and U* 1⁄4 5.1 mWb), as already mentioned in Table II.
The standard total temperature T* 0.75 keV underes- timates the value TR 1.1keV obtained with Eq. (1) because n* 2.3 overestimates nR 1.6. Standard analysis assumes a density maximum at r 1⁄4 R, while the calculation shows a density minimum. This hollowing effect is caused by the fast ions.
There is some evidence for T<T* in C-2 and C-2U data. The deuterium ion temperatures of some C-2 FRCs, estimated by Charge Exchange recombination Spectroscopy (CHERS), are shown in Fig. 6.
The blue and magenta points in Fig. 6 are CHERS data, and the yellow curve is the pressure balance ion temperature T* - Te, assuming an electron temperature of 100eV. The magenta CHERS data, obtained near the FRC field null (R 0.20–0.25 m), are consistent with T T* at t 0.5 ms and with T T*/2 at t 1.5 ms. The latter suggests a rela- tively large fast ion pressure (af 1).
Qualitatively similar results are obtained for oxygen (OV) ion temperatures estimated from Doppler spectroscopy. Some C-2U Doppler data are shown in Fig. 7 as functions of time, for cases with different injected neutral beam powers.
FIG. 6. Deuterium ion temperature as function of time for a C-2 FRC.