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Tuszewski et al.
Phys. Plasmas 24, 012502 (2017)
FIG. 8. Excluded flux radius rises between 1 and 2 ms in C-2U discharges.
feasible, but sensitivity is an issue because of the low inter- nal FRC magnetic fields.
The e-folding decay time sEt of the thermal energy Et
can be significantly overestimated by the standard value sE*.
For hybrid FRCs, E* is approximately the sum of the rising
fast ion energy Ef and of the decaying thermal energy Et.
When E peaks, s exceeds s by a factor 1þE/E. This fE*Et ft
factor is about 2 for C-2 cases. The time history of the fast ion density must be measured to accurately assess the FRC energy confinement. Modeling of Fast Ion D-Alpha (FIDA) measurements20 may permit such an assessment.
The simple hybrid model of Section IV requires some knowledge of the thermal plasma pressure at the FRC mid- plane. The model is only valid for elongated FRCs because it neglects axial variations and magnetic field line curvature. The FRC separatrix radius can be obtained, and the volume and thermal energy content can then be estimated with excluded flux array data. Fast ion and thermal plasma meas- urements would permit to construct more accurate models, and to check the assumed b 1 (b 0) of the core plasma.
The results of the hybrid model are in fair agreement with numerical simulations8,9 of C-2 FRCs. These simula- tions combine fluid transport codes in 1 (r) and 2 (r, z) dimensions with Monte-Carlo calculations of fast ions. The Q1D8 and Q2D9 codes yield very similar results for elon- gated C-2 FRCs. Hence, the 1-D hybrid model of Section IV appears adequate, and can be compared to either Q1D (Tables II and III) or Q2D (Figs. 3 and 4) numerical results.
Q2D simulations of C-2 and C-2U FRCs show broad fast ion axial profiles within the separatrix. The coalescence of fast ions around the field null (z 1⁄4 0) seen in earlier 2-D simulations21 is not observed in Q2D results. This may be explained by differences in neutral beam injection (z 1⁄4 0.5 m rather than z 1⁄4 0), beam cross-section (0.1 m radius rather than a pencil beam), beam divergence (1.2 degree half-angle rather than 0 degree). Neutral beam injection yields some decrease in FRC length in Q2D simulations.
VI. SUMMARY
The equilibrium parameters of thermal FRCs are well estimated by the simple analytical formulae of standard anal- ysis. Strong neutral beam injection yields hybrid FRCs with comparable fast ion and thermal plasma pressures. The fast ions contribute significantly to the radial pressure balance
and to the diamagnetism, and therefore modify the FRC equilibrium. The temperature, separatrix radius, volume, energy, and magnetic flux of hybrid FRCs are overestimated by standard analysis.
New models are required to estimate the equilibrium parameters of hybrid FRCs. A simple model of elongated hybrid FRCs is offered in Section IV. This model requires only thermal plasma pressure information. The magnitude of the fast ion pressure is inferred, and its radial profile is cho- sen consistent with numerical simulations. The results of the model compare well with 1-D and 2-D numerical results.
The fast ion pressure builds up on slowing down time- scales and obscures FRC confinement analysis. The time evolutions of the FRC magnetic flux and thermal energy remain largely unknown. Spatially resolved measurements of either the magnetic field or of the plasma pressure are required to quantify the equilibrium and confinement proper- ties of hybrid FRCs.
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