Page 9 - Anatomy of a field-reversed configuration

Physics of Plasmas ARTICLE
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V. SUMMARY OF OBSERVATIONS
The remarks that follow are primarily applicable to modern FRCs fed by neutral-beam injection and edge-biasing and with signifi- cant magnetic mirrors. These observations follow from properties seen through the lens of the reconstruction tool Grushenka.
Reconstructions exhibit a bifurcation between FRC and mirror (no trapped flux) equilibria. Mirror equilibria, which appear for long plasmas, may have quite large excluded-flux radius, i.e., a “high-b mirror.” FRC equilibria appear within a limited shape domain, (com- bination of {R/, Z/}), appearing as an allowed radius band which rises higher in longer plasmas. Exceeding the upper-radius limit indicates the inability to balance forces, radial and axial. Falling below the lower-radius limit indicates either the transition to a mirror plasma or tearing instability.
Reconstruction allows one to check the accuracy of “standard formulas” for the FRC dimensions, trapped flux, and energy. The reconstructed trapped flux and core thermal energy fall well below the formulas, e.g., 50%–70% as a typical range. Realistic plasma rotation plays a large role in reducing these values below the formulas. Reconstruction shows that the majority of the plasma current and thermal energy are in the peripheral plasma, indicating the prominent role played by the periphery. This is largely a result of coupling between cross field transport and end loss. It also suggests that modern FRCs might be characterized as “hybrids” composed of a core and a substantial periphery. Observation of Grushenka-constructed time- lines shows evidence of strong current drive (build-up of the trapped flux) and heating (rise and sustainment of the plasma energy).
Grushenka has been installed and operates as an automated post- shot-processing tool on the C-2W facility. In terms of future work, it will be augmented from time to time by the inclusion of post-processing features that follow directly from this equilibrium, including among others an interchange stability algorithm. Further, the fully kinetic ion equilibrium model5 can readily be enhanced to include the fast-ion population. This may lead to the next-generation fast FRC reconstructor.
Grushenka may be useful in the design of future FRC facilities. It should steer the designer away from proposing a facility where the {R/, Z/} objective lies outside the allowed shape domain. For example, it appears that a very long FRC (Z/ approaching Zm) is not consistent with a strong mirror field. It could also be used to explore ways to “stretch” the shape domain in some respect by suitably shaping the axial profile of the wall flux ww(z).
ACKNOWLEDGMENTS
The authors acknowledge helpful comments and criticisms by Sean Dettrick and Jesus Romero in this research and manuscript preparation. We are also grateful to the TAE shareholders for their continued support. The data that support the findings of this study are available from the corresponding author upon reasonable request.
APPENDIX A: EXTRACTING WALL FLUX AND SHAPE DIMENSIONS FROM DATA
Wall flux. Magnetic data are extractable from an array of mag-
netic loops mounted at a series of stations “z ” along the wall of the i
Phys. Plasmas 27, 112508 (2020); doi: 10.1063/5.0022663 Published under license by AIP Publishing
plasma vessel. These yield the wall flux ww(zi) and the axial field at the wall Bzw(zi). The adopted fit function for the wall flux builds on the nominal “vacuum” field Bvac,
B ðzÞ1⁄4B þDB e􏰃z2=D2 þDB z2=Z2 (A1) vac 0 0 m max
with three adjustable parameters {B0, DB0, DBm}, namely, the com- mon field, central trim-field increment, and the mirror field incre- ment, respectively. A fixed value D 1⁄4 Rw is adopted for the central trim field length scale. The corresponding wall flux is ww(z) 1⁄4 Bvac(z)Rw2/2 in the straight wall section z < Zstr, and ww(z) 1⁄4 Bvac(Zstr)Rw2/2 in the cone section Zstr 􏰇 z 􏰇 Zm (see Fig. 1). The three “adjustables” are determined by a least squares fit to data in the straight section jzij < Zstr.
Traditional FRC experiments had plasma lifetimes short com- pared to the resistive skin time of the wall. In consequence, the wall flux ww was uniform, DB0 1⁄4 DBm 1⁄4 0 in Eq. (A1), i.e., ww(z) 1⁄4 const both in time and space. However, in modern experiments, notably C-2U20 and even more so in C-2W,6 the skin time is comparable to or somewhat shorter than the plasma lifetime. Moreover, ww can be controlled both in space and in real time by activating trim coils. It is useful then to define a parameter characterizing the wall flux “shape.” For this define, the “field-curvature index” Nf1⁄4Rwjmp where jmp is the vacuum field-line curvature at the wall and mid plane. In terms of the adjustables, this is,
N 􏰆 R2ð􏰃DB =D2 þDB =Z2 Þ=ðB þDB Þ: (A2) fw0 mmax00
Positive index Nf > 0 indicates ww rising with jzj and negative Nf < 0 indicates falling ww. As in Sec. III, traditional FRCs have Nf 1⁄4 0 (flat field). A typical value Nf 1⁄4 0.1 was adopted for the modern FRC class.
Target dimensions. The expression Eq. (1) defines the excluded-flux radius profile R/(z) and the data point values are R/,i 1⁄4 R/(zi). The adopted fit function is a modified “top hat”
hi
R2 ðzÞ1⁄4C exp 􏰃ðz􏰃C ÞC4=CC4 (A3) /;fit1 23
with the adjustable parameter set {C1, C2, C3, C4}. The C2 parameter allows an axial shift of the plasma object relative to the mid plane. A least squares fit of Eq. (A3) to the data (R/,i)2 is found using a Newton’s method. The target parameters follow from the fit. The target excluded flux-radius is R/1⁄4C1. The target excluded-flux half-length Z/ is the value of (z 􏰃 C2) where R/,fit(z) falls to 2/3 its maximum.
APPENDIX B: COMPOSITE RIGID-ROTOR SURFACE FUNCTION
A flexible, multi-parameter modified function P1⁄4P(w; Ci) is needed to accommodate the data. Adopt a two-branch form com- posed of linear sums of rigid-rotor like functions P 􏰁 exp(aw) where each a 1⁄4 qX/kT corresponds to a different ratio X/T. [While X has units of frequency, it is to be distinguished from the X used in the centrifugal correction, Eq. (5)]. The core branch (w < 0) has two components: the main core component with frequency Xc and a “filet” component with frequency 􏰃Xf. The periphery branch (w > 0) also has two components: the main periphery component
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