Page 5 - Anatomy of a field-reversed configuration

Physics of Plasmas ARTICLE
FIG. 3. FRC and mirror equilibrium examples. FIG. 4. R-Z shape domains.
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to satisfy Eq. (4) subject to the boundary condition ww,fit(z). The com- putational burden is modest, requiring about a second or two on an ordinary personal computer. As such it is useful for quickly recon- structing a lifetime sequence of equilibria from experimental data.
Grushenka improves upon the earlier reconstruction method7 in two respects. (1) It finds an entirely new equilibrium for each snapshot {R/, Z/, ww(z)} rather than interpolating within a lookup table of pre- computed equilibria. (2) It accounts the measured wall flux profile ww(z), which is non-uniform, rather than assuming ww 1⁄4 const.
III. GENERAL PROPERTIES OF FRC EQUILIBRIA A. FRC or high-b mirror
With the Grushenka tool in hand, the immediate task is to apply it for analysis to investigate trends in FRCs typical of experiment. After that specific experimental shots will be addressed.
Consider first the assertion that equilibrium solutions can be either an FRC with a closed-flux core surrounded by open flux or a mirror plasma with entirely open flux surfaces. This is a question that has lingered for decades. The issue has remained open largely because internal magnetic measurements are unavailable in hot, high-density FRCs. The reconstruction tool shows that there is indeed a well- characterized bifurcation between FRC and mirror solutions. The dis- tinction depends on the combination of tangible shape dimensions {R/, Z/}. Figure 3 shows w(r,z)1⁄4const contours for both FRC and mirror plasma examples. The red dashed line is the excluded flux pro- file R/(z), see Eq. (3). Note particularly that both examples have the same excluded-flux radius at the mid plane, R/ 1⁄4 0.4 m. What differs is the excluded-flux half-length Z/. The shorter example gives rise to an FRC, while the longer is a mirror, in fact a high-b mirror with bmax 􏰂 0.92.
B. Limited shape domain
Equilibria in the context of the present model can only be com- puted within a limited range of the shape dimensions {R/, Z/}; this region is called the “shape domain.” Its boundary depends on the con- figuration of the main plasma chamber (Fig. 1) as well as the applied
Phys. Plasmas 27, 112508 (2020); doi: 10.1063/5.0022663 Published under license by AIP Publishing
TABLE I. Dimensions of plasma chamber.
Rw Rend
Zstr Zm
0.9 1.0 2.3 2.9
Trad’l FRC C-2W
0.25 0.23 0.8 0.4
magnetics ww(z). The shape domain is examined for two separate clas-
ses of facility. (1) Traditional FRCs have weak mirrors at the ends of
the vessel and fixed wall flux ww 1⁄4 const (flat field-curvature index
Nf 1⁄40, see Appendix A) and modest mirror ratio M 􏰂 1.2. (2)
Modern FRCs have strong mirrors M 􏰂 4, spatially adjustable field-
curvature index; a representative Nf 1⁄4 0.1 is adopted as representative 20 6
of C-2U and C-2W. Table I lists vessel dimensions (meters) for both classes. The main distinction between the two classes arises from the difference in mirror ratio.
Figure 4 shows the R-Z shape domains for the two classes. Both
domains are arbitrarily truncated on the right because few experiments
have larger values of Z//Rw. The shaded regions assume no rotation
(X1⁄40): pink for C-2W and green for traditional FRCs. A limited
computable range of {R/,Z/} was also noticed with the earlier recon- 7
struction method. The heavy blue lines on the C-2W domain show how the shape domain shrinks when realistic rotation X 1⁄4 XJ is included. This effect “squeezes upward” on the lower-radius boundary of the domain.
The limited range of equilibria suggests three explanations. (1) The model may fail to capture essential physics. (2) The failure to find a solution may be a numerical issue. (3) The boundaries may reflect actual operational limits outside which an FRC equilibrium cannot exist.
Regarding (1), the flexibility of the current-density function is considerable (Fig. 2 and Appendix B) and includes several kinetic effects: this casts some doubt on the first conjecture. Regarding (2), the primary equation [Eq. (4)] is highly nonlinear since jh depends in a nonlinear way on w, specifically outside the separatrix. A nonlinear differential equation may have no solution, one solution or multiple solutions. It should be no surprise that conditions exist that have no
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