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Steinhauer, Berk, and TAE Team
Phys. Plasmas 25, 022503 (2018)
FIG. 2. Inferred transport rate in traditional FRCs vs “Bohm” parameter.
experiments, mainly conducted through the early 1990s plus examples from FRX-L around 2010.
Figure 2 compares interpreted core particle transport rates vn,s [Eq. (5)] with vB 1⁄4 kTe/16eB, the Bohm rate.22 The abscissa is a “Bohm-scaling” parameter Tt/Be (Tt is the total temperature, ion þ electron). The shaded blue band is the range from (0.4 to 1.0)vB, assuming Te/Tt 1⁄4 1/3 (typical of traditional FRCs). The band presents a good fit to the data except for the relatively dense FRX-L examples for which vn,s are about a factor of two higher.
Figure 3 shows data from advancing eras in the evolu- tion of C-2.5,6 Here, the abscissa is the total temperature Tt. The red circle is the lone C-2U example shown. The upward tilting dashed line is the Bohm rate (assuming Be 1⁄4 0.1 T and
FIG. 3. Inferred vn,s for C-2 and C-2U: ’09 1⁄4 2009 era (shots #3800–6500); ’12 1⁄4 early 2012 (􏰁#15 700); ’12 g 1⁄4 addition of gun biasing (#18 700 –19 300); and ’12gb 1⁄4 addition of beam injection (#19 300–20 800).
Te/Tt 1⁄4 1/6 as typical of C-2 and C-2U). The early C-2 points are Bohm-like as in traditional FRCs. However, advances in operational methods produced a large reduction in the parti- cle transport rate, approaching the classical rate, shown as the down-sloping dashed red line based on Z1⁄41 (Z is the average ion charge). The C-2U example (at t1⁄43ms) has vn,s1⁄42.6m2/s which is about twice the classical value for Z 1⁄4 1. The observation of near classical particle transport is strong evidence that turbulent fluctuations which drive anomalies have largely been suppressed in these advanced, beam-driven FRCs. This is consistent with measurements of the turbulence level, which show strong suppression in the core.23
It is unwise to infer a temperature scaling from the downward trend of vn,s in the C-2 and C-2U data since they reflect several operational modifications, notably neutral- beam injection, divertor-biasing, and improved plasma hygiene. Even so, these improvements caused vn to approach the classical rate.
III. PERIPHERY INTERPRETATION A. SOL assumptions
Attention now turns to the SOL. The objective is to quantify the intercoupling between the cross-field particle transport rate and along-the-field loss through the mirror. The analysis will yield an expression for the end-loss time sjj in terms of measurable quantities and vn,s found by the core model (Sec. II). As such, it serves as the interpreter for sjj. The idealized geometry is as in Fig. 1. Physics assumptions relevant to the SOL are as follows. (1) Weak collisionality: plasma properties are constant on magnetic surfaces, i.e., pressure and density on a common w are the same in adja- cent and extension sections of the SOL. Thus, end-loss appears as a sink with timescale sjj that applies over the entire half-length ZM of the SOL. This first assumption is valid if the ion mean-free path kmfp is somewhat longer than ZM. In the C-2 and C-2U, the ratio is kmfp/ZM 􏰁 100; this ratio is lower in traditional FRCs but still somewhat larger than unity, with the exception of the high-density FRX-L experiments. (2) Quasi-steady: the SOL structure remains roughly constant in time, i.e., the rates of supply (radial dif- fusion) and loss (end loss) remain roughly in balance.
B. Quasi-steady continuity in the periphery
The continuity equation in the SOL is
1d􏰈 dn􏰉 n 􏰈1 1􏰉
Zsrdr rvndr 1⁄4ZMs 􏰃Zsn s þs ; r􏰅Rs: (6)
jj inv src
The left side is the surface-to-surface diffusion. Here, the effective length is Zs since radial gradients are only apprecia- ble in the “adjacent” section of a thin SOL (Fig. 1). Radial diffusion is much less in the relatively thick “extension” sec- tion of the SOL. The first term on the right side is the end- loss sink of particles from the SOL, which applies over its entire half-length ZM. The second grouping is the inven- tory change rate and the source [as in Eq. (4)]. Again, the