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022503-2 Steinhauer, Berk, and TAE Team
Phys. Plasmas 25, 022503 (2018)
idealized FRC structure, they derived a basic confinement relation [Eq. (16) of Ref. 10], an equivalent form of which is
where vB 1⁄4 kTe/16eB is the Bohm rate (Te, electron tempera- ture; e, electron charge; and B, magnetic field). This led to a scaling law with a relatively strong dependence on sjj
3=4 2 􏰃1=4
sN 􏰁 Rssjj ð‘ivBÞ ; (3)
where ‘i 1⁄4 (mi/l0e2n)1/2 is the ion skin depth (mi 1⁄4 ion mass and l0 1⁄4 permeability of free space). In summary, studies to date support the general idea that particle confinement in FRCs depends on both vn and sjj although the relative contri- bution of each depends on the nature of the presumed parti- cle transport mechanism.
With this as the background, attention returns to inter- pretation. The interpretive task builds upon experimental data to infer transport rates, vn and 1/sjj. As developed here, it relies on a fully transparent model to connect these rates to common measurements, hopefully by simple formulas. Once this is done, the interpreted vn and 1/sjj are compared with prospective ? and jj transport mechanisms. If confidence grows in the applicability of particular mechanisms, the interpretive formulas can be combined to express sN in terms of measured parameters. The last step is the predictive task.
In preview, the results are briefly as follows. (1) Traditional FRCs exhibit Bohm-like vn and inertial end loss for sjj. (2) Advanced beam-driven FRCs break sharply from this, approaching classical vn and mirror-like sjj. Based on classical and mirror mechanisms, the predictive scaling law is quite similar to the empirical scaling observed on the C-2U facility.
The outline of this paper is as follows. Section II devel- ops the core model, based on an idealized geometry and common physics assumptions. The continuity equation leads to an expression for vn, which is applied to a broad experi- mental database. Section III develops the periphery model. The continuity equation leads to an expression for sjj, which is also applied to the database. Section IV combines the core and periphery models to construct a predictive expression for sN in terms of both vn and sjj. Section V summarizes and pro- poses future work.
II. CORE INTERPRETATION
A. Geometry and physics assumptions 1. Diagnostic sources
Both core and periphery particle transport models build on data drawn from two diagnostic tools. (1) Multi-chord interferometry yields the density profile at the mid-plane from which one can extract hni 1⁄4 average density at the mid-plane cross-section, ns 1⁄4 separatrix density, and Ln,s 1⁄4 density gradi- ent length at the separatrix (“edge thickness”). (2) An excluded flux array yields approximate values for Rs and Zs (half-length of FRC). Together, these lead to the global plasma confinement time sN.19 In the present model, particle sources are ignored.
2. Idealized geometry
Adopt the simplified FRC core and periphery structure in Fig. 1, shown in the poloidal (r,z) section of cylindrical
hni R sN1⁄4 ssjj:
ns 2Ln;s
(1)
Here, Rs is the separatrix radius, Ln,s is the density gradient length at the separatrix, and hni and ns are the average and separatrix densities, respectively. (The subscript “s” denotes the separatrix value.) The sjj factor in Eq. (1) highlights the explicit role that end loss plays in global confinement. The factor Rs/2Ln,s is the ratio of the nominal core volume (/pRs2) to the SOL volume (/2pRsLn,s) with Ln,s being the nominal SOL thickness. As such, this ratio acts as a geomet- ric factor multiplying the axial confinement time sjj.
The key recognition in Ref. 10 was the critical impor- tance of the self-consistently coupling core and periphery at the separatrix: particle flux out of the core must match the flux into the periphery. The analysis10 proceeded to construct a particle-confinement scaling law: it assumed a vn based on a lower-hybrid-drift (LHD) transport prediction, and it assumed a sjj based on inertial (free-streaming) end loss. Combining these with suitable approximations led to the scaling
s 􏰂 constR2=q ; (2) N si0
where qi0 is the ion gyroradius based on the external mag- netic field. This expression agreed with an earlier computed result11 but, more importantly, was consistent with experi- mental observations available at the time. For many years, Eq. (2) was the most-quoted confinement scaling. However, the generality of the equation is limited because it obscures the effect of end loss (sjj). A later analysis explicitly included both ? and jj processes12 although it did not recognize the inter-coupling between sjj and Ln,s. A major defect of both Eq. (2) and the formula in Ref. 12 is that they assume a par- ticular particle transport mechanism, LHD. Subsequent experimental evidence showed that LHD transport does not apply in FRCs.9,13–15
Attention then turned to the task of interpretation, i.e., using a simple model to extract active particle transport rates from experimental data. Self-similar solutions to the transport equation9,16 led to analytical formulas for vn and its gross pro- file. As mentioned, these did not correlate with LHD transport. Lacking a viable particle transport mechanism, confinement studies reverted to pure empiricism. One such was “LSX scaling,”4 similar to Eq. (2) but also exhibiting a dependence 􏰁Zs1/2 (Zs is the half-length of the FRC). If the end loss is inertial (sjj / Zs), then sN contains a sjj1/2 factor.
The specter of rapid end loss in the periphery, i.e., inertial or a moderate fraction of it,17 and its deleterious effect on sN has hung over FRC development for a long time. Emerging more recently, however, is the suggestion that the periphery might contribute favorably to confinement if sjj were greatly to exceed the inertial time, perhaps as a consequence of mag- netic mirrors at the ends.18 If so, the periphery (specifically sjj) would act as a second particle transport barrier supple- menting the first barrier (1/vn) provided by the core. The anal- ysis18 presumed drift-transport scaling vn / vB(qi0/Ln)2,