Page 6 - Demo

022503-6 Steinhauer, Berk, and TAE Team
Phys. Plasmas 25, 022503 (2018)
core and periphery effects. In the combined expression, sN depends on both rates, reflecting the importance of both across-the-field and along-the-field particle transport. Of course, the relative importance of the two rates will differ from one regime to another. The analysis leading to an expression for sN combines the core, Eq. (5), and SOL [Eq. (7)] equations while fixing the eigenvalue as k 1⁄4 (3/2)‘i/Ln,s (see the Appendix). Eliminate k and Ln,s between these three equations. Here, it is useful to introduce a reference time- scale reflecting cross-field diffusion
intermediate value sjj/s? 1⁄4 0.5 (similar to the C-2U example cited earlier); at this point, the red curve is nearly a factor of six higher than the lower dashed curve. This reflects the sig- nificant contribution of cross-field particle transport on the overall confinement.
A. Density profile expectations
A remark is in order on expectation values of the density ratio hni/ns, which is treated as a more-or-less known factor in Eq. (11). In Sec. II B was shown that this ratio falls in the range of 1.1–1.3 for typical FRCs based on equilibrium con- siderations. The possibility of much larger density ratio was suggested in Ref. 12; the analysis found that long-lived FRCs require the separatrix density to be relatively small, see Fig. 1 of Ref. 12. However, there are two reasons to believe that large hni/ns never occurs on other grounds rather than it has not yet been observed. First, hni/ns is less than about 3/2 in order not to trigger interchange instability.25 Second, large hni/ns implies steep gradients and rapid drift speed in the separatrix region; this would trigger drift turbu- lence and cause anomalously high vn,s, the effect of which would be to flatten the density profile and raise ns. In fact, observations of turbulence in an advanced, beam-driven FRC show strong suppression of turbulence, especially in the core and to some degree at the separatrix.23
B. Predictive parametric scaling
It is intriguing to consider the trend in future experi- ments and facilities if classical vn and mirror-like sjj can be maintained. Follow Eq. (12a) sN 􏰁 (s?sjj)1/2. Suppose vn / g? / Te–3/2 (classical particle transport), sjj 􏰁 sii / Ti3/2/n (mirror-like end loss), and n / Be2/Tt (Be is the external mag- netic field). Together, these lead to the scaling
s? 􏰆 R2s=8vn;s:
1⁄4 5.1 ms. This analysis leads to a quadratic equation for sN
(10) For example, in the referenced C-2U example, this is s?
􏰈 􏰉2􏰈 􏰉
9hni s?2 s? ZMs? 2nsþs􏰃Zs1⁄40:(11)
s N N sjj
The sN vs sjj relationship is shown in Fig. 5 (red curve) for the case hni/ns 1⁄4 1.25 and Zs/ZM 1⁄4 0.4. Clearly, longer end-loss time has a strong favorable effect on overall particle confinement. The asymptotic limits (dashed lines in the fig- ure) are
􏰈􏰉
Z 1=2 pffiffiffiffiffiffiffiffiffi
sN 1⁄4 ðZs=ZMÞsjj:
For large sjj, the red curve approaches Eq. (12b), reflecting the periphery-dominated regime although the approach to the limit is rather slow. Below about sjj/s? 􏰁 2, the red curve is quite close to the other asymptotic limit, Eq. (12a), exhib- iting the “hybrid” proportionality sN / (sjjs?)1/2. In short, end-confinement (represented by sjj) has a substantial effect even in the low sjj limit. (A qualification on this “limit” will be mentioned shortly.) Even so, core particle transport con- tinues to have a major effect. For example, consider an
FIG. 5. Particle confinement trend.
sjj=s? < 1 sjj=s? 􏰇 1
s 2ns ZM
sjjs?;
(12a) (12b)
3 hni sN 1⁄4 pffiffi
sN
Rs 3=4 3=4 1=2 Rs 2
/ B Te Ti Tt / B T : (13)
ee
If the three temperatures are more-or-less a fixed ratio, then
the latter proportionality holds where T is any of the tem-
peratures. If the ultimate objective is confinement in a
fusion plasma at much higher temperature, this is a favor-
able temperature scaling. It is certainly superior to the
trend in traditional FRCs (Bohmþinertial), which has sN 􏰁 RsZs1/2BeT–3/4.
V. SUMMARY AND DISCUSSION
Both the core and the periphery contribute to particle confinement in comparable degrees. In effect, both raise par- ticle transport barriers, 1/vn in the core and sjj in the SOL. By means of fully transparent interpretive models, both vn and sjj can be inferred from common experimental measure- ments. This has been done for both traditional FRCs and advanced, beam-driven FRCs. The findings are as follows: (1) Traditional FRCs show evidence of full-blown turbulent particle transport with vn comparable to the Bohm rate. Moreover, end loss is at the rapid “inertial” rate of free- streaming. (2) The evolution of experiments and methods in the C-2 and C-2U facilities has shown quite different trends.