Page 10 - Anatomy of a field-reversed configuration

Physics of Plasmas ARTICLE
(B1) w> 0: PðwÞ1⁄4Ps 1􏰃D1ð1􏰃e􏰃apwÞþD2ð1􏰃e􏰃afwÞ ; (B2)
where ac1⁄4qXc/kT, ap1⁄4qXp/kT, and af1⁄4qXf/kT are normalized forms of the frequencies. The smoothness conditions are met when D1, D2 satisfy the following:
2a2f
D11⁄4 2 ; (B3)
Gf 1⁄4
32j2 B 2w R 2s
mfkðTf 􏰃TiÞ2
􏰀a : (C2)
􏰅􏰄
q 2 T f
2a 􏰃apðap􏰃acÞ
f pb
apðap 􏰃 acÞ
D21⁄42 : (B4)
This form has four independent parameters, Ps, ac, ap, af. To accommodate the two target dimensions (R/, Z/), only two parameters are “adjustables”: these are Ps and ac. The other two parameters are prescribed using reasonable values: ap 1⁄4 1 and af 1⁄45ap. The former yields an edge thickness typical of experiment. The latter reflects a relatively slender filet layer. Note that this pre- scription for ap can be changed moderately without appreciable effect on FRC properties.
APPENDIX C: ROTATION RATE
Equation (5) expresses the centrifugal effect in terms of the ratio of the rotation frequency X to the current-drift frequency XJ. It remains to find a reasonable value of this ratio X/XJ. In an end- shorted plasma the ions carrying essentially all current so that X/XJ 􏰂 1. However, it is of interest to consider what this ratio should be in a modern FRC such as C-2W with a fast-ion source (from neu- tral beam injection) which increases the rotation rate) and edge biasing which reduces the rotation rate.
Suppose the edge biasing is sufficient to stop the rotation of the bulk ions (deuterons) but not the fast ions. Then the rotational inertia factor in Eq. (6) becomes (mfnfXf2) where “f” denotes the fast-ions (protons in C-2W). Further, suppose that the fast-ion component accounts for a fraction a of the total pressure, i.e., a 1⁄4 nfkTf/nikTtot, where ni is the total ion density, both majority plus fast ions. Then the centrifugal parameter Eq. (6) becomes
Gf 1⁄4 amf X2f R2s =2kTf : (C1)
It remains to evaluate the fast-ion temperature Tf and rotation fre- quency Xf: first, the former. The beam energy is Eb. As a simple approximation for a typical neutral beam, suppose 2/3 of the beam ions in have energy Eb and 1/3 have energy (2/3)Eb, so that the ensemble average energy of fast ions at birth (upon beam ioniza- tion) is (8/9)Eb. The mean thermal energy of a slowing-down distri- bution is 1/3 of the beam energy so that Tf 1⁄4 (8/27)Eb. For example in C-2W with Eb 1⁄4 15 keV, Tf 1⁄4 4.4 keV. Next, the fast-ion rotation Xf. The ensemble rotation of the fast-ion population is the sum of its diamagnetic drift X􏰄 and the electric drift XE. For the rigid-rotor
Phys. Plasmas 27, 112508 (2020); doi: 10.1063/5.0022663 Published under license by AIP Publishing
tity when the electron density is available from interferometry. A simple analysis yields an expression for X/XJ using Tpb instead of the less commonly available Ti. This uses the sum of the various pressures, neTpb 1⁄4 niTi þ neTe þ nfTf, and charge neutrality ne 1⁄4 ni þ nf. Accordingly, (C3) becomes
sffiffiffiffiffiffiffiffiffi
aTf ðTf þTe 􏰃TpbÞ
X=XJ 1⁄4 !0:99: (C4)
The latter value corresponds to a C-2W-relevant example a1⁄43/4, Tf 1⁄4 4400 eV, Tpb 1⁄4 1500 eV, and Te 1⁄4 200 eV, i.e., a representative value of the ratio X/XJ is unity.
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2a 􏰃apðap􏰃acÞ f
X=XJ 1⁄4 a
ðTf 􏰃 TiÞ= 2TpbTf : (C3)
scitation.org/journal/php
with frequency Xp and a filet component with frequency þXf. The role of the filet is to allow the smoothness condition of continuous P,P,0 andP00 atw1⁄40.Theadoptedformis
2
field structure, the former is X􏰄f 1⁄4 􏰃8jkTf/qBwRs . If the edge bias-
ing is sufficient to stop the majority-ion rotation, then XE 1⁄4 􏰃X􏰄i 1⁄4 þ8jkTi/qBwRs2 (Ti is the majority ion temperature). Accordingly
2
the fast-ion rotation is Xf 1⁄4 􏰃8jk(Tf 􏰃 Ti)/qBwRs . Applying this in
Eq. (6) yields the centrifugal parameter
􏰇 ap 􏰃acw w < 0 : PðwÞ 1⁄4 Ps 1 􏰃 D1 a ð1 􏰃 e
c
þafw 􏰈 Þ 􏰃 D2ð1 􏰃 e Þ ;
Finally, establish an equivalence. Equation (6) is an expression for G. Equate it to Gf Eq. (C1) to establish an equivalent rotation, which is
1=2
The pressure-balance temperature T is a routinely inferred quan-
pffiffiffiffiffiffiffiffiffiffiffiffiffi
6
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