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012502-2 Tuszewski et al.
The relations (1) to (3) permit (standard) estimates of
many FRC parameters. These parameters can be obtained as functions of time with just excluded flux and side-on inter- ferometry measurements. The parameters and their decay rates are inputs to most experimental FRC databases, and to global confinement models.
The standard definitions (*) of the midplane separatrix radius rs, hbi, nR, kTR, of the separatrix volume Vs, of the FRC plasma thermal energy Et, and of the FRC magnetic flux U are given in Table I. Ð
The field null density n* in Table I assumes that ndr is obtained from a single pass interferometer chord aligned along a diameter, and neglects plasma outside of the FRC. The FRC volume Vs is evaluated with an excluded flux array consisting of probe/loop pairs at different axial positions. The separatrix length is often estimated as the 2/3 height of the rDU axial pro- file, in fair agreement with numerical simulations.12 The FRC thermal energy is defined as Et 1⁄4 (3/2)Ð pdVs. The FRC mag- netic flux estimate U* assumes a rigid rotor magnetic field profile satisfying Eq. (2).
III. HYBRID FRC ANALYSIS
A more accurate FRC radial pressure balance must include fast ion pressure and other terms neglected in Eq. (1). Summing equilibrium fluid momentum equations for all spe- cies (s), one obtains
Rmsnsðvs:rvsÞ 1⁄4 j B rp; (4)
where j and p are the total plasma current and pressure, respectively. The electric field cancels out of Eq. (4), assum- ing the quasineutrality relation ne 1⁄4 RnsZs. Using Ampere’s Law, cylindrical coordinates, and assuming axisymmetry, the radial component of Eq. (4) at z 1⁄4 0 can be written as
Phys. Plasmas 24, 012502 (2017) where pfR is the fast ion pressure at the field null, and aZ, dc,
and dh are positive dimensionless correction terms given by aZ 1⁄4 1⁄2TiRnjZj ð RnjTj =1⁄2neRkTR ; (7)
dc 1⁄4 ð2=Bw2Þ ð Bz@Br=@zÞdr; (8) ð
dh 1⁄4 ðBhR =Bw Þ2 –ð2=Bw 2 Þ ðBh 2 =rÞdr: (9)
The impurity term aZ includes all ion species j. The contribu- tions of field line curvature and of possible toroidal magnetic field are given in Eqs. (8) and (9), respectively, where the integrals are from the field null to the wall.
Equation (6) can be rewritten as
kTR 1⁄4 1⁄2Bw2=ð2l0neRÞ ð1 þ dc dhÞ=ð1 aZ þ av þ afÞ;
(10)
where av and af are positive dimensionless flow and fast ion terms given by
av 1⁄4 Rms ð nsvhs2dr=r =1⁄2neRkTR ; (11) af 1⁄4 pfR=1⁄2neRkTR : (12)
Equation (10) reduces to Eq. (1) for an elongated FRC with- out toroidal field, impurities, azimuthal rotation, and fast ion pressure.
22 d=drðpþB=2=l0Þ1⁄4ðBz@Br=@z Bh =rÞ=l0
þ Rmsnsvhs2=r:
(5)
fj 1⁄4 nj/ne. One
(Z 4) or 1% Titanium (Z 8), either concentration being consistent with C-2 Zeff measurements.13 Assuming rigid azi- muthal rotation inside the FRC, one obtains av M2/2 from Eq. (11), where M 1⁄4 vhis/cs is a separatrix Mach number (cs2 1⁄4 2kTR/mi). For C-2 FRCs, Doppler spectroscopy14 sug- gests M 1/3, hence av < 0.1. The terms aZ and av have com- parable magnitudes and tend to cancel each other in Eq. (10). Finally, one estimates dh (BhR/Bw)2/2 from Eq. (9). The cor- rection term dh is a few % for some measured values of Bh.15
The fast ion velocity is assumed sufficiently randomized that the fast ions contribute to the pressure term p in Eq. (5) rather than to the rotational pressure on the right hand side. Monte- Carlo simulations8,9 support this assumption, showing a directed fast ion energy that is about half of the total fast ion energy for C-2 cases. Integrating Eq. (5) between the field null and the wall, one obtains
pfR þ neRkTRð1 aZÞ þ Rms ð nsvhs2dr=r
1⁄4 1⁄2Bw2=ð2l0Þ 1⁄21 þ dc dh ; (6)
TABLE I. Standard values of selected FRC Parameters.
The correction terms aZ, av, and dh in Eq. (10) are rela- tively small for most FRCs. AssumingPequal temperatures for all ion species, Eq. (7) yields aZ fj(Zj 1)Ti/T, with
The curvature term dc can be significant for short FRCs. Numerical results from the Lamy Ridge equilibrium code16 are shown in Fig. 1.
The numerical results (black points) in Fig. 1 are consis- tent with dc 1/E2 (red points) for an FRC inside a long flux conserver. The FRC elongation E is defined as the ratio of the separatrix length to its midplane diameter. The curvature correction term dc is small initially (E > 4), but may become large at late times if energy losses cause FRC axial shrinkage.
Fast ion pressure can modify significantly the FRC
radial pressure balance. Some C-2 and C2-U FRC data6,7
suggest values af 1. For such cases, Eq. (10) indicates that
T might be only about half of the value predicted in Eq. (1). R
Nonetheless, radial pressure balance is still a useful relation. If the fast ion pressure is measured, Eq. (10) permits an esti- mate of the total temperature of hybrid FRCs. If the tempera- tures are measured, Eq. (10) yields an estimate of the fast ion pressure.
estimates aZ 0.1 for either 4% Oxygen
Parameter
rs
hbi
nR
kTR
Vs V*1⁄4ÐprDU2dz
Et E* 1⁄4 (3/2)[Bw2/(2l0)]b*V* U U* 1⁄4 (rDU3/rw) Bw
Standard value (*) r* 1⁄4 rDU
b* 1⁄4 1 (rDU/rw)2/2 n* 1⁄4 Ð ndr/(2rDUb*) kT* 1⁄4 Bw2/(2l0 n*)