Page 3 - Innovative high-gain optically pumped far-infrared laser

012502-3 Tuszewski et al.
Phys. Plasmas 24, 012502 (2017)
FIG. 1. Calculated values of dc as a function of FRC elongation E. Fast ion diamagnetism can also modify significantly the
FRC excluded flux analysis. One can write generally
rDU2 1⁄4 rs2 þ ðð1 bÞ2rdr; (13)
For FRCs with strong neutral beam injection, a signifi- cant fast ion pressure develops up to r 1⁄4 rt, the outer turning radius of a full-energy fast ion. For C-2 and C-2U FRCs, rt exceeds largely rs because low magnetic fields (Bw < 0.1 T) result in large fast ion orbits. This is illustrated in Fig. 2, where 15 keV proton orbits (red curve) are calculated at the midplane of an FRC with rs 1⁄40.3m, rw 1⁄40.7m, and Bw 1⁄4 0.087 T.
Examples of fast ion density and magnetic field radial profiles are shown in Figs. 3 and 4, respectively. These mid- plane profiles are obtained at t 1⁄4 1 ms from a 2-D (Q2D) numerical simulation9 of a C2-U FRC.
FIG. 3. Calculated fast ion density profile of a C-2U FRC.
The fast ion density is zero at r1⁄40 and at r1⁄4rt, and peaks at rm 0.24 m, a value close to rt/2. The fast ion den- sity is substantial at the separatrix and outside of the FRC up to r1⁄4rt. The values of rm, rs, and rt are shown in Fig. 3. The fast ion pressure and density radial profiles are similar, since the fast ion energy distribution has little spatial dependence.
The magnetic field in Fig. 4 increases nearly linearly
from 0.02T at r1⁄4rs to Bw 0.087T at r1⁄4rt. The values
of R, rs, rDU, and rt are shown in Fig. 4. The normalized mag-
netic field b is lower than unity between rs and rt, because of
the combined thermal and fast ion pressures. For such a case,
Eq. (13) predicts that rDU should significantly exceed rs. The
Q2D simulation yields rDU1⁄40.39m and rs1⁄40.29m. The
former value can also be obtained from Eq. (13) with a linear
b 1⁄4 B/Bw, and where the integral is
r 1⁄4 rw. Equation (13) reduces to Eq. (3) if the open field lines are elongated and have negligible plasma pressure (b1⁄41). For FRCs without neutral beam injection, the thermal plasma pressure decreases rapidly away from the separatrix, and rDU exceeds rs by only a few percent.
where
from r 1⁄4 rs to
approximation between rs and rt for the b profile of Fig. 4. Ð
The FRC magnetic flux U ( B2prdr from r1⁄4R to r 1⁄4 rs) is 1.7 mWb for the case in Fig. 4. The standard value (Table I) is U*1⁄47.3 mWb, with rDU 1⁄40.39m, rw 1⁄40.7m, and Bw 1⁄4 0.087 T. The large discrepancy between U and U* arises in part because rs is lower than rDU, and because the FRC internal magnetic field is lower than predicted by a rigid rotor satisfying Eq. (2).
IV. HYBRID MODEL
A primary goal of any FRC hybrid model is to recon- struct rs from the measured rDU value. Equation (13) shows that rs can be calculated if the midplane open-field-line mag- netic field radial profile is known. If internal magnetic field
FIG. 2. Calculated 15 keV proton orbits in the midplane of a C-2U FRC.
FIG. 4. Calculated Bz radial profile of a C-2U FRC.