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012502-4 Tuszewski et al.
Phys. Plasmas 24, 012502 (2017)
measurements are not available, b can still be approximately inferred from pressure measurements.
The right hand side of Eq. (5) is relatively small outside of the separatrix of a sufficiently elongated FRC. Toroidal field and rotational mass are mostly within the FRC, and field line curvature is small at large radii for a straight flux conserver. Integrating the left hand side of Eq. (5) yields
b 1⁄4 ð1 bÞ1=2: (14)
In past FRC experiments without neutral beam injection, Eq. (14) has been used to infer b from measured midplane radial density profiles, assuming isothermal plasmas.1,17 This method is inadequate for hybrid FRCs because interferome- try is dominated by the thermal plasma density, and does not permit to estimate the fast ion pressure that contributes to b. However, the peak fast ion pressure bfm of a hybrid FRC can be inferred from a measurement of the bulk thermal plasma pressure btm, since Eq. (14) implies bm 1⁄4 btm þ bfm 1 inside the FRC where b 1. The open-field-line b radial profile can be modeled as
b 1⁄4 btm exp 1⁄2 ðr rs Þ=d þ ð1 btm Þðr=rm Þx 1⁄2ðrt – rÞ=ðrt rm Þ y ; (15)
where the first term is the thermal plasma pressure bt and the second term is the fast ion pressure bf.
The thermal plasma pressure is assumed to decay expo- nentially on open field lines from its maximum value. The fast ion pressure term in Eq. (15) is chosen to be zero at r 1⁄4 0 and r rt, and to peak at r1⁄4rm. The parameters x and y determine the value of rm and the width of the fast ion pro- file, respectively.
Equation (15) yields b(r, rs) provided that btm and of the decay length d can be estimated from thermal pressure meas- urements, and that x, y, rm, and rt are chosen consistent with separate numerical simulations. Then, Eq. (14) yields b(r, rs), and Eq. (13) yields rs for a given value of rDU.
For example, rDU 1⁄4 0.35 m and btm 1⁄4 0.5 are assumed. These values are those of a C-2 hybrid FRC in a Q1D simula- tion8 at t1⁄41.5ms. Choosing d1⁄40.1m and rt 1⁄40.54m is also appropriate for this C-2 case. Setting y1⁄44 yields a fast ion full-width-half-maximum close to that in Fig. 3. Adopting the value rm 1⁄40.24m of Fig. 3 yields x1⁄4yrm/(rt – rm)1⁄43.2. One calculates rs 1⁄4 0.29 m by iterative procedure from Eqs. (13) to (15). The normalized radial pressure and magnetic field pro- files calculated by the model are shown between rs and rw in Fig. 5.
Once rs is obtained, other FRC parameters can be estimated more accurately, such as Vs V*(rs/rDU)2, and Et E*(btm/b*)(rs/rDU)2. The model is robust to uncertainties in d (rs 1⁄40.2960.01m for d1⁄40.1060.02m). Although b(r) inside the separatrix is unknown, a fair approximation of the FRC magnetic flux is
U 1⁄4 ðpbs=4Þrs2Bw; (16)
where bs 1⁄4 b(rs) is calculated by the model. The values of rs and U obtained by the Q1D simulation8 at t 1⁄4 1.5 ms, by the present hybrid model and by standard analysis, are compared in Table II.
FIG. 5. Model pressure and magnetic field radial profiles of a C-2 FRC.
The hybrid model values in Table II, obtained with Eqs. (13) to (16), are close to those of the Q1D simulation. The standard values in Table II, obtained with formulae in Table I, differ substantially from the Q1D values.
V. DISCUSSION
Standard analysis of FRC equilibrium, reviewed in Section II, describes well elongated FRCs without fast ion pressure. Although Eqs. (1) to (3) are approximate, they have been verified within 10% in the FRX-C device.18 Neutron measurements and Doppler Spectroscopy, combined with Thomson Scattering, support the total temperature esti- mate T*. Multichord side-on interferometry data validates the average beta condition, assuming an isothermal FRC plasma. The midplane separatrix radius rs, apparent from end-on holography, is found close to the measured excluded flux radius rDU.
The temperature T* is always approximate since ion impurities, toroidal magnetic field, and azimuthal flow con- tribute to radial pressure balance. However, as shown in Section III, these effects are relatively small (<10%) in most FRCs that have few impurities (Zeff<1.5), little (ideally zero) internal toroidal magnetic field, and relatively small plasma azimuthal rotations.
Field line curvature can modify significantly the radial pressure balance of short FRCs, as illustrated in Fig. 1. Curvature was negligible for most past FRCs (E > 5) formed in long cylindrical coils with small end mirrors. However, curvature may be important late in the discharges of C-2 and C-2U FRCs because of axial shrinkage.
Substantial modifications of the FRC equilibrium occur when strong neutral beam injection creates comparable ther- mal and fast ion pressures. To illustrate this point, selected
FRC parameters computed TABLE II. Parameter comparison.
by
the
Q1D code8 Model
0.29 1.4
at
t 1⁄4 1.5 ms Standard
0.35 5.1
Parameter rs (m)
U (mWb)
Q1D
0.29 1.6