Page 3 - Compact Toroid Injection Fueling in a Large Field-Reversed Configuration

082512-3
Lau et al.
Phys. Plasmas 24, 082512 (2017)
and ion acoustic speed Cs 1⁄4
TABLE I. Parameters used in simulations of core and SOL.
Quantities ne (cm 3)
Te (eV)
Ti (eV)
qi (cm)
qe (cm)
R0 ðlsÞ Cs
e i
\ i i
Core
4.0 1013 80 400 6.0 0.044 1.8 2.1 0.10
ðTi þ TeÞ=mi.
Temperature and density gradients drive the instabilities. The strengths of these drives are defined by their scale lengths normalized by the machine scale length (where the local minor radius r is the distance measured from the null- point along h 1⁄4 0)
jf 1⁄4R0 1⁄4R0 @ f: (1) Lf f@r
The drives used are the density gradient jn 1⁄4 R0/Ln, ion temperature gradient jTi 1⁄4 R0=LTi , and electron temperature gradient jTe 1⁄4 R0=LTe . In addition, the importance of the drives can be defined by the ratios between the scale lengths of the temperature gradients and the density gradient, gi 1⁄4 jTi =jn and ge 1⁄4 jTe =jn. In most of the simulations presented, the strengths of the three drives are equal, i.e., gi 1⁄4 ge 1⁄4 g 1⁄4 1, and the magnitude of the drive strength would then be referred to as jð1⁄4jn 1⁄4jTi 1⁄4jTeÞ.
Simulations were run both with and without collisions
based on the Fokker-Planck model66 to understand the effects
of collisions. The effective collisionality is the collisional fre-
quency normalized by the transit frequency and is calculated
by 1⁄4 e e=xtr;e; 1⁄4 e i=xtr;e; 1⁄4 i i=xtr;i. The e e e i i i
transit frequency of an electron and ion passing along a field-line is xtr,e1⁄4Vth,e/L and xtr,i1⁄4Vth,i/L, respectively.
FIG. 1. The field-aligned mesh on a poloidal plane of a typical C-2 FRC dis- charge is plotted along with the magnitude of the magnetic field represented by color. The flux surfaces used in simulations are represented by dashed cyan lines. Note that the axes are not proportionally scaled. Arrows denote the directions of the magnetic Boozer coordinate system.
the machine cylindrical axis to the separatrix. These are all measured along the mid-plane (h 1⁄4 0 or Z 1⁄4 0).
The equilibrium used corresponds to an early time in a C-2 discharge, just after the CTs merge to form a single FRC before a significant fast ion population has built up. At this stage, the size of the plasma is large and diagnostics are more robust. In the simulations, temperature and density gradients are input to drive instabilities in plasma composed of deuter- ons and electrons. Parameters are chosen to resemble the con- ditions of recent experiments at TAE35 and are summarized in Table I. The calculated quantities are the ion gyro-radius
Here, Vth i 1⁄4 Ti=mi is the ion thermal velocity, and pffiffiffiffiffiffiffiffiffiffiffiffi
Vth e 1⁄4 Te=me is the electron thermal velocity. In our sim- ulation domain, the field-line length of the core is L 3.6 m, while the field-line length of the SOL is L 5.0m. The effective collisionality is high in both regions for the colder electrons, but low for the hotter ions with a low ion impurity modeled by Zeff 1⁄4 1.5 in both regions.
B. Flux-tube domain
The focus of this study is to characterize the local linear properties of the FRC instabilities in the core and SOL sepa- rately. The large number of simulations is enabled by the reduction of the simulation domain from a full torus [0, 2p] to a partial torus [0, 2p/n] where n is the particular toroidal mode number of interest. In these simulations, the toroidal wavelength is assumed to be much shorter than the radial wavelength of the instabilities, i.e., kr kf.
The radial domain is thus localized to a single flux sur- face where R0 þr1⁄437cm in the core and R0 þr1⁄452cm in the scrape-off layer as shown in Fig. 1. These flux surfaces do not include the magnetic null-point, allowing the guiding- center approximation to remain valid.67 The average gyro- kinetic parameter is qi/LB ’ 0.2 in the core and qi/LB ’ 0.006 in the SOL which are gyro-kinetically valid (about 1.3% difference between the guiding-center and orbit- averaged positions in case of qi/LB ’ 0.2 as shown by Brizard67).
In this gyro-kinetic simulation,68 dynamics faster than ion gyro-period are removed while ion and electron finite Larmor radius (FLR) effects are retained through accurate representation of gyro-averaging on particles via direct cal- culations of Bessel functions for the scattering of charge onto the grid and in the gathering of fields onto the charge.
pffiffiffiffiffiffiffiffi
qi 1⁄4 mi Ti =ðeBÞ, electron
gyro-radius qe pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1⁄4
pffiffiffiffiffiffiffiffiffiffi
me Te =ðeBÞ,
SOL
2.0 1013 40 200 2.2 0.016 2.5 5.7 0.27
pffiffiffiffiffiffiffiffiffiffiffi