Page 9 - Feasibility study of microwave electron heating on the C-2 field-reversed configuration device

012509-9 Fulton et al.
TABLE I. Equilibrium parameters used in both the core and SOL simulation
Phys. Plasmas 23, 012509 (2016) A. Simulation of the FRC core
In the core, ion-scale modes are found to be stable. For a mode with wavenumber kfqi 1⁄4 1:0, the simulation was first run with density and temperature gradients with experimen- tally realistic values, R0 =Ln 1⁄4 R0 =LTi 1⁄4 R0 =LTe 1⁄4 2 4. When no instability was found, gradient drive was artificially increased to the limits of numerical validity of the gyroki- netic model, but still, no linear instabilities were driven. Possible contributors to mode stabilization are ion finite Larmour radius effects and the magnetic field gradient. Detailed stabilization mechanisms for the ion scale turbu- lence are under investigation. In general, we expect that higher wavenumber modes are driven more easily than low wavenumber modes. Electron scale turbulence in the core is still under evaluation and will be reported on, in detail in future work.
B. Simulation of the FRC scrape-off layer
In the scrape-off layer, we find unstable ion-scale and electron-scale modes showing typical exponential amplitude growth with real frequency in the ion diamagnetic direction. An unstable collisional mode with kfqi 1⁄4 5:42 is shown in Fig. 5. The mode is driven by density and temperature gra- dients, R0 =Ln 1⁄4 R0 =LTi 1⁄4 R0 =LTe 1⁄4 4:04, and has real fre- quency xrðR0=CSÞ 1⁄4 9:5 and growth-rate cðR0=CSÞ 1⁄4 2:3. The conservation of momentum and energy for both ions and electrons is shown in the bottom two panels of Fig. 5.
regions.
B0 ne Te Ti qi qe
Core 533:7 G
4:0 1013 cm 3 80 eV
400 eV
5:3 cm 0:039 cm
SOL 2430:5 G
2:0 1013 cm 3 40 eV
200 eV
2:3 cm 0:017 cm
machine axis to the magnetic axis is R0 1⁄4 26:96 cm, and the
distance from the magnetic axis to the separatrix on the outer
midplane is a 1⁄4 11.15 cm. The simulation parameters and
some calculated quantities, including the ion and electron
pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi
gyro-radii (q 1⁄4 m T =ðeBÞ; q 1⁄4 m T =ðeBÞ) and sound
i pffiffiffiffiffiffiiffiffiffiffiiffi e e e velocity (Cs 1⁄4 Te=mi) are listed in Table I.
Boundary conditions for the particles are reflective on the radial boundaries and periodic on the axial boundary. The field solver assumes a fixed value field on the edges of the grid. Simulations are initialized with particles distributed uniformly across computational cells, with random spatial placement within each cell. The velocity distribution of par- ticles is assumed to be Gaussian about the thermal tempera- ture listed in Table I.
The growth-rates and frequencies of the mode shown in Section IV B are listed alongside the transit frequencies and effective collisionalities in Table II. The transit frequencies of electrons and ions passing along a field-line are tr e 1⁄4 Vth e=l and tr i 1⁄4 Vth i=l, respectively, where l is the field line length. In the core, the field-line length is esti- mated by l1⁄4ðR0 þrÞ p, and in the SOL, l 4 m. For each separate region, the effective collisionalities are listed in Table II. Effective collisionality is the collisional fre- quency normalized by the transit frequency
e e 1⁄4 e e= tr e
e i 1⁄4 e i= tr e
1⁄4 i i= tr i: i i
The effective collisionality is quite low for each region due to the short field-line length and the low ion impurity, Zeff 1⁄4 1:5, in both regions.
TABLE II. Real frequencies, growth rates, characteristic transit frequencies, and effective collisionalities for each species in both the collisionless core and collisional SOL.
xrðR0=CSÞ cðR0 =CS Þ
Core SOL
– 9.5 – 2.3
5.47 3.23
tr e R0
tr i
Cs
2:02 10 1:24 10
FIG. 5. Time histories of kfq 1⁄4 5:42 instability in the FRC SOL. The top i
two panels show the electrostatic potential in linear and semi-log plots. The bottom two panels show conservation of energy and momentum for each species, with left and right axes corresponding to energy and momentum, respectively.
e e e i
1.16
2.45 1:20 10 1
3.81
8.61 2:61 10 1
i i
Cs
R0 1 1