Page 10 - Global simulation of ion temperature gradient instabilities in a field-reversed configuration

Physics of Plasmas ARTICLE
FIG. 9. Comparison of 2D poloidal mode structures of ITG instability between differ- ent parallel domain simulations: (a) Z=R0 2 1⁄2􏰂13:6; 13:6􏰃, (b) Z=R0 2 1⁄2􏰂16:2; 16:2􏰃, (c) Z=R0 2 1⁄2􏰂19:2; 19:2􏰃, and (d) Z=R0 2 1⁄2􏰂21:0; 21:0􏰃. The dashed lines show the flux surfaces with the maximum mode amplitude. The blue solid line is the separatrix.
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TABLE I. Parallel domain size effects on the ITG mode.
Domain size
Z=R0 2 1⁄2􏰂9:37; 9:37􏰃
Z=R0 2 1⁄2􏰂13:6; 13:6􏰃 Z=R0 2 1⁄2􏰂16:2; 16:2􏰃 Z=R0 2 1⁄2􏰂19:2; 19:2􏰃 Z=R0 2 1⁄2􏰂21:0; 21:0􏰃
Frequency 􏰂0.0108Xcp
􏰂0.0046Xcp 􏰂0.0022Xcp 􏰂0:0052Xcp 􏰂0:0043Xcp
Growth rate 0.0022Xcp
0.0046Xcp 0.0030Xcp 0.0035Xcp 0.0037Xcp
Parity of real part
Even Even Even Odd Odd
Parity of imaginary part
Even Even Even Odd Odd
we fixed k?qi 1⁄4 2:0 and scanned s; and GTC-X simulation results are
in good agreement with theory as shown in Figs. 5(c) and 5(d). In Fig.
5, the frequency and growth rate are normalized by vth;i=L, where pffiffiffiffiffiffiffiffiffiffiffi
vth;i 1⁄4 Ti =mi is the ion thermal speed, and L 1⁄4 0.4R0 is the parallel length of simulation domain.
B. FRC geometry
Next, we carry out simulation in the middle part of a realistic FRC geometry. The deuterium plasma is simulated in this case which is con-
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is the same as the diamagnetic drift in the outer midplane and opposite to diamagnetic drift in mirror throats.
V. GLOBAL GYROKINETIC SIMULATION OF ITG INSTABILITY
In this section, we study the global effects of FRC on the ITG. Using FRC equilibrium as shown in Fig. 1, we choose the parallel domain Z/R0 as [􏰂13.6, 13.6], [􏰂16.2, 16.2], [􏰂19.2, 19.2], and [􏰂21.0, 21.0], respectively, for simulations of deuterium plasmas. The plasma profile and radial domain are the same with Sec. IV B as shown in Fig. 7(a). We still focus on the n 1⁄4 20 toroidal mode (kfqi 1⁄4 0.36) and compare the mode structure, frequency, and growth rate among simulations with different parallel domain lengths, which are shown in Table I. The 2D poloidal mode structure and 1D parallel mode
26, 042506-10
sistent with experiments.
The inner boundary and outer boundary of
the simulation domain are w0 1⁄4 0.96wO and w1 1⁄4 13:8jwOj, where
wO is the poloidal magnetic flux value at the magnetic axis, R=R0
2 1⁄20:0; 2:45􏰃 in the radial direction at the outer midplane, Z=R0
2 1⁄2􏰂9:37; 9:37􏰃 in the axial direction, R0 1⁄4 26.8 cm, and the region
close to the magnetic axis w 2 1⁄20; w0 􏰃 is excluded for the gyrokinetic
model validation. Both particle and field boundary conditions at the
field line ends (Z/R0 1⁄4 69.37) are periodic since the FRC center is far
away from the divertor region, where the sheath effect becomes impor-
tant. In this paper, we do not take into account the presheath and
sheath effects on the ITG in the FRC center and formation region. In
the radial boundaries, i.e., w 1⁄4 w0 and w 1⁄4 w1, we apply the reflected
boundary condition for particle and zero boundary condition for fields
since the fluctuations of ITG are considered to be zero there. The w, B,
BR, and BZ value are shown in Fig. 6, which are normalized by the mag-
netic field strength B0 1⁄4 531 G at the cross point between the separatrix
and the outer midplane: (R 1⁄4 RX 1⁄4 1.42R0, Z 1⁄4 0). The ion tempera-
ture and density gradients are shown in Fig. 7. We choose to simulate
the ITG mode with the toroidal mode number n 1⁄4 20, and the corre-
sponding kfqi 1⁄4 0:36. The elongated parallel mode structure and finite
radial structure of the ITG mode are found in our simulation as shown
in Fig. 8, i.e., kjj 􏰅 kr and kjj 􏰅 kf, and we can estimate krqi 􏰇 0.57.
According to the parallel mode structure along the field line w
1⁄4 wðR 1⁄4 2:02R0; Z 1⁄4 0Þ as shown in Fig. 8(c), both the real and
imaginary parts of electrostatic potential perturbation of n 1⁄4 20 mode
are even parity. The frequency and growth rates are xr 1⁄4 􏰂5:98vth;i=
LS 1⁄4 􏰂0:0108Xcp and c 1⁄4 1:22vth;i=LS 1⁄4 0:0022Xcp, where LS
1⁄4 18.8R0 is the field line distance between left and right boundaries and
Xcp 1⁄4 eB0=ðcmpÞ is the proton cyclotron frequency. GTC-X
simulation of the ITG mode in the local FRC geometry agrees with
another gyrokinetic particle code ANC, which gives the result
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xr;ANC 1⁄4 􏰂5:30vth;i=LS and cANC 1⁄4 1:17vth;i= LS.
the unstable ITG mode is determined by a balance of temperature drive, gi value, and magnetic drift strength. In FRC, the magnetic gradi- ent drift is opposite to diamagnetic drift, and magnetic curvature drift
Phys. Plasmas 26, 042506 (2019); doi: 10.1063/1.5087079 Published under license by AIP Publishing
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