Page 7 - Global simulation of ion temperature gradient instabilities in a field-reversed configuration

Physics of Plasmas ARTICLE
FIG. 4. Comparison of the trajectory between the fully kinetic particle and the gyro- center for 44.4 eV and 1022 eV deuterium cases. The black solid line represents the separatrix, and green solid lines are the contour of magnetic flux.
lb 􏰁 rB maXca
l 􏰊 @B @B􏰋^ 1⁄4 bZ 􏰂bR f;
In GTC-X, the gyro-average is performed analytically by multi-
plying the fields with Bessel function as hd/i 1⁄4 d/J0 ðkf qi Þ and
hdni i 1⁄4 dni J0 ðkf qi Þ, where qi 1⁄4 vth;i =Xci is the ion gyroradius, pffiffiffiffiffiffiffiffiffiffiffi
vth;i 1⁄4 Ti=mi; Xci 1⁄4 ZiB=cmi; kf 1⁄4 n=R is the toroidal wave vec- tor, n is the toroidal mode number, and R is the radial position, which is only valid for single toroidal mode simulation. The radial compo- nent of perpendicular wave vector kr is not considered for the gyro- average in the current implementation for simplicity. In the future work, a more realistic gyro-average using the 4 point average method21 will be implemented in GTC-X for multiple mode simulation, which is used in another FRC drift wave code ANC.
The comparisons between gyrocenter and fully kinetic particle
trajectories with different particle energies and locations are shown in
Fig. 4. Both gyrocenter and fully kinetic particle pitch angles at the
outer midplane are tan h 1⁄4 vjj =v? 1⁄4 1, which are trapped between
two mirror throats. It is seen that gyrokinetic description is suitable for
SOL region simulation with low temperature and high magnetic field,
and its fidelity decreases with increasing the particle energy and mov-
ing toward the core region. Recently, a theoretical work has illustrated
that the fidelity of the gyrokinetic equation is well achieved in the non-
uniform magnetic field with 􏰑 1⁄4 q/L < 1,40 where q is the gyroradius
and L is the magnetic field gradient scale length. With regard to FRC
geometry, the gyrokinetic description of thermal ion can capture the
"
@hd/i􏰋 RþbZ 􏰂bR f
cb 􏰁 rhd/i
B B R @f @R @Z
c
1⁄4 􏰂bZ
@hd/i ^^
1
vd 􏰀E?C; (30) dt @fi0v?Ti0 Ti0 A
rf 􏰎􏰎 E􏰎Z
􏰋1@T# Ti0 Ti0 @R
fi0 v? |fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl}
wdrive
B0
􏰂bR þ􏰂1:5;
e2ni0 Te0
"
1þk? qi
e2 ne0 Ti
! #
wdrive
wdrift
where
􏰂v􏰀 1⁄4b þ􏰂1:5 f
c 1@hd/i"1@n 􏰊􏰑
C. Poisson solver
Poisson’s equation is solved in a semispectral form. Applying i0􏰎 i0 i0 1
R @f
c 1@hd/i 1@ni0
Pad􏰋e approximation, d/ 􏰇
2 2 d/, Eq. (26) can be written as
ni0 @R "
􏰑 B0 R@f ni0@Z Ti0
􏰊 􏰋
1@Ti0 Ti0@Z
#
2 e2 s
2 ne0 Zi2ni0Te ? ni0
1⁄4Zidni
1
e2n T ;
(31)
scitation.org/journal/php
|fflfflfflfflffl{zfflfflfflfflffl}
maXca
@R @Z @hd/i 􏰊
fgrad􏰂B driftg
#
1 @hd/i ^ þRbR @fZ;
r􏰁b 􏰀ðZ rhd/iþlrBÞ 1 1 vjj
fIg􏰊 􏰋fIIg
and
􏰋
|fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl}
E􏰁B drift
1 1􏰊
jjmBX a
Bvjj a ca
Bþ
v_ 1⁄4􏰂
1⁄4􏰂 b􏰀ðZarhd/iþlrBÞ􏰂 r􏰁b􏰀ðZarhd/iþlrBÞ
ma ma Xca |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
Za @hd/i l @B @B 1⁄4􏰂 􏰂 bR þbZ
ma @S ma @R @Z |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
fIg
Za vjj 1@hd/i􏰄􏰊 @B @B􏰋 􏰊@BR @BZ􏰋􏰉
􏰂 bZ 􏰂bR þ 􏰂 : (29)
ma Xca R @f @R @Z @Z @R |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
gyro-orbit but not betatron and figure-8 orbit.
41
However, a strong
fIIg The weight equation is
finite Larmor radius effect stabilizes the drift wave instability near or
inside the core region, which has been observed in experiment10 and
25–27
0􏰎􏰎 1
by previous local simulations.
nature of ITG instability in the SOL region.
dwi rfi0 􏰎 Zi Zi
1⁄4ð1􏰂wiÞB􏰂vE 􏰀 􏰎 þ
|fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} |fflfflfflfflfflffl{zfflfflfflfflfflffl} |fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl}
vjjEjjþ wpara
Thus, in this paper, we focus on gyrokinetic simulation of ITG instability in FRC as the first step to demonstrate the code capability for GTC-X as well as reveal the global
Zi2
􏰂q1þ rþdw
with􏰑1⁄40:5mv2 þlB, i jj
1þ
e0 i0 Zi2 ni0 Te0
Zi Zi
Zivjj @hd/i Ti0 @S
vjjb􏰀rhd/i1⁄4􏰂
withS1⁄4Sc 􏰁LorS1⁄4SS 􏰁L,andListhefieldlinelength,and
Zi Ti0
Zi Ti0
lb 􏰁 rB jj
r 􏰁 b 􏰀 rhd/i
vd 􏰀 E? 1⁄4 􏰂 |fflfflfflfflfflffl{zfflfflfflfflfflffl}
miXci
þ
Xca
vjjEjj 1⁄4􏰂
;
@B Ti0RB@fZi# B @R @Z
Ti0 Ti0 |fflfflffl{zfflfflffl}
wpara
wdrift
Phys. Plasmas 26, 042506 (2019); doi: 10.1063/1.5087079 Published under license by AIP Publishing
v2 !
"􏰊 mv2􏰋􏰊
Zi 11@hd/i c i jj @B
􏰋
1⁄4􏰂 lþbZ􏰂bR
v2jj 􏰊@BR @BZ􏰋 þ􏰂:
Xci @Z @R
26, 042506-7