Page 6 - Global simulation of ion temperature gradient instabilities in a field-reversed configuration

Physics of Plasmas
ARTICLE scitation.org/journal/php
1 B􏰈
v_ 1⁄4􏰂 􏰀ðZrhd/iþlrBÞ; (20)
i0
ma B􏰈jj
where Za, ma, and fa are the charge, mass, and distribution function of
where ni0 and Ti0 are the equilibrium ion density and temperature. d/ is the double gyrophase average of electrostatic potential for ion spe- cies, which is given as
􏰈􏰈1 þðBv =X Þr􏰁b,andB 1⁄4b􏰀B .d/istheelectrostaticpotential. f
fi0ðR;l;vjjÞhd/idvdRdndðRþqi􏰂xÞ=ð2pÞ; (27)
Ð jj
jj ca
h􏰀 􏰀 􏰀i 1⁄4 ð1=2pÞ dxdnð􏰀 􏰀 􏰀ÞdðR þ qa 􏰂 xÞ represents the gyrophase average, n is the gyrophase angle, x represents the particle position, qa 1⁄4 b 􏰁 v?=Xca is the particle gyroradius, and Xca is the particle cyclotron frequency. vE is the E 􏰁 B velocity, and vd is the magnetic drift velocity, which are given as
vd 1⁄4
d/ðx;tÞ1⁄4
ni0 mi
cb 􏰁 rhd/i vE1⁄4 􏰈 ;
B. Implementation of dynamic equations in cylindrical coordinates
The gyrocenter equation of motion is
_ v2jj lb􏰁rB cb􏰁rhd/i
R1⁄4 vjjb þ r􏰁bþ þ ; (28)
Ldfa 1⁄4 􏰂dLfa0; (22)
where dL 1⁄4 vE 􏰀 r 􏰂 ðZa=ma=B􏰈jjÞB􏰈 􏰀 rhd/ið@=@vjjÞ and L 1⁄4 L0 þdL. Defining particle weight as wa 1⁄4 dfa=fa, we can derive the weight equation from Eq. (22) as
Zavjjb 􏰀 rhd/i Ta0!
dw " rf
a a0
􏰏􏰏􏰏 1⁄4ð1􏰂waÞ􏰂vE􏰀 􏰏􏰂
dt fa0 v?
#
􏰂
Za Ta0
lb 􏰁 rB0 maXca
v2jj
þ b􏰁ðb􏰀rbÞ 􏰀rhd/i ; (23)
FIG. 3. The green þ symbols denote the orthogonal grids in the SOL and core regions with respect to the shared grids at the separatrix, and the dotted line in
^
We use the electrostatic Vlasov-Poisson system for physics simu- lation in this paper. Particle dynamics is described by the gyrokinetic equation using gyrocenter position R, magnetic moment l, and paral- lel velocity vjj as independent variables in five dimensional phase space
!
Xca
where we have used the chain rule rfa0 j 1⁄4 rfa0 j þ ðlfa0 rB=Ta0 Þ
magenta is along the perpendicular direction b0 􏰁 f from the shared grids as shown by the black stars. The grids shown here are only for illustrating the algo- rithm, which are much sparser than the ones used in realistic simulation.
III. PHYSICS MODEL A. Formulation
v? l in the derivation of Eq. (23) from Eq. (22).
The gyrokinetic Vlasov equation is used for ion species, and its perturbed density is
ð
ne0: (25)
ed/ @@ Te0
_ 14
þR􏰀rþv_ f ðR;v ;l;tÞ1⁄40; (18) @t jj@vajj
The gyrokinetic Poisson’s equation is
a species. B is the equilibrium magnetic field, b 1⁄4 B=B; B􏰈 1⁄4 B
jj
R1⁄4vbþvþv; (19)
dfiðR; l; vjj; tÞdvdRdndðR þ qi 􏰂 xÞ=ð2pÞ: (24) tron perturbed density is
hdniðx; tÞi 1⁄4
Electron dynamics is assumed as adiabatic for simplicity, and the elec-
􏰎
jja f
_
Zi2ni0 􏰍 T
f
jj E d
d/ 􏰂 d/
1⁄4 Zihdnii 􏰂 edne; (26)
ð
where Ð dv 1⁄4 2pB Ð dvjjdl.
dneðx; tÞ 1⁄4
Bjj
cma v2jj cl
b􏰁ðb􏰀rbÞþ
􏰈􏰈 |{z}XmXB
b􏰁rB:
ZB ZB ca a ca |fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl}
a jj a jj
fparallel motiong |fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl}
In order to minimize the particle noise, the perturbative df simula- tion method16,17 is applied. The particle distribution is decomposed into equilibriumandperturbedpartsasfa 1⁄4fa0ðR;l;vjjÞþdfaðR;l;vjj;tÞ, and the equilibrium fa0 satisfies the following equation:
Lf1⁄40; (21) 0 a0
where L0 1⁄4 @=@t þ ðvjjb þ vdÞ 􏰀 r 􏰂 ðl=maÞB􏰈 􏰀 rB=B􏰈jjð@=@vjjÞ. ^
1:5 Þ
fcurvature driftg
fgrad􏰂B driftg
fE􏰁B driftg
Because vd is only in the f direction, the particle drift orbit width is
r􏰁b 1⁄4 |fflfflfflfflfflffl{zfflfflfflfflfflffl}
1 mav2jj 􏰌 maXca B
bZ
@B @R
􏰂bR #
@B􏰋 @Z
zero in FRC geometry, then fa0 1⁄4 na0ð equation for perturbed distribution dfa
Phys. Plasmas 26, 042506 (2019); doi: 10.1063/1.5087079 Published under license by AIP Publishing
exp1⁄2􏰂
exact solution of Eq. (21), and na0ðwÞ and Ta0ðwÞ are the 1D function of magnetic flux surface. Subtracting Eq. (18) by Eq. (21), we have the
ma 2pTa0
ma v2jj þ2lB 2Ta0
􏰃 is the
fcurvature driftg
􏰌􏰋
with
vjjb |{z}
fparallel motiong
v2jj Xca
1⁄4 vjjðbRR^þbZZ^Þ;
"
v2jj
XcaB @Z @R
@ BR @ BZ
þ 􏰂 f;
^
26, 042506-6