Page 3 - Global simulation of ion temperature gradient instabilities in a field-reversed configuration

Physics of Plasmas ARTICLE
where i 2 1⁄21; LSR􏰄 and j 2 1⁄21; LSZ􏰄 are the radial and axial indexes of equilibrium grids, and DR1⁄4R􏰃Ri;DZ1⁄4Z􏰃Zj;Ri 􏰅R<Riþ1 andZj 􏰅Z<Zjþ1.
It is straightforward to show that r 􏰁 B 1⁄4 0 is guaranteed theo- retically and numerically
(4)
1@ @B r􏰁B1⁄4 ðRBRÞþ
Z 1⁄40: R@R @Z
B. Field line coordinates for the perturbed field calculation
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wðR;ZÞ1⁄4wð1;i;jÞþwð2;i;jÞDRþwð3;i;jÞDR2
þ wð4; i; jÞDZ þ wð5; i; jÞDRDZ þ wð6; i; jÞDR2 DZ þwð7;i;jÞDZ2 þwð8;i;jÞDRDZ2 þwð9;i;jÞDR2DZ2;
(3)
field line length at each w grid and range from 0 to 1. In the core region, Sc starts at the outer midplane with Z 1⁄4 0 and increases along the clockwise direction, and grids at Sc 1⁄4 0 and Sc 1⁄4 1 are overlapped. In the SOL region, the parallel coordinate SS starts on the left boundary Z0 and ends on the right boundary Z1.
Next, considering the properties of geometry topology of core
and SOL regions in FRC, forward spline functions Sc 1⁄4 Sc1⁄2wðR; ZÞ;
hðR; ZÞ􏰄 and SS 1⁄4 SS1⁄2wðR; ZÞ; Z􏰄 are created for the transformation
from cylindrical coordinates to field line coordinates, where sinðhÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1⁄4 Z= Z2 þ ðR 􏰃 R0Þ2 is the geometric angle with respect to the magnetic axis position ðR 1⁄4 R0 ; Z 1⁄4 0Þ. The uniform scale of geomet-
ric angle h for creating spline function Sc ðw; hÞ is hðjcÞ1⁄4dh􏰂ðjc 􏰃1Þ;
where dh 1⁄4 2p=ðlstc 􏰃 1Þ, lstc is the spline resolution in the h direc- tion, and 1 􏰅 jc 􏰅 lstc. By tracing each field line and calculating the Sc value on the uniform scales of w and h, the spline function Scðw;hÞ can be derived as
Scðw;hÞ1⁄4Scð1;ic;jcÞþScð2;ic;jcÞDwþScð3;ic;jcÞDw2 þScð4;ic;jcÞDhþScð5;ic;jcÞDwDhþScð6;ic;jcÞDw2Dh
2222 þScð7;ic;jcÞDh þScð8;ic;jcÞDwDh þScð9;ic;jcÞDw Dh :
(5)
The uniform scale of Z for creating spline function SSðw; ZÞ is ZðjSÞ 1⁄4 dZS 􏰂 ðjS 􏰃 1Þ;
where dZS 1⁄4 ðZ1 􏰃 Z0Þ=ðlszs 􏰃 1Þ, lszs is the spline resolution in the Z direction, and 1 􏰅 jS 􏰅 lszs. By tracing each field line and calculat- ing the SS value on the uniform scales of w and Z, SSðw; ZÞ is given as
SSðw;ZÞ1⁄4SSð1;iS;jSÞþSSð2;iS;jSÞDwþSSð3;iS;jSÞDw2 þSSð4;iS;jSÞDZþSSð5;iS;jSÞDwDZþSSð6;iS;jSÞDw2DZ
2222 þSSð7;iS;jSÞDZ þSSð8;iS;jSÞDwDZ þSSð9;iS;jSÞDw DZ :
(6)
Then, we create the inverse spline functions Rcðw;ScÞ and Zcðw; ScÞ for the core region, and RSðw; SSÞ and ZSðw; SSÞ for the SOL region. The uniform scales of Sc and SS for the spline function are
ScðjcÞ 1⁄4 dSc 􏰂 ðjc 􏰃 1Þ;
where dSc 1⁄4 1=ðlssc 􏰃 1Þ, lssc is the spline resolution in the parallel
direction in the core region, and 1 􏰅 jc 􏰅 lssc SSðjSÞ 1⁄4 dSS 􏰂 ðjS 􏰃 1Þ;
where dSS 1⁄4 1=ðlsss 􏰃 1Þ, lsss is the spline resolution in the parallel direction in the SOL region, and 1 􏰅 jS 􏰅 lsss. It is straightforward to get the values Rc 1⁄2wðic Þ; Sc ðjc Þ􏰄; Zc 1⁄2wðic Þ; Sc ðjc Þ􏰄; RS 1⁄2wðiS Þ; SS ðjS Þ􏰄, and ZS 1⁄2wðiS Þ; SS ðjS Þ􏰄 on the uniform scales of ðw; Sc Þ and ðw; SS Þ, and the quadratic spline functions Rcðw;ScÞ; Zcðw;ScÞ; RSðw;SSÞ, and ZSðw; SSÞ can then be created as
In magnetized plasmas, the gyrocenter drift motion across the
magnetic field is much slower than the parallel motion along the field
line; thus, the wave pattern is always anisotropic in the parallel and
perpendicular directions with kjj 􏰆 k?ðkjj and k? are the parallel and
perpendicular wave vectors). In order to improve the numerical effi-
ciency and accuracy, the field aligned mesh is widely adapted for parti-
cle-in-cell simulation of magnetized plasmas, i.e., the grids are aligned
along the magnetic field direction with only a small number in the par-
allel direction, which can dramatically suppress the high kjj noise and
save computational cost without sacrificing key physics dominated by
23,38
In global FRC simulation, we setup a field aligned mesh in both core and scrape-off layer (SOL) regions across the separatrix in cylindrical coordinates. Due to the fact that the magnetic field is not uniform in FRC, the field aligned mesh is not regular in cylindrical coordinates. For solving perturbed fields as well as particle-grid gather-scatter operation, we create the field line coordinates for core ðw; ScÞ and SOL ðw; SSÞ regions on the poloidal plane, separately, where Sc and SS represent the normalized field line distances along the magnetic field line direction in core and SOL regions, and the mesh is regular in the corresponding field line coordinates. Thus, in GTC-X, we use two different coordinate systems: cylindrical coordinates and field line coordinates to represent the location.
small kjj.
The simulation domain is different from equilibrium shown in Fig. 1. Because drift wave instabilities and associated transports are anisotropic in perpendicular and parallel directions, we choose the simulation domain based on perpendicular coordinates w (we do not need to consider about f domain because it is toroidally symmetric from ð0;2pÞ), i.e., the inner boundary in the core and the outer boundary in SOL are labeled by poloidal magnetic flux: w0 and w1. Furthermore, left and right boundaries in the SOL region are given by Z0 and Z1, where Z0 1⁄4 􏰃Z1 is symmetric with respect to outer mid- planeZ1⁄40.
First, we define Sc and SS by tracing each field line on each w grid in the core and SOL regions as
wðicÞ1⁄4w0 þdwc 􏰂ðic 􏰃1Þ;
where dwc 1⁄4 ðwX 􏰃 w0Þ=ðlspc 􏰃 1Þ, wX is the value of w at the separa-
trix, lspc is the spline resolution in the core region, and 1 􏰅 ic 􏰅 lspc.
And þRcð4;ic;jcÞDSþRcð5;ic;jcÞDwDSþRcð6;ic;jcÞDw2DS
wðiSÞ1⁄4wX þdwS 􏰂ðiS 􏰃1Þ;
where dwS 1⁄4 ðw1 􏰃 wX Þ=ðlsps 􏰃 1Þ, lsps is the spline resolution in
SOL region, and 1 􏰅 iS 􏰅 lsps. Both Sc and SS are normalized by the
Phys. Plasmas 26, 042506 (2019); doi: 10.1063/1.5087079 Published under license by AIP Publishing
Rcðw;ScÞ1⁄4Rcð1;ic;jcÞþRcð2;ic;jcÞDwþRcð3;ic;jcÞDw2 2222
þRcð7;ic;jcÞDS þRcð8;ic;jcÞDwDS þRcð9;ic;jcÞDw DS ; (7)
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