Page 4 - Global simulation of ion temperature gradient instabilities in a field-reversed configuration

Physics of Plasmas ARTICLE
FIG. 2. (a) Global field aligned mesh in cylindrical coordinates. Field aligned mesh mapping from cylindrical coordinates to magnetic coordinates: (b) core region grids in ðw; Sc Þ coordinates and (c) SOL region grids in ðw; SS Þ coordinates. The black stars represent the shared grids at the separatrix. The grids shown here are only for illustrating the algorithm, which are much sparser than the ones used in realistic simulation.
and ZSðw;SSÞ1⁄4ZSð1;iS;jSÞþZSð2;iS;jSÞDwþZSð3;iS;jSÞDw2
þZ ð4;i ;j ÞDSþZ ð5;i ;j ÞDwDSþZ ð6;i ;j ÞDw2DS SSSSSSSSS
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and
Zcðw;ScÞ1⁄4Zcð1;ic;jcÞþZcð2;ic;jcÞDwþZcð3;ic;jcÞDw2 þZcð4;ic;jcÞDSþZcð5;ic;jcÞDwDSþZcð6;ic;jcÞDw2DS þZcð7;ic;jcÞDS2 þZcð8;ic;jcÞDwDS2 þZcð9;ic;jcÞDw2DS2;
(8) whereDw1⁄4w􏰃wðicÞandDS1⁄4Sc 􏰃SðjcÞ.wðicÞ􏰅w<wðic þ1Þ,
and SðjcÞ 􏰅 Sc < Sðjc þ 1Þ.
RSðw;SSÞ1⁄4RSð1;iS;jSÞþRSð2;iS;jSÞDwþRSð3;iS;jSÞDw2
þRSð4;iS;jSÞDSþRSð5;iS;jSÞDwDSþRSð6;iS;jSÞDw2DS
þR ð7;i ;j ÞDS2þR ð8;i ;j ÞDwDS2þR ð9;i ;j ÞDw2DS2; SSSSSSSSS
By using Eqs. (7)–(10), we could compute the field aligned mesh in cylindrical coordinates with a given regular mesh in the core region ðw;ScÞ and in the SOL region ðw;SSÞ, respectively. An example of global field aligned mesh is given in Fig. 2(a). The field aligned grids are irregular in cylindrical coordinates. For the overlap part of core and SOL regions at the separatrix (R > 0), the grid positions are deter- mined by using Rcðw;ScÞ and Zcðw;ScÞ, which are shared by both core and SOL regions with the scopes of two coordinates: ðw;ScÞ and ðw; SS Þ as shown by the black stars. In magnetic coordinates, the grids in SOL and core regions are regular inside each domain, respectively. The shared grids at the separatrix are designed regularly with the inte- rior grids in the core region (keep the same parallel coordinate Sc value), which are shown in Figs. 2(b) and 2(c).
In GTC-X, particle dynamic equations are evolved in cylindrical coordinates ðR; f; ZÞ to avoid the singularity at the separatrix. However, the field aligned mesh is irregular in (R, Z) space, which is difficult to carry out particle-grid gather-scatter operation for PIC sim- ulation. It is noted that the field aligned grids are labeled by both cylin- drical and magnetic coordinates: (R, Z) and ðw; Sc Þ in the core region and (R, Z) and ðw; SS Þ in the SOL region. The mesh is regular in mag- netic coordinates in the core and SOL regions, respectively, as shown in Fig. 2. Thus, we can transform the particle coordinate from (R, Z) to ðw;ScÞ and ðw;SSÞ by using Eqs. (3), (5), and (6), and then a simple linear interpolation between the particle location and the regular rect- angular mesh can be used for particle-grid gather-scatter operation,
(9)
þZSð7;iS;jSÞDS2 þZSð8;iS;jSÞDwDS2 þZSð9;iS;jSÞDw2DS2;
(10)
whereDw1⁄4w􏰃wðiSÞandDS1⁄4SS 􏰃SðjSÞ.wðiSÞ􏰅w<wðiS þ1Þ and SðjSÞ 􏰅 SS < SðjS þ 1Þ.
Phys. Plasmas 26, 042506 (2019); doi: 10.1063/1.5087079 Published under license by AIP Publishing
26, 042506-4