Page 5 - Global simulation of ion temperature gradient instabilities in a field-reversed configuration

Physics of Plasmas ARTICLE
where we implement 3 1D linear interpolations along w, f, and Sc=SS for the simulations in 3 dimensional ðw; f; Sc =SS Þ space.
C. Laplacian operator
In the global simulation of FRC geometry with the field aligned mesh, the perpendicular Laplacian operator is discretized in ðw; f; Sc Þ space for the core region and in ðw; f; SS Þ space for the SOL region, respectively, since the irregular mesh in cylindrical coordinates becomes regular in magnetic coordinates.
The Laplacian operator can be expanded in a generalized coordinates
r2f1⁄4gww@2fþ2gwS @2f þgSS@2fþgff@2f S @w2 S@w@SS S@S2S @f2
and it becomes antisymmetric when the indices change.
@S
gS1⁄4rw􏰁rSS1⁄4 þ þ :
where
XX
In the core region with ðw; f; S Þ, c
gc 1⁄4rSc􏰁rSc1⁄4
;
1⁄4 SS 1⁄2wðR;
r?cfn􏰈g 2þ2gc @ w
þgc
gwS 1⁄4rw􏰁rS cc
2
ww @2fn
2 ww@2fn􏰏􏰏 2 ff
􏰋􏰌
@w^ @w^ 1^ @SS @w^ @w^ @SS ^ 1⁄4RþZ􏰂f􏰁 RþZþZ
2 1@ a b@f rf1⁄4aJrn􏰁rnb; (11)
@R @Z
J @n @n ab1⁄4;
where f represents an arbitrary scalar field, n ; n refer to the three dimensional coordinates, and J is Jacobian and defined as J 􏰃1 􏰀abc
1@w@SS R @R @Z
􏰋 􏰌
􏰋 􏰌
a^a^a^ 1⁄4rna􏰂rnb􏰁rnc;rna1⁄4@n Rþ@n 1fþ@n Z, and 􏰀abc1⁄41,
@S@w2 gSS1⁄4rS􏰁rS1⁄4 S
@S@w @S 2 þ S þ S ;
@R @fR @Z 39
@w@Z @Z @w 2 @w 2 @w@S
the perpendicular Laplacian operator is expanded as
r2 f 1⁄4 gww @2f 􏰏􏰏􏰏 þ 1 @JXgww @f 􏰏􏰏􏰏 þ gff @2f ; (14)
?;X @w2b0􏰂f JX @w@wb0􏰂f @f
where jb0􏰂f represents the partial derivative with respect to w among
the orthogonal grids along the b0 􏰂 f direction, and J􏰃1 1⁄4rw􏰂rf􏰁rS
considering r2f1⁄4gww@2fþ2gwS @2f þgSS@2fþgff@2f
@w @R @Z @Z @Z
For the shared grids between core and SOL regions at the separatrix,
Sc 1⁄4Sc1⁄2wðR;ZÞ;hðR;ZÞ􏰄,theLaplacianiswrittenas
c @w2 c@w@Sc c@S2c @f2
1 "@􏰃J gww􏰁 @􏰃JcgwS􏰁# @f þcþc
1 "@􏰃JcgwS􏰁 @􏰃JcgSS􏰁# @f þcþc;(12)
S S
S
wS S S
scitation.org/journal/php
@􏰍J gwS􏰎3
S S 5@f
þ
þ14 SS þ@JSgS 5@f; (13)
2 􏰃 ww􏰁 þ14@ JSg
@SS 3@w SS􏰁
JS2 @w @􏰍J gwS􏰎
􏰃
JS @w @SS @SS J􏰃1 1⁄4rw􏰂rf􏰁rS
where
S 􏰋 S􏰌 " 􏰋 􏰌 #
R @w@R @Z @Z
@w@R
"􏰋 􏰌 􏰋 􏰌#
ww
1⁄4 rw 􏰁 rw 1⁄4
􏰋@w􏰌2 @R
þ þ
􏰋@w􏰌2 @Z
Jc @w @Sc @w
Jc @w J􏰃1 1⁄4rw􏰂rf􏰁rS
@Sc
@Sc
c c" @R@ZR@R@Z 􏰋@w @w 􏰌 1 @Sc􏰋@w @w 􏰌 1 @w @SX 1 @w @SX
^^^^^ 1⁄4􏰃:
1⁄4RþZ􏰂f􏰁RþZ @R @Z R @w@R @Z
R@R @Z R@Z @R
Taking advantage of the toroidal symmetry of FRC, we could
transform the Laplacian operator into Fourier space with respect to
the toroidal angle f, which can avoid solving the 3 dimensional matrix.
In this paper, we can also simplify Eqs. (12)–(14) assuming k 􏰇 j ?
and kjj 􏰆 k?, where j 1⁄4 rB0=B0. Thus, for each toroidal mode n (@=@f 1⁄4 inf), the Laplacian operators in the core region, SOL region, and at the separatrix can be written as
@Sc 􏰋@h ^ @h ^􏰌􏰊 þ@h @RRþ@ZZ
1@w@Sc @h 1@w@Sc @h 1⁄4􏰃;
R@R @h @Z R@Z @h @R
g
SS cccc 2wwnwSnSSn2ff
􏰋@S @w @S @h􏰌2
; 􏰋@S @w
þ
@S @h􏰌2 @h @Z
@2f @2f @2f
þ
"􏰋􏰌2􏰋􏰌2#􏰅 􏰊 ?S @w2 @w@SSS@S2S
@w @R
@h @R
@w @Z
@ w @ S c wS @2fn
@ S 2c SS @2fn
@Sc @w @w @Sc @w @h @w @h 1⁄4þþþ; @w @R @Z @h @R@R @Z@Z
þg @w b0􏰂f
ZÞ; Z􏰄, the Laplacian is written as
Phys. Plasmas 26, 042506 (2019); doi: 10.1063/1.5087079 Published under license by AIP Publishing
ff 1 g 1⁄4rf􏰁rf1⁄4 2:
R
In the SOL region with ðw; f; SS Þ, considering SS
􏰋@w^ @w^􏰌 1^ 􏰅@SX ^ @SX ^􏰊 1⁄4RþZ􏰂f􏰁RþZ
􏰃ng fn; (15)
2 ff ; (16) 􏰃ng fn
(17)
where fn is the n toroidal mode component of f.
The grids shown in Figs. 2 and 3 are only for illustrating the field
aligned grid algorithm in cylindrical coordinates clearly, which are much sparser than the realistic simulation.
26, 042506-5
r fn􏰈g
þ2gS
r?Xfn􏰈g 2􏰏 􏰃ng fn;