Page 3 - Electrostatic quasi-neutral formulation of global cross-separatrix particle simulation in field-reversed configuration geometry

Physics of Plasmas ARTICLE
m2R2v2 2f
a2ðv Þ 1⁄4 􏰁
f q2
􏰅 þ􏰁5þ
􏰁
;
T 2 n@wT@w E 15􏰊􏰉1 @T􏰊2
q n@w T 2 T@w
scitation.org/journal/php
a 0 ð v 0f Þ 1⁄4 1 ; 􏰉 􏰊
mRvf 1@n E 3 1@T a1ðvfÞ 1⁄4 􏰁 þ 􏰁
In ANC, the mesh is field-aligned while the coordinate system is cylindrical with an unconventional signing, i.e.,
R^ 􏰅 Z^ 1⁄4 ^f: (6) Vector operations are obtained via chain rule and are numerically dis-
cretized via an assumed quadratic interpolant. In Secs. II B 1–II B 3, mesh operations are detailed, where Xw and XS refer to physical spac-
^^^
frequency, and mode structure of a linear instability was performed and found to be consistent with results obtained via the cylindrically
7
1. Partial derivatives
The mesh is generated by finding points along the magnetic field direction and along the direction, that is perpendicular to the magnetic field. The gradient, in this mesh, is then
~ @f^ @f^ 1@f^
rf 1⁄4 w þ S þ f: (7)
1
! "􏰉E 3􏰊􏰉1 @n 1 @T􏰊
(4)
For the time-advance of the weight, i.e., Eq. (19), the quantity used is actually the partial derivative with respect to velocity. Assuming an expansion
1􏰉E2
ing in the directions of w and S 􏰂 b0.
Using the field-aligned mesh, a comparison of the growth-rate,
2T2 T4T@w!#
1 1@2n 􏰉E 3􏰊1@2T þþ􏰁:
2 n@w2 T 2 T@w2
~h􏰋 􏰂i @F0ðw;EÞ􏰄 @ f0ðw;EÞ a0ðv0fÞþa1ðv1fÞþ􏰀􏰀􏰀
@~v
@~v
􏰄ð1þa1 þa2 þ􏰀􏰀􏰀Þ@f0
@v 􏰉􏰊
(5)
@a1 @a2
þ @v þ@v þ􏰀􏰀􏰀 f0:
@Xw @XS @R@f
Using this relation, the real space partial derivatives can be calcu-
regular mesh. Some benchmarks regarding the field-aligned mesh operations are also shown in the Appendix.
ff
To estimate the comparative largeness of the different terms of the
expansion, the azimuthal velocity can be assumed comparable to the
thermal velocity, and the average ratio for ions is jqwj=mivth;iR 􏰄 2,
while the average ratio for electrons is jqwj=me vth;e R 􏰄 300. @R Currently, ANC uses only the lowest order expansion, which reduces
to the expected form in the drift-kinetic limit as described in Sec. III. For ions, the large second order term is a departure from the usual gyrokinetics due to large finite Larmor radius, and further exploration of this effect is in progress.
B. Field aligned mesh
Although the physics of the code is formulated in cylindrical coordinates in order to couple regions across the magnetic separatrix, the simulation mesh itself is generated such that grid points lie along the magnetic field-lines. This allows for a coarser resolution in the direction parallel to the field-lines.
The field-aligned mesh is generated by picking points along the separatrix, then using the magnetic field directional unit vectors to
^
parallel to the magnetic field (along S). Because of the differences between the two regions, the coordinate parallel to the field-line is defined differently for the core and SOL. These coordinates are only used in the interpolation of particles-to-mesh or mesh-to-particles.
For the core, S 1⁄4 61 is the location of the inner mid-plane where the mesh is periodically bound both physically and numerically while S 1⁄4 0 is the location of the outer mid-plane. The field-aligned coordi- nate Sðw; hgeoÞ 􏰂 SðwðR; ZÞ; hgeoðR; ZÞÞ is defined by the two inde- pendent variables, w (the poloidal flux label) and hgeo (the geometric angle about the null-point).
For the SOL, S1⁄4􏰁1 and S1⁄41 are the locations of the left and right boundaries, respectively, and S 1⁄4 0 is the location of the outer mid-plane. The field-aligned coordinate Sðw;ZÞ􏰂SðwðR;ZÞ;ZÞ is defined by the two independent variables, w (the poloidal flux label) and Z (the position along the axial direction).
Phys. Plasmas 27, 082504 (2020); doi: 10.1063/5.0012439 Published under license by AIP Publishing
@Z @X Z @X Z wS
assemble lines perpendicular to the magnetic field (along w) and lines ^
@Xw @XS @R@f 3. Laplacian
lated by taking the scalar dot product
2. Gradient
@f^~@f @f
1⁄4 R 􏰀 rf 1⁄4
@Xw
wR þ SR; @XS
@f^~@f @f 1⁄4Z􏰀rf1⁄4 wþ S:
(8)
Using the definitions of the partial derivatives, the real space gra- dient in cylindrical coordinates is
~@f^@f^1@f^􏰉@f @f􏰊^ rf1⁄4RþZþ f1⁄4 wRþSRR @R @Z @R@f @Xw @XS
􏰉 @f @f 􏰊^ 1 @f ^
þ wZþ SZ Zþ f: (9)
Using the definition of the gradient, the Laplacian can be calcu- lated from the following relation:
~~1@􏰋^~􏰂 r2f 1⁄4r􏰀rf 1⁄4 RðR􏰀rfÞ
R@R
@ 􏰋^ ~ 􏰂 1 @ 􏰋^ ~ 􏰂
þ@Z Z􏰀rf þR@f f􏰀rf : (10) Using the real space gradient operator given by (9) and some math,
this relation becomes r2f!@2fþ@2fþ1@2fþð2wS þ2wSÞ @2f
@Xw2 @XS2 R2@f2 R R Z Z @Xw@XS 􏰉@wR @wZ wR􏰊 @f 􏰉@SR @SZ SR􏰊 @f
þ @R þ @Z þ R @X þ @R þ @Z þ R @X : (11) wS
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