Page 4 - Electrostatic quasi-neutral formulation of global cross-separatrix particle simulation in field-reversed configuration geometry

Physics of Plasmas ARTICLE
@2f 1 @2f 􏰉@w @w w 􏰊 @f 2RZRE0
r?f!@X2þR2@f2þ @Rþ@ZþR @X : ww
C. Toroidal wedge
(12)
velocity (composed of the drifts due to electric fields ~v 􏰂 ~E 􏰅 B~ = 2 mv2?~~ 2
^ Taking advantage of the symmetry of the FRC toroidal direction f (azimuthal about the machine axis), ANC simulations use a trun- cated toroidal wedge based on the lowest common denominator, nLCD, of the toroidal mode numbers kept in the simulation, i.e., going from a full torus 1⁄20;2p􏰆 to a truncated toroidal wedge 1⁄20;2p=nLCD􏰆. This allows for physics of short toroidal wavelength modes to be fully resolved without waste in the case where simulations are focused on subsets of toroidal modes.
D. Magnetic field
To ensure that numerical heating or cooling of particles due to magnetic field quantities does not occur, a divergence-free magnetic field is ensured by representing all magnetic field related quantities through the derivatives of the poloidal flux. Through cubic spline rep- resentation of the poloidal flux, the magnetic field, defined as
R@Z (13) ~B Z 􏰂 1 @ w ;
1
jB0j , magnetic field gradients~vg 􏰂 2qB0 ðB0 􏰅 rB0Þ=jB0j , and mag- neticfieldcurvature~vc 􏰂mv2k ð~Rc 􏰅~B0Þ=R2cjB0j).Theblendedparam-
qB0
eter used in the position and velocity update equations is defined as
scitation.org/journal/php
Finally, a perpendicular Laplacian operator can found by drop- @f ~2 ~ ^ ~
due to the magnetic mirror force, Dxk 􏰂~vkDt and Dx? 􏰂~v?Dt are the changes in position due to parallel and perpendicular velocities, andDxd􏰂ð~vEþ~vgþ~vcÞDtisthechangeinpositionduetodrift
pingtheparallelderivatives,i.e.,@XS !0,orbyr?􏰂r􏰀ðb0􏰅rÞ,
~ 1 @w BR 􏰂 􏰁 ;
where the effective velocity and acceleration are shown in (14) and (15).
In the perturbed df model,13,14 the distribution function is split
into f 􏰂 f0 þ df , where f0 is a known analytic equilibrium distribution
such as the Maxwellian distribution and df is the perturbation to this
equilibrium distribution. When considering the moments of the distri-
bution function as the main quantities of interest, the PIC method is
essentially a series of Monte Carlo calculations, where the df approach
15
will have a divergence that is analytically zero.
III. PARTICLE DYNAMICS
The particle-in-cell (PIC) method is conceptually simple: repre- sent physical particles with simulation marker particles and follow their trajectories in the electrostatic (or electromagnetic) fields self- consistently calculated from these particle distributions. Mathematically, the marker particles follow the characteristics such that the phase-space volume represented by each particle does not change. Different models can be used to describe the particle dynamics suitable for different particle species. In ANC, particles can be described by a blended drift-Lorentz perturbative df model or by a gyrokinetic perturbative df model.
A. Blended drift-Lorentz particle model
Based on blended drift-Lorentz particle pusher introduced by Cohen et al.,12 a drift-Lorentz perturbative df particle model has been developed and implemented in ANC. Briefly described, the drift- Lorentz particle pusher is a Boris-push algorithm with corrections for time-steps large relative to the local cyclotron oscillation. These modi- fied velocity and position updates are defined as
nþ1=2 n􏰁1=2 n n
v 1⁄4v þDvLþð1􏰁aÞDvlrB; (14)
xnþ1 1⁄4 xn þ Dxnþ1=2 þ aDxnþ1=2 þ ð1 􏰁 aÞDxnþ1=2 ; (15) k?d
~~ whereDvL􏰂qðEþ~v􏰅B0ÞDtisthechangeinvelocityduetothe
normal Lorentz force, DvlrB 􏰂 􏰁l @B Dt is the change in velocity @ ~x k
Phys. Plasmas 27, 082504 (2020); doi: 10.1063/5.0012439 Published under license by AIP Publishing
df1⁄4􏰁 f0 dt dt
R@R dd
dt
eff eff v
a 􏰂sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi: s 􏰉􏰊2
1 þ Xc;sDt (16) 2
Together, the particles described by this model follow a modified Vlasov equation:
d􏰌@~~􏰍
fð~x;~v;tÞ􏰂 þv~􏰀rþa~􏰀r f; (17)
@t
is a case of the control variates method. reformulated into the weight equation,
Equation (17) can then be
df =g
f ð0Þ=gð0Þ
(18) and g(t) is the marker distribution. Substituting
dw~
! 1⁄4􏰁ð1􏰁w~Þ ;
1 df0 dt f0 dt
where w~ 􏰂
from (17) and noting that portions due to the equilibrium distribution
function are zero by definition, the weight equation becomes
dw~ q~ 1@f0
1⁄4􏰁ð1􏰁w~Þ E1􏰀 : (19)
dt mf0@~v
Assuming an expansion of the equilibrium distribution function
based on the local Maxwellian via Eq. (5),
~
􏰉􏰊􏰉􏰊
@F0ðw;EÞ􏰄 @fM þ@a1 f þ a @fM þ@a2 f þð􏰀􏰀􏰀Þ
@v @v M 1 @v @v M ff
@~v
whereb0 aretermsofO ðqw Þ ,b1 aretermsofO ðqw Þ ,and
so on, with ordering defined in Sec. II. More explicitly, the first order term is
􏰄 fMðb0 þb1 þb2 þ􏰀􏰀􏰀Þ; (20) 􏰋 mRvf 0􏰂 􏰋 mRvf 1􏰂
mj􏰁 􏰀 mjR 1@nj;0
􏰉􏰊!
3 1@Tj;0 ^
of b0, it can be seen that the contributions to the change in the
27, 082504-4
b0􏰂􏰁 v~kþajv~?􏰁
Tj qj
Ej
2 Tj;0 @w where the subscript j refers to the particle species. In the explicit form
nj;0 @w
Tj;0
þ 􏰁
f; (21)