Page 2 - Electrostatic quasi-neutral formulation of global cross-separatrix particle simulation in field-reversed configuration geometry

Physics of Plasmas ARTICLE
FIG. 1. The simulation mesh in (a) real-space (b) field-aligned space. The SOL region is represented by red, and the core region is represented by green. The two regions are topologically separated by the separatrix, represented by the black dashed line. Only a subset of the usual grid is plotted for clarity in visualization. When the gyrokinetic model is used, a gap in the core region is made to avoid the magnetic null region due to gyrokinetic validity; however, this gap is not required when using the blended particle model and is not shown.
where w 􏰂 wðR; ZÞ 􏰁
q
available experimentally relevant pressure profiles,
scitation.org/journal/php
GTC, the simulation code ANC is a microturbulence code tailored to investigate transport in FRC geometry using a first-principles model.
In follow-up non-local nonlinear ANC and GTC simulations
using a gyrokinetic ion model with an adiabatic electron response,
instability was shown to only grow in the SOL,7–9 consistent with
the local GTC results. These ANC simulations use a non-local sim-
ulation domain extending from the FRC core to the SOL within the
confinement vessel with an axial periodic boundary condition.
The non-local capability was shown to be important for allowing
the fluctuations, which arise due to SOL instability, to non-linearly
spread into the core, resulting in a fluctuation spectrum that is
qualitatively consistent and quantitatively within expectations with
2
II. EQUILIBRIUM AND COORDINATE SYSTEM A. Equilibrium
ANC is built to use equilibrium data such as wpol, ne, ni, Te, and Ti on a 2D R-Z plane with quantities assumed to be azimuthally sym- metric. Simulations use equilibria generated using the LReqMI code,11 and experimentally relevant equilibria are obtained by comparing syn- thetic diagnostics with experimental results.
Currently, simulations are confined to a nonlocal domain span- ning the confinement vessel region of the reactor with periodic bound- aries in the axial directions and neglect parallel outflow effects. This assumption limits the model to investigations of perpendicular trans- port only. As shown in Fig. 1, the magnetic field null region is excluded in simulations using the gyrokinetic particle model to preserve gyroki- netic validity.
A true equilibrium distribution function is of the following form:
F0ðw; EÞ 1⁄4 ~ mvfR
q
function, which is a constant of motion due to its dependence on only
the canonical angular momentum. Currently, equilibria generated by
LReqMI are spatially dependent but not velocity dependent. While
questionable for ions in the FRC, this function can be expanded with
the assumption of wðR; ZÞ 􏰃 mvf R, a necessary step to match currently
experimental measurements. As the next generation of high per- formance computers enters exascale computing regimes, algo- rithms and models can also be updated to accommodate more realistic physics. Because of the low magnetic fields, the gyrokinetic model is inadequate to fully describe the possible particle trajecto- ries in the FRC,10 such as Fig. 8 or betatron orbits. To overcome this, the blended particle model, described in Sec. III, is imple- mented in ANC. Furthermore, in order to calculate particle diffu- sivity and electron thermal conductivity, the non-adiabatic electron response must be represented. Using the blended model, the non- adiabatic electron response is now implemented, and it is now pos- sible to self-consistently calculate electron transport from ANC simulations. At present, ANC is electrostatic and nonlinear, based on the Vlasov–Poisson system of equations. With the goal of understanding transport scaling applicable toward future reactor- grade plasmas, effective collisionality is assumed to be low and no collisional effects are present in the model. The particle model is generalized, allowing gyrokinetic (GK) or blended drift-Lorentz (“blended”) particle species. Toward the goal of experimental vali- dation, it accepts arbitrary numerical FRC equilibria, spanning closed and open magnetic field-line regions. In this paper, the for- mulation of the simulation model implemented in ANC is described: the equilibrium and coordinate system, including the field-aligned mesh, are presented in Sec. II; the blended and gyroki- netic particle models are described in Sec. III; Sec. IV is devoted to solving for the self-consistent fields; various benchmarks are dis- cussed in Sec. V; and preliminary FRC simulation results are shown in Sec. VI.
~
~ nðwÞ ~
e􏰁E=TðwÞ; (1) 1⁄4 􏰁Pf=q is the modified poloidal flux
~ 3=2 ð2pTðwÞ=mÞ
@f0ðw;EÞ􏰉 mvfR􏰊 F0ðw; EÞ 􏰄 f0ðw; EÞ þ 􏰁
~
􏰋@w q􏰂 􏰄 f0ðw;EÞ a0ðv0fÞþa1ðv1fÞþ􏰀􏰀􏰀 ;
þ 􏰀 􏰀 􏰀
where f0 is an analytic distribution function dependent on only the spatially dependent w and the particle energy E. Taking f0 to be a local Maxwellian with spatially dependent temperatures and densities,
nðwÞ 􏰁E=TðwÞ fM ðw; EÞ 1⁄4 e
; (3) the first three terms of the expansion from (2)can be written as
(2)
ð2pT ðwÞ=mÞ3=2
Phys. Plasmas 27, 082504 (2020); doi: 10.1063/5.0012439 Published under license by AIP Publishing
27, 082504-2