Page 8 - Electrostatic quasi-neutral formulation of global cross-separatrix particle simulation in field-reversed configuration geometry

Physics of Plasmas ARTICLE
FIG. 4. Analytic theory (dashed line) of the so-called neutralized Ion Bernstein mode is plotted on top of spectral power found from ANC simulation. The simulation results are consistent with theory, tracking along the frequencies which are shifted away from the cyclotron harmonic resonances.
FIG. 5. Analytic theory (range depicted by cyan shaded region) of the lower-hybrid wave is plotted on top of plasma response found from ANC simulation (represented by fits to the initial linear rate of increase in potential due to resonance with the antenna). The simulation results are consistent with theory with strongest response in the range of frequencies expected by analytic theory.
k2 ! x2 1⁄4 x2 1 þ mi k ;
L H m e k 2? where the lower-hybrid frequency is defined by
!􏰁2 pi pe c;i c;e
211 xLH 􏰂 x2 þx2 þX X
scitation.org/journal/php
first branch at longer wavelengths. This verifies the capability to prop- erly simulate waves available only in the fully kinetic model in ANC.
Note that the same benchmark was also performed with the fully-kinetic ion model. In that benchmark, a much smaller time step must be used; otherwise, numerical instability arises from the under- resolved higher frequency branches. On the other hand, in the blended ion model benchmark, larger time steps can be taken because the blended ion model becomes more drift-kinetic-like and the physics of the Bernstein mode will be unavailable to under-resolve. This high- lights a benefit of the blended model in which the relevant physics can be chosen by choosing the simulation time step.
D. Lower-hybrid wave
Using a simple straight magnetic field with non-uniform density, the lower-hybrid wave is probed by external antenna excitation to compare the plasma response from simulation with analytic theory.
With the fully kinetic ion model with non-adiabatic kinetic elec- tron model and assumption of kk 􏰇 k?, the dispersion of the lower- hybrid wave is
(38)
: (39)
An external electrostatic potential (/ext / cosðx0tÞ) is imposed on the plasma and causes the self-consistent electrostatic potential to grow linearly when the antenna frequency is near an eigenfrequency, much like forced harmonic oscillation. In this case, a range of eigenfre- quencies exists and the response is not as sharp as might be expected, though the largest response lies in the range defined by the standard deviations of the mean frequency from (38). Due to the non-uniform
Phys. Plasmas 27, 082504 (2020); doi: 10.1063/5.0012439 Published under license by AIP Publishing
density and non-local domain (kf 1⁄4 n=R), there is a range of possible solutions to the dispersion (38), depicted as the shaded region in Fig. 5. In the same figure, plasma response from external antenna exci- tation at five different frequencies is shown. This verifies the capability to properly simulate waves available only in the fully kinetic model in ANC with both blended ions and electrons.
VI. INITIAL BLENDED FRC SIMULATION RESULTS
Recently, simulations using blended thermal ions and electrons have been performed, and the preliminary results are reported here. Eight toroidal modes are kept (n 1⁄4 1⁄20; 10; ...; 60; 70􏰆) in the simula- tions shown. Similar results were obtained (but not shown here) for simulations with only four toroidal modes kept (n 1⁄4 1⁄20; 25; 50; 75􏰆). As a first step, no equilibrium electric field profile is used in these simu- lations despite the existence of equilibrium electric fields in experiment.
The density profile used in these simulations is shown in Fig. 6.
The temperature profiles follow the shape of the density profiles, i.e.,
g 1⁄4g 1⁄41 where g 􏰂L􏰁1=L􏰁1 and g 􏰂L􏰁1=L􏰁1, and the i e i T;i n e T;e n
maximum ion and electron temperatures are Ti;max 1⁄4 400 and Te;max 1⁄4 80 eV. Although the density and temperature profiles are assumed to be flux-functions in these simulations, true FRC equilib- rium profiles are surface-functions which may differ from flux- functions after accounting for toroidal rotation.
From the simulation parameters, ions are effectively represented as fully kinetic particles with ai 1⁄4 1 throughout the simulation domain such that the correction terms are negligible. For non- adiabatic electrons, however, ae is much more interesting and shown in Fig. 7: in the core along the region of lowest magnetic field, ae 􏰈 1 such that electrons are represented as fully kinetic particles; in the SOL, a 􏰄 0–0:5 such that electrons are represented as drift-kinetic particles.
A. Initial results
Consistent with previous work, instability only forms in the SOL. This instability has higher toroidal mode numbers (n1⁄41⁄260;70􏰆,
27, 082504-8