Page 6 - Electrostatic quasi-neutral formulation of global cross-separatrix particle simulation in field-reversed configuration geometry

Physics of Plasmas ARTICLE
e/ Te
x < kkvcut􏰁off ; (30) where the condition is based on kk of the mode structure and the
selected cutoff velocity. For the turbulence studies of interest, this cut- off frequency is unlikely to affect the possible wave-particle resonances, and variation of the assumed cutoff velocity will be used to confirm convergence on resultant physics. With this partial adiabatic assump- tion and assuming the equilibrium particle densities to be equal, the form of the quasi-neutrality equation changes to
ðv > vcut􏰁off Þ 1⁄4 cutoff frequency based on
dfe fe
With (1) the gyro-kinetic ion model and drift-kinetic electron model; (2) flat temperatures but non-uniform density; and (3) a simple straight uniform magnetic field, the dispersion of the drift wave is
f0: (29) The assumption of a cutoff velocity is essentially an assumption of a
"􏰉1 þ b􏰊 b
0
2Te 2kkqe
􏰋􏰂
x=Xc;i ! pffiffi
􏰅
􏰁 ð1􏰁aeÞq2er2? þð1􏰁aiÞq2i r2? 􏰁cA
􏰉dn􏰊 􏰉dn􏰊 ie
2kk qi
expð􏰁bÞ is the exponentially scaled modified Bessel function of the
􏰁􏰀 􏰎 e/ kk 61⁄4 0
Te
where
x=Xc;i
Z pffiffi is the plasma dispersion function, K0 ðbÞ 􏰂 I0 ðbÞ
0 1⁄4 1 􏰁 1
jn 􏰁k?qi pffiffi
K0ðbÞZ jn Ti
2kkqi
x=Xc;i ! pffiffi
2kkqe
(35)
þ K0ðbÞZ 2
x=Xc;i !
2kkqi 2kk Te
Z
1 Ti 0 x=Xc;i !!# 􏰁Zpffiffi ;
2kk
pffiffi þ k?qe pffiffi
scitation.org/journal/php
1⁄4􏰁; (31) n0 n0
first kind, b 􏰂 k2?q2i , and jn 􏰂 􏰁rn=n is the density gradient inverse scale length.
In Fig. 2, the numerical solutions of the analytic theory (35) is plotted as the dashed line, while the normalized 2D spectral power shows the ANC simulation results. Three models in ANC are tested and shown: (a) gyrokinetic ions with drift-kinetic electrons, (b) gyroki- netic ions with adiabatic electron response, and (c) blended drift- Lorentz ions with adiabatic electron response. The simulation results agree with the theoretical dispersion relation at long wavelengths due to better numerical resolution and because shorter wavelengths (for the blended drift-Lorentz case (c), jkk qi j > 4:2 􏰅 10􏰁5 ) are numeri- cally smoothed. This verifies the capability to simulate drift-waves in ANC.
B. Ion acoustic wave
Using a simple straight magnetic field without pressure gradients, the ion-acoustic wave dispersion is found from simulation and com- pared to analytic theory.
With (1) the fully kinetic ion model with an adiabatic electron response; (2) flat temperatures and density; (3) a simple straight uni- form magnetic field; and (4) the assumption of k? 1⁄4 0, the dispersion of the ion acoustic wave is
where cA and cNA are the fractions of the electrons assumed to be adia- batic and non-adiabatic, calculated by
cNA 1⁄4 erfðvcut􏰁off =vthÞ; cA 1⁄41􏰁cNA;
(32)
where vth is the local thermal velocity. Note that this equation is only valid for the component of electrostatic potential with finite kk due to the adiabatic response. A second quasi-neutrality equation without the adiabatic response must now be solved for the component of electro- static potential with kk 1⁄4 0,
􏰇􏰅 2 2 2 2 􏰎􏰃 e/􏰁kk 1⁄4 0􏰀 􏰁 ð1􏰁aeÞqer?þð1􏰁aiÞqir? Te
􏰇􏰉dni 􏰊 􏰉dne 􏰊􏰃
1⁄4􏰁; (33)
where h􏰀 􏰀 􏰀i denotes a field-line average. Because the magnetic field is not uniformly small along the field-lines in the problematic null mag- netic field regions of the FRC, this field-line average is sufficient to ensure that the LHS remain finite. The scenario that the LHS falls to zero due to small simulation time-steps requires the Debye shielding term but is outside of the interest of this model. The total electrostatic potential is then found by combining the results of the two equations:
n0 n0
􏰁􏰀􏰁􏰀 x=Xc;i
where Z pffiffi kkk 2kkqi
is the plasma dispersion function.
/ðkÞ1⁄4/k 1⁄40 þ/k 61⁄40: (34) V. BENCHMARKS
Several benchmarks against analytic theory have been performed with ANC. These benchmarks use a simple uniform magnetic equilib- rium for analytic simplicity. The routines and functions called within ANC are almost exactly the same set as in more complicated FRC equilibria; thus, these benchmarks provide some measure of confi- dence in the implementation of the physics outlined in Secs. I–IV.
A. Drift-wave
Using a simple straight magnetic field with varying density, the drift-wave dispersion is found from simulation and compared to ana- lytic theory.
Phys. Plasmas 27, 082504 (2020); doi: 10.1063/5.0012439 Published under license by AIP Publishing
In Fig. 3, the numerical solution of the analytic theory (36) is plotted as dashed lines, while the normalized 2D spectral power shows the ANC simulation results. The simulation results using the blended ion and adiabatic electron response agree with the theoretical disper- sion relation. This verifies the capability to properly simulate waves in the parallel direction in ANC.
C. Ion Bernstein wave
Using a simple straight magnetic field without pressure gradients, the ion Bernstein wave dispersion is found from simulation and com- pared to analytic theory.
With (1) the fully kinetic ion model with an adiabatic electron response; (2) flat temperatures and density; (3) a simple straight
27, 082504-6
􏰋􏰂
!
01⁄41􏰁 Z pffiffi ; (36)
1Te 0 x=Xc;i 2Ti 2kkqi