Page 5 - Electrostatic quasi-neutral formulation of global cross-separatrix particle simulation in field-reversed configuration geometry

Physics of Plasmas ARTICLE
d"@_@# fð~X;l;v;tÞ􏰂 þ~X􏰀r~þv_ f (22)
where the first term on the LHS is Debye shielding, the second and third terms on the LHS are electron and ion polarization densities, respectively, and the two terms on the RHS are the electron and ion densities. Note that the blended parameter modifies the polarization densities because the polarization densities are self-consistently con- tained within the RHS densities when the blended model approaches the fully kinetic representation.
B. Quasi-neutral blended Poisson equation
For the turbulence studies that is the focus of ANC, space-charge waves, such as the electron plasma wave, constrain the simulation time step but are not physically interesting. In gyrokinetics, the quasi- neutrality equation, sometimes called the gyrokinetic Poisson equa- tion, is solved instead of the full Poisson equation
dt k @t k@vk with the velocity given by
_^
and the acceleration given by
1 􏰉^ vk
~ 􏰂 v b þ v~ þ v~ þ v~ ; X k0 E rB Rc
(23)
􏰉􏰊􏰉􏰊 􏰅 2 2 2 2􏰎e/ dni dne
scitation.org/journal/php
perturbed distribution function is due to three effects: (1) the first term relates to the motion of the particles, most strongly the parallel direction; (2) the second term relates to the density gradient; (3) the third term relates to the temperature gradient. The current usage in ANC only includes the lowest order b0 as a first step. Understanding the effect of the next order b1, which is relatively large for ions, is in progress. It is also important to note the aj factor in the velocity, which allows for the weight equation to reach its correct form in the two limits: (1) in the fully kinetic limit, the velocity is just the normal particle velocity; (2) in the drift-kinetic limit, the velocity reduces to just the parallel velocity.
B. Gyrokinetic particle model
Due its lineage from GTC, ANC has also implemented the stan- dard gyrokinetic particle model. The dynamics of gyrokinetic particles are described by the Vlasov equation for guiding centers
The form of the Poisson equation, complementary to the blended mover,12 is
􏰅2 2 2 2 2 2􏰎e/ 􏰁 kder /þð1􏰁aeÞqer? þð1􏰁aiÞqi r? T
􏰉n þdn􏰊 􏰉n þdn􏰊 e i;0 i e;0 e
1⁄4􏰁; (26) n0 n0
^ 􏰊
v_ 􏰂􏰁 b þ r􏰅b 􏰀ðlrB þr/Þ: (24)
􏰁ð1􏰁aeÞqer?þð1􏰁aiÞqir? 1⁄4 􏰁 ; (27) Te n0 n0
km0X00 c
Again, splitting the distribution function into perturbed and equilibrium components, the electrostatic gyrokinetic weight equation can be found:
where the Debye shielding term is dropped and the equilibrium ion and electron densities are assumed to be equal due to quasi-neutrality. This removes the space-charge waves from the simulation model and is sufficient only if there is a particle species away from the fully kinetic limit to provide a finite polarization term. This equation is insufficient when this is not the case and is discussed in Sec. IV C.
In the case, where the gyrokinetic model, described in Sec. III, is used for the ions, the quasi-neutrality equation3 solved depends on the ion polarization term only. That is,
dw~ " rf 􏰉􏰉 vk 􏰊 􏰊11@f# 0^^0
1⁄4􏰁ð1􏰁w~Þ􏰁v~􏰀 þ bþ r􏰅b 􏰀r/ : dt Ef0X0mf@v
0c0k (25)
e/ 􏰇dni 􏰃 1⁄4
􏰇dne 􏰃
􏰁 ; (28)
3
static approximation. The linear trajectories of gyrokinetic and drift-
kinetic particles are equivalent. A difference between the gyrokinetic
and drift-kinetic models comes from the particle-to-grid interpolation
to calculate density. For gyrokinetic particles, the gyro-averaged density
is calculated by using a four-point average approach, based on a ring
16,17
in (28).
IV. SELF-CONSISTENT FIELDS
Aside from the particle model, the other component of the PIC method is the self-consistent calculation of the electrostatic (or electro- magnetic fields) fields arising from the particle distributions. Currently in ANC, only the electrostatic approximation is considered.
A. Blended Poisson equation
To obtain the electric fields used to advance particle velocities, the electrostatic potential is found through solving the Poisson equa- tion in ANC.
Phys. Plasmas 27, 082504 (2020); doi: 10.1063/5.0012439 Published under license by AIP Publishing
ThisisequivalenttotheformgivenbyHolodetal. intheelectro-
􏰁q2i r2?
where the h􏰀􏰀􏰀i represents the gyro-averaged (via gyro-sampling) quantities. This equation can be seen as (27) in the limit of relevant time-scales being slower than cyclotron oscillations (XceDt;XciD 􏰃 1 ! ae;ai 􏰄 1) and relevant length scales being larger than electron gyroradii (k2? q2e 􏰇 k2? q2i ).
C. Quasi-neutral blended Poisson equation with partial adiabatic response
In the FRC, there are regions of extremely low magnetic field strength. As such, there are locations where the blended particle model will effectively be a fully kinetic representation for all simulated particle species. In such case, the polarization terms on the LHS of (27) will decrease to zero due to the blended parameter. On the other hand, there is no guarantee for the densities on the RHS to cancel to zero as well. Without the Debye shielding term, trying to solve (27) for the electrostatic potential / in this limit is a source of numerical issues. To avoid this problem while maintaining quasi-neutrality, a portion of the electron response is assumed adiabatic:
27, 082504-5
For drift-kinetic particles, the ring is reduced to the guiding center, and no gyro-averaging is performed. A second difference between the two models arises in the inclusion of a polarization density for gyrokinetic particles in the field solver as seen
with radius equal to gyroradius.
Te n0 n0