Download Particle-in-Cell Plasma Simulation Model: Properties and Applications δ f R. D. Sydora

Advanced Methods for Space Simulations, edited by H. Usui and Y. Omura, pp. 47–60.
c TERRAPUB, Tokyo, 2007.
δf
Particle-in-Cell Plasma Simulation Model:
Properties and Applications
R. D. Sydora
Department of Physics, University of Alberta
Edmonton, Alberta, Canada
The δ f particle-in-cell (δ f -PIC) method can extend the range of conventional
PIC methods where the entire distribution function is represented by particles. By
separating the total distribution into a background and perturbed part, f = f o +
δ f , it is possible to formulate a PIC method where only the perturbed part of the
distribution is represented by particles. This leads to orders of magnitude in noise
reduction and allows for the study of weakly-driven instabilities where δ f << f o .
Several examples of phase space structure formation in perturbed Vlasov equilibria
such as plasma echoes, BGK modes and sideband instability are used to illustrate the
technique.
1
Introduction
The particle-in-cell (PIC) approach is widely used in the modeling of nonlinear processes in the collisionless dynamics of space plasmas (Birdsall and Langdon,
1985; Hockney and Eastwood, 1988). These include diverse phenomena such as
collisionless shocks, magnetic reconnection, beam plasma instabilities, mirror and
firehose instabilities, just to name a few. Parallel PIC algorithms based on domain
decomposition (Liewer and Decyk, 1989) have allowed for multidimensional simulations and resolution of a larger range of spatial and temporal scales. However, there
are a number of regimes where the standard PIC approach of representing the total
distribution function with particles leads to statistical uncertainty and masking of the
physical effects due to noise arising from particle discreteness.
In recent years the Vlasov simulation approach (Cheng and Knorr, 1976; Ghizzo
et al., 1990) has been advancing to the point where one and two dimensional electrostatic and magnetostatic simulations are possible. These are noiseless simulations
where the main numerical issue is the accuracy of the representation of small scales
on the phase space grid, which is problematic in a high dimensional phase space. The
PIC method has a spatial accuracy issue as well in addition to the statistical accuracy
of the discrete particle distribution function, which is problematic, particularly in the
tail of the distribution where there are very few particles.
In the δ f method (Denton and Kotschenreuther, 1995) each particle in the simulation is assigned an initial weight which evolves such that the collection of simulation
particles (markers) together with their weights, provides a proper Monte Carlo representation of δ f rather than f at that moment. This typically reduces the number
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R. D. Sydora
of particles used in a ‘full- f ’ simulation by a factor of δ f / f , which can be of order
10−3 − 10−5 . By using the Monte Carlo interpretation of the PIC algorithm, Aydemir
(1994), extended these ideas to include the δ f PIC method and demonstrated that it
is an example of the control variates method of variance reduction (Fishman, 1999).
It is now rigorously established that the δ f method offers a substantial improvement in accuracy over the standard PIC simulations where the particle distribution is
only weakly modified (Sydora, 1999). A systematic kinetic theory has been developed (Hu and Krommes, 1994) demonstrating that the δ f separation results in a numerical noise intensity reduction of order |δ f / f |2 . In this work several test problems
are considered which have been previously investigated with Vlasov simulations but
are difficult to treat with the full- f PIC simulation approach. The problem of weakly
perturbed Vlasov equilibria including echoes, BGK modes and sideband instabilities
are considered using the δ f -PIC simulation method.
2 δ f Basic Equations
In this section the basic equations for the δ f -PIC algorithm are presented using
the generalized weighting scheme (Hu and Krommes, 1994). The advantage of this
formulation is that it provides a consistent set of equations for the case of sources,
such as thermal and density gradients, and avoids mathematical difficulties in defining
the particle weight as a ratio of δ f and the numerically loaded distribution.
We begin with the collisionless Vlasov equation which describes the evolution of
the kinetic distribution
function,
f (
x , v, t), in the presence of electric and magnetic
q
v
×
B
forces, F = m E + c given as
∂f
∂f
∂f
+ v ·
+ F ·
=0
∂t
∂ x
∂ v
(1)
The total distribution function evolves along the characteristics
d x
dt
d v
dt
= v
= F
(2)
which are also the particle trajectories, such that
df
=0
dt
(3)
where dtd is the total derivative evaluated along the characteristics. We may now split
the distribution function and the force into background and perturbation pieces
f = fo + δ f
F = Fo + δ F
(4)
and upon substitution into Eq. (1) leads to the equation
∂δ f
∂f ∂δ f
∂δ f
∂ fo
∂ fo o
+ v ·
+ Fo + δ F ·
=−
+ v ·
+ Fo + δ F ·
(5)
∂t
∂ x
∂ v
∂t
∂ x
∂ v
δ f Particle-in-Cell Plasma Simulation Model
We can express the right hand side of this equation as
∂ fo
∂ fo
o + δ F · ∂ fo = f o ∂ ln( fo ) + v · ∂(ln( fo ) + Fo + δ F ·
+
v
·
+
F
∂t
∂ x
∂ v
∂t
∂ x
= f o d lndt fo
49
∂ ln( f o )
∂ v
(6)
and therefore
d ln( f o )
dδ f
= − fo
dt
dt
Now a dual set of weights is introduced such that
f o = pg
δ f = wg
(7)
(8)
where p and w are marker weights for each piece of the distribution function and g
represents the marker distribution function. From the total f evolution equation
df
dt
= d(fodt+δ f ) = d( pg+wg)
dt
=0
= g ddtp + dw
dt
(9)
and
dp
dw
=−
dt
dt
which also leads to the constant of motion p(t) + w(t) = constant.
The nonlinear characteristics for the δ f evolution are
d x
= v
dt
d v
o + δ F
=
F
dt
fo )
dw
= − p d ln(
dt
dt
d ln( f o )
dp
= p dt
dt
(10)
(11)
The last 2 equations can be combined into a single weight evolution equation if we
use the condition w(0) = 0 (or δ f (t = 0) = 0) and for a marker distribution matching
f o (or g = f o ) we have p(0) = 1. This gives
p(t) + w(t) = constant = p(0) + w(0) = 1
(12)
leading to the combined weight evolution
dw
d ln( f o )
= −(1 − w)
dt
dt
(13)
For the electrostatic case, considered in this work, the force consists of the external
and self-consistent electric fields
q Fo + δ F =
E o + δ E
(14)
m
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R. D. Sydora
where δ E is obtained from
· δ E = 4π s ρ̃s
∇
(15)
and s is a sum over the charged particle species, s, and ρ̃s = qs δ f d v.
It should be noted that the electric force and distribution function splitting is quite
arbitrary and one may use nonlinear kinetic equilibria as an initial condition for the
simulation.
3
Algorithm and Computational Cycle
The time integration steps for the δ f − P I C equations, along with the selfconsistent electrostatic fields is presented here. The finite-difference form of the particle equations of motion can be expressed in the time-centered, second order accurate,
leapfrog scheme (Birdsall and Langdon, 1985) as
n+1/2
n−1/2
xi
= xi
+ t vin
n+1/2
vin+1 = vin + t Fi
(16)
and the time-centered weight evolution equation
n+1/2
wi
n−1/2
= wi
− t 1 − win Rin x, v, F
(17)
where
∂ ln( f )
∂ ln( f o ) o
R x, v, F ≡ v ·
+ Fo + δ F ·
(18)
∂ x
∂ v
is evaluated along the fully nonlinear trajectory (or characteristic). We assume that
f o changes on a slower time scale compared with δ f and we neglect its time rate of
change.
The computational cycle can proceed as a 2-step method with the following steps
n−1/2
• Initially we have: xi
n−1/2
, vin , wi
n−1/2
• Compute ρ n−1/2 from xn−1/2 , wi
(15)
n−1/2
and solve for Fi
using Eqs. (14) and
∗n+1/2
• Advance weights (predictor step): wi
xn−1/2 , vn , F n−1/2
n+1/2
• Update the positions xi
n−1/2
= xi
n−1/2
n−1/2
−t 1 − wi
Ri
+ t vin
∗n+1/2
• Compute ρ ∗n+1/2 from xn+1/2 , wi
and (15)
n−1/2
= wi
∗n+1/2
and solve for Fi
using Eqs. (14)
∗n+1/2
• Update the velocities vin+1 = vin + t Fi
∗n+1/2
• Obtain Ri
used.
•
δ f Particle-in-Cell Plasma Simulation Model
n+1/2
xn+1/2 , vi
51
≈ vin+1 + vin /2 is
, F ∗n+1/2 where vi
n+1/2
n+1/2
n−1/2
Advance weights(corrector step): wi
= wi
−t
∗n+1/2
n−1/2
n
/2 is used.
where wi ≈ wi
+ wi
1−
win
∗n+1/2
Ri
The perturbed charge density is obtained from the δ f weights, wi , using
ρ̃ =
qi wi S (
x − xi )
n−1/2
+Ri
2
(19)
i
where S is the spatial shape factor associated with the phase space volume element of
the marker particles. The perturbed charge density is formed at discrete grid points
using interpolation, and then Eq. (15) is solved using a Fast Fourier Transform (FFT)
method to obtain the self-consistent perturbed electric field at the grid points. This
electric field is used to determine the electric force at the particle position, also by
interpolation. For both charge density and force interpolations, the second order
quadratic method is used (Birdsall and Langdon, 1985).
4
Conservation Relations and Statistical Properties
The δ f -PIC simulation model has several important conserved quantities which
can be used to monitor the accuracy of the calculations. A fluctuation-dissipation
theory for thermal equilibrium can be developed to obtain the electrostatic energy
partition per mode with discrete marker particles as the source of excitations (Hu and
Krommes, 1994). An inherent feature of the δ f -PIC model is the diffusion of the
particle weights in phase space which can lead to a buildup of noise in the moments
over time. A summary of the theoretical description of this diffusion is presented in
this section.
4.1 Conserved quantitites
In the absence of particle sources and sinks the particle number should be conserved and the zeroth velocity moment of Eq. (7) gives
∂
δ f d x = 0
(20)
∂t
or
N
wi (t) = 0
(21)
i=1
if the sum of the weights is nearly zero initially. The first velocity moment of Eq. (1)
leads to total momentum conservation
α
mα
N
i=1
vαi wαi (t) = 0
(22)
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R. D. Sydora
where α=electron and ion species. The second velocity moment gives the total energy conservation relation
N
mα α
2
2
vαi
wαi (t) +
i=1
1
8π
2 d x = 0
| E|
(23)
if the particle weights and fields are zero initially.
4.2 Equilibrium thermal fluctuations
A plasma sustains fluctuations of various modes of oscillation in thermal equilibrium. For the case of unmagnetized electrostatic oscillations in a collisionless
plasma, the full- f PIC simulation model gives an electric field energy equipartition
per wavenumber as
E2L
T /2
< k >=
(24)
8π
1 + k 2 λ2D S 2 (k)
where S(k) is the finite-sized particle shape factor, taken to be Gaussian, S(k) =
2 2
ek a /2 .
For the δ f -PIC model the fluctuation-dissipation theory gives a similar equipartition
E2L
(T /2)w̄ 2
< k >=
(25)
8π
1 + k 2 λ2D S 2 (k)
where w̄2 is the spatially averaged |δ f / f |2 .
4.3 Phase space marker diffusion
In the δ f -PIC method the weight, w, of the marker particles obeys an evolution
equation of the form
dwi
(26)
= Ri (t)
dt
where
∂
∂
Ri (t) = vi ·
ln( f o ) + Fi ·
ln( f o )
(27)
∂ xi
∂ vi
This equation can be formally integrated to give
t
wi (t) =
dt Ri (t )
(28)
0
One can form the positive definite quantity (Krommes and Hu, 1994)
2
x , v, t) x
δ f (
1
1
2
S(t) =
wi (t)
d v
→
2
f o (
2 i
v , t)
(29)
which is related to the fluctuation intensity. This has a time evolution determined by
the two-time correlation function
t
t
2 wi (t) =
dt
dt Ri (t )Ri (t )
(30)
0
0
δ f Particle-in-Cell Plasma Simulation Model
53
and exhibits a temporal increase as the phase space markers diffuse, since Ri2 ∼ |δ Fi |2
and |
v − v(t = 0)|2 . This temporal increase of the fluctuation intensity grows approximately linearly in time. It is a weak noise source inherent in the model; however, with
a sufficiently large number of markers it does not strongly influence the signal to noise
ratio.
5
Dynamics of Perturbed Vlasov Equilibria
In this section several types of linear and nonlinear phenomena associated with
weakly perturbed Vlasov equilibria are considered. These include temporal plasma
echoes, BGK mode formation and trapped particle or sideband instabilities.
5.1 Temporal echoes
One stringent test of kinetic plasma models is the numerical simulation of plasma
echoes. The temporal echo is formed by the nonlinear beating between two electric
field pulses separated by a time, τ . The distribution function contains small scale
correlations until the time of the echo and it is important for the nonlinear kinetic
simulation to retain this information throughout the calculation. There are very few
studies of this phenomenon using standard full- f PIC models due to the inherently
high noise level (Baker and Oldam, 1969). Vlasov simulations have successfully
demonstrated plasma echo formation (Brackbill, 1972; Manfredi and Feix, 1996) and
we show that the δ f -PIC model is also capable of accurately simulating them as well.
For the temporal plasma echo simulation we consider a 1D simulation configuration, unmagnetized, with electrons only; the ions are considered a charge neutralizing
background. We excite two pulses using an external electric field of the form
E(x, t) = E 1 cos(k1 x)δ(t) + E 2 cos(k2 x)δ(t − τ )
(31)
The two pulses both excite density modulations which exponentially decay at approximately the Landau damping rate. After a time longer than the inverse Landau
damping rate of the first two pulses, a third echo wave appears as a density modulation with wavenumber kecho = k2 − k1 . The echo time is given as (O’Neil and Gould,
1968)
k2
τecho =
τ
(32)
k2 − k1
For the δ f -PIC simulation we chose physical parameters: k1 λ D = 0.483, k2 λ D =
2
0.966, τ = 20ω−1
pe , and E 1 = E 2 = 0.18 where E is normalized as eE/m e ω pe and
= 1.64λ D . N=2560000 particles were used.
The results of the plasma wave echo excitation are presented in Fig. 1 which illustrates the electrostatic energy evolution with and without the second pulse. Without
the second pulse the initial perturbation damps exponentially at roughly the Landau
damping rate predicted by theory (Nicholson, 1983). When the second pulse is applied at ω pe τ = 20 it also damps away until ω pe τ ≈ 40 where the echo fluctuation
appears. This is consistent with the value of τecho = 40ω−1
pe predicted by Eq. (32).
The overall shape of the echo and amplitude compare well with a kinetic Eulerian
54
R. D. Sydora
Fig. 1. Total electrostatic energy (arbitrary units) versus time. Green corresponds to single pulse at t=0
and red is with pulses at t=0 and t=20.
Fig. 2. Total electrostatic energy (arbitrary units) versus time. Red curve is for N=2560000 particles and
the green curve is for 25600 particles.
code which solves the Vlasov equation by direct discretization of the phase space
(Manfredi and Feix, 1996).
The plasma echo can be masked by collective effects and collisions. In the δ f -PIC
simulation model the echo can be also strongly affected by an insufficient number of
phase space marker particles. This is illustrated in Fig. 2 where the results of using
2560000 particles versus 25600 particles are shown. Clearly in the latter case, the
δ f Particle-in-Cell Plasma Simulation Model
55
overal shape of the echo amplitude is modified due to the poor statistical sampling
of the perturbed distribution function. For N=256000 particles the echo pulse shape
matched well with the N=2560000 case.
5.2 BGK modes
One class of solutions to the Vlasov-Poisson system with zero electric field is the
spatially homogeneous distribution, F(v), known as the Vlasov equilibria. Linear
theory well describes the time evolution of the exact solutions to the Vlasov-Poisson
system when the initial condition is chosen sufficiently close to F(v). Linear theory,
however, does not account for particle trapping effects and the long-time behavior of
Landau damping leads to a removal of the decay. In fact, recent numerical simulations of the 1D Vlasov-Poisson system indicate that the time-asymptotic beahavior
of small perturbations to Vlasov equilibria can be characterized by the presence of
periodic traveling structures in phase space and electric fields that exhibit standingwave oscillations with constant amplitude (Demeio and Zweifel, 1990). It has been
shown that these structures can be described by some combination of BGK modes,
which are the nonlinear undamped traveling waves discovered by Bernstein, Greene
and Kruskal (BGK) in 1957. In 1993 Buchanan and Dorning introduced a nonlinear
superposition principle for two small amplitude BGK modes of arbitrary phase velocity. These modes have been simulated using Vlasov codes and demonstrated that the
time-asymptotic states are well described by a superposition of BGK modes traveling
at wave speeds given by the linear dispersion relation. In this section it is shown that
the δ f -PIC simulation is also capable of generating these traveling BGK modes and
resolving the fine scale phase space structure.
The two counterpropagating BGK modes can be obtained from the stationary
solution to the Vlasov-Poisson system and by applying two Galilean transformations
with velocities V and -V which gives distributions f 1 (x − V t, v − V ) and f 2 (x +
V t, v + V ) and electric fields E(x − V t) and E(x + V t). The single particle energies
in the two moving frames are K 1,2 = (v ± V )2 /2 − φ(x ± V t) and the distributions
can be expressed in terms of these energies as f 1 (x − V t, v − V ) = g(K 1 ) and
f 2 (x + V t, v + V ) = g(K 2 ). The electric potential divides the electrons into three
populations consisting of trapped particles, untrapped particles with velocities larger
than the BGK phase velocity V and untrapped particles smaller than the BGK phase
velocity.
We initialize the system with a perturbed Vlasov equilibrium of the form
f o = F(v)(1 + cos(kx))
(33)
where F(v) = (m e /2π Te )1/2 exp −m e v 2 /2Te . The other parameters are: particle
number N=256000, system length L = 12.6λ D , grid size = 0.76, kλ D = 0.5
and = 0.2. In Fig. 3 the total electrostatic energy versus time is shown and we
can observe the formation of the undamped mode asymptotically in time. Figure 4
displays the phase space taken at the ω pe t = 200 and the existence of two phase space
vortices traveling in opposite directions is clearly visible. The particles in the phase
space around the origin are not displayed. These results compare well with Vlasov
simulations (Demeio and Zweifel, 1990).
56
R. D. Sydora
Fig. 3. Total electrostatic energy (arbitrary units) versus time for undamped plasma wave formation.
Fig. 4. Marker particle phase space illustrating counter-propagating BGK modes. Markers in the middle
part of the distribution function not displayed.
5.3
Sideband instability
For the investigation of BGK modes in the previous section, wavelengths which
were on the order of the simulation box length were considered. When the system
length is increased it is possible to excite longer wavelength perturbations via the
sideband instability mechanism (Kruer and Dawson, 1970; Shoucri, 1978) which can
dissipate the coherent, undamped BGK mode.
δ f Particle-in-Cell Plasma Simulation Model
57
Fig. 5. Electrostatic energy versus time for three modes. Green curve corresponds to the main excited
BGK mode (km = 0.5), red curve is the lower sideband mode (kl = 0.4) and the blue curve is the upper
sideband mode (ku = 0.6).
Fig. 6. Growth rate (normalized to ω pe ) versus distribution function perturbation strength.
The sideband instability is due to a nonlinear resonance between the frequency
of a plasma wave and the electron bounce frequency ω B = (eE o k/m e )1/2 . The presence of a finite-amplitude traveling electron plasma wave sets up a spatially periodic
potential in the wave frame. The trapping of electrons in the wave trough produces
a bounce frequency resonance. The coupling of the finite-amplitude wave with the
58
R. D. Sydora
bounce frequency produces upper and lower sidebands about the traveling wave frequency. It is clear that the growth rate should be proportional to the number of particles trapped in the main wave which can be related the the main wave amplitude. As
a crude estimate, we can relate the maximum kinetic energy of a trapped particle in
the wave frame to the potential drop in a wave of amplitude E o
mv 2 /2 2eE o /k
(34)
Therefore, the number of trapped particles per wavelength around the phase velocity
is
NTrapped ≈ 2v f o (v ph ) = 4 (ω B /k) f o v ph ∝ E o1/2
(35)
1/2
The scaling of the sideband growth is then γ ∝ NTrapped ∝ E o . This scaling behaviour was approximately observed in particle simulations performed by Kruer and
Dawson in 1970; however, due to high noise levels a very large amplitude wave was
used. The signal-to-noise ratio for the sideband growth rate is substantially improved
using the δ f -PIC method and much lower and realistic wave amplitudes can be used.
We consider the following parameters: particle number N=256000, system size
L = 64λ D , grid spacing = λ D , = 0.18 and main perturbation wavenumber
km λ D = 0.5. Figure 5 shows the time evolution of the electrostatic energy for the
main perturbation wavenumber and the lower (kl = 0.4) and upper (ku = 0.6) sideband modes. The growth of the upper and lower sidebands are clearly visible and
after t=100 they reduce the amplitude of the primary mode. The lower sideband is
observed to have a slightly larger growth rate because it has a phase velocity further
away from the bulk of the electron distribution function. The upper side band is closer
to the thermal velocity and is more heavily damped. Figure 6 illustrates the scaling of
the growth rate with the distribution function perturbation amplitude. The growth rate
1/2
scaling follows an approximate E o scaling as predicted by the theoretical analysis.
6
Summary
In this paper an efficient and accurate technique for the study of the linear and
nonlinear evolution of kinetic microinstabilities in collisionless plasmas has been presented. This is achieved by following the evolution of the nonlinear perturbed plasma
distribution, formulated in a Lagrangian approach, which is suitable for particle simulation. An algorithm for the electrostatic simulation was presented using an explicit
time-centered form of the finite difference equations. For strongly unstable plasmas
with γ ≥ ω and where δ f / f ∼ 1, the conventional or total-f simulation approach is
viable with large numbers of particles. In the opposite limit where instabilities are
weak and saturation amplitudes low, the δ f -PIC method can accurately simulate the
nonlinear dynamics. Several examples were used to illustrate this: temporal plasma
echo formation, dynamics of undamped BGK plasma modes and the sideband instability from particle trapping.
The total-f method is particularly difficult for investigating weak instabilities
in multi-dimensional plasmas. Even if billions of particles can be used to repre-
δ f Particle-in-Cell Plasma Simulation Model
59
sent the total distribution the integration time scales are prohibatively long to follow
the growth and long-time saturation dynamics. Therefore, only a limited parameter
regime can be investigated. Finally, the discrete nature of the δ f -PIC method allows
for straightforward parallelization of the algorithm, just as in the conventional PIC
method. Hence, a large number of discrete markers can be used to cover the various
regions of phase space and detailed particle convergence studies can be carried out.
Acknowledgments. This work is supported by a research grant from the Natural Sciences and
Engineering Research Council (NSERC) of Canada.
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