Electromagnetic Particle-In-Cell Model With Adaptive Mesh Refinement

D21-0045-08

1
2
Keizo Fujimoto and Richard Sydora(E-mail: keizo@stelab.nagoya-u.ac.jp)

1. Solar-Terrestrial Environment Laboratory, Nagoya University, Nagoya, Aichi 464-8601, Japan
2. Department of Physics, University of Alberta, Edmonton, Alberta T6G 2G7, Canada

1. Introduction

2. Data Structure

Local kinetic processes in space and laboratory plasmas are, in many cases, considered to have a significant impact on global-scale energy transport and conversion processes. Magnetic reconnection is a typical case in which microscopic processes in thediffusion region, formed around the magnetic X-line, can control the MHD-scale dynamics. Generally, it is very difficult to describe in numerical approach such a phenomenon that includes a number of physical scales which can couple with each other.
The adaptive mesh refinement (AMR) technique is one of the promising methods toovercome the difficulties. It subdivides the computational cells locally in space anddynamically in time. In this study, we apply the AMR to the electromagnetic particlein-cell (EM-PIC) model for both the 2D and 3D systems, and successfully achieve
efficient high-resolution simulations on the nonlinear evolution of the plasma sheet.

The AMR technique subdivides only cells that satisfy some refinement
criteria and enhances the local spatial resolution. The data sets for thechild cells are added onto the parent cell and develop a hierarchical treestructure.

1/2
r=/N
∆
∆L p

v1,2,3,4 = v0
m1,2,3,4 = m0/4
q1,2,3,4 = q0/4
0



0∆r



Bx(z) = -B0 tanh(z/λ) ; n(z) = nps sech2(z/λ) + nb tanh2(z/λ)
mi/me = 100; Ti,ps/Te,ps = 5.0; c/VA = 20
λ = 0.5 λi; nb = 0.044 nps∆L >2.0 λDe or Vey > 2.0 VA

−vqv
nn
=E (x ) +∆tmxn+1 − xn= v n+1/ 2∆tv

n+1/ 2+v
2n−1/ 2+Bn+1/ 2=c ∇×− jT/ε0 ;∆t2n
B n+1 − B n
E n+1+ET= −∇ × T∆t22

Bn+1E n+1 = E n+1+ELTn

inp

inp

inp

AMR
No
Yes
Yes
Yes
Yes

Particle splitting
NoNo
Yes
Yes
Yes

System size
15.4 ×15.415.4×15.4
15.4 ×15.4
15.4 ×0.24× 15.4
15.4 ×1.92× 15.4

Nct
1.0 ×106
6.7 ×104
7.1 ×104
5.0 ×105
3.3 ×106Npt
1.9×107
1.9×107
4.9×106
4.8×107
5.4×108

Simulation Time

Run1Run2
Run3
Run4
Run5

0.6
0.4
0.2

2.7
Elapsed walltime
Number of cells
Number of particles

0.8
0.6

0.59

1.6

0.4

1.1
0.25

0.2
0510tωci

15

200

2.1

0.18

Run1

0.06

0.07

Run2Run30.5
0.0

We have successfully developed a new EM-PIC code with the AMR both in the 2D and 3D systems. In order to control the number of particles per cell, the present code also performs the particle splitting and coalescence.

E L = −∇φ; ∇2 φ= −ρ/ε0n+1

Oct

6. SummaryElectric & Magnetic FieldnE n+1−ETT

iNb

1Jy/enpsVA

× B n (xn )

iNb

pNb

0-3
0 0.20.40.6

n−1/ 2

inp-2Particle Motions

OctChiPr

Oct

0.8

L= 7
L= 8
L= 9
L=100

4. Basic Equations

Cell

Cell Cell Cell Cell

Reconnection Electric Field-1n+1/ 2iNb

Run#Run1Run2Run3Run4Run5Refinement Criteria1OctCh
iPr

Cell

Simulation information.

2

5. Evolution of the Plasma Sheet

30Cell(b)Cellinp (to a particle belongingto Oct)
Level L+1
Cell pointers
OctCh (to child Oct)
Particle pointers
ParticlespNb (to a neighboring particle)



Oct

Elapsed walltime (103 min.)(b) Splitting in 3D(c) Coalescence
1/3
m1 = m2; q1 = q2
r=/N
L
p∆
∆
x0 = (x1 + x2)/2
v1, ... ,8 = v0
v0 = (v1 + v2)/2
m1, ... ,8 = m0/8; q1, ... ,8 = q0/8
m0 = m1 + m2∆rq0 = q1 + q2
Level LiNb (to parent cell of theneighboring Oct)

-Ey/VAB0(a) Splitting in 2D

Oct

Normalized value

The number of particles per cell is controlled by splitting particles in fine cells andcoalescing them in coarse cells.

Oct pointers
iPr (to parent cell)

Cell Level(a)3. Particle Splitting & Coalescence

The hierarchical cell structure is supported by a set of pointers.

n+1nE−En+1/ 2LjT= j n+1/ 2 + ε0 L

∆tWe have demonstrated that the AMR and particle splitting-coalescence techniques, combinedwith the EM-PIC code, enable efficient high-resolution simulations of the plasma sheet, andcan be a promising method for studying physical phenomena which include a number of physical scales that can be coupled with each other.
Ref. K. Fujimoto, and S. Machida, J. Comput. Phys., 214 (2006) 550.
K. Fujimoto, and R. Sydora, Comput. Phys. Comm., 178 (2008) 915.