Full Particle Simulation of the Plasma Sheet
Using Adaptive Mesh Refinement (AMR) TechniqueK. Fujimoto and S. Machida
Department of Geophysics, Graduate School of Science, Kyoto University
It is widely believed that magnetic reconection plays an important role in the magnetospheric substorm and the solar flare. However, physical processes around the diffusion region are
not well understood. Recently, it has been suggested that multi-scale coupling process should be important in the reconnection triggering and the anomalous plasma heating and acceleration around the diffusion region. Now, it is necessary to conduct a self-consistent large-scale simulation including phenomena with various scales to describe multi-scale coupling.
However, a realization of such a simulation with an ordinary PIC technique is still difficult because electron-scale phenomena are very localized in ion-scale or MHD-scale system.
To overcome this difficulty, we have made a new 2-1/2 dimensional electromagnetic particle code with adaptive mesh refinement (AMR) technique. The AMR technique dynamicallysubdivides the cells that satisfy a certain refinement criterion, which is defined by local electron Debye length and electron flow velocity in the present case, and it is quite effective toachieve high-resolution simulations such as those around the X-type neutral line.
In this paper, we show our recent results of test simulations and preliminary results on the time developing of the Harris plasma sheet using the newly developed electromagnetic
full-particle code with AMR technique.

AMR Techique used in Electromagnetic Particle Code[Calculations of Electromagnetic Field][Data Structure]
In order to realize an effective high-resolution simulation, the AMR technique subdivides only cells thatsatisfy a refinement criterion and add data sets for finer cells (with higher level) onto the uniform basecells (with the lowest level) hierarchically. The data structure that we use to organize the hierarchical cellstructure is similar to the implemented fully threaded tree structure [Khokhlov, 1998]. Each cell is neededto access the parent, child, and neighboring cells. In our code, we group four cells togather that have asame parent cell in order to save memory
Oct 1
Oct 2
for pointers and facilitate parallel com122Level L 1puting. We call this group an oct (afteran octet in the case of a cubic cell).
Each cell has a pointer to the child octOctCh
iPr OctCh iPr(OctCh). Each oct has pointers to theiNb
iNb
iNb
parent cell (iPr), the parent cells of12
12
Level L+1
neghboring octs (iNb), and a representative particle belonging to the oct (inp).
Oct 3
Oct 4
inp
inp
inp
inp
Each particle has a pointer to any otherparticle in the same octs, which leads
Particles
to a beaded structure of particles.
pnb

Level LIn addition to the above refined cellsthat satisfy the refinement criterion,
we also split the buffer cells that are
adjacent to the refined cells.
Calculations of electromagnetic field
are performed first on the coarsest
cells, which are the root cells in thehierarchical structure. The solutions
are projected onto the buffer cells in
the next level cells as their boundary
conditions. Then the elecromagnetic
field is calculated on the cells withthis level. This process is recursively
repeated to reach the dynamic range
level. Eventually the solutions ineach cell are replaced by those in
the finest cells after proper smoothing operations to avoid the aliasing.

Level L+1

Refined cells

Buffer cells

Test Simulations

Simulation Results of the Plasma Sheet

Our code is checked against the Landau damping of the Langmuir
waves that propagate across the regions with different size of cells.
We find the Langmuir waves propagate properly across the boundaries,that is, neither remarkable noise nor wave reflection are shown inthe boundaries.

We have performed simulations of the plasma sheet using our AMR code. We adopt the Harris current sheet as an initialcondition. The initial parameters are as follows;Ex Propagationty

mi/me = 100, Ti,ps/Te,ps = 8.0, c/ve,th = 3.3, λ = 0.5 λi,64.1 x 10 particles for each species.
The initial magnetic fields are given by,Bx(x,z) = -B0 tanh(z/λ) - 2δ/λ cos(2πx/Lx)sech2(z/λ)tanh(z/λ)
2Bz(x,z) = 2πδ/Lxsin(2πx/Lx)sech (z/λ)δ = 0.02 B0λi.
10.30.08

System size is Lx x Lz = 20.5 λi x 41.0 λi.

x

The theoretical profiles deduced from the linear analysis of the Landau
resonance are plotted in red lines. The simulation result provides a good
fit to the theoretical profile.
Field Energy Ex
We also compare the results
1using the AMR technique with 0.5
λ=16∆B
those without the AMR. We
0.2
λ=12∆Bfind 'With AMR' run is in good 0.1
Without AMR
agreement with 'Without AMR' 0.05
With AMR
run, which indicates the appli- 0.02
λ=8∆B
0.01
cation of the AMR techniqueto the ordinary PIC code is
00.001
0.002
0.003
0.004
tωci
successful.

SummaryWe have developed a new electromagnetic full particle codewith the AMR technique, which is able to increase a spatial
resolution dynamically by subdividing cells that satisfy arefinement criterion.
The simulations of the Landau damping and Harris-type plasma
sheet indicate that the AMR technique is successfully applied tothe ordinary PIC code and high-resolution simulations areeffectively realized.
We are now parallelizing our code and will achieve a particle
simulation with the spacial dynamic range of 104 to 105, whichenables us to perform a global simulation including multi scales
from the electron scale to the MHD scale.

0.04

Z-6.1

-10.2
0.0

4.1

8.2

X12.3

16.4-0.10

20.5

t= 28.16

1.0Z

The most significant advantage of the AMR
technique is to effectively reduce the numberof cells in the simulation box. In the case ofplasma particle simulations, the AMR technique is also useful to reduce the number ofbackground particles, which is needed to suppress the numerical noise that arises where
the number of particles par cell is small.
Without AMRWith AMR

64> 2.1 x 10-0.06
-0.08

1.9

This condition must be satisfied in all cells.

Particles-0.02-0.04

λDe < ∆ <2λDeorVe > 1.8 VA

70.00

-2.0

The refinement criterion is simply defined by
the local electron Debye length and electron
flow velocity as

2.1 x 10

0.02

2.1

Grid spacing is ∆B = 0.32 λi in the base layerand ∆D = 0.02 λi in the dynamic range layer.
Time step is ∆t = 0.0016 ωci-1, which is thesame in all layers.

Cells

By

0.06

6.2

Boundary conditions are periodic boundaries
in x and conductive walls in z.

x0.10

5.0 x 10

4.1 x 1060.0-1.0-1.9
4.2-0.10

-0.08

1.9

-0.04-0.02

X
0.00

9.0
0.02

10.6
0.04

0.06

12.2
0.08

0.10

1.0

42

We confirmed the cells that satisfy the refinement criteria are dynamically and adequately
divided in our code . We realized effectively
high-resolution simulations of the plasma sheet.

-0.06

7.4

t= 28.16

Ratio

55.8

Z

0.0
-1.0
-1.9
4.2

-0.10

-0.08

5.8
-0.06

7.4
-0.04

-0.02

X
0.00

9.0
0.02

10.6
0.04

0.06

12.2
0.08

0.10