PIC Simulation of Magnetic Reconnection
with Adaptive Mesh Refinement
Keizo Fujimoto
National Astronomical Observatory of Japan

Overview Introduction
 Particle-in-Cell (PIC) model with adaptive meshrefinement (AMR) Wave activities in the reconnection region
Electrostatic solitary waves (ESWs)
Whistler waves
EM waves around the x-line
 Summary

ISSS11@Taiwan

2Magnetic Reconnection in Space
Auroral Substorms[NASA]

Solar Flares
ISSS11@Taiwan

3

Multi-Scale Nature of Reconnection
1km

Dissipation

λDe

103km

10km

Full PICTopology

Hall effectρe, λe(Particle-In-Cell)

1011km105km

ρ i, λ i∬

LMHD(Magnetohydrodynamics)

ISSS11@Taiwan

LcCoulomb mean
free path

βi ~ 1[NASA]
4

Restrictions of Explicit PIC ModelCell size∆x ≲ 3λDe
Num of particles per cellNp ≳ 102
Memory requirement per cell
Field (ns, Js, E, B)
14×4 Byte<<Particle (xs, vs)(12×4 Byte)×102

ISSS11@Taiwan

5

AMR-PIC Model [Fujimoto, JCP, 2011](Adaptive Mesh Refinement – Particle-in-Cell)∆x ≲ 3λDe∝ ne-1/2
λDe,lobe ~ 6×103 m
λDe,ps ~ 3×102 m
Removing
unnecessary cells.

Electron scaleParticle splitting-coalescence

ISSS11@Taiwan

6

AMR-PIC Simulations

ISSS11@Taiwan

7

Data Structure
Similar to a fully
threaded tree
(FTT) structure(Khokhlov, 1998).

[Fujimoto & Machida, JCP, 2006]

ISSS11@Taiwan

8Basic Equations
Superparticle
S: Shape function
Superparticles (Buneman-Boris method)

EM Field (Yee-Buneman scheme)

ChargeConservation
Method[Villasenor &Buneman, 1992]〇 Local operations
Facilitate parallel computing
× No numerical damping for any wave numbers!
ISSS11@Taiwan

9Radiation of EM Waves

ESM=fSM(E)
BSM=fSM(B)

ISSS11@Taiwan

10Load Balancing* Block = Decomposition domain

Example using 8 nodes
Fixed block case

Adaptive block case

ISSS11@Taiwan

11Adaptive Block Technique [Fujimoto, JCP, 2011]
Base-level cells in the entire domain are sorted in an
appropriate order: That is similar to Morton order,
 So that the block surface is as
small as possible,
 Especially in the central current
sheet, the surface must be small.

ISSS11@Taiwan

12Performance of the AMR-PIC Model[Fujimoto, JCP, 2011]

Fujitsu FX1@Nagaya Univ.

NodeＣＰＵ(4 cores)ＣＰＵ(4 cores)

Memory（32G）

Memory（32G）
Parallel fraction:

ＣＰＵ(4 cores)

ＣＰＵ(4 cores)

Memory（32G）

Memory（32G）

Load balancing

99.8%
Removing
Poisson Eq.

ISSS11@Taiwan

13

Examples of the AMR-PIC simulation:
Wave activities in the reconnection region

ISSS11@Taiwan

14

Electrostatic Solitary Wave (ESW)
ESWs are often observed in thePSBL in the Earth magnetotail(Matsumoto et al., 1994).

Magnetic reconnection isclosely related to the generationof the ESWs (Deng et al, 2004;
Cattell et al., 2005; Viberg et al.,2013).

[Matsumoto et al., GRL, 1994]ISSS11@Taiwan

15

Electrostatic Solitary Wave (ESW)[Fujimoto & Machida, JGR, 2006]

ISSS11@Taiwan

16

Electrostatic Solitary Wave (ESW)[Fujimoto & Machida, JGR, 2006]

Wave spectrum +Linear dispersion

Electron heating

B

Electron 2-stream instability
rather than Buneman.

Consistent withobservations.
[Shinohara et al., 1998]
ISSS11@Taiwan

17

Whistler Waves
Cluster observations[Wei et al., JGR, 2007]

High frequency waves
with 0.1ωce ~ ωce.
Right-hand polarizartion

0.1ωce ~ ωceRight-handpolarization

ISSS11@Taiwan

18

Whistler Waves[Fujimoto & Sydora, GRL, 2008]

Transverse E||ISSS11@Taiwan

19

Whistler Waves[Fujimoto & Sydora, GRL, 2008]

TimeB
ξB
ISSS11@Taiwan

20Wave Activities in Separatrix Region
ExEy

ISSS11@Taiwan

21

EM Waves near the X-line
Magnetotail

Lab. Exp.

ωLH

ωLH
ωci
[Zhou et al, JGR, 2009]
ISSS11@Taiwan[Ji et al, PRL, 2004]
22

EM Waves in 3D magnetic reconnection[Fujimoto & Sydora, PRL, 2012] Surface: |J|, Line: Field line
# of particles ~ 1011
Color on the surface: Ey, Cut plane：Jy

ISSS11@Taiwan

23

Dissipation Mechanism [Fujimoto & Sydora, PRL, 2012]
3D reconnection

2D reconnection

Anomalous effects
ISSS11@Taiwan

24

EM Waves in 3D Magnetic Reconnectionω = ωr + iγ

Simulation resultsωci < |ωr| < ωＬＨ
Vph ≈ VA
Linear analyses
Profiles taken fromsimulation

ωＬＨ

Electronωci

Two-fluidequations
Ion[Fujimoto & Sydora, PRL, 2012]
ISSS11@Taiwan

25Summary
Adaptive mesh refinement (AMR) has been implemented in
the electromagnetic particle-in-cell (PIC) model to achieve
large-scale simulations of magnetic reconnection.
Main differences from the usual PIC models are Tree-type data structure, Smoothing of the EM fields to avoid the wavereflection,
 Particle splitting-coalescence to control thenumber of particles per cell, Adaptive block technique to keep load balancing.
ISSS11@Taiwan

26

Summary2
We found wave activities in the reconnection region of Electrostatic solitary waves (ESWs) generated due tothe electron 2-stream instability in the PSBL,
 Whistler waves excited by the temperature
anisotropy in the downstream region, and EM waves maintained by the velocity shear aroundthe x-line in 3D magnetic reconnection.

ISSS11@Taiwan

27