AMR-PIC Simulation of Collisionless
Magnetic Reconnection

Keizo Fujimoto
National Astronomical Observatory, Japanlines, Contour： ne
Magnetic Reconnection Line：Field
Arrow：Ve

Anti-parallel B-field

Particle acceleration

 Change in field line topology
 Plasma acceleration andheating

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Magnetic Reconnection in Space
Auroral Substorms[NASA]

Solar Flares
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Multi-Scale Nature of Reconnection
1km

10km

103km

Topology

Dissipation Hall effectDee, eFull PIC(Particle-In-Cell)

i, i

1011km

105km
∬L
MHD
(Magnetohydrodynamics)

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Coulomb meanfree pathi ~ 1

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Kuznetsova et al., J. Geophys. Res., 2007
Numerical
resistivity only

Nongyrotropic
correction case

Microscopic physics can have an impact onlarge-scale dynamics!
 Slow reconnection
 Quasi-steady
configuration
 Fast reconnection
 Quasi-periodic
process

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Restrictions of Explicit PIC Model
Cell sizex ≲ 3De
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

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AMR-PIC Model [Fujimoto, JCP, 2011](Adaptive Mesh Refinement – Particle-in-Cell)x ≲ 3De∝ ne-1/2
De,lobe ~ 6×103 m
De,ps ~ 3×102 m

Electron scale

Removing
unnecessary cells.

Particle splitting-coalescence

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AMR-PIC Simulations

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

[Fujimoto & Machida, JCP, 2006]

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Basic Equations of PIC model[Birdsall and Langdon, IOP, 1991]
Superparticle
S: Shape function
Superparticles (Buneman-Boris method)

EM Field (Yee-Buneman scheme)

ChargeConservation
Method[Villasenor &Buneman, 1992]

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von Neumann Stability Analysis

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Radiation of EM Waves

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

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Parallelization on Distributed Memory System
Case with 8 nodes

More than 99% of the total computation
is devoted to the particle calculations.
Imbalance of the number ofparticles per cell.

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Adaptive Block Technique[Fujimoto, JCP, 2011]

Case with 8 nodes

Morton-like orderingfor the base octs.

Imbalance of the number ofparticles per cell.

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Performance of the AMR-PIC Model
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.

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Large-Scale 3D Simulation[Fujimoto & Sydora, PRL, submitted]

Fujitsu FX1, 1024 cores
Bx = -B0 tanh(z/)Jy = -J0 sech2(z/)81 imi/me = 100
Max resolution:
4096×512×4096 ~ 1010
Max number of particles
Ion + Electron ~ 1011

10i ~ 1Re

81i ~ 10Re

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Max memory used ~ 6TB

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Time Evolution of the Current Sheet
Surface: |J|, Line: Field line
Color on the surface: Ey, Cut plane：Jy

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EM Turbulence at the X-line

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Summary
Adaptive mesh refinement (AMR) has been implemented
in the electromagnetic particle-in-cell (PIC) model toachieve 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 (Lapenta, JCP, 2002),
 Adaptive block technique to keep load balancing.
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