AMR-PIC Model and Application to
Magnetic Reconnection

Keizo Fujimoto
Computational Astrophysics Laboratory, RIKEN

Magnetic Reconnection in Space[Solar Flares][Magnetospheric Substorms](http://www2.nict.go.jp/dk/c232/)(http://vestige.lmsal.com/TRACE/)(Carmichael, 1964)
ISSS10@Banff, Canada(Kivelson and Russel,1995)

Multi-Scale Nature of Reconnection
i ~ 1MHD

Full PIC
Dissipation

Hall eff.

e, ei, iDe∬

Topology

L

Coroniti [1985]

L~105 km
ISSS10@Banff, CanadaLcCoulomb mean
free pathe~10 kmi~103 km

Multi-Scale Nature of Reconnection

MHD

Full PICDeDissipation

Hall eff.

e, ei, i∬
AMR-PICTopologyLLc

MHD simulations
 The reconnection rate depends on the resistivity model. (Biskamp,1986;Ugai, 1995)
 Global responses in substorms and flares are sensitive to theparameterization of the resistivity. (Raeder et al.,2001; Kuznetsova et al., 2007)
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[Fujimoto & Machida, JCP, 2006;
Fujimoto & Sydora, CPC, 2008]

AMR-PIC Model(Adaptive Mesh Refinement – Particle-in-Cell)e scale structure

Restriction in explicit
methodx < De, pet < 1x/t > c

De,ps ~ 3×102 m
De,lobe ~ 6×103 m

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AMR-PIC Model[Fujimoto & Machida, JCP, 2006;
Fujimoto & Sydora, CPC, 2008]

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Finite-Difference Equations
Yee-Buneman scheme (Staggering grid scheme)
Charge Conservation
Method (CCM)

Ey, jy[Villasenor & Buneman, 1992]

Ez, jzLocal operations

Facilitates parallel
computation

Buneman-Boris method

ISSS10@Banff, CanadaBz
ByEx, jxBx

Inter-Level Communications
Fine-to-coarse operations
 Deliver the data from fine cells tocoarser cells,
 Give appropriate smoothingwhich removes the aliasing.

Coarse-to-fine operations
 Deliver the data from coarse cells
to finer cells,
 Give the boundary conditions forthe refinement regions.

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Electromagnetic Wave in Vacuum
Staggering grid scheme
No numerical damping
for any wave numbers

Von Neumann stability analysis
g: Amplification factor|g| = 1 (x/t > c; Courant condition)
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Electromagnetic Radiation Test

It is very difficult to apply
the AMR to the staggering
grid scheme!

ISSS10@Banff, CanadaSmoothing Function: fSM

x

For A = E and B,(1+2)xASM,jAj-1 Aj Aj+1

 Selectively damps the shortwavelength modes.
• Very simple (Very fast)• Easy parallelization• No wave dispersion changes• Flexible about damping rate
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j-1

jj+1

Electromagnetic Radiation Test 2
 The case without the smoothing The case with the smoothing ( = 0.002)

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Load Balancing* Block = Decomposition domain

Example using 8 nodes
Fixed block case

Adaptive block case

ISSS10@Banff, CanadaAdaptive Block TechniqueBase-level cells in the entire domain are sorted in an appropriate
order:
 That is similar to Morton order,
 So that the block surface is assmall as possible,
 Especially in the central current
sheet, the surface must be small.

ISSS10@Banff, Canada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%
RemovingPoisson Eq.

ISSS10@Banff, CanadaLarge-Scale 3D Simulation
Bx = -B0 tanh(z/)Jy = -J0 sech2(z/)81 i m /m = 100i e
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 ~ 5TB

Large-Scale 3D Simulation
Side surface：JySurface contour: |J|

Solid curves：Field lines

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Wave Spectra (Y Direction)
neX

Z|Ve |YZLHDI
kye ~ 1 ~ lhY

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Scale of Electron Meandering Orbit
m|Ve |[Speiser, 1965]

m[Fujimoto, 2009]

Wavelength of EM mode ~ Electron meandering scale
Possibility of electron scattering and resultant
anomalous resistivity
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Comparisons With Observation
 Frequency~ a few 100km

 Observation of Cluster[Zhou et al., JGR, 2009] lh-range EM waves near thecentral current sheet.
 Wavelength  ~ 352km
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Summary
A new electromagnetic particle-in-cell (PIC) model with adaptive meshrefinement (AMR) has been developed for high-performance parallelization
and applied to 3D magnetic reconnection.
 Staggering grid scheme + Charge conservation method + Smoothing Adaptive block technique More than 80% of parallel efficiency with 128 cores has beenachieved for reconnection test simulations.
 Electromagnetic waves arise along the X-line and are enhanced
associated with the splitting of the electron diffusion region.
 The wavelength is in the scale of the electron meandering orbit.
⇒ Electron scattering and resultant anomalous resistivity.
ISSS10@Banff, CanadaAdaptive Block Technique
Buffer region and private octs

Buffer region
Private oct
Allocated in each node

Global octShared by all the nodes
grank: Rank
gOctNb: Neighboring oct

pOct

rank: Rank
iNb: Parent cell of neighboring oct

iPr

ipr: Parent cell
OctCh: Child oct
・・・
Physics data

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