Electromagnetic Particle-in-Cell Model with
AMR and Application to Current Sheet
Evolution in 2D and 3D Systems

Keizo Fujimoto
宇宙科学セミナー＠NiCTon December 3, 2007

Introduction
MHD scale~105 kmDynamic range~ 104
Electron scale~10 kmElectromagnetic
particle code isrequiredCoroniti [1985]

宇宙科学セミナー

宇宙科学セミナー

Introduction
Restriction in full particle code:
Magnetotail Lobe

[Birdsall & Langdon, 1995]

Grid spacing can becoarser in the lobe region
than in the plasma sheet.
Central Plasma Sheet

宇宙科学セミナーIntroduction
Restriction in full particle code:[Birdsall & Langdon, 1995]

宇宙科学セミナー

AMR法の使用例
9 多階層格子を偏微分方程式系に適用
Berger and Oliger (1984), Berger and Colella (1989)
9 MHD + AMR
Groth et al. (2000)
9 PIC + AMR (N-body code)
Villumsen (1989), Jessop et al. (1994), Suisalu and Saar(1995), Gelato et al. (1997), Kravtsov et al. (1997), Yahagiand Yoshii (2001)
9 EMPIC + AMR
Fujimoto and Machida (2006)
宇宙科学セミナーData Structure
Cells are treated as
independent units
organized inrefinement trees rather
than elements of arrays,
so that a very flexible
cell hierarchy isachieved.
The hierarchical cellstructure is supported bya set of pointers, which isbasically same as the
fully threaded tree (FTT)
structure [Khokhlov,1989].

宇宙科学セミナーBasic Equations
Equation of motion

Electromagnetic field

宇宙科学セミナーCalculation of Electromagnetic Field
EM solver on thecoarsest cells
Interpolation to thebuffer cells
EM solver on thefiner cells
Projection of the solution
on the finer cells onto thecoarsest cells
宇宙科学セミナーMesh Refinement Criteria
Electron Debye length alone is really enough?

Aliasing

λDe < ∆ < 2λDe

λDe < ∆ < 2λDe, Vey > 2VA0

Refinement meshes arerequired in the region
where the electron scale
physics is expected to besignificant.
宇宙科学セミナーThe Number of Meshes and Simulation Cost
¾ The number of meshes¾ Walltime¾ Simulation cost

96 % of the total simulation time is devoted
to the routines related to the particle data.
In order to perform an efficientsimulation, it is inevitable to reduce
the number of superparticles.

宇宙科学セミナーParticle Splitting and Coalescence (Lapenta, 2002)

Conserving through the splitting
Moment on the grids（ρc, J）、Total charge andmass （Σρc, Σm）、Total energy of particles（Σmv2/2）、Distribution function （f(v)）¾ Wall time

宇宙科学セミナーInitial Setting for the Test Simulations
Harris-type current sheet
Bx(z) = B0 tanh[z/λ]Jy(z) = J0 sech2[z/λ]宇宙科学セミナーTest Simulations in the Non-tearing System
Linear dispersion of the lower hybrid drift instability[Davidson et al., 1977]

γωr

宇宙科学セミナーTest Simulations on Magnetic Reconnection

In 3D system with the lower hybrid
drift instability (LHDI), the current
density is enhanced at the center ofthe current sheet, which facilitates
the onset of a fast reconnection.
宇宙科学セミナーTearing and Kink Modes in 3D System
System size: Lx×Ly×Lz = 30.7λi×7.7λi×30.7λi
Maximum resolution: Nx×Ny×Nz = 1024×256×1024
mi/me = 25

宇宙科学セミナー

Parallelization Efficiency(The test was performed using the FUJITSU HPC2500.)
宇宙科学セミナーSummary and Conclusions
We have successfully developed new 2D and 3D electromagnetic
particle code with the adaptive mesh refinement technique.
¾ Refinement meshes are required not only in the region with small
Debye length, but also in the region where the electron scale
physics is expected to be significant.
¾ In order to perform an efficient simulation, it is inevitable toreduce not only the number of meshes, but also the number ofsuperparticles.
¾ Numerical errors associated with cell refinement and particle
coalescence are small.
¾ The code is checked against the LHDI and Tearing instability.
¾ The computing performance and efficiency are well enhanced byparallelizing the code, using the OpenMP and MPI.
宇宙科学セミナー