Electromagnetic Particle Code With Adaptive Mesh Refinement Technique:
Application to the Plasma Sheet
K. Fujimoto* and S. Machida
Department of Geophysics, Graduate School of Science, Kyoto University(* E-mail : keizo@kugi.kyoto-u.ac.jp)
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 developed a new 2-1/2 dimensional electromagnetic particle code with adaptive mesh refinement (AMR) technique. The AMR technique dynamically subdivides the cells that satisfy a certain refinement criterion, and it is quite effective to achieve high-resolution simulations such as those around the X-line.

AMR Techique used in Electromagnetic Particle Code[Calculations of Electromagnetic Field]

Level LIn order to realize an effective high-resolution
simulation, the AMR technique subdivides only
cells that satisfy a refinement criterion and add
data sets for finer cells (with higher level) onto
the uniform base cells (with the lowest level)
hierarchically. We also split the buffer cells thatare adjacent to the refined cells to calculate
electromagnetic field. Calculations of electromagnetic field are performed first on thecoarsest 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 in each cell are replaced by thosein the finest cells after proper smoothing operations to avoid the aliasing.

Level L+1

[Particle Splitting & Coalescence Algorithm]
The AMR technique in our particle code resorts to the particle splitting and coalescence algorithm
developed by Lapenta [2002] in order to control the number of particles per cell.
Processes of particle splitting and coalescence should conserve
1. the grid moments (ρc , j),
2. the charge and mass of particles (Σq, Σm),,
3. the total momentum
and energy of particles(Σ mv, Σ mv2/2)
4. the velocity distributions ( f(v) ).

* Particle Splitting∆r : Small value in comparison
with grid spacing
x1,2 = x0 ± ∆r; x3,4 = x0
y1,2 = y0; y3,4 = y0 ± ∆r
Buffer cells

Refined cells* Particle Coalescence0∆r0∆r

v1,2,3,4 = v0
q1,2,3,4 = q0 /4
m1,2,3,4 = m0 /4

q1 = q2; m1 = m2
x0 = (x1 + x2)/2
v0 = (v1 + v2)/2
q0 = q1 + q2
m0 = m1 + m2

The process of particle coalescenceconserves the grid moments and thetotal charge and momentum of particles, but perturbs the energy conservation and velocity distribution ofparticles. We choose two particles
close in space and velocity to reduce
the perturbation.

Test Simulations

Simulation Results of the Plasma Sheet

One of the sources causing numerical noise in the AMR simulation is thewave reflections at the boundaries of the refined regions. The reflections
are attributed to the difference in the dispertion relation, which differsfrom the physics for point particles and is dependent on the grid spacing.

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;

In order to check whether the wave reflections at the boundaries are significant, we test the Langmuir waves that propagate across the boundaries.

(a)(b) Ex Propagation

ymi/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.

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

xx(c)ω/ωpeω/ωpe(b) The Langmuir waves seems to prop4-6agate properly.
3. 102(c) Wave specrum is shown. The strong
0spectra around ω = ± ωpe are thoseof the normal Langmuir waves and-2
the weak spectra around ω = ± 2ωpe-40.
are those of the higher harmonics.
-75 -50 -25 0
25 50 75(d)If the waves reflect at the boundaries,
4 ‘With AMR’ - ‘Without AMR’
the spectra around ω = ± ωpe, ω/k < 0-8
1. 10
2should be present. However, such
spectra are not detected in this case0(λ = 32∆).
-2(d) Differences of the spectra in the casesof 'With AMR' and 'Without AMR' are -4-8-1. 10-75 -50 -25 0752550
much smaller than the peak value of the���
normal Langmuir waves.

Summary
We have developed a new electromagnetic full particle code with
the AMR technique, which is able to increase a spatial resolution
dynamically by subdividing cells that satisfy a refinement criterion.
The particle splitting and coalescence algorithm allows us to subdivide cells around the X-line, where the plasma number density
is very small, and realize effectively high-resolution simulationsof the plasma sheet.
We will extend the simulation box and achieve a particle simulation with the spacial dynamic range of 104 to 105, which enables
us to perform a global simulation including multi scales from theelectron scale to the MHD scale.

(a)Boundary conditions are periodic boundaries
in x and conductive walls in z.
Grid spacing is ∆B = 0.16 λi in the base layerand ∆D = 0.02 λi in the dynamic range layer.
-1Time step is ∆t = 0.0016 ωci , which is thesame in all layers.
The refinement criterion is defined by thelocal electron Debye length, the out-of-planeflow velocity of electrons, and the in-palne
electron current density.

λDe < ∆ <2λDe
Vey > 2 VA
jx2+jz2 > 1.2 enpsVA

oror(b)This condition must be satisfied in all cells.
The number density and flow vector ofelectrons are shown in (a). In this time, thevelocity of the electron outflow reach nearly
the electron Alfven velocity.
Structure of the electron diffusion region,
which is characterized by the strong out-ofplane flow of electrons (b), is quasi-equilibrium.
Overall evolution of the plasma sheet isbasically same as that conducted by theconventional PIC codes, even though in ourcode the finest cells are distributed onlyaround the electron diffusion region (c).
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 plasmasheet.

(c)