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Particle acceleration in a
turbulent electric field
produced by 3D reconnection
Marco Onofri
University of Thessaloniki
Presentation of the work
An MHD numerical code is used to produce a turbulent electric
field form magnetic reconnection in a 3D current sheet
Particles are injected in the fields obtained from the MHD
simulation at different times
The fields are frozen during the motion of the particles
Incompressible cartesian code
We study the magnetic reconnection in an incompressible
plasma in three-dimensional slab geometry.
Different resonant surfaces are simultaneously present in
different positions of the simulation domain and nonlinear
interactions are possible not only on a single resonant surface,
but also between adjoining resonant surfaces.
The nonlinear evolution of the system is different from what
has been observed in configurations with an antiparallel
magnetic field.
Geometry
The MHD incompressible equations are solved to study magnetic
reconnection in a current layer in slab geometry:
Periodic boundary conditions
along y and z directions
Dimensions of the domain:
-lx < x < lx, 0 < y < 2ly, 0 < z < 2lz
Description of the simulations: equations and geometry
Incompressible, viscous, dimensionless MHD equations:
V
1 2
 ( V   ) V   P  (  B)  B   V
t
Rv
B
1 2
   ( V  B) 
 B
t
RM
B  0
V  0
B is the magnetic field, V the plasma velocity and P the
kinetic pressure.
RM and Rv are the magnetic and kinetic Reynolds
numbers.
Description of the simulations: the initial conditions
Equilibrium field: plane current sheet (a = c.s. width)
Vx  0
Vy  0
Vz  0
Bx  0
By  B y 0
x
x/a
Bz  tanh 
a cosh 2 (1 / a)
Incompressible perturbations superposed:
v x  bx 
v y  by 
k y max k z max
   cos(

2
2
x ) k ( k y  k z )sin (k z z  k y y)
2
k y min k z min
k y max k z max
2



cos(
x
)
sin
(
x
)
k
cos( k z z  k y y)

2
2
k y min k z min
v z  bz 
k y max k z max
2



cos(
x
)
sin
(
x
)
k
cos (k z z  k y y)

2
2
k y min k z min
Description of the simulations: the numerical code
Boundary conditions:
• periodic boundaries along y and z directions
• in the x direction:
dP
Bx  0 V  0
0
dx
B
y
x
0
Bz
0
x
Numerical method:
• FFT algorithms for the periodic directions (y and z)
• fourth-order compact difference scheme in the
inhomogeneous direction (x)
• third order Runge-Kutta time scheme
• code parallelized using MPI directives
Numerical results: characteristics of the runs
Magnetic reconnection takes place on resonant surfaces
defined by the condition:
k
B

0
0
where
k is the wave vector of the perturbation.
The periodicity in y and z directions imposes the
following conditions:
m
ky 
ly
k  B0  0 
n
kz 
lz
Bz l y
m

By l z
n
Testing the code: growth rates
The numerical code has been tested by comparing the growth
rates calculated in the linear stage of the simulation with the
growth rates predicted by the linear theory.
Two-dimensional
modes
Testing the code: energy conservation
The conservation of energy has been tested according to the
following equation:
 V 2
V 2  B 2 

V 2
1



     V
 2  P
  V  B  B  R 
 2
2



v

 
1 V j V j
1 B j B j


0
Rv xk xk
RM x k xk

t

 B 2 
1

  R 
 2 
 

M


Numerical results: instability growth rates
Parameters of the run: N x  128
N y  32
RM  5000 Rv  5000
ly / lx  1
N z  128
a  0.1
lz / lx  3
Perturbed wavenumbers: -4  m  4, 0  n  12
Resonant surfaces on both sides of the current sheet
Numerical results: B field lines and current at y=0
Numerical results: time evolution of the spectra
Spectrum anisotropy
The energy spectrum is anisotropic, developping mainly in one
specific direction in the plane, identified by a particular value
of the ratio, which increases with time. This can be expressed
by introducing an anisotropy angle:
  tan 1
k z2
where
2
y
k z2
and
k y2
k
are the r.m.s. of the wave vectors weighted by the
specrtal energy:
k z2 
2
(
n
/
L
)
m ,n z Em ,n

m, n
Em , n
k 2y 
2
(
m
/
L
)
Em , n
m ,n
y

m, n
Em , n
Spectrum anisotropy
Spectrum anisotrpy
Contour plots of the total energy spectrum at different times. The
straight lines have a slope corresponding to
the anisotropy angle.
Three-dimensional structure of the electric field
Isosurfaces of the current at t=400
Time evolution of the electric field
The surfaces are drawn for E=0.005 from t=200 to t=300
Time evolution of the electric field
The surfaces are drawn for E=0.01 from t=300 to t=400
Distribution function of the electric field
P(E)
γ = 1.56
γ = 1.55
t=200
E
γ = 1.48
t=300
t=400
Fractal dimension of the electric field
Particle acceleration
Relativistic equations of motions:
dp
e
= eE + v  B
dt
c
dr
=v
dt
p = γmv
γ=
1
v2
1 2
c
The equations are solved with a fourth-order Runge Kutta
adaptive step-size scheme.
The electric and magnetic field are interpolated with local 3D
interpolation to provide the field values where they are needed
Parameters of the run:
Number of particles: 10000
Maximum time: 0.2 s
Magnetic field: 100 G
Particle density:
Size of the box:
9
10 cm
l
3
9
x
10 cm
Particle temperature: 100 eV
Kinetic energy as a function of time
<E> (erg)
t=400
t=300
t (s)
Total final energy of particles:
34
10 erg
Magnetic energy:
30
10 erg
Trajectories of particles in the xz plane
The trajectories of the
particles are simpler
near the edges than in
the center, where they
are accelerated by the
turbulent electric field
Initial and final distribution function of particles
for different electric fields
P(E)
t=300
t=400
E (erg)
E (erg)
Summary
•We observe coalescence of magnetic islands in the center of the
current sheet and prevalence of small scale structures in the lateral
regions
•We have developped a MHD numerical code to study the evolution of
magnetic reconnection in a 3D current sheet
•The spectrum of the fluctuations is anisotropic, it develops mainly in
one specific direction, which changes in time
•The electric field is fragmented and its fractal dimesion increases
with time
•The particles injected in the fields obtained from the MHD
simulation are strongly accelerated
•The results are not reliable when the energy gained by the particles
becomes bigger than a fraction of the energy contained in the fields