Particle acceleration in a Particle acceleration in a turbulent electric field turbulent electric field
produced by 3D reconnectionproduced by 3D reconnection
Marco Onofri
University of ThessalonikiUniversity of Thessaloniki
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
Presentation of the workPresentation of the work
IncompressibleIncompressible cartesiancartesian codecode
We study the magnetic reconnection in an incompressibleplasma 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.
The MHD incompressible equations are solved to study magnetic reconnection in a current layer in slab geometry:
Periodic boundary conditionsalong y and z directions
GeometryGeometry
Dimensions of the domain:-lx < x < lx, 0 < y < 2ly, 0 < z < 2lz
Description of the simulations: equations and geometryDescription of the simulations: equations and geometryIncompressible, viscous, dimensionless MHD equations:
0
0
1)(
1)()(
2
2
V
B
BBVB
VBBVVV
M
v
Rt
RP
t
B B is the magnetic field, is the magnetic field, VV the plasma velocity and the plasma velocity and PP the the kinetic pressure.kinetic pressure.
MR vRand are the magnetic and kinetic Reynolds are the magnetic and kinetic Reynolds numbersnumbers.
Description of the simulations: the initial conditionsDescription of the simulations: the initial conditionsEquilibrium field: plane current sheet (a = c.s. width)
Incompressible perturbations superposed:
)/1(cosh/
tanh0
0
00
2
0
aax
ax
BV
BBV
BV
zz
yyy
xx
max
min
max
min
max
min
max
min
max
min
max
min
)(cos)(sin)cos(
)(cos)(sin)cos(
)(sin)()cos(
222
222
222
y
y
z
z
y
y
z
z
y
y
z
z
k
k
k
kyzzz
k
k
k
kyzyy
k
k
k
kyzzyxx
ykzkkxxbv
ykzkkxxbv
ykzkkkkxbv
Description of the simulations: the numerical codeDescription of the simulations: the numerical codeBoundary conditions:• periodic boundaries along y and z directions• in the x direction:
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
000 dx
dPBx V
0
x
B y0
x
Bz
Numerical results: characteristics of the runsNumerical results: characteristics of the runs
Magnetic reconnection takes place on resonant surfacesdefined by the condition:
00 B k
where is the wave vector of the perturbation.kThe periodicity in y and z directions imposes the following conditions:
y
yl
mk
z
zl
nk
nm
lB
lB
zy
yz 00B k
Testing the code: growth ratesTesting the code: growth rates
Two-dimensional modes
The numerical code has been tested by comparing the growthrates calculated in the linear stage of the simulation with thegrowth rates predicted by the linear theory.
Testing the code: energy conservationTesting the code: energy conservation
The conservation of energy has been tested according to the following equation:
011
2
1
2
1
22
22222
k
j
k
j
Mk
j
k
j
v
Mv
x
B
x
B
Rx
V
x
V
R
B
R
V
RP
VBV
tBBVV
Numerical results: instability growth ratesNumerical results: instability growth rates
Parameters of the run:
Perturbed wavenumbers: -4 m 4, 0 n 123/1/
1.050005000
12832128
xzxy
vM
zyx
llll
aRR
NNN
Resonant surfaces on both sides of the current sheetResonant surfaces on both sides of the current sheet
Numerical results: B field lines and current at y=0Numerical results: B field lines and current at y=0
Numerical results: time evolution of the spectraNumerical results: time evolution of the spectra
Spectrum anisotropySpectrum anisotropy
The energy spectrum is anisotropic, developping mainly in one The energy spectrum is anisotropic, developping mainly in one specific direction in the plane, identified by a particular value specific direction in the plane, identified by a particular value of the ratio, which increases with time. This can be expressed of the ratio, which increases with time. This can be expressed by introducing an anisotropy angle:by introducing an anisotropy angle:
2
2
1tany
z
k
k wherewhere
2zk andand
2yk
are the r.m.s. of the wave vectors weighted by the are the r.m.s. of the wave vectors weighted by the specrtal energy:specrtal energy:
nm nm
nm nmz
z E
ELnk
, ,
, ,2
2)/(
nm nm
nm nmy
y E
ELmk
, ,
, ,2
2)/(
Spectrum anisotropySpectrum anisotropy
Contour plots of the total energy spectrum at different times. The Contour plots of the total energy spectrum at different times. The straight lines have a slope corresponding tostraight lines have a slope corresponding tothe anisotropy angle.the anisotropy angle.
Spectrum anisotrpySpectrum anisotrpy
Three-dimensional structure of the electric field Three-dimensional structure of the electric field
Isosurfaces of the current at t=400
Time evolution of the electric fieldTime evolution of the electric field
The surfaces are drawn for E=0.005 from t=200 to t=300
Time evolution of the electric fieldTime evolution of the electric field
The surfaces are drawn for E=0.01 from t=300 to t=400
P(E)
t=200t=300
t=400
Distribution function of the electric fieldDistribution function of the electric field
1.48=γ1.55=γ1.56=γ
E
Fractal dimension of the electric fieldFractal dimension of the electric field
Particle accelerationParticle acceleration
Relativistic equations of motions:
vr
=dt
dBvE
p
c
e+e=
dt
d
The equations are solved with a fourth-order Runge Kuttaadaptive step-size scheme.
The electric and magnetic field are interpolated with local 3Dinterpolation to provide the field values where they are needed
vp γm=2
2
1
1
cv
=γ
Parameters of the run:Parameters of the run:
Number of particles: 10000
Maximum time: 0.2 s
Magnetic field: 100 G
Particle density:
Size of the box:
109cm 3
lx
109
cm
Particle temperature: 100 eV
Kinetic energy as a function of timeKinetic energy as a function of time
<E> (erg)
t (s)
t=400
t=300
Total final energy of particles: Magnetic energy:1034
erg 1030
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
E (erg)
t=300
t=400
P(E)
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
