Transcript Evoluzione di particelle sopratermiche durante eventi di

Electron behaviour in three-dimensional
collisionless magnetic reconnection
A. Perona1, D. Borgogno2 , D. Grasso2,3
1 CFSA,
Department of Physics, University of Warwick, UK
2Burning
Plasma Research Group, Politecnico di Torino, Italy
3Istituto
dei Sistemi Complessi-CNR, Roma, Italy
Abstract
Aim of this project, still in progress, is to investigate the behaviour of an
electron population during the evolution of a spontaneous collisionless
magnetic reconnection event reproduced by a fluid formulation in a
three-dimensional geometry. In this 3D setting the magnetic field lines
become stochastic when islands with different helicities are present1.
The reconstruction of the test electron momenta, in particular, can assess
the small scale behaviour shown by the fluid vorticity and by the current
density during the nonlinear phase of the reconnection process even in
the presence of chaoticity.
We present here preliminary results of the numerical tool developed on
this purpose.
1D.
Borgogno et al., “Aspects of three-dimensional magnetic reconnection”, Phys.
Plasmas, Vol. 12, 032309, (2005).
14th European Fusion Theory Conference, Frascati, September 26-29, 2011
The fluid model for collisionless
magnetic reconnection
We
consider a fluid model that describes drift-Alfvén perturbations
in a plasma immersed in a strong, uniform, externally imposed
magnetic field.
Effects
related to the electron temperature, through the sonic Larmor
radius, and to the electron density, through the electron inertia, are
retained.
Modes
with different helicities can be present. They evolve
independently during the linear phase, while a strong interaction
occurs during the nonlinear stage of the process.
The
set of equations is closed by assuming constant temperatures for
both the ions and the electrons.
14th European Fusion Theory Conference, Frascati, September 26-29, 2011
The fluid model for collisionless
magnetic reconnection
B = B0 ez+ (x,y,z,t)  ez magnetic field with B0=const. and |B0|>>|B|
( = 8p2/B2 <<1),  magnetic flux function,  stream function.

   de  
2
2

t
  
   ,  
2
d e  
2
2
t
   s 
2
2
 
,   

z
 s
2
 
2

z
 
2
   ,         ,   
2
2
z
T. J. Schep, F. Pegoraro, B. N. Kuvshinov, Phys. Plasmas, Vol. 1, 2843, (1994)
Fluid velocity: v=ez+
Two scale lengths:
de 
m e /   0 ne e
2
  c /
electron skin depth
pe
S 
2
Te M i / ( eB )
sound Larmor radius
14th European Fusion Theory Conference, Frascati, September 26-29, 2011
The magnetic reconnection model:
simulation of the reconnection process
nonlinear 
We adopt a static, linearly unstable
magnetic equilibrium
ln 

 eq ( x )   ln(cosh( x ))
linear
Single-helicity initial perturbation
 J  x , y , z , t      x , y , z , t 
 Jˆ  x  exp  ik y  ik z 
Island width vs time
2
y
k y   m / Ly
z
k z   n / Lz
d e  0.16  n e  10 m
19
 m , n  m ode w ave num bers along y
and z
 s  0.32  Te 1 keV
14th European Fusion Theory Conference, Frascati, September 26-29, 2011
3
Single-helicity case (kz0):
the ‘fluid’ parallel electric field
Linear phase:
E ||
monopole structure peaked at the
X-point of the magnetic island.
V /m
E ||
V /m
14th European Fusion Theory Conference, Frascati, September 26-29, 2011
The electron model
In order to verify whether the
parallel electric field generated
during the reconnection leads
to suprathermal energetic
generation, a relativistic Hamiltonian formulation* of the electron
guiding-center dynamics has been chosen.
The unperturbed relativistic guiding-centre phase-space Lagrangian
can be written in terms of the guiding-centre coordinates as
L   p ||b - e A   R 
m
  W
e
p ||  p 
2
where b  B / B
p ||  m 0  v  b
 
1
2
2
0
m c
2
m  m 0
* A. J. Brizard, A. A. Chan, Phys. Plasmas 6, 4548 (1999)
14th European Fusion Theory Conference, Frascati, September 26-29, 2011
W  e
The electron model
The relativistic Hamiltonian is
magnetic moment:
2
H 
 
p || c  2  B m c  m 0 c  e 
2
2
2
2
4
p
2m 0 B
The equations of motion follow from the Hamiltonian in the usual manner
y 
H
z 
p y
H
p z
py  
H
y
pz  
H
z
where the conjugate momenta to the y and z coordinates are given by
py 
L
y
 p ||b y  eA y ,
pz 
L
z
 p ||b z  eA z
14th European Fusion Theory Conference, Frascati, September 26-29, 2011
The electron equations

  Az  Az
  bz  bz
 Az 
 bz  
by  pz  e 

y
z   p || 

y
z

t

y

z

t

y

z





x
b y g z  bz g y

Ay 
by  
 Ay Ay
 by by
bz  p y  e 

y
z   p || 

y
z

t

y

z

t

y

z





b y g z  bz g y
y
 Az
m  g z b y  g y bz 
by
Ay
B
 

p
 p || e
 by  m0 
 me

x
x
x
x 

2
||
z
bz
B
 

p
 p || e
 bz  m 0 
 me

x
x
x
x 

2
||
m  g z b y  g y bz 
Ay 
 by
g y   p ||
e


x

x


 Az 
 bz
g z   p ||
e

x
x 

  B z
Ax  0
Ay  Bz x
A z   eq  

  by by
by 
Ay  
 Ay Ay
  Az  Az
 Az 
g z p y  g y pz  g ye 

y
z   g z  p || 

y
z e

y
z 

t

y

z

t

y

z

t

y

z





 
p || 
b y g z  bz g y
14th European Fusion Theory Conference, Frascati, September 26-29, 2011
The f approach


A f approach has been adopted in order to reduce the number of
particles required while resolving small fluctuations in the electron
distribution function.
The distribution function is decomposed in an analytically described
background component and the remaining component

f  f0 
 p
   f 
analytic
p
,t

m arkers
• In each cell labeled by i in the real space, we calculate
 ne i
• the electron density
• the current density

 w
j
(c)
j
 /   
(s)
i
j i
N
J 

  ep
w j
|| j
j 1
c
j
 /  m  
(s)
i
• and the mean longitudinal kinetic energy of electrons
N
Wi 
p
||
 p ||0

2
w j
(c)
j

/ 2m i
j 1
14th European Fusion Theory Conference, Frascati, September 26-29, 2011
(s)

Kinetic simulations





The electron equations have been implemented in a 3-D code,
which reads at each time step the fluid fields provided by the
reconnection code1.
The kinetic code evolves the spatial coordinates, the parallel
velocity and the change in the weight of the markers according
to the fluid fields.
We assume periodic boundary conditions along y and z for the
flux and for the stream function.
The results presented have been obtained loading 2.5x106
electrons in the 5-D phase space (x,y,z,p||, p).
The code has been parallelized by distributing markers among
processers using Message Passing Interface (MPI) librairies.
1
A. Perona, L-G Eriksson, D. Grasso, Phys. Plasmas 17, 042104 (2010)
14th European Fusion Theory Conference, Frascati, September 26-29, 2011
Kinetic simulations:
single particle trajectory
The Poincaré plot of a single test
electron crossing the X-point maps the
magnetic island as expected.
magnetic surface
electron
(kz=0 case)
electron trajectory
14th European Fusion Theory Conference, Frascati, September 26-29, 2011
Numerical simulations: kinetic results (kz0)
During the linear phase the
electron and fluid current density
evolve in good agreement, while
the velocity distribution becomes
thinner and the amount of
electrons in the tails increases.
Linear phase:
Kinetic current density
J
Fluid current density
J
14th European Fusion Theory Conference, Frascati, September 26-29, 2011
Numerical simulations:
kinetic current density
Linear growth rate:
2  10
4
A/ m
2
Linear phase
6
A/ m
2
Nonlinear phase

5  10
I t  2

a
0
J 0 1  r / a
2
2
 rdr   J
2
a /2
0

fluid current
* kinetic current
J 0  2 It /  a
2

6.4  10 A /m
6
2
Good agreement with the toroidal
current of small Tokamaks such as T-3
(a ≈ 10-1 m, It=100 kA).
14th European Fusion Theory Conference, Frascati, September 26-29, 2011
Further steps
 The behaviour of the electron population will be followed during the
nonlinear phase of the single-helicity case (kz0) in order to confirm the
agreement with the fluid results, as already observed in the 2D case.
 The test electron distribution function will be reconstructed in the
presence of multiple-helicity fluid fields. This will allow us to assess the
structures of the electron and current density in the stochastic fields that
develop during the nonlinear phase of the reconnection process.
 In order to analyse the influence of the magnetic chaoticity on the
electron transport we plan to compare the test electron distribution and
the magnetic barriers detected through the analysis of the finite-time
Lyapunov exponent ridges1.
1D.
Borgogno et al., “Barriers in the transition to global chaos in collisionless
magnetic reconnection”, in press Phys. Plasmas (2011).
14th European Fusion Theory Conference, Frascati, September 26-29, 2011