Non linear evolution of 3D
magnetic reconnection in
slab geometry
M. Onofri, L.Primavera, P. Veltri, F. Malara
University of Calabria, 87036 - Rende - Italy
Summer school on Turbulence - Chalkidiki, September 23rd-28th 2003
Magnetic reconnection as a driver for turbulence
The presence of current sheets and magnetic reconnection
are seen to enhance the level of turbulence both in
astrophysical and plasma machines context
Earth magnetotail (Savin et al.)
Earth magnetosphere (Hoshino et al.)
Understanding turbulence dynamics during the
reconnection process
Open questions:
• are the growth rates foreseen by the linear theory still
valid when several modes are initially excited?
• saturation levels of the instability?
• nonlinear dynamics of the 3D reconnection: inverse
cascade, coalescence of islands, etc.(Malara, Veltri,
Carbone, 1992)
Our approach: numerical simulations
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
Magnetic reconnection in a current layer in slab geometry with
the plasma confined between two conducting walls:
Periodic boundary conditions
along y and z directions
Dimensions of the domain:
-lx < x < lx, 0 < y < 2ply, 0 < z < 2plz
Description of the simulations: the initial conditions
Equilibrium field: plane current sheet (a = c.s. width)
vx  0
Bx  0
vy  0
By  By 0
x
x/a
Bz  tanh 
2
a cosh (1 / a )
vz  0
Incompressible perturbations superposed:
v x  bx 
k y max k z max
2

cos(
2
p
x
)
(k y  k z )sin (k z z  k y y )

k y min k z min
v y  b y 
v z  bz 
k y max k z max
  p cos(2px)sin (k z  k
z
y
y )cos(k z z  k y y )
y
y )cos(k z z  k y y )
k y min k z min
k y max k z max
  p cos(2px)sin (k z  k
z
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, conducting walls give:
Bx  0 v  0
dP
0
dx
Numerical method:
• FFT algorithms for the periodic directions (y and z)
• fourth-order compact differences scheme along the
inhomogeneous direction (x)
• third order Runge-Kutta time scheme
• code parallelized using MPI directives to run on a 16processor Compaq a -server
Numerical results: characteristics of the runs
Magnetic reconnection takes place on resonant surfaces
defined by the condition:
modenumber along y
k  B0  0 
Bz l y
m
   q( x)
Bylz
n
safety factor
modenumber along z
The growth rates of the instability depend on the
position of the resonant surfaces
According to the linear theory, the m=0 (bidimensional) modes
are the most unstable ones!
Numerical results: instability growth rates
Parameters of the run: N x  128
N y  32
RM  5000 Rv  5000
l y / 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 domain!
Numerical results: spectrum along z for m=0
Numerical results: spectrum along z for m=1
Numerical results: B fieldlines and current at y=0
Numerical results: B fieldlines and current at y=0.79
Numerical results: B fieldlines and current at y=3.14
Numerical results: B fieldlines and current at y=15.70
Numerical results: time evolution of the spectra
Conclusions
• The two-dimensional modes (m=0) are not the most unstable
ones
• Initially,the modes with n=3 (m=0,1) grow faster
• At later times an inverse cascade transports the energy
towards longer wavelengths
• This corresponds, in the physical space, to a cohalescence of
the magnetic islands
• The spectrum of the fluctuations, which is initially growing
mainly along the z direction rotates towards higher values of
m/n.