Solar Flares and Magnetic Reconnection
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Transcript Solar Flares and Magnetic Reconnection
Lab.-Hinode Reconnection Workshop
2008.11.29. Hayama
MHD Simulations of 3D Reconnection
Triggered by Finite Random Resistivity
Perturbations
T. Yokoyama
Univ. Tokyo
in collaboration with H. Isobe (Kyoto U.)
1
“Size-gap” problem in solar flares
(c.f. Tajima Shibata 1997)
For a rapid energy release by the fast
reconnection, an anomalous resistivity is
expected near the neutral point.
Spatial size necessary for the
anomalous resistivity:
d = ri ~ 1 m
d; thickness of the current sheet
ri ; ion-gyro radius
Spatial size of a flare
–
104 –105 km
Enormous gap with a ratio of 107 !
It is hard to believe that a laminar flow structure
is sustained in a steady state manner.
2
Fractal Current Sheet model
Tajima & Shibata (1997)
Global current sheet
>1 km
~104 km
~1-10 m
“Turbulent reconnection”:
Matthaeus & Lamkin (1986), Strauss (1988), Lazarian & Vishniac (1999),
Kim & Diamond (2001) ...
3
Our final goal (but far away …)
To investigate the physical nature (evolution, saturation,
steadiness etc. ) of the 3-dimensional turbulent
magnetic reconnection
This study
By using the resistive MHD simulations, the evolution is
studied about a current sheet with initially imposed
finite random perturbations in the resistivity.
We focus on:
– (1) Evolution of 3D structures
– (2) Energy release rate
4
32d
Periodic condition on all
the boundaries
Model
0.1
CA / CS 3.4
d
128d
• 3D MHD eqs
with the uniform resistivity
Rm CAd / 4300
plus finite resistivity perturbations
spatially random; 50% max. amp.
during t/(d/Cs)<4
grid: 256 x 256 x 256
y
z
x
10d
5
xy-plane (z=0) r
Vx
Vy
Bx
By
Vz
y
x Jz
Bz
Tiny structures evolve first. They are overcome by the larger wavelength mode, whose size are
roughly consistent with the most unstable mode of the tearing instability. In later phase, the
magnetic islands in the sheet coalesce with each other to form a few dominant X-points.
6
zy-plane, x=0
r
Vx
Vy
Jz
Bx
By
Vz
y
z
Bz
• "Turbulent" structures
7
Evolution in the current sheet (zy-plane, x=0)
r
Vx
Vy
Vz
Jz
Bx
By
Bz
y
z
• Many tiny concentrations in the current are all X-points. They are associated
with bipolar structure of the reconnected magnetic fields (Bx) and outflows
(Vy).
8
Reconnection X-point
contour:Jz
Jz
P
Bx
y
y
y
Vz
y
x
z
•A pair of z-directional flows into the X-point evolve for each of the current concentrations. It
controls the evolution of 3D structure in the z-direction.
9
Power spectrum
Bx(x=0)
t/ts=1, 10, 50, 100, 200
IK
K
time
ky(Ly/2p)
time
kz(Lz/2p)
•Inverse cascade
•Evolution of “power-law”-like spectrum
10
Comparison with the Sweet-Parker reconnection
random perturbation
MA
SP
dE m
MA
dt
•The energy release rate is
slightly larger than the SweetParker reconnection.
B2
2 S
CA
4p
t / (d/Cs)
random perturbation
Vx
t=90
Vy
Sweet-Parker
Vx
Vy
t=90
11
Discussion: divided-sheet vs. Sweet-Parker
(e.g. Lazarian & Vishniac 1999)
• single current sheet
L
M A Rm
1 / 2
1/ 2
LC A
• N-divided multi current sheet
L L / N
M A Rm
1/ 2
1 / 2
LC A
N 1/ 2 Rm
1 / 2
When the current sheet is divided into N parts, the length of each part becomes shorter.
Then, keeping the aspect ratio of the diffusion region, the total amount of the released
energy increases like this proportional to the square-root of the dividing number N.
12
Comparison with the 2D turbulent case
2D
MA
• The energy release is weaker
in the 3D case than 2D.
３D
t / (d/Cs)
3D perturbation
Vx
t=110
Vy
2D perturbation
Vx
Vy
t=110
13
Comparison with the 2D turbulent case
2D
MA
• The energy release is weaker
in the 3D case than 2D.
３D
t / (d/Cs)
3D perturbation
Jz
Bx
t=110
2D perturbation
Jz
Bx
t=110
14
Discussion: 3D vs. 2D
Why the energy release became less effective in the 3D case than in
the 2D case ?
In case of 3D, the distribution of the current density is not uniform with
scattering concentrations here and there in the sheet. The filling
factor is smaller than the 2D case. As a result, the efficiency of the
energy release might be smaller.
Jz
Current sheets
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Dependence on diffusivity
M A; ki
dEki
dt
B2
2S
CA
4p
1.d-2
MA /
1.d-1
1.d-3
t / (d/Cs)
1/ 2
• The reconnection rate is: M A Rm
i.e.,
has a Sweet-Parker like dependence on Rm.
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Summary
•
•
•
•
3D structures in the z-direction evolve in time. A pair
of Inflows toward the z-direction play role for this
evolution.
Slight increase (3%) of energy-release rate above
the Sweet-Parker rate is observed. We interpreted it
is due to the spatial division of the current sheets by
the magnetic islands.
Energy-release rate is reduced in 3D random case
compared with 2D random case, presumably
because of the reduction of the filling factor of the
diffusion regions in the additional dimension.
The dependence on the magnetic Reynolds number
is consistent with that of the Sweet-Parker model.
17
Case with anomalous resistivity
anomalous
MA
uniform
t / (d/Cs)
•uniform resistivity
Vx
t=140
Vy
• In case with the anomalous resistivity,
Petschek like structures (i.e. pairs of slowmode shocks) appear, leading to the
enhanced reconnection rate.
• anomalous resistivity
0
0 anom ( D 1) 2
0 anom
D
Vx
J /r
VD0
for D 1
for1 D 2
for 2 D
VD0 3( J 0 / r0 )
Vy
t=140
18
Guide field (preliminary)
Bz0/By0=0.1
MA
Bz0/By0=0
t / (d/Cs)
Bz0/By0=0
Vx
t=150
Bz0/By0=0.1
Vy
Vx
Vy
t=228
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r
Vx
Vy
Vz
Bx
By
Bz
Bz0/By0=
0.1
xy-plane
(z=0)
y
x
Jz
20
r
Bz0/By0
=0.1
zy-plane
(x=0)
Vx
Vy
Vz
Bx
By
Bz
y
z
Jz
21
Guide field
Bz0/By0=2
Bz0/By0=0
MA
t / (d/Cs)
Bz0/By0=1
Vx
t=110
Bz0/By0=0.1
Vy
Vx
Vy
t=180
22
Comparison with the simple diffusion
Random
MA
dE m
MA
dt
B2
2 S
CA
4p
simple diffusion
t / (d/Cs)
random perturbation
Vx
t=130
Vy
simple diffusion
Vx
Vy
t=130
23
Tanuma et al. (2001)
•2D MHD simulations: Resistivity: uniform
(Rm=150) + current-dependent
• Current-sheet thinning by the secondary
tearing
• Impulsive nature of the inflow induced by
the ejections of mag. islands
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In the context of emerging flux study
(Shimizu & Shibata 2006 private commu.)
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