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Magnetic Reconnection and its
Applications
Zhi-Wei Ma
Zhejiang University
Institute of Plasma Physics
Chengdu, 2007.8.8
Outline
1. Numerical Scheme
2. Steady-state reconnection
A. Sweet-Parker model
B. Petschek model
3. Time-dependent force reconnection
A. Harris sheet
B. Magnetotail
C. Solar corona
4. Magnetic reconnection with Hall effects
A. Harris sheet
B. Magnetotail
C. Solar corona current dynamics
D. Coronal mass ejection
5. Summary
Numerical Scheme
Consider the general first-order ordinary differential equation,
y
 f ( y, t )
t

y
Euler's method
xn 1 xn xn 1
t
The standard fourth-order Runge-Kutta method
takes the form:
round-off error =O( /h)
dy
  y
dt
 ~ 1016 for double precision
 ~ 107
for single precision
Global integration errors associated with Euler's method (solid
curve) and a fourth-order Runge-Kutta method (dotted curve)
plotted against the step-length. Double precision calculation.
The equation for the shock propagation:
*
*
n+1/2
Hall MHD Equations

d / dt    V  0
  

dV / dt    P  J  B
B / t    E


p /  t  V  p  p V
   
 (  1) J  ( E  V  B)
  

 
E  V  B  J  d i ( J  B  P) / 
Rouge-Kutta Scheme (4,4)
Dispersion Properties for Different Schemes
Fast rarefaction wave (FR),
Slow compressional wave (SM),
Contact discontinuity (CD)
Slow shock (SS)
What is magnetic reconnection?
t  t0
t  t1
Another key requirement:
Time scale must be much
faster than diffusion time
scale.
Magnetic energy converts into kinetic or thermal energy
and mass, momentum, and energy transfer between
two sides of the central current sheet.
1. Steady-state Reconnection

A. Sweet-Parker model (Y-type geometry)
Reconnection rate
 Time scale

 ~
 ~
1/ 2
1 / 2

B. Petschek model (X-type geometry)
 ~ ln
Reconnection rate and time scale are
weakly dependent on resistivity.

Difficulties of the two models

For Sweet-Parker model
– The time scale is too slow to explain the
observations.
– Solar flare
 ~ 10  10
 sp ~ years
14
 ~ hour
12
 Substorm
in the magnetotail
 ~ 10
8
~ 10
 sp ~ days
 ~ hour
10
For Petschek model

The time scale for this model is fast enough to explain
the observation if it is valid. But the numerical
simulations show that this model only works in the high
4
resistive regime. For the low resistivity   10 , the Xtype configuration of magnetic reconnection is never
obtained from simulations even if a simulation starts
from the X-type geometry with a favorable boundary
condition.
Basic problem in both models is due to the steady-state
assumption. In reality, magnetic reconnection are timedependent and externally forced.
2. Time-dependent force reconnection

A. Harris Sheet
v( x)  v0 (1  cos kx)
Resistive MHD Equations

d / dt    V  0

 
dV / dt  p  J  B
B / t    E
  

E  V  B  J


p /  t  V  p  p V
   
 (  1) J  ( E  V  B)
New fast time scale in the nonlinear phase
(Wang, Ma, and Bhattacharjee, 1996)
3
1/ 5
 N  ( 0 A R ) or
 N 
1/ 5
B. Substorms in the magnetotail

Observations (Ohtani et al. 1992)
Time evolution of the cross tail current density at the nearEarth region (Ma, Wang, & Bhattacharjee, 1995)
C. Flare dynamics in the solar corona
Time evolution of maximum current density
(Ma and Bhattacharjee, 1996)
(Ma and Bhattacharjee, 1996)
Brief summary for time-dependent force
reconnection




1. New fast time scale is obtained for timedependent force reconnection.
2. The new time scale is fast enough to explain
the observed time scale in the space plasma.
3. The weakness of this model is sensitive to the
external driving force which is imposed at the
boundary.
4. The kinetic effects such as Hall effect are not
included, which may become very important
when the thickness of current sheet is thinner
than the ion inertia length.
3. Magnetic reconnection with Hall
effects
E  v  B  J
2
de
dJ

 dt

di

(p  J  B)
Resistive term  ~ 1/ 2
Inertia term ~ d e
Hall term ~ d i
Spatial scales




If   di , the resistivity term is retained
(resistive MHD).
If  ~ di  de , both the resistivity and Hall terms
have to be included (Hall MHD).
If di  de   , we need to keep the Hall and
inertia terms and drop the resistive term
(Collisionless MHD).
For solar flare,
di 5  10m   1/ 2 a 1  10m
where  1014  1012 , a 104 km

For magnetotail,
di
50  500km   1/ 2 a  1km
where  1010  108 , a 104 km
A. Harris Sheet
(Ma and Bhattacharjee,
1996 and 2001,
Birn et al. 2001)
1.
2.
3.
4.
5.
6.
7.
X-type vs. Y-type
Decoupling
Separation
Quadruple B_y
Time scale
Reconnection rate
No slow shock
Time evolution of the current density in the hall
(dash line) and resistive MHD (solid line)
The GEM challenge
results indicate that the
saturated level from Hall
MHD agrees with one
obtained from hybrid and
PIC simulation.
B. Hall MHD in the
magnetotail (Ma and
Bhattacharjee, 1998)
1.
2.
3.
4.
5.
6.
Impulsive growth
Quite fast disruption
Thin current sheet
Strong current density
Fast time scale
Fast reconnection rate
Explosive trigger of substorm onset

With increasing computer capability, we are able
to further enhance our resolution of the
simulation to reduce numerical diffusion. In the
new simulation, explosive trigger of substorm
onset is observed due to breaking up extreme
thin current sheet.
The tail-ward propagation
speed of the x-point or
Disruption region ~ 50km/s
Zhang H., et al., GRL, 2007
Reconnection rate ~ 0.1
Density depletion and
heat plasma around
the separatrices
C. Flare dynamics
1. Geometry
2. Electric field
(Bhattacharjee,
Ma &Wang, 1999)
Time evolution of current density and
parallel electric field
D. Coronal mass ejection or flux rope
eruption

Initial Geometry
Catastrophe
Or loss
equilibrium
Hall MHD Run
MHD run
Flux rope
region
Total energy
Thermal energy
Magnetic energy
Kinetic energy
Comparison between Hall and Full PIC simulation

Spontaneous Reconnection
– Periodic boundary condition
– Open boundary condition
Periodic boundary condition
(Hall MHD)
Open boundary condition
(Hall MHD)
Periodic boundary condition (PIC)
Open boundary condition (PIC)
[Daughton and Scudder; Fujimoto; PoP, 2006)]
Summary
Hall MHD vs. Resistive MHD
1.
1.
2.
3.
4.
5.
6.
Time scale and reconnection rate:
Fast with very weak dependence of the resistivity
vs. Fast with a suitable boundary conditions
Geometry:
X-type vs. Y-type
Decoupling Motion of ions and electrons: yes vs. no
Spatial scale separation of electric field and current
density:
Yes vs. No
Magnitude and distribution of parallel electric field:
strong and broad vs. weak and narrow
Quadruple distribution of B_y: yes vs. no
No slow shock for both cases, which is different from
Petschek’s model
Hall MHD vs. Full Particle
1. Periodic boundary condition: Nearly identical
Fast, time-dependent, x-type.
2. Open boundary condition:
Slow and steady vs. Fast and unsteady in the transition
period & Slow and steady in the late phase
Thanks!
!!