Non-Equilibrium and Current
Sheet Formation in Line-Tied
Magnetic Fields
C. S. Ng and A. Bhattacharjee
Center for Magnetic Reconnection Studies
The University of Iowa
2002 APS Division of Plasma Physics, November 12
Outline
• Introduction
Solar corona: heating problem
Parker's Model (1972)
• A theorem on Parker's model
• Constraints on the geometry of the
current sheet
• Evidence from numerical simulations
• Reconnection without magnetic nulls
• Conclusion
Solar corona: heating problem
photosphere
Temperature
Density
Time scale
3
~ 10 K
3
12
~ 5 10 K
23
10
m
4
~ 10 s
corona
6
10 m
~ 20s
Magnetic fields (~100G) --- role in heating?
3
Solar corona: heating problem
6
2 1
4
Ideal MHD: ~ 10 km s ~ 10 years
Require sharp gradient (current sheets)
Line-tying in photosphere
Quasi-equilibrium --- most of the time
Parker's model: enough heating for quiet corona
Parker's Model (1972)
Objections:
van Ballegooijen
(1985)
Longcope &
Strauss (1994)
Cowley,
Longcope &
Sudan (1997)
and others
Reduced MHD equations
J
2
+ [ , ] =
+ [ A, J] + t
z
A
+ [ , A] =
+ 2 A
z
t
B =zˆ + B = zˆ + A zˆ --- magnetic field,
v = zˆ --- fluid velocity,
2
= --- vorticity,
2
J = A --- current density ,
--- resistivity, --- viscosity,
[ , A] y Ax x Ay
Magnetostatic equilibrium
(current density
J
+ [ A, J] = 0 ,or B J = 0 fixed on a field line)
z
with = = 0. c.f. 2D Euler equation. (but has two
points b. c.)
Field-lines are tied at z = 0 and z = L .
Footpoint Mapping:
x (z) = X[x (0), z], x (L) = X[x (0), L],
dY
dX A
A
=
(X,Y, z) ,
= ( X,Y,z)
with
x
dz
dz y
Can we have more than one smooth equilibrium
for a given smooth footpoint mapping?
A theorem on Parker's model
For any given footpoint mapping connected
with the identity mapping, there is at most
one smooth equilibrium.
A proof for RMHD, periodic boundary condition in x
(Ng & Bhattacharjee, phys. plasma 1998)
Implication: an equilibrium cannot relax to a
second smooth equilibrium after it becomes
unstable; hence a current sheet.
Simulations of Parker's model
Start with a uniform B field
Apply constant footpoint twisting
(x ,0) = 0, (x , L) = 0 (x ) with
[ 0 , 2 0 ] 0
Current layers appear after large distortion.
Quasi-equilibrium at first, becomes unstable
J grows faster in the middle non
equilibrium
Simulations of Parker's model
-
+
Simulations of Parker's model
-
+
Simulations of Parker's model
t
J max
J max 0 q 2
d
2
qmax
dmax
118.5 9.91 9.86
0.233 0.00028
7.946 0.708
131.6 16.5 10.9
0.930 0.00238
16.04 2.272
145.6 28.3 14.6
4.361 0.1471
32.94 5.997
150.5 35.3 16.4
21.98 13.486
83.39 79.01
with q J / z + [ A, J ], d q + 2 .
No simple current sheet
Cowley, Longcope & Sudan (1997):
Force-free equilibrium ( B dl = 0 )
ˆ (l)dl = 0
B (l)dl B (l) B
1
1
1
2
2
ˆ (l)dl = 0
B
(l)dl
B
(l)
B
1
2
2
2
1
ˆ (l) B
ˆ (l)]dl = 0
B(l)[1
B
2
1
1+2
therefore, no discontinuity.
van Ballegooijen (1985)
Current sheet => singular footpoint mapping
More general topology
Parker's optical
analogy (1990)
Possible topology
B1(l)dl B1 (l) Bˆ 2 (l)dl = 0
1
2
2
1
ˆ (l)dl 0
B
(l)dl
B
(l)
B
2
2
1
B(l)[1 Bˆ 2 (l) Bˆ 1 (l)]dl 0
1+2
May have tangential
discontinuities
Possible topology
z=0
1
Main current sheet
Current layers in simulations
An example using 3D coalescence instability
Initial equilibrium:
A0 = A sin( 2x) sin(2y)
= 0 (line-tied) at z = 0 and z = L
2
Unstable for 4 A L > 10.81
Energy dissipated by viscosity
J max increases with t and resolution
J max larger in middle
--- non-equilibrium increases with t
3D coalescence instability
A
J
Top
Middle
-
Bottom
+
3D coalescence instability
J contours
z=0
1
3D coalescence instability
J contours
z=0
1
3D coalescence
instability
J contours
c.f.
z=0
1
Reconnection without nulls
Magnetic
field lines
before
reconnection
Magnetic
field lines
after
reconnection
z=0
1
Reconnection without nulls
top view
Magnetic
field lines
before
reconnection
Magnetic
field lines
after
reconnection
z=0
1
Reconnection without nulls
top view
Magnetic
field lines
before
reconnection
Magnetic
field lines
after
reconnection
Conclusion
Parker's model for solar corona heating is
realizable when an equilibrium becomes unstable.
For RMHD and periodic boundary condition, there
is only one smooth equilibrium for each mapping.
Formation of current layers and trend toward nonequilibrium seen in numerical simulations.
Possible topology of current sheets proposed and
seen in simulation of 3D coalescence instability.
3D magnetic reconnection possible without
magnetic null points or neutral lines.