• Good conductor: E = 0
• Good moving conductor: E + v B = 0
• Ez + (v B)z = 0
• = const.
B = 0
Ez = d /dt
+Bi
-Bi
0J B 2Bx(x,0 ) (y) ˆ z
Magnetic field discontinuity: tangential discontinuity
J B B B 12
ˆ y y
B20
Magnetic equilibrium
Minimum magnetic energy state
Ohm’s law• Q: What makes a conductor “good”? (E=0)
(e.g. copper or gold? a plasma?)
• A: electrons move to eliminate E• Q: What might limit “goodness”? (E≠0)
• A: electrons cannot respond effectively
momentum eq. of electron fluid
mene
dve
dt
t P e eneE eneve B mene ei(vi ve)
E vi Bme
e
dve
dt
1
ene
t P e
1
ene
J Bme ei
e2ne
J
drag
Hall term 1/ eelectron inertia
J ene(vi ve)
Generalized Ohm’s law
E vi Bme
e
dve
dt
1
ene
t P e
1
ene
J B eJ
(i) (iv) (v)
(i)
(iv)
vB
B2 / 0enel
vl
B / 0ene
v
vA
l
c / pi
(i)
(v)
vB
eB / 0l
l v 0
e
Rm
(ii) (iii)
(i)
(ii)
vB
mevB / 0e2nel
2
l 2
me / 0e2ne
l
c / pe
2
Importance of term dependson length scaleof solution
On large scalesplasma is ideal conductor*
What’s really important?
(i)
(iii)
vB
pe /enel
vl
kBT /eB
v
cs
l
i * except where B=0
How large is “large”?Earth’s
magnetotailSolar
corona
Externalscale
L 108 m 108 m
Collisionlessskin depth
c/ pe 104 m 0.1 m
ion skin depth
c/ pi 106 m 10 m
Ion gyro-radius i 105 m 0.1 m
LL
All small :E+v B=0
Good vs. Fair conductance
Scale of ``external’’ solution: large
Solution develops smallerscale spontaneously
Good vs. Fair conductance
E + v B ≠ 0
Ideal region
diffusion region
E = - v BExternal Eset by inner solution
≠ 0
Work done on wires
Ez ≠ 0
W
Ez = 0
separate wires slowly
Build current, store energy
release energy
IcsE dt Ics
d
dtdt IcsdW Spontaneous & irreversible
Ez = -d /dt = vy Bx
2L
Total reconnection
time
LBxrx
d /dt
L Bx
vyBx
rx
L
vi
vA
vi
A
A
MAi
MAi
v i
vA
A
L
vA
MAi << 1 : Slow reconnection
MAi ~ 1 : Fast reconnection
Classic Ohm’s law
E vi B me
dve
dt
1
ene
t P e
1
ene
J B eJ
E z eJz ~eBi
0
Ez viBi
2
v i ~e
0
MAi
vi
vA
e
vA 0
e
0LvA
LLu 1 L
MAi
MAi Lu 1/ 2 L
2
2L
Ideal region
diffusion region
Lu 1/2 L
advection balancesdiffusion
from mass conservation
&
Rmv i 0
e
~ 1
2 = 2L
2
vi
vo
Special case: Sweet-Parker
uniform: =L
MAi Lu 1/ 2 LLu 1/ 2 1
LLu1/ 2
LLu1/ 2 1
v. slow reconnection
v. great aspect ratio
Collisionless reconnection
E vi B me
dve
dt
1
ene
t P e
1
ene
J B eJ
2
MAi ~12
2L
from mass conservation
c/ pi from Hall term
Drake et al. 2008
c/ pi
SMSSMSCS
FMRWFMRW
v vv v
Response to bendRiemann problem for 1D current sheet (CS)
Lin & Lee 1994
t=0
hot & dense
Petschek reconnection
vA vA
SMS
SMS
SMS
SMS
vi
External solution requires current -current appears in SMSs
Q: Will resistivity always result in slow (Sweet-Parker) reconnection?
Bis
kam
p&
Scw
artz
20
01
A: Yes, if is uniform in space…
But not, when (x) is locally enhanced*
JzSMS
enhanced
*as by micro-instability
Q: Is slowly reconnecting sheet stable?
Bhattacharjee et al. 2009 (vertical scale is expanded)
Lu = 6.3 X 105
A: No. Subject to resistive instability: tearing mode
• Solution becomes time-dependant• Smaller diffusion region(s) develop w/ larger ~ MAi
SeparatricesThe X-point (null point): B = 0
B(x,y)0 B'
B' 0
x
yL
d
ds
x
y
x(s)
y(s)
1
1emB 's
approaches null as s ±∞
e-values = B’
Bold new world: 3d
B(x,y,z)
Bx
x
Bx
y
Bx
z
By
x
By
y
By
z
Bz
x
Bz
y
Bz
z
x
y
z
Ld
ds
x
y
z
B(0,0,0) 0
Null point (@ origin)
Mij
det(M) < 0: “positive” null point
2 pos. e-values (fan surface)
1 neg. e-value (spines)
fan = separatrix
det(M) > 0: “negative” null point
(reverse all arrows)
B 0 Tr(Mij ) i
July 31, 2007
A Real Example
PHOTOSPHERE
CORONA
TRACE 171A
movie
SOHO/MDI
Solar-B Science Meeting Kyoto, Nov. 9, 2005
Reconnection observed
24 hour delay
Reconn’n burst1017 Mx/sec
Heating ~ 5 x 1026 erg/sec
emergence
= 1021 Mx
0.1 GV1 GV