Reconnection: Geometry and Dynamics A. Bhattacharjee, N. Bessho, J. Dorelli, S. Hu, J.
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Reconnection: Geometry and Dynamics
A. Bhattacharjee, N. Bessho, J. Dorelli, S. Hu, J. M. Greene, Z. W. Ma, C. S. Ng, and P. Zhu
Space Science Center Institute for the Study of Earth, Oceans, and Space University of New Hampshire
Isaac Newton Institute, Cambridge University, August 19, 2004
Geometry of 3D Reconnection
• Brief review of 2D and 2.5D models: “anti parallel” and “component” reconnection. Applicable to toroidal devices with closed field lines.
• 3D reconnection in the presence of magnetic nulls. Applicable to the Earth’s magnetosphere.
• 3D reconnection without nulls or closed field lines. Applicable to Parker’s model of the solar corona.
Resistive Tearing Modes in 2D/2.5D Geometry Equilibrium magnetic field assumed to be either infinite or periodic along z x
B
tanh
y
/
a
BT
ˆ Neutral line at y=0 Time Scales Tearing modes
A
R
a
/
VA
4
a
2 y
a
4 1/2 /(
c
2 ) /
BP S
R
/
A
S
3 / 5 (slower) or
S
1 / 3 (faster) (Furth, Killeen and Rosenbluth, 1963)
Collisionless Tearing Modes at the Magnetopause (Quest and Coroniti, 1981) • Electron inertia, rather than resistivity, provided the mechanism for breaking field lines, but the magnetic geometry is still 2D/2.5D.
• Growth rates calculated for “anti-parallel” ( ) and “component” ( ) tearing. It was shown that growth rates for “anti-parallel” merging were significantly higher.
• Model provided theoretical support for global models in which global merging lines were envisioned to be the locus of points where the reconnecting magnetic fields were locally “anti-parallel” (e.g. Crooker, 1979)
Magnetic nulls in 3D: Building blocks for separatrices Spine Fan (Lau and Finn, 1990)
Towards a fully 3D model of reconnection • Greene (1988) and Lau and Finn (1990): in 3D, a topological configuration of great interest is one that has magnetic nulls with loops composed of field lines connecting the nulls. • The null-null lines are called separators, and the “spines” and “fans” associated with them are the
global
3D separatrices where reconnection occurs (cf. Priest and Forbes, 2000).
• For the magnetosphere, the geometrical content of these ideas were implicit in the vacuum superposition models of Dungey (1963), Cowley (1973), Stern (1973). However, the vacuum model carries no current, and hence has no spontaneous tearing instability. The real magnetopause carries current, and is amenable to the tearing instability.
Dungey’s Model for Southward and Northward IMF
Equilbrium
B
-field
Perturbed
B
-lines produced by the Spherical Tearing mode
Equilibrium Perturbed Field lines penetrating the spherical tearing surface
Equilbrium Current Density,
J
Features of the spherical tearing mode
• The mode growth rate is faster than classical 2D tearing modes, scales as
S -
1/4 (determined numerically from compressible resistive MHD equations).
• Perturbed configuration has three classes of field lines: closed, external, and open (penetrates into the surface from the outside).
• Tearing eigenfunction has
global
support along the separatrix surface, not necessarily localized at the nulls.
• The separatrix is global, and connects the cusp regions. Reconnection along the separatrix is spatially inhomogeneous. Provides a new framework for analysis of satellite and ground-based observations at the dayside magnetopause.
Solar corona
astron.berkeley.edu/~jrg/ ay202/img1731.gif
www.geophys.washington.edu/ Space/gifs/yokohflscl.gif
Solar corona: heating problem
Temperature Density Time scale photosphere
~ 5 10 3 K
~
10 23 m 3 ~ 10 4 s
corona
~ 10 6 K
~
10 12 m 3 ~ 20s
Magnetic fields (~100G) --- role in heating?
Alfvén wave current sheets
Parker's Model (1972)
Straighten a curved magnetic loop Photosphere
Parker’s original model has been questioned
• van Ballegooijen (1985) • Zweibel and Li (1987) • Longcope and Strauss (1994) • Cowley, Longcope, and Sudan (1997) Did all these papers, which are technically correct (given their assumptions), consider current sheets of sufficiently complex topology?
Reduced MHD equations
B v
J
t
A t
[ [ ˆ , ] ,
B
2 2
A
]
A
ˆ
J
z
z
ˆ [
A
,
A
2 --- vorticity,
J
]
A
ˆ --- fluid v elocity, --- current density , 2 low limit of MHD --- magnetic field, [ --- resistivity, --- viscosity, ,
A
]
y A x
x A y
with
Magnetostatic equilibrium
J
z
[
A
, =
J
] 0 ,or
B
J
0 (current density fixed on a field line) 0. Field-lines are tied at
z
0 ,
L
.
c. f. 2D Euler equation
t
[ , ] 0
A
, J
, z
t
Existence theorem: If is smooth initially, it is so for all Time. However, Parker problem is not an initial value problem, but a two-point boundary value problem.
x
(
z
)
Footpoint Mapping
X
[
x
(0),
z
],
x
(
L
)
X
[
x
(0),
L
], with
dX dz
A
y
(
X
,
Y
,
z
) ,
dY dz
A
x
(
X
,
Y
,
z
)
Identity mapping:
x
(
L
) e. g. uniform field
B x
(0) ˆ For a given smooth footpoint mapping, does more than one smooth equilibrium exist?
A theorem on Parker's model
For any given footpoint mapping connected with the identity mapping, there is at most one smooth equilibrium.
Caveat
:A proof for reduced MHD, periodic boundary condition in
x
(Ng & Bhattacharjee,1998)
Remark
solution.
: Aly(2004) has shown recently, using the full resistive MHD equations, that the identity mapping has only one smooth
Implications of our theorem
An unstable but smooth equilibrium cannot relax to a second smooth equilibrium, hence must have current sheets .
Reconnection without nulls or closed field lines
Formation of a true singularity (current sheet) thwarted by the presence of dissipation and reconnection.
Classical reconnection geometries involve: (i) closed field lines in toroidal devices (ii) magnetic nulls in 3D Parker’s model is interesting because it fall under neither class (i) or (ii). Questions: • Where do singularities form? What are the geometric properties of singularity sites and reconnection? Are these Quasi-Separatrix Layers (QSL’s)? How do we characterize mathematically such singularities globally? • Need more analysis and high-resolution simulations.
In 3-D torii (such as stellarators), current singularities are present where field lines close on themselves
• • Solutions to the quasineutrality equation,
J
0,
B
J
B B
2
J
mn
J
||
mn e im
B
in
,
mn
(
m
n
i )
mn e im
in
p
'
mn
mn a mn e im
in
General solution has two classes of singular currents at rational surface [ iy mn ) = n/m] (Cary and Kotschenreuther, 1985; Hegna and Bhattacharjee, 1989).
mn
p
'
mn a mn
(
m
n
i )
mn
( y y
mn
) – Resonant Pfirsch-Schluter current – Current sheet
Impulsive Reconnection: The Trigger Problem
Dynamics exhibits an impulsiveness, that is, a sudden change in the time-derivative of the reconnection rate.
The magnetic configuration evolves slowly for a long period of time, only to undergo a sudden dynamical change over a much shorter period of time. Dynamics is characterized by the formation of near singular current sheets in finite time.
Examples
Sawtooth oscillations in tokamaks and RFPs Magnetospheric substorms Impulsive solar and stellar flares
Magnetospheric Substorms
Current Disruption in the Near-Earth Magnetotail (Ohtani et al., 1992)
Inward Boundary Flows Magnetic field
B
ˆ
P
tanh
y
/
a
ˆ
T
Inward flows at the boundar ies
v
V
0 (1 cos
kx
) ˆ Two simulations (Ma and Bhattacharjee 1996): Resistive MHD versus Hall MHD Two cases: Quasi-steady and triggered
B
v A /c
T i
E x yb
B
0 tanh (
z
0.2
a
/
a
) L
z
/2 5
T e
0.01
d i B
0 /
a
(v A 0.1
/
c
)(1
m i
cos /
m e
25
x
/
L x
) 0.02
0.01
0 0 2 1 0 0 10 t/ 20 A 30 40 10 t/ 20 A 30 40 J y y
Observations (Ohtani et al. 1992) Hall MHD Simulation Cluster observations: Mouikis et al., 2004
Ux Uz t=5.8
Highly Compressible Ballooning Mode in Magnetotail (Voigt model) x = -1 to -16 R E z = -3 to 3 R E ky= 25*2 e = 126 growth rate:0.2
t=29
k y and Dependence
Possible Scenario of Substorm Onset: Near-Earth Ballooning Instability Induced by Current Sheet Thinning and/or Reconnection
The start of large magnetic fluctuations and the modulation of energetic ion fluxes observed in Earth's magnetotail plasma sheet coincides with the onset of an isolated substorm.
Note the impulsive relaxation of the radial pressure gradient signified by the sudden reduction of the differences between the duskward (90) and dawnward (270) fluxes -- consistent with the feature of ballooning instabilities.
[L.-J. Chen, et al., 2003]
Power spectrograms of the magnetic field show impulsive enhancement of the wave power in the ballooning frequency range.
The enhancement is broadband, bursty, and simultaneous with the impulsive pressure gradient reduction.
[L.-J. Chen, et al., 2003]
Simulations of Parker's model
Start with a uniform
B
field [ Apply constant footpoint twisting (
x
,0) 0 , 2 0 ] 0, (
x
0 ,
L
) 0 (
x
) with Current layers appear after large distortion.
Quasi-equilib rium at first, becomes unstable
J
grows faster in the middl e non equilib rium
bottom
Simulations of Parker's model
middle top -
+
bottom
Simulations of Parker's model
middle top -
+
t
Simulations of Parker's model
J
max
J
max 0
q
2
d
2
q
max
d
max 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
150.5 35.3
16.4
21.98
13.486
32.94
5.997
83.39
79.01
with
q
J
/
z
[
A
,
J
],
d
q
2 .
More general topology
Parker's optical analogy (1990)
Main current sheet
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