Magnetic Reconnection: Some Answers and
Open Questions
Amitava Bhattacharjee
Space Science Center and Center for Magnetic SelfOrganization
University of New Hampshire
NESSC Meeting, Durham, New Hampshire,
January 9, 2007
Collaborators
Naoki Bessho, John Dorelli, Terry Forbes, Kai Germaschewski, Chung-Sang Ng,
David Pontin, Joachim Raeder, Hong-Ang Yang, UNH
John Greene, GA
Shuanghui Hu, UC-Irvine
Zhiwei Ma, Unversity of Iowa
Xiaogang Wang, Beijing University
CMSO : John Finn, Chris Hegna, Hantao Ji, Yi-Min Hwang, Vladimir Mirnov,
Stewart Prager, Dalton Schnack, Carl Sovinec, Masaaki Yamada, Ping Zhu,
Ellen Zweibel
Outline
• Classical steady-state models of Sweet-Parker and Petschek.
• Impulsive, time-dependent reconnection dynamics:
comparative studies of magnetospheric substorms, and solar
flares.
• Scaling of the reconnection rate.
• 3D reconnection:
Separator reconnection
Current sheet formation and reconnection in geometries
without nulls or closed field lines.
I will not discuss the important subjects of particle acceleration,
and turbulent reconnection.
Classical (2D) Steady-State Models of Reconnection
Sweet-Parker [Sweet 1958, Parker 1957]
Geometry of reconnection layer : Y-points [Syrovatskii 1971]
Length of the reconnection layer is of the order of the system
size >> width 
Reconnection time scale
 SP   A R 
1/2
 S1/2 A
12
Solar flares: S ~ 10 ,  A ~ 1s

  SP ~ 10 6 s

Too long to account for solar flares!
Q. Why is Sweet-Parker reconnection so slow?
A. Geometry
Conservation relations of mass, energy, and flux
VinL  Vout, Vout  VA
Vin 


L
VA ,

L
 S1/ 2
Petschek [1964]

Geometry of reconnection layer: X-point
Length  (<< L) is of the order of the width 
 PK   A ln S
2

~
10
s
Solar
flares:
PK


Numerical Simulations of the Petschek Model
[Sato and Hayashi 1979, Ugai 1984, Biskamp 1986, Forbes and Priest 1987,
Scholer 1989, Yan, Lee and Priest 1993, Ma et al. 1995, Uzdensky and
Kulsrud 2000, Breslau and Jardin 2003, Malyshkin, Linde and Kulsrud
2005]
Conclusions
• Petschek model is not realizable in high-S plasmas, unless the
resistivity is locally strongly enhanced at the X-point.
• In the absence of such anomalous enhancement, the
reconnection layer evolves dynamically to form Y-points and
realize a Sweet-Parker regime.
2D coronal loop : high-Lundquist number resistive MHD simulation
T=0
T = 30
[Ma, Ng, Wang, and Bhattacharjee 1995]
Biskamp’s Critique of the Petschek Model
Magnetic energy accumulates upstream
of the current sheet (flux pileup).
increases with increasing Lundquist number; this directly contradicts the
Petschek model, which requires 
Flux pile-up: Litvinenko, Forbes and Priest [1996], Craig, Henton, and
Rickard[1993], Dorelli and Birn [2001], Simakov, Chacon, and Knoll
[2006]
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 nearsingular current sheets in finite time.
Examples
Sawtooth oscillations in tokamaks
Magnetospheric substorms
Impulsive solar flares
Sawtooth events in MST
[Almagri and the MST group 2003]
Current Disruption in the Near-Earth
Magnetotail
[Ohtani, Kokubun, and Russell 1992]
Time Profile of Reconnection Rates for X3 Flare
observed by TRACE
Linear Plot
Log Plot
[Saba, Tarbell, and Gaeng 2003]
Hall MHD (or Two-Fluid) Model and the
Generalized Ohm’s Law
In high-S plasmas, when the width of the thin current sheet ( )
satisfies
c /  pe    c /  pi
(or  c / pi if there is a guide field)
“collisionless” terms in the generalized Ohm’s law cannot be
ignored.

Generalized Ohm’s law (dimensionless form)
1
2 dJ di
E  v  B  J  de
 J  B   epe 
S
dt n
Electron skin depth
Ion skin depth
Electron beta
de  L1c /  pe 
di  L1 c /  pi 
e
Forced Magnetic Reconnection Due to Inward
Boundary Flows
Magnetic field
B  xˆ BP tanh z /a  zˆ BT
Inward flows at the boundaries
,  0

Two simulations: Resistive MHD versus Hall MHD [Ma and
Bhattacharjee 1996]
v  V0 (1 coskx)yˆ
 perspective, with similar conclusions,

For another
see Horiuchi and Sato
[1994] and more recently, Cassak, Shay, and Drake [2005]

….. Hall
___ Resistive
dln  /dt
Transition from Collisional to Collisionless Regimes in MRX
“Standard
Model”
QuickTime™ and a
Video decompressor
are needed to see this picture.
Scaling of Reconnection
• Observations: Yokoyama et al. (2001), Isobe et al. (2002)
This Workshop: Lin et al. (2004), Noglik et al, (2004)
Reconnection rates : inflow velocity/Alfven velocity~0.001-0.1.
Hall MHD reconnection scaling depends on the ion skin depth, the
Lundquist number (weakly), and system size. The range of
reconnection rates in Hall MHD and PIC simulations ~ 0.01-0.1.
Observations (Ohtani et al. 1992)
Hall MHD Simulation
[Ma and Bhattacharjee 1998]
Scaling of the collisionless reconnection rate
How does it scale with ion/electron skin depth, resistivity, plasma
beta, and system size?
y
Vin
i
e
e
i
out-of-plane magnetic field, Bz
x
Scaling is a controversial subject
• Is the reconnection rate insensitive to the details of the
electron layer (current sheet layer), and controlled by ions?
The GEM Challenge Perspective
[Birn et al. 2001; Mandt et al. 1994, Shay and Drake 1998, Hesse et al. 1999,
Rogers et al. 2001, Pritchett 2001, Ricci et al. 2002]
Reconnection is insensitive to the mechanism that breaks field
lines (electron inertia or resistivity).
• In the presence of Hall currents, whistler waves mediate
reconnection. The characteristic outflow speed is the whistler
phase speed (based on the upstream magnetic field).
• The inflow velocity Vin  ee2 /i where i ~ k  L (= system
size). This rate is independent of me .
Scaling is a controversial subject
• The whistler waves generate an out-of plane quadrupolar
magnetic field (seen in MRX/PPPL, SSX/Swarthmore as well as
in situ satellite observations in space).
• The ratio of the horizontal electron outflow to the horizontal
magnetic perturbation scales as k for the dispersive whistler (or
kinetic Alfven) wave.
• How does the reconnection rate scale with the system size?
The length of the reconnection layer i ~ 10di. Reconnection rate
is a “universal constant”, Vin  0.1VA [Shay et al. 2001, 2004,
Huba and Rudakov 2004].

Scaling is a controversial subject
The GEM perspective is not universally accepted. An alternate point
of view provides evidence that:
• Reconnection is not a universal constant, and depends on system
parameters (such as ion/electron skin depth, plasma beta,
boundary conditions)
• Reconnection rate is not independent of the system size, and in
fact, often decreases as the system size increases.
[Wang et al. 2001, 2006, Fitzpatrick 2004, Bhattacharjee et al. 2005, Daughton
et al. 2006]
Three examples : (1) Forced reconnection without guide field
(2) Undriven reconnection with guide field
(3) Undriven reconnection with open boundaries
Scaling of steady-state reconnection rate in a
driven Harris sheet (without guide field)
Two-fluid simulations as di is varied [Wang et al., 2006]
Confirms analytical scaling law [Wang, Bhattacharjee, and Ma 2001]
Effect of Open Boundary Conditions and System Size
[Daughton, Scudder and Karimabadi 2006]
Hall currents cancel exactly in electronpositron plasmas: no whistlers
Generalized Ohm’s law does not contain the Hall term.
vB
me J
  (Pe  Pi )
E

  (Jv  vJ)

2
c

2ne
2ne t
[Bessho and Bhattacharjee 2006]
Open questions on scaling
• While collisionless reconnection models do provide a clear mechanism
for the onset of impulsive and fast reconnection, issues of scaling
remain wide open.
• Is reconnection controlled entirely by ions? Or by electrons? Or is it a
hybrid of ion and electron parameters, as well as boundary conditions?
• What is the role of periodic and open boundary conditions?
• Is the reconnection rate independent of the system size? If it isn’t, the
theory faces great challenges in working for solar coronal plasmas
while it may be adequate for magnetotail and fusion plasmas.
• Is the Hall current essential for fast reconnection?
• Role of turbulence--the Eyink-Aluie theorem (2006): “Frozen-in-flux
condition in a turbulent quasi-ideal plasma will occur if the sites of
current and vortex sheets intersect one another.”
3D Reconnection:
Geometry and Dynamics
Resistive Tearing Modes in 2D/2.5D Geometry
B  xˆ BP tanhy / a  BT zˆ
x
Neutral line at y=0
y
 A  a /VA  a41/2 /BP
Time Scales
R

Lundquist Number

Tearing modes

 4a 2 /(c 2 )
S   R / A
  S3/ 5 (slower) or S1/ 3 (faster)

[Furth, Killeen and Rosenbluth 1963, Coppi, Galvao, Pellat,
Rosenbluth, and Rutherford 1976]
Magnetic nulls in 3D play the role of X-points in 2D
Spine
Fan
[Greene 1988, Lau and Finn 1990]
Dungey’s Model for Southward and Northward IMF
“Magnetopause phenomena are more complicated as a result of merging.
This is why I no longer work on the magnetopause.” -- J. W. Dungey
[Dungey 1961, 1963]
Observation of magnetic null in space
[Xiao, Pu, Wang et al., Nature, 2006]
Equilbrium Current Density, J
[Greene and Miller 1995]
Equilbrium B-field
B-lines in a spherically tearing state
[Hu, Bhattacharjee, Dorelli, and Greene 2004]
Equilibrium
Perturbed
Field lines
penetrating the
spherical tearing
surface
Magnetopause reconnection topology in global MHD
Greene, J., Locating three-dimensional roots by a bisection method, JCP, 1992.
[Dorelli, Bhattacharjee, and Raeder 2006]
Magnetopause reconnection geometry
[Dorelli,
Bhattacharjee,
and Raeder, 2006]
Current density is concentrated at the intersection of the two separatrix
surfaces; the X line has collapsed into a double Y line topology.
Such ribbons were discussed by Longcope and Cowley [1996], but for
force-free fields.
Solar corona
astron.berkeley.edu/~jrg/ ay202/img1731.gif
www.geophys.washington.edu/
Space/gifs/yokohflscl.gif
Solar corona: heating problem
photosphere
Temperature
Density
Time scale
3
~ 5  10 K
23
~ 10 m
4
~ 10 s
3
corona
6
~ 10 K
12
~ 10 m
~ 20s
Magnetic fields (~100G) --- role in heating?
 Alfvén wave
 current sheets
3
Parker's Model (1972)
Straighten a
curved magnetic loop
Photosphere
Reduced MHD equations

J
 [ ,  ]   [ A, J]    2 
t
z
A

 [ , A] 
 2 A
t
z
low 
limit of
MHD
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
J
 [ A, J]  0 ,or B J  0
z
with  =   0. Field-lines are tied at z  0 , L .
Key Question: What is the nature of the solutions, given
a sufficiently complicated footpoint mapping?
Objections to Parker’s claim:
van Ballegooijen [1985], Longcope and Strauss [1994].
Cowley et al. [1997] and others.
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 based on reduced MHD equations,
periodic boundary condition in x
[Related results by Aly 2005 and Low 2006]
Implication
An unstable but smooth equilibrium cannot relax to a
second smooth equilibrium, hence must have current sheets.
Possible current sheet topology in line-tied geometry at
Quasi-Separatrix Layers (QSL)
Summary
•
•
•
•
Collisionless magnetic reconnection, governed by a generalized Ohm’s law holds
the promise to resolve a number of outstanding questions pertaining to impulsive
reconnection dynamics in laboratory and astrophysical plasmas, such as sawtooth
oscillations in tokamaks and reversed-field pinches, magnetotail subtorms, or
solar flares. We have elucidated the role of two-fluid effects in triggering fast
reconnection.
The question of scaling of reconnection rates in driven as well as undriven
2D/2.5D systems remains open. The standard picture suggests that there is a
“universal” fast reconnection rate, but there is now a strong body of evidence that
suggest that reconnection rates depend on plasma parameters (ion/electron skin
depths, guide field, plasma beta) as well as the system size.
3D reconnection calls for a new topological framework. We have discussed the
role of magnetic nulls and null-null lines in defining magnetic skeletons. Current
sheets can form in 3D in ideal quasi-separatrix layers, but the reconnection rate in
the presence of such singularities remains an open question.
Confluence of experimental and theoretical results in laboratory, magnetospheric,
and solar physics is key to answering fundamental questions.