The Diffusion Region of Asymmetric
Magnetic Reconnection
Michael Shay – Univ. of Delaware
Bartol Research Institute
Collaborators
• Paul Cassak
• Our Asymmetric reconnection publications (no guide field):
– General Scaling theory and resistive MHD:
• Cassak and Shay, Physics of Plasmas, 14, 102114, 2007.
– Hall MHD simulations
• Cassak and Shay, GRL, (In press)
Semantics
• Diffusion region
– A non-MHD region where at least one species
is not frozen-in
– Not necessarily irreversible dissipation
• Example: Hall region of regular collisionless
reconnection.
Review: Reconnection
Magnetic Reconnection
Vin
d
CA
Process breaking the frozen-in
constraint determines the width of
the dissipation region, d.
Y
Z
X
Magnetic Reconnection Simulation
Jz and Magnetic Field Lines
Y
QuickTime™ and a
BMP decompressor
are needed to see this picture.
X
• d
Reconnection drives
convection in the Earth’s
Magnetosphere.
Kivelson et al., 1995
Reconnection in Solar Flares
•
X-class flare: t ~ 100 sec.
•
B ~ 100 G, n ~ 1010 cm-3 , L ~ 109 cm
•
tA ~ L/cA ~ 10 sec.
F. Shu, 1992
Calculating Reconnection Rate
Vin
d
cA
D
• Reconnection rate  Vin
Z
• Conservation of mass: Flow into and out of dissipation region:
Vin ~ (d /D) cA
•
d determined by the process breaking the
frozen-in constraint.
=> The spatial extent of the dissipation region is of
key importance to determining the reconnection rate.
Y
X
Two Types of 2D Reconnection
Out of Plane Current
•
d << D
D
Vin << cA => Slow
Y
Out of Plane Current
•
d ~ D
Vin ~ cA => Fast
D
Z
X
Kinetic Reconnection (cont.)
• dissipation region in hybrid model ( Shay, et al., 1999)
• Effect of Hall Physics

• Ion dissipation region
– Controls R. Rate
Vi
Vin ~ (c/pi /Di) cA
(c/pi /Di) ~ 1/10
Ji
c/pi
No system size Dependence!
Di
• Electron dissipation region
– No impact on R. Rate
Je
c/pe
Vine ~ (c/pe /De) cAe
Y
De
Z
X
Whistler signature
•
Magnetic field from particle simulation (Pritchett, UCLA)
•Self generated out-of-plane field is whistler signature
•Confirmed with satellite and laboratory measurements.
Overview: Asymmetric Reconnection
• What is Asymmetric Reconnection?
• Diffusion region analysis
• Resistive MHD Simulations
– No guide field
• Hall MHD Simulations
– No guide field
• Conclusions
Asymmetric Reconnection
• Different B,n on either side of diffusion
region.
• Dayside magnetosphere
• Solar reconnection?
• Heliopause reconnection
High
• dn
Low B
Low n
High B
Intense currents
• MHD not valid
• No frozen-in
Kivelson et al., 1995
Observation
• Asymmetric
– Reconnecting B-field
– Density
– Temperature
Previous Work
• Shock structure
– Petschek slow shocks => Intermediate wave+expansion fan (Levy et al., 1964)
– Further work
• Petschek and Thorne, 1967; Sonnerup, 1974; Cowley, 1974; Semenov et al., 1983, MHD
(Hoshino and Nishida, 1983; Scholer, 1989; Shi and Lee, 1990; Lin and Lee, 1993; La Belle-Hamer et al., 1995;
Kinetic - Hybrid: (Lin and Lee, 1993; Lin and Xie, 1997; Omidi et
al, 1998; Krauss-Varban et al., 1999; Nakamura and Scholer, 2000; …), Particle: - Okuda, 1993.
Ku and Sibeck, 1997; Ugai, 2000; …),
– Other relevant studies: Ding et al., 1992; Karimabadi et al., 1999;
Siscoe et al., 2002; Swisdak et al., 2003; Linton, 2006; many dayside studies
• Scaling studies undertaken only recently
–
–
–
–
–
–
Diamagneticd Stabilization (Swisdak et al., 2003)
Orientation of X-line, outflow speed (Swisdak and Drake, 2007)
MHD studies: (Borovsky and Hesse, 2007, Birn et al., 2008)
Global MHD (Borovosky et al., 2008)
PIC: (Pritchett, 2008), Tanaka, 2008; Huang et al., 2008
PIC-Satellite comparisons (Mozer, Pritchett et al., 2008)
Conservation Laws
• Write MHD in conservative form ( = mass density, v = flow velocity,
B = magnetic field, P = pressure, E = electric field,
B2
1
P
2
E  v 

2
  1 8
= total energy)
– Integrate over closed surface.
r
r

    v
t
r r
r  rr 
B 2  t BB 
r

 v      vv   P 

I
t
8

4





r r
2

r


v
B r
B
r
E
    E  P 
B
v
t
8 
4
 

r
r
r
B
 c  E
t
 
 dS  v  0
 





B2 
BB 
dS


vv

P

I


0


 
8

4






vB 
B2 
 dS    E  P  8  v  4 B   0


 dS E  0


More General Diffusion Region
• Steady state diffusion relation
• Integrate conservation relations
 dS  v  0
Conservation of mass


B2 
BB 
 dS   vv   P  8  I  4   0



vB 
B2 
 dS    E  P  8  v  4 B   0


 dS E  0
Conservation of momentum

Conservation of Energy
B/t = 0
B1
v1 1
out
2d
2L
vout
B2
v2 2

More General Diffusion Region
• Steady state diffusion relation
• Integrate conservation relations
 v
1
Conservation of mass
Conservation of momentum
Conservation of Energy
B/t = 0
1


Pressure Balance
 B12
B22 
1

2
v

v
L
~

v
v 2d

1
2
out out  out

8 
2

 8
v1 B1 ~ v2 B2
B1
v1 1
out
2d
2L
vout
B2

 2 v2 L ~ out vout 2d
v2 2
Asymmetric Scaling Relations
• Solving gives
Outflow speed
Reconnection Rate
• Need out
2
out
v
~
B1 B2
B1  B2
4 1 B2  2 B1
1  out B1 B2 
2d
E~ 
v
c  1 B2  2 B1  out L
Outflow Density?
•
Assume reconnected flux tubes mix and conserve total volume.
– Each flux tube contains same amount of flux:
• B1A1 ~ B2A2
out
L
M 1 A1 L  2 A2 L
~
~
V
A1 L  A2 L
out ~

2
out
v
~
1 B2  2 B1
B1 B2
B1  B2
4out
and
1
A1
2
L
1  2B1 B2 
d
E~ 
v
c  B1  B2  out L
A2
Structure of the Dissipation Region
• Since v1 B1 ~ v2 B2, the stronger magnetic field flows in slower
– So it makes sense that the
X-line is displaced toward
the strong field side of
the dissipation region.
But this is incorrect!
Weak field B1
Strong field B2
• The X-line is actually shifted toward the weak field side!
– Why? While the flow coming in the strong
field side is slower, the flux of energy is larger.
Weak field B1
 B2  B12
v2 ~
v1 ~ 
v1

8
8
 B1  8
B22
B1 B2
X
Strong field B2
dX1
dX2
Calculation of Location of X-line
• Evaluate conservation of energy for volume from edge to X-line
B1
v1
1
out
dX1
2L
vout
B2
X
v2
 B12 
1

2
v1  L ~  out vout  voutd X 1

2

 8 
 B22 
1

2
v2  L ~  out vout  voutd X 2

2

 8 
dX2
2
Their ratio gives:
d X 2 B2
~
d X 1 B1
Location of the Stagnation Point
• Similar argument for mass flux
2 v2 ~ 2
v1 B1
B2
 2 B1 
~
 1 v1

B
 1 2
d S2 2 B1
~
d S1 1 B2
• Stagnation point offset toward side with smaller B/.
v1
B1, 1
BS2
dS1
dS2
v2
B2 ,  2
X-line and Stagnation Point
are not colocated!
• There is a flow across the X-line
– Generic to asymmetric reconnection!
– Previous magnetopause simulations (Siscoe, 2002; Dorelli et al., 2007, …)
– Quantitative predictions of the location of X-line and stagnation point (Cassak and
Shay, 2007) have been questioned (Birn et al., 2008)
Which plasma flows across the X-line?
• Inflow Alfven speeds control flow across X-line
d S1  d X 1

B22
42

B12
41
Since cAsp > cAsh  there is a flow of magnetosheath
plasma in to magnetosphere. (Matches observations.)
Results are General
• These relations give E and vout in terms of upstream
parameters.
– No specificaion of diffusion mechanism or Hall term.
– General applicability
• Require diffusion (non-MHD) mechanism to determine absolute values:
– Sets diffusion region widths d d and L
– Determines actual reconnection rate
Resistive MHD
• To find an absolute reconnection rate, we need to specify a dissipation
mechanism. For asymmetric Sweet-Parker,
v1
c
v1
 B1 :  J 1
c 2
~
d 1 4d 12
• Uniform resistivity Sweet-Parker reconnection
1
c 2
E~
B1 B2 vout
c
4 vout L
Fluid Simulations
• Double current sheet
configuration
• x = outflow
y = out-of-plane
z = inflow
•
•
B, T tanh functions
n balances B2
Resistive MHD Simulations
•
•

•

•
•
V normalized to cA, Length normalized to L0
Size: 409.6 X 204.8, 4096 X 2048 grids
  0.05
(Lundquist number = 8,192-40,960)
min = 1 initially
n1 = n 2
[B1,B2] = [1,1], [2,1], [3,1], [4,1], [5,1], [4,2]
Resistive MHD
Simulations
• [B1,B2] = [1,3]
• [n1,n2] = [1,1]
• x = outflow
y = out-of-plane
z = inflow
MHD Results
Out-of-plane
current density J
Cut across X-line
along inflow
S X
Cut across X-line
along inflow
Decoupling of X-line
and stagnation point
borne out in MHD
simulations.
MHD Results
• Color = out-of-plane current
White = magnetic field lines
• Initial field asymmetry = 3,
no density asymmetry
• Signatures
– Typical “bulge” into
low-field region
– Particles flow across
X-line
Energy and Mass Flux Check
• Determined geometry of
diffusion region from
simulations.
• Energy and Mass Flux balances
in each sub-region
Flux out
– Non-trivial
Flux in
Verification of Scaling
•
Scaling laws for outflow speed vout and reconnection rate E in terms of geometry
and upstream parameters tested
vout
E
B1 B2 / 4
•
•
E
2d vout B1 B2
L c B1  B2
Very good agreement
Other studies find agreement:
–
–
–
–
Borovsky and Hesse, 2007 (anomalous resistivity MHD)
Birn et al., 2008 (Anomalous resistivity MHD)
Borovsky et al., 2008 (Global MHD)
Pritchett, 2008 (Kinetic PIC)
c 2
v BB
4 L out 1 2
Hall MHD Simulations
• Two dimensional Hall-MHD simulations
• Anti-parallel magnetic fields
• Three sets of runs
– Asymmetric fields [B01,B02] = [1,1], [2,1], [3,1], [0.5,1], Symmetric density
– Asymmetric density [n01,n02] = [1,1], [2,1], [3,1], [0.5,1], Symmetric field
– Asymmetric density and field [B01(n01),B02(n02)] = [2(1),1(2)], [1(1),0.5(4)]
• Asymmetric initial temperature to balance pressure
• Box size = 204.8 x 102.4 c / pi
• Grid scale = 0.05 c / pi
• me = mi / 25 (density asymmetry not included in electron inertia term)
 min = 4 initially
Hall MHD
Simulations
• [B1,B2] = [1,2]
• [n1,n2] = [2,1]
• x = outflow
y = out-of-plane
z = inflow
Hall-MHD Results
• Electron and ion stagnation
points different!
Cuts across x-line along inflow
Top - field lines (white) and out-of-plane
magnetic field (color)
Bottom - electron (black) and ion (white)
flow lines and out-of-plane current (color)
Initial field asymmetry = 3
Verification of Scaling
• Generalized Sweet-Parker like scaling is satisfied for both electrons and
ions.
vout (ions)
vout (electrons)
E
theory
B1 B2 (B1  B2 )
4 ( 2 B1  1 B2 )
B1 B2 (B1  B2 )
4 ( 2 B1  1 B2 )
The Big Picture
Magnetospheric Applications?
•
Agreement of the scaling of E for Hall reconnection
 d / L ~ 0.1 is independent of the asymmetry in B and 
•
Are the results applicable to dayside magnetopause reconnection?
– Yes (Borovsky)
• In global MHD simulations, the reconnection rate at the nose of the magnetopause agreed
with E based on local parameters rather than the solar wind electric field (Borovsky et
al., 2008).
– No (Dorelli)
• The analysis is manifestly two-dimensional, whereas 3D effects
(such as flows) are important at the magnetopause.
• The orientation of the X-line between arbitrary fields not predicted.
– Critical Question:
• Can significant portions of dayside reconnection be characterized as quasi-2D?
• Does a fluid element traversing the diffusion region see 3D effects?
Solar Wind-Magnetospheric
Coupling Models
• Newell et al., 2007
– Best model to date,
but it uses ad hoc fitting
to achieve performance
• Borovsky (2008) used
our scaling result to derive
a coupling function from
first principles
– It performed as well as
Newell’s
Scaling of reconnection is a
potential starting point for a
quantitative understanding of
solar wind-magnetospheric coupling
Conclusion
• We have derived the scaling of the reconnection rate and outflow speed
with upstream parameters during asymmetric reconnection [Cassak and
Shay, Phys. Plasmas 14, 102114 (2007)]
– Numerical simulations agree with the theory for
collisional and collisionless (Hall) reconnection
• Signatures of Asymmetric Reconnection
– X-line and stagnation point not coincident for asymmetric B field
– There is a bulk flow across the X-line
• Potential applications to the dayside magnetosphere (Borovsky, 2008;
Turner et al., in prep), though future work is needed
Future Directions
• Much work to be done
• Effect of guide field
– Diamagnetic stabilization (Swisdak et al., 2003)
– Orientation of X-line (Swisdak et al, 2007)
• More realistic two-scale diffusion region
– Requires Kinetic PIC
– Pritchett, 2007
• Separatrix structures
– Mozer et al., 2007
• Linking separatrix structures with diffusion region
structure.