SECTP LANL GSFC UMD
The Collisionless Diffusion Region:The Collisionless Diffusion Region:An IntroductionAn Introduction
Michael Hesse
NASA GSFC
SECTP LANL GSFC UMD
Overview:
Diffusion region basics
The (electron) diffusion region for anti-parallel reconnection
The (electron) diffusion region for guide-field reconnection
An avenue toward fast MHD reconnection without Hall terms
Acknowledgements: J. Birn, M. Kuznetsova, K. Schindler,M. Hoshino, J. Drake
SECTP LANL GSFC UMD
Magnetic Reconnection: Dissipation Mechanism(How does it work?)
time
DR
DR
DR
Conditions:IMPOSSIBLE (for species s) if
E
v s
B 0
SECTP LANL GSFC UMD
Electric Field Equations
E
v e
B
1
nee
�P e
me
e
v et
v e
v e
v i
B 1
nee
j
B 1
nee
�P e
me
e
v et
v e
v e
ye
yeyyyzxy
ey vv
t
v
e
m
y
P
z
P
x
P
enE
1
Electron eqn. of motion
At reconnection site
small, limited by me?important?
x
z
SECTP LANL GSFC UMD
Results for anti-parallel reconnection:
Brief review
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Magnetic field and ion-electron flow velocities
P. Pritchett
M. Hoshino
SECTP LANL GSFC UMD
F Bz
X
0
(x, z 0)dx
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
0 5 10 15 20 25 30 35 40
normal magnetic flux
mi/me = 9mi/me = 25mi/me = 64mi/me = 100
i t
evolution electron-mass independent!
Normal Magnetic Flux:
=> Local electron physics adjusts to permit large scale evolution
SECTP LANL GSFC UMD
mi/me=9, i t = 18
mi/me=100, i t = 16
Compare extremes along dashed lines
- ion quantities- electron quantities
SECTP LANL GSFC UMD
-> Ion scale features approx invariant.
-0.80
-0.60
-0.40
-0.20
0.00
0.20
0.40
0.60
0 5 10 15 20 25
Bz magnetic field
Bz(9)Bz(100)
shifted x
-1.0 100
-5.0 10 -1
0.0 100
5.0 10-1
1.0 100
0 5 10 15 20 25
ion v x
vix(9)vix(100)
shifted x
Large (ion) Scale Features
SECTP LANL GSFC UMD
Small (electron) Scale Features
-6.0 100
-4.0 100
-2.0 100
0.0 100
2.0 100
4.0 100
6.0 100
0 5 10 15 20 25
electron v x
vex(9)vex(100)
vex(
9)
shifted x
SECTP LANL GSFC UMD
-6.0 10 -2
-4.0 10 -2
-2.0 10 -2
0.0 100
2.0 10-2
4.0 10-2
6.0 10-2
0 5 10 15 20 25
Pxye
pxye(9)pxye(100)
shifted x
Pressure Tensor
x
vpP x
zxy
~
SECTP LANL GSFC UMD
xyeP
yzeP
SECTP LANL GSFC UMD
10.0<x< 11.0 -0.5<z< 0.5
0.076
-0.739
-1.555
-2.370
-3.185
-4.000
log f
-0.4
0.2
0.0
0.4
-0.2
uy
-0.4 -0.2 0.0 0.2 0.4ux
Sample Electron Distribution (Pxye)
Thermal inertia (nongyrotropic pressure)-based dissipationseems key to anti-parallel reconnection
SECTP LANL GSFC UMD
4/1
22 )/(
2
zBe
Tm
x
eez
E 2meTe vx'
[Biskamp and Schindler, 1971]
Can be explained by trapping scale:
=> Estimate of reconnection electric field
[Hesse et al., 1999][Kuznetsova et al., 2000]
“bounce motion” [Horiuchi and Sato, 1996]
SECTP LANL GSFC UMD
realistic electron massRicci et al.
3D – no LHD, kink, …Zeiler et al.
SECTP LANL GSFC UMD
But, some questions remain…
Sausage mode,Buechner et al.
Kink, LHD,Ozaki et al. Ion sound mode…
SECTP LANL GSFC UMD
…and other limitations, such as
-Finite (small) system size-Finite (small) ion/electron mass ratio-Finite (small) speed of light-Periodicity
…there is work to be done!
SECTP LANL GSFC UMD
What changes in the presence of guide field?
if guide field strong enoughelectrons are magnetizedno bounce orbitsno nongyrotropic pressures(?)bulk inertia dominant(?)
Method: Theory and PIC simulations
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Simulation Setup
- 1-D “Harris” Equilibrium, Lx= 2Lz= 25.6 c/pi
- Flux function: A = -ln cosh(z/) - normal magnetic field perturbation (X type, 2.5% of lobe field) - 0, 40, 80% guide field - Sheet Full-Width = c/pi
- Ti/Te = 5
- mi/me=256
- 100x106 particles - 800x800 grid
Results averaged over 60 plasma periods
SECTP LANL GSFC UMD
SECTP LANL GSFC UMD
By
P. Pritchett
Change of symmetry
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Parallel electric field it=16
…also analytic theory by Drake et al.
SECTP LANL GSFC UMD
Electric Field Equations
E
v e
B
1
nee
�P e
me
e
v et
v e
v e
v i
B 1
nee
j
B 1
nee
�P e
me
e
v et
v e
v e
ye
yeyyyzxy
ey vv
t
v
e
m
y
P
z
P
x
P
enEE
1
ll
Electron eqn. of motion
At reconnection site
small, limited by me?important?
x
z
SECTP LANL GSFC UMD
Magnitude of Bulk Acceleration Contribution
Time derivative of (negative) electron velocity in y direction:
-3
-2
-1
0
1
2
0
5
10
15
0 5 10 15 20 25 30
d/dt vey_max vey_max
time
SECTP LANL GSFC UMD
Pxye
Pyze
SECTP LANL GSFC UMD
-(vezBx-vexBz)
-me(ve.grad vey)/e
z
P
x
P
enyzexye
e
1
SECTP LANL GSFC UMD
Electron Distribution Functions
F(vx,vy) F(vx,vz) F(vy,vz)
vx
vy
vx
vz
vy
vz
SECTP LANL GSFC UMD
..pressure tensor nearly(?) gyrotropic
BBB
pppeg
engege
2
||
1P
PPP
But:0|
2
||
2
||
yyyeg BB
B
pp
B
ppBB
P
if Bx, Bz=0
-> nongyrotropy important. How to estimate?
SECTP LANL GSFC UMD
QPPPPPP
T
eee
Teeeeee
e BBm
evvv
t
Scaling the pressure tensor evolution equation
P
L
vP Pe
L
Q
Assume
ijii
e
PP
vL
/,1
ignore heat flux…
SECTP LANL GSFC UMD
y
xzzeyye
ey
e
zzexye B
BPP
z
vPP
y
zxxeyye
ey
e
xxeyze B
BPP
x
vPP
Hesse, Kuznetsova, Hoshino, 2001
Pressure tensor approximations
SECTP LANL GSFC UMD
Electron Pressure Tensors
from simulation approximation
Pxye Pxye
Pyze Pyze
critical difference at reconnection site!
SECTP LANL GSFC UMD
-0.003
-0.002
-0.001
0
0.001
0.002
0.003
0.004
-0.4 -0.2 0 0.2 0.4
Pyze
at x=13.15, t=16
z coll. skin depth
SECTP LANL GSFC UMD
z
Q
B
BPP
x
vP
z
Q
x
Q
B
BPP
x
vPP
xyze
ey
zxxeyye
ey
e
xxe
xyzexxye
ey
zxxeyye
ey
e
xxeyze
1
1
QxxyeQxyze
Pyza approximation
SECTP LANL GSFC UMD
ss fudm ))()((3 vuvuvuQ�
0
][
][
][
)(
rsisjkrrjks
rsjsikrriks
rsksijrrijs
srs
s
jl
likl
il
ljkl
kl
lijl
lijkkijlkjiljiklijklll
ijk
BQBQ
BQBQ
BQBQ
m
e
vx
Qvx
Qvx
Q
vQvvPvvPvvPx
Qt
Heat Flux Tensor Time Evolution
lots of work
SECTP LANL GSFC UMD
022
)2(
xxxzzxxxyxyzzxyye
yl
lxxl
xl
lyxl
xl
lxyl
lxyxxxylyxxlll
BQBQBQBQm
e
vx
Qvx
Qvx
Q
vQvvPvvPx
22 5.05.0
1xyzyxxzxxyyxxx
yxyz vPvvP
zvPvvP
xQ
x
vvPvvP
xQ x
y
yxxyxxx
yxyz
)(
1
Approximations for Qxyze
neglect ;0t
Assume near gyrotropy, By>>Bx, Bz
Leading order, Pii>>Pjk
x,y,x component:
SECTP LANL GSFC UMD
-0.008
-0.006
-0.004
-0.002
0
0.002
0.004
0.006
0.008
-0.4 -0.2 0 0.2 0.4
Q xyze and approximation, x=13.15
Q xyze
Q xyze approximation
z
Approximations for Qxyze
From simulation:
Approximation:
Ok in center, differencedue to 4-tensor?
SECTP LANL GSFC UMD
21
2
2
21
2
2
020
02
1
1||
11~|~|
L
cE
L
cvB
ne
mvB
Lx
vv
e
mE
pe
convection
pe
ze
ezyz
einertial
x
vvP
zB
BPP
x
vPP x
y
yxx
ey
zxxeyye
ey
e
xxeyze
1
22
2
222
2
2
2
2
||11
|~|
11~
11|~|
L
rE
mn
P
LE
z
vvP
zenx
vvP
zenE
Linertial
eey
xxinertial
z
y
yxx
ee
x
y
yxx
eepressure
Scaling of diffusion region
=> 2 Scale lengths: Collisionless skin depthElectron Larmor radius in guide field
SECTP LANL GSFC UMD
Physical Mechanism:
Larmor orbit interacts with “anti-parallel” B components
SECTP LANL GSFC UMD
3D Modeling
M. Scholer et al.: Formation of“2D” channel
J. Drake et al.: Buneman modes,electron holes, anomalous resistivity
SECTP LANL GSFC UMD
P. Pritchett: inertia important
SECTP LANL GSFC UMD
…and other limitations, such as
-Finite (small) system size-Finite (small) ion/electron mass ratio-Finite (small) speed of light-Periodicity
…there is work to be done!
SECTP LANL GSFC UMD
Results from GEM reconnection challenge:
•Hall effect (dispersive waves) speeds up reconnection rate
•Reconnection rate otherwise independent on model
•MHD models with simple resistivity show only slow reconnection rates
Question:
Are Hall effects the only way to include fast reconnection in MHDmodels?
SECTP LANL GSFC UMD
Approach:
•Hall effect result of ion-electron scale separation
•Eliminate scale separation by- Choosing equal ion and electron mass- Choosing equal ion and electron temperatures
•Simple and cheap…, includes ion and “electron” kinetic physics
•“Small” GEM runs with and without guide field
•“Large” runs, with and without guide field
SECTP LANL GSFC UMD
GEM-size run, no By
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0
0.02
0.04
0.06
0.08
0.1
0.12
0
0.4
0.8
1.2
1.6
2
2.4
0 5 10 15 20 25 30
small, no By
reconnection E recflux
time
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0
0.5
1
1.5
2
2.5
3
3.5
0 5 10 15 20 25 30
mi/me=256
rec. E
recflux/0.0
time
GEM-size run, no By
me=1 me=1/256
SECTP LANL GSFC UMD
GEM-size run, By=0.8
SECTP LANL GSFC UMD
GEM-size run, By=0.8
0
0.05
0.1
0.15
0
0.5
1
1.5
2
2.5
3
0 5 10 15 20 25 30
typical
d flux/dt
recflux
time
0
0.05
0.1
0.15
0.2
0
0.5
1
1.5
2
2.5
3
3.5
4
0 5 10 15 20 25 30
typical
rec. E recflux
time
me=1 me=1/256
SECTP LANL GSFC UMD
large run, By=0.
SECTP LANL GSFC UMD
large run, By=0.8
SECTP LANL GSFC UMD
large run, By=0.
0
0.05
0.1
0.15
0.2
2
3
4
5
6
7
8
9
10
0 10 20 30 40 50 60
typical
rec. E recflux
time
2
3
4
5
6
7
8
9
10
0
0.05
0.1
0.15
0.2
0 10 20 30 40 50 60
typical
recflux reconnection E
time
large run, By=0.8
Reconnection rates similar to GEM problem
SECTP LANL GSFC UMD
initial By=0.8 initial By=0.
By, both large runs, t=40
no quadrupole or quadrupolar modulation!
SECTP LANL GSFC UMD
large run, By=0., t=40
Pxye
Pyze
vix
jiy
SECTP LANL GSFC UMD
large run, By=0.8, t=40
Pxye
Pyze
vix
jiy
SECTP LANL GSFC UMD
Electric Field Equations
E
v e
B
1
nee
�P e
me
e
v et
v e
v e
v i
B 1
nee
j
B 1
nee
�P e
me
e
v et
v e
v e
Electron eqn. of motion
x
z
Approximate representation in MHD:
ii
i Pen
BvE�
1
SECTP LANL GSFC UMD
Additional slides
SECTP LANL GSFC UMD
Pxye Pyze
jyi jye By
A tour of the reconnection region…
SECTP LANL GSFC UMD
Mass Dependence of Electron Diffusion Region:Simulation Setup
- 1-D “Harris” Equilibrium, Lx= 2Lz= 25.6 c/pi
- Flux function: A = -ln cosh(z/) - normal magnetic field perturbation (X type, 5% of lobe field) - Sheet Full-Width = c/pi
- Te/Ti = 0.2 - me/mi=1/9-1/100
- pe/ce=5
- 50x106 particles - 800x400 grid
SECTP LANL GSFC UMD
0
0.5
1
1.5
2
2.5
3
0
0.05
0.1
0.15
-5 0 5 10 15 20 25 30 35
typical
recflux d flux/dt
time
mi=me, By=1
rate slightly reduced due to higher plasma mass
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Additional Material
SECTP LANL GSFC UMD
Pyze
18.028
25.0
35.0
1~
1|
1
E
E
z
P
enenE yz
eye
ey P
Magnitude of Pressure Tensor Contribution
ne
SECTP LANL GSFC UMD
Particle Picture: Straight Acceleration and Thermalization
Question: Are electrons transiently accelerated while crossingthe diffusion region, or is some of the energy thermalized?
Approach: Integrate 104 electron orbits in vicinity of reconnection region
Relevance: straight acceleration ->
thermalization ->e
e
eyee
en
ve
m
P
v
1
SECTP LANL GSFC UMD
-0.5
0
0.5
1
1.5
2
-12 -10 -8 -6 -4 -2 0 2
kinetic energy change as function of delta y
delta Ek
y = -2.5605e-05 - 0.17785x R= 0.98882
delta y
-0.5
0
0.5
1
1.5
2
-12 -10 -8 -6 -4 -2 0 2
delta y-component of kinetic energy vs. delta y
delta Eyk
y = -0.027939 - 0.16877x R= 0.9873
delta y
yE
yE
ykin
kin
67.1
78.1
,
Approximately 6% of energyis thermalized
y
kin
Ee
Ey
SECTP LANL GSFC UMD
-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
13.15 13.2 13.25 13.3 13.35 13.4 13.45
orbit( 6293): x-z plane
x
0
0.1
0.2
0.3
0.4
0.5
0.6
0
0.05
0.1
0.15
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
orbit( 6293): Ekin y and x-z
Ekin_y
Ekin_xz
time
x-z energization
-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
13.15 13.2 13.25 13.3 13.35 13.4 13.45
orbit( 6293): z-x acceleration phase
z
x
SECTP LANL GSFC UMD
Contours of Poloidal Magnetic Field
Scale length related to electron Larmor radius
SECTP LANL GSFC UMD
Vmax= 0.65
Vmax= 2.8
SECTP LANL GSFC UMD
Scaling the pressure tensor evolution equation
0T
eee
Teeeeee BB
m
evvv
PPPPP
zxy
,
0
yzyxxyyzxzx
xzyzxxxyxxxyyzxzyxxxxye
PPPP
vPvPvPvPvPPv
xy component
near reconnection site:
ijii
zxy
PP
BBB
,
SECTP LANL GSFC UMD
0
0.5
1
1.5
2
2.5
3
3.5
0 5 10 15 20 25 30
magnetic flux normal to current sheet
recflux/0.0recflux/0.4recflux/0.8
time
Reconnection faster for smaller guide fields
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