William Daughton Plasma Physics Group, X-1 Los Alamos National Laboratory Presented at: Second Workshop on Thin Current Sheets University of Maryland April 19, 2004 The Onset of Magnetic Reconnection
William Daughton
Plasma Physics Group, X-1
Los Alamos National Laboratory
Presented at:
Second Workshop on Thin Current Sheets
University of Maryland
April 19, 2004
The Onset of Magnetic Reconnection
Motivation for this work Current sheet geometry is often employed to study the
basic physics of collisionless magnetic reconnection
Kinetic Simulations are typically 2D with large initial perturbation:
a. Does not allow instabilities in direction of currentb. Avoids the question of onset completely
www-spof.gsfc.nasa.govwww-spof.gsfc.nasa.gov
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rB
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rJ Courtesy of Hantao Ji (PPPL)
Basic Approach
€
cost ∝mi
me
⎛
⎝ ⎜
⎞
⎠ ⎟
1+n / 2
n → dimensionsFor a given problem with fixed box size
Explicit PIC must resolve all relevant scales
€
cΔt < Δx ωpeΔt <1 ΩceΔt <1 Δx ≈ λ D
3D Simulations - Must choose very artificial parameters
2D Simulations - More realistic parameters are possible
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mi
me
,ωpe
Ωce
, etc
€
Bx
€
z
€
Jy
€
x
€
y
€
z − x plane → Tearing →γ
Ωci
~ 0.05
z − y plane → LHDI →γ
Ωci
~ 5
Harris Current Sheet
€
fs =n(z)
π 3 / 2V||sV⊥s2
exp −vx
2
V||s2
−vy −Us( )
2+ vz
2
V⊥s2
⎡
⎣
⎢ ⎢
⎤
⎦
⎥ ⎥
MainDistribution
€
fbs =nb
π 3 / 2v th3
exp −vx
2 + vy2 + vz
2
Vbs2
⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥
BackgroundDistribution
€
n(z) = no sech2 z
L
⎛
⎝ ⎜
⎞
⎠ ⎟
€
V||s =2T||s
m s
V⊥s =2T⊥s
m s
Us =2cT⊥s
qsBoL
Anisotropy
€
T⊥s
T ||s
Thickness
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ρi
L=
U i
V⊥s
€
Bx (z) = Bxo tanh(z /L)
€
Jy (z) =cBo
4π Lsech2 z
L
⎛
⎝ ⎜
⎞
⎠ ⎟
€
x€
z
2D Simulations of Tearing
Consider 3 simulations - Only change the box length
1. Single island saturation
2. Two island saturation
3. Four island saturation
€
ρi
L=1
mi
me
=100Ti
Te
=1ωpe
Ωce
= 5Equilibrium Parameters
€
γΩci
≈ 0.11 kxL ≈ 0.5
Reduced by 30% for
€
mi
me
=1836
€
Box Size → 4πL × 4πL 640 × 640 grid 50 ×106 particles
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Box Size → 8πL × 4πL 1280 × 640 grid 100 ×106 particles
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Box Size →16πL × 4πL 2560 × 640 grid 200 ×106 particles
Single Island Tearing Saturation
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γΩci
€
kxL€
T⊥e
T||e
=1
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z
L
€
x /L
€
T⊥e /T||e
€
T⊥e
T||e
= 0.95
Linear Growth Rate Mode Amplitude
€
tΩci
PIC Simulation
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Ay
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Two Island Coalescence
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z
L
€
x /L
€
T⊥e /T||e
Mode Amplitudes
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Ay
€
kxL
€
tΩci
€
γΩci
Linear Growth Rate
M=1
M=2
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T⊥e
T||e
=1
€
0.95
€
0.9
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Four Island Coalescence
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z
L €
x /L€
z
L
Onset Stage• Central region of box
• Linear tearing islands
• Coalescence
• Very slow process
Fast Reconnection
• Show entire box
• Large scale reconnection
• Saturation limited by box
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tΩci = 0 →190
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tΩci =190 → 244
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x /L
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Reconnection Onset from Tearing
How might this change in 3D?• LHDI is much faster than tearing
• 2D simulations in oblique plane
• Can the LHDI modify onset physics ?
• Single island tearing saturates at small amplitude
• Onset requires coalescence of many islands
• Finite Bz is stabilizing influence
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1−Te⊥
Te||
⎛
⎝ ⎜
⎞
⎠ ⎟>
ρ e
LLaval & Pellat 1968Biskamp, Sagdeev, Schindler, 1970
Scholer et al, PoP 2003
Horiuchi
Shinohara & Fujimoto
Pellat, 1991Pritchett, 1994Quest et al, 1996
Sitnov et al, 1998 -> can go unstable?
Tearing is stablein magnetotail
Lower-hybrid Drift Instability (LHDI)• Driven by density gradient
• Fastest growing modes
• Real frequency
• Growth rate
• Stabilized by finite beta
• Primarily electrostatic and localized on edge
€
kyρ e ~ 1
€
β =8π n(z)(Te + Ti)
Bx2(z)
€
ω ≤Ωlh =ωpi
1+ ωpe2 /Ωce
2( )
1/ 2 ≈ ΩciΩce( )1/ 2
€
γ≤Ωlh
€
ˆ φ (z) = ˜ φ (z) exp −iωt + iky y[ ]€
z /L
Example Eigenfunction
GoodAgreement
Carter, Ji, Trintchouck, Yamada, Kulsrud, 2002Davidson, Gladd, Wu & Huba, 1977
Huba, Drake and Gladd, 1980Theory Experiment
€
˜ φ (z)
Bale, Mozer, Phan 2002 Observation
€
U i < Vthi ⇒ kinetic (dissipative)
U i > Vthi ⇒ fluid - like (reactive)
Established Viewpoint on LHDI
€
z
€
y€
˜ φ (z)• Localized on edge of layer
• Small anomalous resistivity
• Wrong region to modify tearing
• Not relevant to reconnection
New results challenge this conclusion
1. Direct penetration of longer wavelength linear modes
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ky ρ iρ e ~ 1
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ρi
L>1
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ρi
L≤12. Nonlinear development of
short wavelength modes
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kyρ e ~ 1
Penetration of LHDI
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mi
me
=1836ρ i
L= 2
€
Δx ≈1.4λ D Δt Ωce ≈ 0.08 Box Size =12L ×12L
1280 ×1280 cells 500 ×106 particles
€
z
L
€
z
L
€
z
L
€
yL
tΩci=3
tΩci=11
tΩci=8
€
Bx (y,z) − Bxo tanh(z /L)
€
yL
€
kyL = 2.62 ⇒ ky ρ iρ e ≈ 0.8
tΩci=13
tΩci=13
€
Bx€
Jy€
z
L
€
z
L
2D Simulation of Lower-Hybrid
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ρi
L=1
mi
me
= 512Ti
Te
= 5ωpe
Ωce
= 4nb
no
= 0.02Tb
Te
=1Equilibrium Parameters
€
Box Size →12L ×12L 2560 × 2560 grid 1.6 ×109 particles
ΔtΩce = 0.08 Δz = Δy ≈ λ D 256 processors
Simulation Parameters:
Thicker Sheet Colder Electrons
More relevant to magnetospheric plasmas
Background
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Electrostatic Fluctuations
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ωr /Ω lh = 0.54 γ /Ωci =1.93
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z /L
€
˜ φ (z)
€
ωr /Ω lh = 0.57 γ /Ωci = 2.26
€
˜ φ (z)
€
z /L
Two fastest Growing modes
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kyρ e ≈ 0.75
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z
L
€
y /L
Lower-Hybrid Drift Mode
Lower-Hybrid Drift Mode
Fluctuations are confined to the edge of the sheet
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˜ φ (z)
€
Ey (y,z)
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Evolution of Current Density
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z
L
€
y /L
€
Jy
Jo
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Jy (z) = Jo sech2(z /L)Initial
Y-averaged
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z
L
€
Jy = −eneVey + eniViy
Contours of
€
Jy (z,y)
€
Jy =1
Ly
Jy∫ (z,y) dy
€
z
L
€
y /L
€
ni(z) = no sech2(z /L)Initial
Y-averaged
€
z
L
Contours of
€
ni(z,y)
€
ni =1
Ly
ni∫ (z,y) dy
Evolution of Ion Density
€
ni
no
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z
L
€
y /L
Initial
Y-averaged
€
z
L
Contours of
€
Viy (z,y)
€
Viy =1
Ly
Viy∫ (z, y) dy
€
Viy
Vthi
Evolution of Ion Velocity
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Viy (z) =U i
1+ nb cosh2(z /L)
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€
z
L
€
y /L
Initial
Y-averaged
€
z
L
Contours of
€
Vey (z,y)
€
Vey =1
Ly
Vey∫ (z,y) dy
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Vey
Vthe
Evolution of Electron Velocity
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Vey (z) =Ue
1+ nb cosh2(z /L)
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€
z
L
€
y /L
Y-averaged
€
z
L
Contours of
Evolution of Electron Anisotropy
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T⊥e
T ||e
€
T⊥e
T ||e
€
T⊥e
T ||e
Resonant Scattering of Crossing Ions
€
z
€
y
€
Bx (z) = Bo tanhz
L
⎛
⎝ ⎜
⎞
⎠ ⎟
€
δi ≈ 2ρ iL
Scale forCrossing Orbit
€
vy
€
vz
€
U i
€
ωky
≈U i
2
Noncrossing
Crossing
Crossing Example of scattering
Lower-hybrid fluctuations
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˜ φ (z)
€
zlh
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€
z
L
€
y /L€
z
L
Contours of
€
φ(z, y)
Electrostatic Potential
€
−e φTe
Net gain + + + + + + + + +
Net gain + + + + + + + +
Net loss - - - - - - - - -
Electron Acceleration
€
mene
dVe
dt= −∇ • Pe − ene E+
Ve ×B
c
⎛
⎝ ⎜
⎞
⎠ ⎟
Neglect
€
Vey =c
eBxne
∂Pe
∂x−
c
Bx
∂φ
∂z
Use EquilibriumProfiles
€
z
L
€
y /L€
z
L
€
Vey /Vthe
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Inductive Heating of ElectronsEvolution of current profile modifies magnetic field
€
Jy
€
Bx
For electrons, magnetic field changes slowly
Changes on the ion time scale
€
pdq∫
€
μ =mv⊥
2
2B
€
z
€
y
€
δe ≈ 2ρ eL
€
p = mv⊥r
dq = dθ
€
T⊥e (t)
T⊥e0
≈Bx (t)
Bx 0
How to construct adiabatic invariant for these orbits?
Magnetic Moment
Inductive Heating
Adiabatic Invariant
€
x
€
Λ(x)
Anisotropic Electron Heating
€
z
L
€
y /L
€
T⊥eT⊥e 0
€
T⊥e
T⊥e 0Contours of Y-averaged
€
T⊥e (t)
T⊥e 0
Y-averaged
€
Bx (t)
Bx 0
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Physical Mass
€
mi
me
=1836
€
5120 × 5120 grid
6 ×109 particles
Plasma parameters are same butnumerical requirements increase
€
tΩci = 7
Results show same basic physics
Details are described in preprint
How big of a mass ratio is needed?
What about lower mass ratio?
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z
L
€
y /L
€
Jy
€
z
L
€
mi
me
= 512
€
mi
me
=100
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1. Critical thickness for process to occur
2. Potential structure accelerates electrons
3. Enhances tearing mode
New Model for Fast Onset of Reconnection
€
zlh ≈ (1− 2)L
€
δi ≈ 2ρ iL
€
ρi
L≈ 0.5
Lower-hybrid drift instability
Lower-hybrid drift instability
1. Current density2. Anisotropy
€
kxL€
γΩci
= 0.035
€
γΩci
€
T⊥e
T||e
=1
€
T⊥e
T||e
=1.1
€
γΩci
= 2.2
4. Rapid onset of reconnection
Critical Scale
Tearing Growth RateForslund, 1968
J. Chen and Palmadesso, 1984
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Test this idea at reduced mass ratio
€
γΩci
€
kxL€
T⊥e
T||e
=1
Tearing Growth Rate
€
T⊥e
T||e
=1.5
€
z
L
€
z
L
€
x /L
Factor of 17 increase in growth rate
Fastest mode shifts to shorter wavelength
Growth of small islands --> Coalescence
Rapid onset of large scale reconnection
Initialize previous 2-Mode case with
€
T⊥e
T||e
=1.5
€
mi
me
=100
Electron Anisotropy Instabilities?Theory of Space Plasma Microinstabilities, S.P Gary
€
T⊥e
T||e
<1 Ωci < ω << Ωce
kc
ωpi
>1 k × Bo = 0€
T⊥e
T||e
>1 Ωci < ω < Ωce
kc
ωpe
≤1 k × Bo = 0
1. Whistler Anisotropy Instability
2. Electron Firehose Instability
1. Edge region is low beta2. Center has complicated orbits3. Does not appear in simulations?
Should these occur in neutral sheet?
Neutralization of Electrostatic Potential
€
γ>>Vthe
D
€
γDVthe
=γ
Ωci
⎛
⎝ ⎜
⎞
⎠ ⎟Vthi
Vthe
⎛
⎝ ⎜
⎞
⎠ ⎟ΩciL
Vthi
⎛
⎝ ⎜
⎞
⎠ ⎟D
L
⎛
⎝ ⎜
⎞
⎠ ⎟>>1
€
1
20
€
1
€
5
€
D >> 4L
€
D
Growth of LHDI
Time scale for electrons to flow in and neutralize
Future Work
Working with collaborators to simulate in 3D
However, many things left to examine in 2D:
1. Does predicted critical thickness hold?
2. Role of guide field and/or normal component
3. Influence of background (lobe) plasma
4. More realistic boundary conditions
Possible relevance to recent Cluster observations Runov et al, Cluster observation of a bifurcated current sheet, GRL, 2003