william daughton plasma physics group, x-1 los alamos national laboratory presented at:
DESCRIPTION
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. www-spof.gsfc.nasa.gov. Motivation for this work. - PowerPoint PPT PresentationTRANSCRIPT
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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
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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
€
rB
€
rJ Courtesy of Hantao Ji (PPPL)
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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
€
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
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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
€
ρi
L=
U i
V⊥s
€
Bx (z) = Bxo tanh(z /L)
€
Jy (z) =cBo
4π Lsech2 z
L
⎛
⎝ ⎜
⎞
⎠ ⎟
€
x€
z
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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
€
Box Size → 8πL × 4πL 1280 × 640 grid 100 ×106 particles
€
Box Size →16πL × 4πL 2560 × 640 grid 200 ×106 particles
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Single Island Tearing Saturation
€
γΩci
€
kxL€
T⊥e
T||e
=1
€
z
L
€
x /L
€
T⊥e /T||e
€
T⊥e
T||e
= 0.95
Linear Growth Rate Mode Amplitude
€
tΩci
PIC Simulation
€
Ay
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Two Island Coalescence
€
z
L
€
x /L
€
T⊥e /T||e
Mode Amplitudes
€
Ay
€
kxL
€
tΩci
€
γΩci
Linear Growth Rate
M=1
M=2
€
T⊥e
T||e
=1
€
0.95
€
0.9
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Four Island Coalescence
€
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
€
tΩci = 0 →190
€
tΩci =190 → 244
€
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
€
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
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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)
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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
€
ky ρ iρ e ~ 1
€
ρi
L>1
€
ρi
L≤12. Nonlinear development of
short wavelength modes
€
kyρ e ~ 1
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Penetration of LHDI
€
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
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2D Simulation of Lower-Hybrid
€
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
€
ωr /Ω lh = 0.54 γ /Ωci =1.93
€
z /L
€
˜ φ (z)
€
ωr /Ω lh = 0.57 γ /Ωci = 2.26
€
˜ φ (z)
€
z /L
Two fastest Growing modes
€
kyρ e ≈ 0.75
€
z
L
€
y /L
Lower-Hybrid Drift Mode
Lower-Hybrid Drift Mode
Fluctuations are confined to the edge of the sheet
€
˜ φ (z)
€
Ey (y,z)
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Evolution of Current Density
€
z
L
€
y /L
€
Jy
Jo
€
Jy (z) = Jo sech2(z /L)Initial
Y-averaged
€
z
L
€
Jy = −eneVey + eniViy
Contours of
€
Jy (z,y)
€
Jy =1
Ly
Jy∫ (z,y) dy
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€
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
€
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
€
Vey
Vthe
Evolution of Electron Velocity
€
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
€
T⊥e
T ||e
€
T⊥e
T ||e
€
T⊥e
T ||e
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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
€
˜ 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 - - - - - - - - -
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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)
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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?
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What about lower mass ratio?
€
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
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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?
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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
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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