William Daughton Plasma Physics Group, X-1 Los Alamos National Laboratory Presented at: Second Workshop on Thin Current Sheets University of Maryland April

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

€

rB

€

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

€

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

€

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

€

Box Size → 8πL × 4πL 1280 × 640 grid 100 ×106 particles

€

Box Size →16πL × 4πL 2560 × 640 grid 200 ×106 particles

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

QuickTime™ and aYUV420 codec decompressor

are needed to see this picture.

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





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



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

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

€

ky ρ iρ e ~ 1

€

ρi

L>1

€

ρi

L≤12. Nonlinear development of

short wavelength modes

€

kyρ e ~ 1

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

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



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)



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

€

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



€

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)





€

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)



€

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

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



€

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



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



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?

€

z

L

€

y /L

€

Jy

€

z

L

€

mi

me

= 512

€

mi

me

=100





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



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

Documents

William Daughton Plasma Physics Group, X-1 Los Alamos National Laboratory Presented at: Second Workshop on Thin Current Sheets University of Maryland April