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

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Results for anti-parallel reconnection:

Brief review

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Magnetic field and ion-electron flow velocities

P. Pritchett

M. Hoshino

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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

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mi/me=9, i t = 18

mi/me=100, i t = 16

Compare extremes along dashed lines

- ion quantities- electron quantities

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-> 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

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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]

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realistic electron massRicci et al.

3D – no LHD, kink, …Zeiler et al.

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But, some questions remain…

Sausage mode,Buechner et al.

Kink, LHD,Ozaki et al. Ion sound mode…

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…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!

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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

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By

P. Pritchett

Change of symmetry

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Parallel electric field it=16

…also analytic theory by Drake et al.

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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

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Pxye

Pyze

SECTP LANL GSFC UMD

-(vezBx-vexBz)

-me(ve.grad vey)/e

z

P

x

P

enyzexye

e

1

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Electron Distribution Functions

F(vx,vy) F(vx,vz) F(vy,vz)

vx

vy

vx

vz

vy

vz

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..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

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Electron Pressure Tensors

from simulation approximation

Pxye Pxye

Pyze Pyze

critical difference at reconnection site!

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-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:

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-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?

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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

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Physical Mechanism:

Larmor orbit interacts with “anti-parallel” B components

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3D Modeling

M. Scholer et al.: Formation of“2D” channel

J. Drake et al.: Buneman modes,electron holes, anomalous resistivity

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P. Pritchett: inertia important

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…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!

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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?

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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

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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

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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

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large run, By=0.

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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

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initial By=0.8 initial By=0.

By, both large runs, t=40

no quadrupole or quadrupolar modulation!

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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

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Pxye Pyze

jyi jye By

A tour of the reconnection region…

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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

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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

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Contours of Poloidal Magnetic Field

Scale length related to electron Larmor radius

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Vmax= 0.65

Vmax= 2.8

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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