Semenov, V.St. Petersburg University, Russia

Divin, A. K. U. Leuven, Belgium

Thanks to N. Erkaev, I. Kubyshkin, G. Lapenta, S. Markidis, and H. Biernat

Effective conductivity of collisionless plasma

Motivation

Magnetic reconnection is one of the most important energy conversion process in various plasmas.

The process is determined by the presence of some sort of diffusivity in plasma, which breaks the magnetic field lines frozen-in constraint.

In collisionless plasma environments the problem of dissipation is not (yet) clear and several mechanisms are proposed (turbulent or laminar)

From global scales to EDR / History

Collisionless plasma: no collisional dissipation, but…

Turbulent: anomalous resistivity, wave-particle interaction

Laminar: electron inertia

dide ?From: [Hesse, 2001]

~10…500 km (magnetotail)

Harris equilibrium, L=0.5di , Ti/Te = 5, localized X-point perturbation

Ay =Ay0 cos(2x/Lx )cos( z/Lz )exp(-(x2+z2)/a2)

Variables are normalized to initial CS parameters: n0, B0, VA, di, etc.

Bz(t=0)=0

Runs reported:

mi/me=1836, c/vA= 274, L= 40dix20di, (1024x512) 2048 p.

mi/me=256, c/vA= 102, Ti/Te = 5, L= 200dix30di (2048x386) 940 p.

mi/me=64, c/vA= 51, Ti/Te = 5, L= 20dix10di (512x256), 32 p.

We use the following magnetotail parameters as a reference (e.g. Pritchett, 2004): B0~20 nT, n0~0.3 cm-3 , Ωci

-1~0.5s, c/pi~400km,vA~800km/s, EA=(vA/c)B0~16mV/m

)/(tanh)( 0 zBzBx 2.01.0,)/(cosh)( 20 ornnznzn bb

Model parameters

Diffusion region: Ohm’s law

yPneE yzz /)/1(

Anisotropy of electron pressure (mostly Peyz) supports Ez near X-point (in agreement with, see e.g. [Kuznetsova, 2000], [Pritchett, 2001])

Center of the EDR, Ey ~ -(1/ne)(dPyz/dz)y

Edges of the EDR, Ey = (ve X B)y

z

vv

x

vv

e

m

z

P

x

P

enBvBv

cE ey

ezey

exeeyzexy

ezexxezy

1)(

1

y

eyzee

eyexe

exy

e

eyz

e

ye

E

zvve

m

xvve

m

x

P

en

z

P

en

Bv

/

/

1

1

][

Generation of Pyz

Vy

Vz

n1 v1z

nacc vy

acc

21

1122

21 d))()((

nn

vnvnm

vffvvm

yze

zye

VVPyz

Vy

Vzn1 v1z

nacc vy

acc

yzee vvnm

Sweet-Parker analysis: Results

2/1

22

2

2

22

)π4(

,π4

e

xAexy

epe

e

nm

BVvv

dc

ne

mc

e

e

Ae

zz

A

y

L

d

V

v

B

B

E

E

0

EDR width: electron inertial length

Outflow velocity vx, electron Alfven

Reconnection rate (Ey/EA) connects all other parameters.

yy Ej xz B

cen

B

cen

1

~

Pressure-tensor based dissipation scales as Bohm diffusion !

Scaling: comparison to simulations (1)

x/di t/tA

mi/me=256

Scaling: comparison to simulations (2)

mi/me=1836

Rescaling

dide

ez

AeeLe L

eB

cVm

3.0~/ LeeLk

Rescaling

The scaling implies that Larmor gyroradius

Simulations reveal that This ratio is introduced into scaling:

Ae

yz

A

y

V

v

B

B

E

E

0

2/1)π4( e

xAexy nm

BVvv

eLe L

3.0~/ LeeLk

AexAey

e

kVvVv

d

,

,

A

y

Ae

zz

e

e

E

E

V

v

B

kB

L

kd

0

Scaling: comparison to simulations (2)

mi/me=1836

mi/me=256 k=0.3Rescaling

mi/me=1836 k=0.3

Rescaling

Summary & Results

1. Magnetic reconnection is investigated by means of PIC simulations.

2. Study of EDR structure is performed and model for the pressure anisotropy is developed.

3. Sweet-Parker-like analysis of the EDR is performed;

4. All typical EDR parameters are recovered, diffusivity scales as Bohm diffusion. Rescaled equations are presented

Vx

=kVAe

VA

e

ee kL

Vy =VAe

0BL

dB

e

ez

de

xB

cen

1

~z

Ey

vy