The Hall Current in Collisionless Reconnection andReconnection in an Electron-Positron Plasma

Manfred Scholer

Max-Planck-Inst. Extraterrestrische Physik, Garching, GermanyQueen MaryCollege, University London, UK

Claus H. Jaroschek

Rudolf A. Treumann

1. Hall physics and the importance for collisionless reconnection

2. Onset of reconnection in thin current sheets (PIC simulations)

3. PIC simulations of collisionless reconnection in an electron – positron plasma and particle acceleration

4. Recent PIC simulations of reconnection in large systems (Daughton et al.)

in outu L u

22

2 2out

out Ao

uBu V

2in

o

u B B

Mass conservation

Energy conservation

Ohm‘s+Ampere‘s law

Sweet-Parker Reconnection

1/ 2

1/ 2

1in A A

A m

u V VV L R

Sweet-Parker reconnection rate

/m AR LV is magnetic Reynolds number (Lundquist number)

L is of ambient system size and small: Sweet/Parker rate is very small

Petschek Reconnection

Petschek reconnection rate

1

ln( )in Am

v VR

Because of logarithmic dependence on Rm Petschek rate is much larger

Outer circle is the ion inertial domain. Inner circle (light blue) is the electron inertial region. Red arrows indicateinflow (of electrons) and reconnection jet outflow. Ions are unmagnetized on the ion inertial scale. Thin blue arrows are the Hall currents which generate quadrupolar Hall magnetic field.

Schematic of Hall Current Systen in Reconnection

Two-Fluid-Simulation (left) and Cluster Observations (right) of Reconnection on Ion Scale

ReconnectingB-component

Out-of-planeB-component

NormalB-component

Vaivads et al., PRL 2004

Color-coded is out-of-plane magnetic field component (2D two-fluid simulation)

Spaceraft configuration

Geospace Environmental Modeling Reconnection Challenge Results

Dissipation region of the order of the electron skin depth and thus much smaller than the ion inertial length

At this distance the Hall term in Ohm‘s law becomes important and introduces dispersion

Below ions and electrons decouple. Electrons are frozen in. Whistler waves(and not nondispersive Alfven waves) control dynamics

Electron frozen in condition boken at

Electric field at X line supported by nongyrotropic electron pressure or electron inertia

i e2pe

4 dj 1 1 1E v B j B p j

dt c nec ne

��������������������������� ��

e2pe

4 dj 1E p

dt ne

�

/ pec / pic Scales

Electron inertia Hall termwhistler waves

/ pic

/ pec

in outv L v

Why is Wave Dispersion Important?

Quadratic dispersion of whistler wave

2; phask v k Smaller scales have higher velocities

/ 1/in outv v L L

Shay and Drake, GRLett 1998

or

since 1/out phasv v k

Electron diffusion region

When electron diffusion region of order of electron inertial scale de (skin depth)electron outflow velocity is of order of electron Alfven velocity

(Ion) Alfven velocity and length of ion diffusion region determines reconnection rate.

2-D fluid, hybrid, and PIC simulations show that length of ion diffusion region isabout 10 di (ion inertial length). Thus reconnection rate (inflow velocity) is about

0.1 vA

(Claim of universal reconnection rate)

Reconnection is independent of and therefore of the mechanism by which the electron frozen in condition is broken (no bottleneck on electron scale)

Shay and Drake, GRLett 1998

GEM Result – Reconnection Rate in Various Numerical Simulations

2-D Simulation - current sheet withanti-parallel magnetic field.

In GEM challenge initial reconnection isenforced at by superposition of magnetic field disturbance

Reconnection rate independent of dissipation mechanism. Whistler phase speedlimits outflow speed. When diffusion region has electron scale the outflow velocityshould be whistler speed based on electron skin depth = electron Alfven speed.

MHD reconnection is too slow by orders of magnitude

Reconnection rate is slope of reconnected flux vs time

Birn et al., JGR 2001

VlasovCodes

Full particle codes PIC

Hybrid Code MHD Code

Kinetic Description Fluid Description

Classification of Computer Simulation Models of Plasmas

2 Fluid MHDCode

Electronsmassless fluid

Finite masselectron fluid

Reconnection simulations needan artificial resistivity

Reconnection electric fieldsupported by either electroninertia or pressure tensor

PIC Simulation Industry

Bhattacharjee, ………, Univ. New Hampshire

Büchner,…., MPI Lindau, Germany

Daughton, Scudder, Karimabadi, Univ. Iowa, UC San Diego

Drake, Sitnov, Swisdak, Shay, Univ. Maryland

Grauer, Schmitz,…, Univ. Bochum, Germany

Hesse, Kutzentsova, Winske, NASA Goddard, Los Alamos Natl. Lab.

Horiuchi, Pei, Sato, Kyoto Univ., Japan

Hoshino, Shinohara, Fujimoto, ..,Tokyo Univ., ISAS, Tokyo Inst. Techn., Japan

Lapenta, Brackbill, Ricci, Los Alamos Natl. Lab.

Pritchett, Coroniti, UCLA

(MPI Garching)

3-D Full Particle Simulations (PIC) of Reconnection

Double Harris-sheetCurrent sheet width = 1 ion inertial lengthPeriodicity in all three directions

x y z i

i pi

L L L 6

c /

i em /m 150(64)

A

i e

c /V 15

T /T 2.7

6180 10 particles of each species

3-D Particle-in-Cell code (relativistic):Multigrid algorithm for Poisson equationmassively parallel

Investigate reconnection onset

Thin Current Sheet with Antiparallel Magnetic Field

Right: Reconnected flux versus time. The whole flux between thetwo current sheets is reconnected when 0

Left: Magnetic field pattern at four different times (Isointensitycontours of

2z zJ (x, y)

Scholer et al., Phys. Plasmas 2004

Explosive reconnection within a few ion times!

Lower Hybrid Drift Instability at the Current Sheet Edge

Color coded electron density (left) and electric field (right) in the current directionin the plane perpendicular to the magnetic field.

z is in the current direction – perp to magneticfield

Cuts of Various Parameters BEFORE Reconnection Starts

Cuts of electron contribution to currentdensity (top) and electron density acrossthe current sheet. Profiles at t=0 are shown dashed for reference.

Reduced electron distribution function f(v_z) verusus v_z in the current sheet gradientregion (top) and in the center (bottom) ofthe current sheet exhibiting electron acceleration in the electric field of the LHDwaves.

t=0 t=4

Thin Current Sheet with Antiparallel Field plus Guide Field(Sheared Field Configuration)

B = 1

In the guide field case the LHDI develops as well, but it takes considerbly longertime for reconnection to set in. After reconnection onset the reconnection rate is about the same as in the exactly antiparallel case.

z

Reconnection in a Pair Plasma: 2-D and 3-D Full Particle Simulations

Jaroschek et al., Phys. Plasmas 2004

Simulation:

Initial state: 1-D curent sheet, relativistic Maxwell-Juettner plasma (100 keV)Reconnection initialized by disturbance in center of current sheet

Parameters: current sheet width (1 – 2 inertial lengths do) c / vAo between 1 and in a 3-D run about 2 x 109 particles

Pair dominance in plasma of

a) Relativistic extragalactic jetsb) Pulsar outflows (Crab)c) Core of AGN‘s

Ez

n

Current sheet and quasi-static accelerationregion begin to sparate

X-line at early times

Ez

Reconnection rate after onset phase(electric field along X line) (E > 0.2)

Multiple X line and island coalescence phase

Field lines

Ez

Particle acceleration ()along center line

Schematic offield topology

y

x

y

Temporal development

0 ton tcoa

tsep tequ

Single X line Separation of X linesalong current sheet

Islandcoalescence

tc = 40

80 140

X line buildup

Temporal development of the distributionfunction in the 2-D run

More (2-D) PIC Simulations of Reconnection in an Electron-Positron Plasma

(From Bessho and Bhattacharjee, PRL 2005)

Reconnection rate

Bessho and Bhattajarchee: High reconnection rate in pair plasma (E > 0.2)

Note: there is no Hall current in a pair plasma!

Results from a 3-D PIC Simulation

x

yEz /B

z

x

Ez

z

x

density

Fast onset of relativistic driftkink instability due to RTSI

Acceleration regionabout 20 electroninertial lengths in current direction (z).Limits to 30.

vz

z

Heating by a relativisticBuneman-type Instability (RTSI)

Trapping

Total spectral synchrotronoutput as a function of co

(co = cyclotron frequency)

Application to Pair-Dominated Active Galactic Nucleus Core Regions(Extremely hard radio spectra, Power output of P ~ 1047 ergs/sec)

Jaroschek et al., ApJ Lett. 2004

Model assuming stochastic distribution of reconnection zones over the entire coronalsouce region can explain power output and spectra

There is no Hall current system in a pair plasma,yet reconnection is fast

Is Hall physics really the key mechanism for fast reconnection in an electron – proton plasma?

Large Scale PIC Simulation of Reconnection (Electron-Proton Plasma)

Daughton et al. Phys. Plasmas 2006

Extent of electron diffusion region > 10 di

Island formation

Daughton et al.:

The results are not consistent with the standard model of Hall mediated fastreconnection.

Fast reconnection may still be possible so long as the process for generating secondary islands remains vigorous…

y-component of ion vorticityBy – field (out-of-plane)

B B (m / e) u

is frozen into ion fluid u. In the inflow region both termsare zero. Occurrence of vorticity has to be cancelled by By

Hybrid Simulation of Tail Reconnection

Arzner and Scholer, JGR 2001

Reconnection in a pair plasma (no Hall physics involved) is fast

In the late phase reconnection in a pair plasma is violent and particlesare accelerated to high energies (>30 , also in 3-D)

PIC simulations of (undriven) reconnection in an electron – proton plasma in a long system show that the electron diffusion region becomesvery large, more than 10 ion inertial lengths long

Hybrid simulations (massless electrons) with a large (resistive)diffusion region exhibit quadrupolar (Hall-type) magnetic fields.(Hall magnetic fields may only tell us that the plasma is collisionless)

Summary

Physics of collisionless reconnection continues to be an open question

Electron Acceleration in Collisionless Reconnection

PIC Simulations of Reconnection – Electron Acceleration

Drake et al., PRL 2005

Electron distibution at three times

(Large Guide-Field Simulation)

Stong parallel electric fields

en

eT

E

Electron holes

2-D PIC Simulations of Reconnection – Electron Acceleration by Surfing

(Strongly driven inflow)

Ez polarization electric field near sepatarix

zeE/y xev B c

Electric force can balance

Lorentz force

yE

Hoshino, JGR 2005

Electron stays there and gains energy by moving inthe y direction

Development of electron spectrum

Acceleration in Contracting Magnetic Islands

Drake et al. Nature 2006

Test particle simulation of electron Fermi acceleration in squashed flux bubblesPIC reconnection simulation producing magnetic

islands