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The Hall Current in Collisionless Reconnection
and Reconnection in an Electron-Positron Plasma
Manfred Scholer
Max-Planck-Inst. Extraterrestrische Physik,
Garching, Germany Queen MaryCollege, University
London, UK
Claus H. Jaroschek Rudolf A. Treumann
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• Hall physics and the importance for collisionless
  reconnection
• 2. Onset of reconnection in thin current sheets
  (PIC simulations)
• PIC simulations of collisionless reconnection in
  an electron positron
• plasma and particle acceleration
• Recent PIC simulations of reconnection in large
  systems
• (Daughton et al.)

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Sweet-Parker Reconnection
Mass conservation
Energy conservation
OhmsAmperes law
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Sweet-Parker reconnection rate
is magnetic Reynolds number (Lundquist number)
L is of ambient system size and h small
Sweet/Parker rate is very small
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Petschek Reconnection
Petschek reconnection rate
Because of logarithmic dependence on Rm Petschek
rate is much larger
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Schematic of Hall Current Systen in Reconnection
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.
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Two-Fluid-Simulation (left) and Cluster
Observations (right) of Reconnection on Ion Scale
Vaivads et al., PRL 2004
Reconnecting B-component
Out-of-plane B-component
Normal B-component
Spaceraft configuration
Color-coded is out-of-plane magnetic field
component (2D two-fluid simulation)
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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 Ohms 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
Hall term whistler waves
Electron inertia
Scales
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Why is Wave Dispersion Important?
Quadratic dispersion of whistler wave
Smaller scales have higher velocities
Electron diffusion region
or
since
Shay and Drake, GRLett 1998
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Shay and Drake, GRLett 1998
Reconnection is independent of d and therefore
of the mechanism by which the electron frozen in
condition is broken (no bottleneck on electron
scale)
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 is about
10 di (ion inertial length). Thus reconnection
rate (inflow velocity) is about
0.1 vA
(Claim of
universal reconnection rate)
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GEM Result Reconnection Rate in Various
Numerical Simulations
Birn et al., JGR 2001
2-D Simulation - current sheet
with anti-parallel magnetic field. In GEM
challenge initial reconnection is enforced at by
superposition of magnetic field disturbance
Reconnection rate is slope of reconnected flux vs
time
Reconnection rate independent of dissipation
mechanism. Whistler phase speed limits outflow
speed. When diffusion region has electron scale
the outflow velocity should be whistler speed
based on electron skin depth electron Alfven
speed. MHD reconnection is too
slow by orders of magnitude
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Classification of Computer Simulation Models of
Plasmas
Kinetic Description
Fluid Description
Full particle codes PIC
Vlasov Codes
MHD Code
2 Fluid MHD Code
Hybrid Code
Electrons massless fluid
Finite mass electron fluid
Reconnection electric field supported by either
electron inertia or pressure tensor
Reconnection simulations need an artificial
resistivity
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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)
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3-D Full Particle Simulations (PIC) of
Reconnection
Investigate reconnection onset
Double Harris-sheet Current sheet width 1 ion
inertial length Periodicity in all three
directions
particles of each species
3-D Particle-in-Cell code (relativistic) Multigri
d algorithm for Poisson equation massively
parallel
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Thin Current Sheet with Antiparallel Magnetic
Field
Explosive reconnection within a few ion times!
Right Reconnected flux versus time. The whole
flux between the two current sheets is
reconnected when
Left Magnetic field pattern at four different
times (Isointensity contours of
Scholer et al., Phys. Plasmas 2004
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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
magnetic field
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Cuts of Various Parameters BEFORE Reconnection
Starts
t0
t4
Cuts of electron contribution to current density
(top) and electron density across the current
sheet. Profiles at t0 are shown dashed for
reference.
Reduced electron distribution function f(v_z)
verusus v_z in the current sheet gradient region
(top) and in the center (bottom) of the current
sheet exhibiting electron acceleration in the
electric field of the LHD waves.
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Thin Current Sheet with Antiparallel Field plus
Guide Field (Sheared Field Configuration) B 1
z
In the guide field case the LHDI develops as
well, but it takes considerbly longer time for
reconnection to set in. After reconnection
onset the reconnection rate is about the same as
in the exactly antiparallel case.
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Jaroschek et al., Phys. Plasmas 2004
Reconnection in a Pair Plasma 2-D and 3-D Full
Particle Simulations

• Pair dominance in plasma of
• Relativistic extragalactic jets
• Pulsar outflows (Crab)
• c) Core of AGNs

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 Ö2 in a 3-D run about 2 x 109
particles
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Reconnection rate after onset phase (electric
field along X line) (E gt 0.2)
Ez
n
Ez
Current sheet and quasi-static
acceleration region begin to sparate
X-line at early times
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Multiple X line and island coalescence phase
Schematic of field topology
Field lines Ez Particle acceleration
(g) along center line
y
y
g
x
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Temporal development
0 ton
tcoa
tsep tequ
Island coalescence
Separation of X lines along current sheet
Single X line
X line buildup
80
140
t wc 40
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Temporal development of the distribution function
in the 2-D run
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More (2-D) PIC Simulations of Reconnection in an
Electron-Positron Plasma
(From Bessho and Bhattacharjee, PRL 2005)
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Reconnection rate
Bessho and Bhattajarchee High reconnection rate
in pair plasma (E gt 0.2)
Note there is no Hall current in a pair
plasma!
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Results from a 3-D PIC Simulation
Ez /B
y
x
Acceleration region about 20 electron inertial
lengths in current direction (z). Limits g to 30.
Ez
z
Fast onset of relativistic drift kink instability
due to RTSI
x
Heating by a relativistic Buneman-type
Instability (RTSI)
z
vz
density
Trapping
x
z
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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
Total spectral synchrotron output as a function
of w / wco (wco cyclotron frequency)
Model assuming stochastic distribution of
reconnection zones over the entire coronal souce
region can explain power output and spectra
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• 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?

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Large Scale PIC Simulation of Reconnection
(Electron-Proton Plasma)
Daughton et al. Phys. Plasmas 2006
Island formation
Extent of electron diffusion region gt 10 di
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Daughton et al.
The results are not consistent with the standard
model of Hall mediated fast reconnection. Fast
reconnection may still be possible so long as the
process for generating secondary islands remains
vigorous
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Hybrid Simulation of Tail Reconnection
Arzner and Scholer, JGR 2001
y-component of ion vorticity
By field (out-of-plane)
is frozen into ion fluid u. In the inflow region
both terms are zero. Occurrence of vorticity has
to be cancelled by By
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Summary
Reconnection in a pair plasma (no Hall physics
involved) is fast In the late phase reconnection
in a pair plasma is violent and particles are
accelerated to high energies (gt30 g, also in
3-D) PIC simulations of (undriven) reconnection
in an electron proton plasma in a long system
show that the electron diffusion region
becomes very 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)
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Physics of collisionless reconnection continues
to be an open question
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Electron Acceleration in Collisionless
Reconnection
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PIC Simulations of Reconnection Electron
Acceleration
(Large Guide-Field Simulation)
Electron holes
Electron distibution at three times
Stong parallel electric fields
Drake et al., PRL 2005
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2-D PIC Simulations of Reconnection Electron
Acceleration by Surfing
(Strongly driven inflow)
Ez polarization electric field near sepatarix
Electric force
can balance
Lorentz force
Electron stays there and gains energy by moving
in the y direction
Hoshino, JGR 2005
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Development of electron spectrum
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Acceleration in Contracting Magnetic Islands
Drake et al. Nature 2006
Test particle simulation of electron Fermi
acceleration in squashed flux bubbles
PIC reconnection simulation producing
magnetic islands