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Kinetic Modeling of Magnetic
Reconnection in Space and Astrophysical
Systems
J. F. Drake
University of Maryland
Large Scale Computation in
Astrophysics
Newton Institute 2004
Recent Collaborators
•
•
•
•
Marc Swisdak (NRL)
Mike Shay (U. Maryland)
Michael Hesse (GSFC)
Cindy Cattell (U. Minn.)
Collisionless reconnection is ubiquitous
• Inductive electric fields typically exceed the Dreicer
runaway field
– classical collisions and resistivity not important
• Earth’s magnetosphere
– magnetopause
– magnetotail
• Solar corona
– solar flares
• Laboratory plasma
– sawteeth
• astrophysical systems?
Resistive MHD Description
• Formation of macroscopic Sweet-Parker layer
V ~ ( /L) CA ~ (A/r)1/2 CA << CA
•Slow reconnection
•sensitive to resistivity
•macroscopic nozzle
• Petschek-like open outflow configuration does not appear in resistive MHD
models with constant resistivity (Biskamp ‘86)
• Why Sweet-Parker?
Singular magnetic island equilibria
• Equilibria that form as a consequence of reconnection are
singular
– Sweet-Parker current layers reflect this underlying singularity
• Allow reconnection to produce a finite magnetic island (   0 )
• Shut off reconnection ( = 0) and evolve to relaxed state
– Formation of singular current sheet
• Consequence of flux conservation and requirement that

magnetic energy is reduced (Waelbroeck, 1989)
Overview
• MHD Reconnection rates too slow to explain observations
– solar flares
– sawtooth crash
– magnetospheric substorms
• Some form of anomalous resistivity is often invoked to explain
discrepancies
– strong electron-ion streaming near x-line drives turbulence and
associated enhanced electron-ion drag
– observational evidence in magnetosphere
• Non-MHD physics at small spatial scales produces fast
reconnection
– coupling to dispersive waves critical
• Electron-positron reconnection
– No dispersive waves
– Fast reconnection from turbulent outflow jet??
• Mechanism for strong particle heating during reconnection?
Kinetic Reconnection
• Coupling to dispersive waves in dissipation region at small
scales produces fast magnetic reconnection
– rate of reconnection independent of the mechanism which breaks
the frozen-in condition
– fast reconnection even for very large systems
• no macroscopic nozzle
• no dependence on inertial scales
Generalized Ohm’s Law
• Electron equation of motion
4 dJ
1
1
1
 E  vi  B 
J  B    pe  J
2
 pe dt
c
nec
ne
c/pe
Electron
inertia
c/pi
whistler
waves
•MHD valid at large scales
•Below c/pi or s electron and ion motion decouple
•electrons frozen-in
•whistler and kinetic Alfven waves control dynamics
•Electron frozen-in condition broken below c/pe
•Non-gyrotropic pressure tensor dominates
s
kinetic
Alfven
waves
scales
Kinetic Reconnection: no guide field
• Ion motion decouples from that of the electrons at a
distance c/pi from the x-line
– coupling to whistler and kinetic Alfven waves
• Electron velocity from x-line limited by peak speed of
whistler
– exceeds Alfven speed
GEM Reconnection Challenge
• National collaboration to explore reconnection with a
variety of codes
– MHD, two-fluid, hybrid, full-particle
• nonlinear tearing mode in a 1-D Harris current sheet
Bx = B0 tanh(x/w)
w = 0.5 c/pi
• Birn, et al., JGR, 2001, and companion papers
Rates of Magnetic Reconnection
Birn, et al., 2001
• Rate of reconnection is the slope of the  versus t curve
• All models which include the Hall term in Ohm’s law yield essentially
identical rates of reconnection
– Reconnection insensitive to mechanism that breaks frozen-in condition
• MHD reconnection is too slow by orders of magnitude
Reconnection Drive
• Reconnection outflow in the MHD model is driven by the expansion of the
Alfven wave
– Alfvenic outflow follows simply from this picture
• Coupling to other waves in kinetic and two-fluid models
– Whistler and kinetic Alfven waves drive outflow from x-line
• Dispersive waves
Why is wave dispersion important?
• Quadratic dispersion character
 ~ k2
Vp ~ k
– smaller scales have higher velocities
– weaker dissipation leads to higher outflow speeds
– flux from x-line ~vw
» insensitive to dissipation
Wave dispersion and the structure of nozzle
• Controlled by the variation of the wave phase speed with
distance from the x-line
– increasing phase speed
•Closing of nozzle
•MHD case since Bn and CA increase with distance from the x-line
- decreasing phase speed
•Opening of the nozzle
•Whistler or kinetic Alfven waves v ~ B/w
Whistler Driven Reconnection: weak guide
field
• At spatial scales below c/pi whistler waves rather than
Alfven waves drive reconnection. How?
•Side view
•Whistler signature is out-of-plane magnetic field
Whistler signature
• Magnetic field from particle simulation (Pritchett, UCLA)
•Self generated out-of-plane field is whistler signature
Coupling to the kinetic Alfven wave: with a
guide field
• Signature of kinetic
Alfven wave is odd
parity density
perturbation
Kleva et al, 1995
Structure of plasma
density
Bz0=0
• Even parity with no
guide field
• Odd parity with
guide field
– Kinetic Alfven
structure
Bz0=1.0
Tanaka, 1996
Pritchett, 2004
Fast versus slow reconnection
• Structure of the dissipation region
– Out of plane current
With dispersive waves
No dispersive waves
Fast Reconnection in Large Systems
•Large scale hybrid simulation
T= 160 -1
T= 220 -1
•Kinetic models yield Petschek-like open outflow configuration
even in very large systems
•Rate of reconnection insensitive to system size vi ~ 0.1 CA
•Many simulations in the literature are not large enough to
enter the asymptotic regime  confusion
Positron-Electron Reconnection
• No decoupling of the motion of the two species
– No dispersive whistler waves
– Displays Sweet-Parker structure
– Reconnection rate in large systems?
Turbulent outflow jet in electron-positron
reconnection
• Outflow jet goes unstable and becomes fully turbulent
– Broadens outflow region to Petchek-like open outflow geometry
– Another mechanism for producing fast reconnection?
• a laVishniac and Lazarian??
3-D Magnetic Reconnection
• Turbulence and anomalous resistivity
– strong electron streaming near x-line leads to Buneman
instability and evolves into nonlinear state with strong
localized parallel electric fields produced by “electronholes” and lower hybrid waves
– resulting electron scattering produces anomalous resistivity
that is sufficient to break the frozen-in condition
Observational evidence for turbulence
• There is strong observational support that the dissipation
region becomes strongly turbulent during reconnection
– Earth’s magnetopause
• broad spectrum of E and B fluctuations
• fluctuations linked to current in layer
– Sawtooth crash in laboratory tokamaks
• strong fluctuations peaked at the x-line
– Magnetic fluctuations in Magnetic Reconnection eXperiment
(MRX)
3-D Magnetic Reconnection: with guide field
• Particle simulations with up to 1.4 billion particles
• Bz=5.0 Bx, mi/me=100
• Development of strong current layer
– Buneman instability evolves into electron holes
y
x
Buneman Instability
• Electron-Ion two stream instability
• Electrostatic instability
– g~~(me/mi)1/3 pe
– k lde ~ 1
– Vd ~ 1.8Vte
Ez
z
Initial Conditions:
Vd = 4.0 cA
Vte = 2.0 cA
x
Formation of Electron holes
• Intense electron beam generates Buneman instability
– nonlinear evolution into “electron holes”
• localized regions of intense positive potential and associated bipolar
parallel electric field
Ez
z
B
x
Electron Drag
Electron Distribution Functions
vz
B
Scattered electrons
Accelerated electrons
vx
Anomalous drag on electrons
• Parallel electric field scatter electrons producing effective
drag
• Average over fluctuations along z direction to produce a
mean field electron momentum equation
p ez
 en 0 E z  en˜E˜ z 
t
– correlation between density and electric field fluctuations yields
drag
• Normalized electron drag
cn˜E˜ z 
Dz 
n0 v A B0
Electron drag due to scattering by parallel
electric fields
y
• Drag Dz has
complex spatial
and temporal
structure with
positive and
negative values
• Results not
consistent with the
quasilinear model
x
Energetic electron production in nature
• The production of energetic electrons during magnetic
reconnection has been widely inferred during solar flares and
in the Earth’s magnetotail.
– In solar flares up to 50% of the released magnetic energy appears in
the form of energetic electrons (Lin and Hudson, 1971)
– Energetic electrons in the Earth’s magnetotail have been attributed to
magnetic reconnection (Terasawa and Nishida, 1976; Baker and
Stone, 1976).
• The mechanism for the production of energetic electrons has
remained a mystery
– Plasma flows are typically limited to Alfven speed
• More efficient for ion rather than electron heating
Wind magnetotail
observations
• Recent Wind spacecraft
observations revealed
that energetic electrons
peak in the diffusion
region (Oieroset, et al.,
2002)
– Energies measured up to
300kev
– Power law distributions
of energetic electrons
Requirements of a theoretical model
• Energetic electrons seem to be produced in the dissipation
region near the x-line
– Not coherent acceleration at a single x-line
• Electron energy exceeds the cross-tail potential drop
• Why power law distribution of energetic particles?
– Fermi mechanism produces power law distributions at shocks
– Does such a mechanism exist for electron heating during
reconnection?
• Need repetitive interaction with magnetic neutral line(s)
– Turbulence seems unable to produce sufficient scattering
– Trapping or scattering in magnetic islands can lead to repetitive acceleration
– Power law energy distribution?
• Need to accelerate large numbers of electrons. How?
– Solar observations
Electron
acceleration
during
reconnection
Bz0=1.0
vparallel
• Strongest bulk
acceleration in low
density cavities
– Not at x-line!!
– Pritchett 2004
• Length of density
cavity increases with
system size
• Maximum vparallel
increases with system
size
– Longer acceleration
region
ne
Structuring of the parallel electric field
along separatrix: 2-D
• The parallel electric field remains non-zero in the low density cavities
that parallel the magnetic separatrix
– Drive strong parallel electron beams
• Strong electron beams break up Ep into localized structures
– Electron holes and double layers
– Most intense in density cavities
By=1.0
Electron-holes and double layers
• Structure of Ep along field line
– Electron holes and double layers
– Structures predominate in low
density cavity remote from the xline
Electron heating
• Electron cooling in cavity accelerators
– Well known from accelerator theory
• Cooling along direction of acceleration
• Overall large number of accelerated electrons
• Strong acceleration within secondary island
– Multiple passes through acceleration region
Scaling of electron acceleration cavities
• Acceleration cavities are limited in length
– Constraint from maximum in-plane electron currents and
associated magnetic fields
• Limit on the electron velocity in a single pass through the
acceleration cavity
V|| = CAe(1 + )1/2
Production of energetic electrons
• Bz=1.0, three times in the simulation
• High energy tail from multiple interactions with x-line in secondary
island
Location of energetic electrons
• Electrons with energy greater than 1.4mec2
• Most energetic electrons have multiple interactions with
acceleration cavities
• Magnetic islands rather than turbulent scattering facilitates
multiple interactions with acceleration regions
Electron acceleration in a secondary island
• Test particle
acceleration in the
secondary island is
consistent with the
large electron heating
seen in the full
simulation in this
region
– Island facilitates multiple
accelerations
– Too much energy and the
particle escapes
Conclusions
• Fast reconnection requires either the coupling to dispersive
waves at small scales or a mechanism for anomalous
resistivity
• Coupling to dispersive waves
– rate independent of the mechanism which breaks the frozen-in
condition
– Open Petschek-like magnetic configuration
– Supported by magnetospheric satellite observations
• Turbulence and anomalous resistivity
– strong electron beams near the x-line drive Buneman instability
– nonlinear evolution into “electron holes” and lower hybrid waves
• seen in the ionospheric and magnetospheric satellite measurements
Conclusions (cont.)
• Production of energetic electrons
– Large scale density cavities that develop during reconnection with a
guide field become electron accelerators
– All electrons crossing into these cavities undergo significant
acceleration
• A multi-island reconnection picture will leader to acceleration of
essentially all electrons crossing magnetic separatrices
– Requires many interacting islands and not a single large island
– Secondary islands facilitate multiple interactions of electrons with
acceleration cavities and the production of very energetic electrons
• Explanation for observed powerlaw energy distributions
remains an outstanding problem
Dispersive waves
• Geometry
• whistler
=
c
  ky
k CA
pi
=
• kinetic Alfven
k 
=
c
  ky
k Cs
pi
By0
B0
ky
Parameter space for dispersive waves
• Parameters
•For sufficiently
large guide field
have slow
reconnection
Rogers, et al, 2001
 y  4nT / B
2
0y
B02 m e
  (1  ) 2
B0 y m i

none
kinetic Alfven
1
whistler
kinetic Alfven
whistler
1
y
Dissipation mechanism
• What balances Ep during guide field reconnection?
• In 2-D models non-gyrotropic pressure can balance Ep
even with a strong guide field (Hesse, et al, 2002).
4 dJz
1
1
 E z  (v e  B) z  (  pe ) z
2
 pe dt
c
ne
Bz=0
Bz=1.0

y
y
Satellite
observations
of electron
holes
• Magnetopause
observations
from the Polar
spacecraft
(Cattell, et al.,
2002)