Multiscale issues in modeling magnetic
reconnection
J. F. Drake
University of Maryland
IPAM Meeting on Multiscale Problems in Fusion Plasmas
January 10, 2005
Magnetic energy dissipation in the universe
• The conversion of magnetic energy to heat and high speed flows underlies
many important phenomena in nature
– solar and stellar flares
– magnetospheric substorms
– disruptions in laboratory fusion experiments
• More generally understanding how magnetic energy is dissipated is
essential to model the generation and dissipation of magnetic field energy
in astrophysical systems
– accretion disks
– stellar dynamos
– supernova shocks
• Known systems are characterized by a slow buildup of magnetic energy
and fast release
– trigger?
– mechanism for fast release?
– Mechanism for the production of energetic particles?
Magnetic Free Energy
• A reversed magnetic field is a source of free energy
B
xxxxxxxxxxxxxxxxxxxxxxxxx x
J
•Can imagine B simply self-annihilating
•What happens in a plasma?
•How does magnetic reconnection work?
Frozen-in Condition
• In an ideal plasma (=0), the fluid moves so that the magnetic flux
through any fluid element is preserved.
Energy Release from Squashed Bubble
2
B
1
F  (p  ) 
B  B
8
4
magnetic tension
• Magnetic field lines want to become round
Energy Release (cont.)
w
L
• Evaluate initial and final magnetic energies
– use conservation law for ideal motion
• magnetic flux conserved
• area for nearly incompressible motion
Wf ~ (w2/L2) Wi << Wi
•Most of the magnetic energy is released
R
Flow Generation
• Released magnetic energy is converted into plasma flow
1 2 B2
v 
2
8
2
B 1/ 2
v  vA  (
)
4
A  L / v A
•Alfven time A is much shorter than observed energy release time
Magnetic Reconnection
• Strong observational support for this general picture
Resistivity and the multiscale problem
• The frozen-in condition implies that in an ideal plasma (=0) no
topological change in the magnetic field is possible
– tubes of magnetic flux are preserved
– Breaking of magnetic field lines requires resistivity or some other
dissipation process
• As in fluid systems, dissipation can only be important at small spatial scales
• Breaking of field lines occurs at very small spatial scales where the magnetic
field reverses  dissipation region
• Release of energy in a macroscopic system depends on the complex
dynamics of a boundary layer
– Typically kinetic and turbulent
– Reconnection is inherently a multiscale problem whose description is a
computational challenge
Expulsion of the core temperature during
sawteeth in tokamaks
• Reconnection is broadly important in fusion experiments
• The “sawtooth crash” is an important example
– Periodic expulsion of the plasma from the core of tokamaks
Yamada, et al, 1994
Characteristic Times
Laboratory Tokamaks
Resistive Time
Alfven Time
1 - 10 sec
~ 1 sec
~ 104 years
Solar Flares

Magnetosphere
r  4a / c 
2
2
resistive time
~ 0.1 sec
100 sec
Release Time
50 sec
~ 20 min
~ 30 min
Resistive Magnetohydrodynamic (MHD)
Theory
• Formation of macroscopic Sweet-Parker layer
V ~ ( /L) CA ~ (A/r)1/2 CA << CA
  (A r )1/ 2
•Slow reconnection
•sensitive to resistivity
•macroscopic nozzle
Failure of the MHD model
• Resistive MHD reconnection rates are 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
• Non-MHD physics at small spatial scales produces fast reconnection
– coupling to dispersive waves critical
• Mechanism for strong particle heating during reconnection?
Role of dispersive waves
• Coupling to dispersive waves at small scale is key to
understanding magnetic reconnection
– rate of reconnection insensitive to the mechanism that breaks the
frozen-in condition
– fast reconnection even for large systems
• no macroscopic nozzle
Generalized Ohm’s Law
• Electron equation of motion


4 d J  1  
1   1
 E  vi  B 
J  B  p e  J
2
pe dt
c
nec
ne
c/pe
Electron
inertia
c/pi
whistler
waves
•MHD valid at large scales
•Below c/pi electron and ion motion decouple
•electrons frozen-in
•Whistler and kinetic Alfven waves are dispersive
•Electron frozen-in condition broken below c/pe
s
kinetic
Alfven
waves
scales
Kinetic Reconnection
• Ion motion decouples from that of the electrons at a
distance c/pi from the x-line
– ion outflow width
c/pi
p
• electron current layer and outflow width c/
e
• Whistler and kinetic Alfven waves control the dynamics in
the inner region
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(z/w)
w = 0.5 c/pi
Birn, et al., 2001
Rates of Magnetic Reconnection
• Rate of reconnection is the slope of the  versus t curve
• All models that include the Hall term in Ohm’s law yield
essentially identical rates of reconnection
– Consequence of dispersive waves
• MHD reconnection is too slow by orders of magnitude
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
Fast reconnection in large systems
•Large scale hybrid simulation (Shay, et al., 1999)
T= 160 -1
T= 220 -1
•Rate of reconnection insensitive to system size vi ~ 0.1 CA
•No large scale nozzle in kinetic reconnection
3-D Magnetic Reconnection
• Turbulence and anomalous resistivity
– 2-D models produce strong electron streaming around the
magnetic x-line
• Can such streams drive turbulence?
• Electron-ion streaming instability (Buneman) evolves into nonlinear
state with strong wave turbulence
• Electron scattering produces enhanced electron-ion drag,
(anomalous resistivity) that is sufficient to break magnetic field lines
even without classical resistivity
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
– 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 (PIC) with up to 1.4 billion particles
• Development of strong current layer
• Current layer becomes turbulent
– Electron-ion streaming instability (Buneman) evolves into electron holes
y
x
Turbulence and the formation of electron
holes
• Intense electron beam generates Buneman instability
– nonlinear evolution into “electron holes”
• localized regions of depleted electron density
• Seen in satellite observations in the magnetosphere
Ez
z
B
x
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
• Drag Dz has complex
spatial and temporal
structure with positive
and negative values
• Sufficient to break
magnetic field lines
during reconnection
y
x
The computational challenge
• Modeling reconnection in plasma systems (solar corona, fusion plasmas,
the Earth’s magnetosphere) requires the description of the dynamics of
the largest spatial scales
– describes the buildup and storage of magnetic energy
– MHD description adequate
• At the same time must include the dynamics of a microscale boundary
layer
– This dissipation region is both kinetic and turbulent
• Modeling the dissipation region
– Including the coupling to dispersive waves to model fast reconnection
requires a two-fluid or kinetic (PIC, gyrokinetic) description
• Modeling turbulence and anomalous resistivity
– Kinetic (PIC) description down to Debye scales
• Modeling the production of energetic particles
– Kinetic (PIC) description
Range of spatial scales
Spatial Scales
Macro
c/pi c/pe
L
Fusion plasma 200cm 5cm 0.1cm
Solar Corona
104km
Earth’s
105km
magnetosphere
10m
0.2m
50km 1km
L/(c/pe)
2000
5107
105
• Modeling kinetic turbulence requires even smaller spatial scales!!
•Even AMR codes will not be able to treat such disparate scales
•The development of innovative multiscale algorithms for handling
such problems is an imperative
Conclusions
• Magnetic reconnection causes an explosive release of
energy in plasma systems
– similar to other types of explosions
• sonic flows
– a difference is that the explosion is non-isotropic
• Fast reconnection depends critically on the coupling to
dispersive waves at small scales
– rate independent of the mechanism which breaks the frozen-in
condition
– rate independent of all kinetic scales ~ 0.1 CA
– rate consistent with observations
• Modeling magnetic reconnection in a macroscale system
requires the simultaneous treatment of a microscale
boundary layer that is both collisionless and therefore
inherently kinetic and turbulent
– Describing the dynamics is a multiscale challenge
Outstanding Issues
•
•
•
•
Onset
Structure of slow shocks
Electron heating
Role of turbulence and anomalous resistivity