Multiscale issues in modeling magnetic reconnection

1
Multiscale issues in modeling magnetic
reconnection

• J. F. Drake
• University of Maryland
• IPAM Meeting on Multiscale Problems in Fusion
  Plasmas
• January 10, 2005

2
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?

3
Magnetic Free Energy

• A reversed magnetic field is a source of free
  energy

• Can imagine B simply self-annihilating
• What happens in a plasma?
• How does magnetic reconnection work?

4
Frozen-in Condition

• In an ideal plasma (?0), the fluid moves so that
  the magnetic flux through any fluid element is
  preserved.

5
Energy Release from Squashed Bubble
magnetic tension

• Magnetic field lines want to become round

6
Energy Release (cont.)
w
L
R

• Evaluate initial and final magnetic energies
• use conservation law for ideal motion
• magnetic flux conserved
• area for nearly incompressible motion

Wf (w2/L2) Wi ltlt Wi

• Most of the magnetic energy is released

7
Flow Generation

• Released magnetic energy is converted into plasma
  flow

• Alfven time ?A is much shorter than observed
  energy release time

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Magnetic Reconnection

• Strong observational support for this general
  picture

9
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

10
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
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Characteristic Times

• Resistive Time
  Alfven Time Release Time
• Laboratory Tokamaks 1 - 10 sec
  1 ?sec 50 ?sec
• Solar Flares 104
  years 0.1 sec 20
  min
• Magnetosphere ?
  100 sec 30
  min

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Resistive Magnetohydrodynamic (MHD) Theory

• Formation of macroscopic Sweet-Parker layer

V (? /L) CA (?A/?r)1/2 CA ltlt CA

• Slow reconnection
• sensitive to resistivity
• macroscopic nozzle

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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?

14
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

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Generalized Ohms Law

• Electron equation of motion

• 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

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Kinetic Reconnection

• Ion motion decouples from that of the electrons
  at a distance from the x-line
• ion outflow width
• electron current layer and outflow width
• Whistler and kinetic Alfven waves control the
  dynamics in the inner region

c/?pi
c/?pi
c/?pe
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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

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Rates of Magnetic Reconnection

• Rate of reconnection is the slope of the ? versus
  t curve
• All models that include the Hall term in Ohms
  law yield essentially identical rates of
  reconnection
• Consequence of dispersive waves
• MHD reconnection is too slow by orders of
  magnitude

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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

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Fast reconnection in large systems

• Large scale hybrid simulation (Shay, et al., 1999)

• Rate of reconnection insensitive to system size
  vi 0.1 CA
• No large scale nozzle in kinetic reconnection

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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

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Observational evidence for turbulence

• There is strong observational support that the
  dissipation region becomes strongly turbulent
  during reconnection
• Earths 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)

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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
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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
x
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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
• correlation between density and electric field
  fluctuations yields drag
• Normalized electron drag

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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
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The computational challenge

• Modeling reconnection in plasma systems (solar
  corona, fusion plasmas, the Earths
  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

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Range of spatial scales

• 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

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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

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Outstanding Issues

• Onset
• Structure of slow shocks
• Electron heating
• Role of turbulence and anomalous resistivity