Non-linear Force-Free Models of the Sun

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Non-linear Force-Free Models of the Suns
Magnetic Field

• Duncan H Mackay
• Solar Physics Group
• University of St. Andrews

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Contents.

• Part1 Introduction.
• Overview of Solar
  Atmopshere
• MHD Equations and
  Force-Free Fields.
• Part 2 Construction of the New Global Model.
• Test 1 Hemispheric Pattern of
  Filaments.
• Test 2 Open Magnetic Flux.
• (Aad
  van Ballegoijen, Dr Anthony Yeates)?
• Part 3 Model for a Decaying Active Region
• . (
  Lucie Green, Aad van Ballegoijen)?
• 
  

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Solar Structure
Interior Generation of
Magnetic Fields Solar Dynamo.


Corona T gt 106 K Coronal Loops, CMEs Open Flux
Photosphere T 6000 K
Sunspots. Measure B
Chromosphere 4300 K lt T lt 105 K Transition
Region 105 K lt T lt 106 K
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MHD Equations.

• On the Sun material in plasma state B field
  important (Electromagnetism Fluid Mechanics).
• Corona Force-Free Fields.

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2. Global Non-Potential Magnetic Field Model for
the Solar Corona
Development and Application van Ballegooijen,
Priest and Mackay2000 Mackay, Gaizauskas and van
Ballegooijen 2000 Mackay and van Ballegooijen
2006a,b Yeates, Mackay and van Ballegooijen
2007, 2008a,b, 2009a,b.
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Global Model Key Features

• Long Term simulations (months years).
• Retain memory of previous magnetic field
  interactions.
• Build up of coronal currents (free
  magnetic energy)?
• Transport of energy, magnetic
  helicity across the solar surface.
• Not possible through single extrapolation
  approach.
• Coupled evolution of photospheric and coronal
  fields
• Without resetting the
  photo/coronal field
• Include flux emergence.
• Accurately reproduce the large-scale photospheric
  field of the Sun (based on observations).

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Coupled 3D Model.

• Coronal Model Magneto-Frictional Relaxation
  (velocity ? lorentz force.)?
• 
  Coronal field relaxes to a non-linear
  
  force-free field, j x B 0.
  Relaxation time scale not physical

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3D Inserting Bipoles

• Bipoles are inserted as an isolated field
  containing either zero, ve or ve helicity
  (alpha) both in the photosphere and corona.

Day 250
Day 251
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3D Non-Potential Coronal Field

• Initial Condition Potential field
• Coronal field evolves through a series of nlfff.

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Test 1 Hemispheric Pattern of Filaments.

• Solar Limb - Prominence
  Solar Disk - Filament
• H? Emission
  H? Absorption
• Indicators of non-potential fields
• coronal flux
  flux ropes (weakly twisted)?
• free magnetic
  energy.

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Filament Chirality

• Two types of chirality Sinistral and
  Dextral.
• 
  Northern
  Hemisphere
• 
  -
  Dextral
  Southern Hemisphere
  - Sinistral
  (Martin et al. 1995, Leroy 1983,1984)
  
• Differential rotation produces the opposite
  results.
• 
  What other global effects could cause
  the hemispheric pattern ?
  
  As exceptions to hemispheric
  pattern
• 
  occur model must predict them as
• 
  well.

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Skew Comparison
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Results with Hemispheric Distribution of Twist
109 filaments
Shapes observed chirality
dextral

sinistral
Colours correct wrong
Up to 96.9 correct

• Results improve the longer the simulation is
  run.
• Yeates, Mackay and van Ballegooijen
  2007,2008,2009

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Test 2 Open Magnetic Flux

• Part of Suns magnetic field that fills the
  heliosphere surrounds the Earth and directly
  interacts with the magnetosphere.
• Current potential field models underestimate
  level of open flux particularly true at cycle
  maximum.
• Four main sources of open flux
• - Background level
  (location of flux sources).
• - Enhancement due to 1)
  radial outflow (small)?
• 
  2) inflation due to electric currents.
• 
  3) sporadic flux rope ejections.

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3. Model of a Decaying Active Region

• Use observations of Bn to directly simulate the
  dispersal of an active region
• 
  - NOAA 8005
• 
  - evolution over 4 days
  (16th-19th Dec).
• 
  - use 96 min MDI full disk
  magnetograms.
• 
  - No strong shear/converging
  flows
• 
  small scale motions due to
  convection.

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The Model

• Coronal evolving non-linear force-free field.
• Photospheric direct input of observational
  data.
• ?
  ?
• Bz(t) ? Ax(t), Ay(t) Bz(t1) ? Ax(t1),
  Ay(t1)?
• Time evolution of Bz ? Interpolate between Ax,
  Ay at different time

- OBS
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• Observations
  Simulation

• Technique produces an accurate representation
  of observed magnetograms.

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Coronal Field

• Initial Potential Field
  NLFFF 4 days
• 
  Potential Field
• 4
  days.
• Non-potential fields are seen to develop
• along some locations.

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Magnetic Energy

• Left Total Mag Energy
• (solid Pot, dotted NLFFF)
• Free Mag Energy
• 4 days 81030 ergs
• (10 that of pot. field)?
• Rate of input 2.31025 ergs/s.
• Location of free energy (white) low corona.

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Conclusions

• First long term continuous simulation of coronal
  field (rather a independent extrapolations).
• - Convincing explanation for the
  hemispheric pattern of filaments
• through flux emergence, surface
  transport and reconnection of
• large scale active region fields.
• - Predicts IMF strength better than PFSS
  Models.
• - Transport of helicity from low to high
  latitudes over many months
• is a fundamental element of the
  coronal evolution agreement
• gets better the longer the
  simulations are run (Sun has long term
• memory).
• Used observations to directly model the dispersal
  of an active region
• (new technique) see large amount of free
  magnetic energy built up due to small scale
  random motions.

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Future Applications/Improvements

• Immediate improvements
• Better description
  of flux emergence.
• Include observed
  active region twist.
• Applications.
• Open Flux
  Variation.
• Relationship to
  CMEs
• Helicity
  Transport

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Formation of flux rope along a PIL

• Day 3 Day
  22 Day 35
• Day 40
  Day 42
• Mackay and van Ballegooijen 2006 (a,b) - APJ

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Magnetic Helicity

• Left Mag Helicity
• (solid Pot, dotted NLFFF)
• Right - Relative Mag Energy ve
• PDF for a on base.
• solid day 0, symmetrical
• dashed day 2, skewed
• dotted day 4, skewed

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Two Component Model

• Evolve, Suns large-scale field, B, through the
  induction equation.
• Photospheric BC Flux Transport Model
• Differential Rotation
• Meridional Flow
• Surface Diffusion
• 
  Shears the surface fields coronal
• 
  field diverges from equilibrium.
  
  
• 
  Physical time scale.

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Test 3 Coronal Mass Ejections

• Many models have been proposed to explain CMEs
  flux rope ejection (Lin 2003).
• Global simulations contain these effects applied
  to observations.
• Yeates and Mackay 2009 (ApJ)?
• Comparison
• 1) Rate of ejections to rate of
  CMEs model produces 50.
• 2) One-to-one comparison between
  ejections and EIT coronal dimmings
• good match with dimmings
  outside active regions.
• (Yeates et al. 2010).

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Emerging flux

• Use a semi-automated procedure
• compare successive magnetograms
• find new bipolar regions
• measure key properties
• insert as ideal bipoles into simulation.

CR1948
CR1948 rotated
CR1949
Total 118 bipolar regions
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Coronal Evolution
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Flux Transport Model(2).

• Form of Coronal Diffusion.
• Outflow Velocity.
• Resolution nx 361, ny293,nz53
• Bipole Description.

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Statistical Test for Filament Chirality

• T-test used to classifify chirality from
  individual barbs.
• n no. of
  barbs (x1, x2, x3, .., xn)?
• xi 1
  (dextral) -1 (sinistral)?
• The number of dextral barbs is
  ????????????????????????ns n nd
• Now assume nd following a binomial
  distribution with parameters (n,p) and assume p
  0.5
• is 0 if neither chirality is significant. The
  classification scheme is then
• where we choose T 1.5
• (For large n, t should approximate a normal
  distribution with mean n and variance 1)?

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Modelling Coronal Magnetic Fields

• Observations generally only supply Bn at
  photosphere.
• Deduce coronal field magnetic extrapolations
  (static).
• Potential
  Non-linear fff
• Aim Construct a global nlfff model
• Based on observed
  magnetograms
• Field evolves through
  related sequences of equilibria
• Model coronal field for
  years.

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