Slides - Karen Meyer_ Duncan Mackay and Aad van Ballegooijen

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					Karen Meyer, Duncan Mackay and Aad van Ballegooijen
             Outline of talk
 Introduction
    Magnetic Carpet
    Corona
 Model
    Photospheric model
    Coronal model
 Results
 Summary and future work
                 Magnetic Carpet
 Magnetogram
                                  Hinode: SOT/SP

                                  ~ 120x120 arcsec²
                                  ~ 87,000x87,000km²

 Supergranular cells

                        Provide dominant flow pattern on solar surface at
                        small scales.

                        Diameter: 10,000-50,000km (~14,000km)

                        Flow: Up-flow at centre, radial outflow ~0.5km/s,
                              down-flow at edges.

                        Strong down-flow where 2 or more cells meet:
                        leads to build-up of magnetic flux.
                 Magnetic Carpet Fluxes
 3 main classifications: ephemeral regions, network fields and intranetwork
 Ephemeral Regions (Harvey & Martin 1973)
    Clusters of newly emerging bipolar pairs, emerge within s/g cell centers.
                    - average total flux: ~10¹⁹Mx (Schrijver et al. 1997)
                    - diameter: 3,000-5,000km (Harvey 1993)

 Network Fields (Martin 1988)
    Typically at sites of strong down-flow, at the edges of 2 or more supergranular cells.
                      - flux: 1x10¹⁸-1x10¹⁹Mx
                      - diameter: 1,000-10,000km
    Produced from the residuals of other flux concentrations
      - Composition: ~ 90% ephemeral regions, ~ 10% intranetwork elements (Martin 1990)

   Intranetwork fields
    Smallest of concentrations, originate within s/g cells as emerging bipoles.
                    - diameter: ~ 2000km
                    - flux: 1x10¹⁶-1x10¹⁸Mx
 Evolution of Magnetic Carpet Fields
Four main processes (as observed):                          Emergence

1)   Flux Emergence: appearance as pairs of new
     magnetic features opposite polarity (equal flux).

2)   Cancellation: two features of opposite polarity come
     into contact and mutually lose flux.
3)   Coalescence: joining of two features of the same       Fragmentation
     polarity.                                              Cancellation

4)   Fragmentation: splitting of a large feature into two
     or more elements (perhaps due to granulation or
       Magnetic Carpet Coronal Field
 Connections between magnetic fragments exist
  within the corona.

 Photosphere is constantly evolving
   - recycle time for photospheric field ~ 1-2hrs
                                                                  Hinode: spicules on the limb (Y. Suematsu 2009)
                           (Hagenaar et al. 2008)

 Surface motions lead to energy being built up
  within the coronal field
   - coronal heating, X-ray bright points.

 Coronal field cannot be measured directly, so we
  may wish to model it and compare to observed

                                             TRACE loops (2005)
       Magnetic Carpet Coronal Field
 Close et al. (2002)
   - 12.5hrs hi-res MDI magnetograms
   - 264x264Mm² regions
   - simulated potential field

 50% of connections close below 2.5 Mm
   - most of quiet Sun flux is contained in low
     lying flux tubes.

                                                   (Close et al. 2002, 2004)
 Coronal remap time: 1.4hrs (Close et al. 2004)
   - 15min cadence magnetograms

 Quiet Sun corona is highly dynamic.
                            Thesis Aims
 Previous theoretical models have mostly considered potential fields and
  independent extrapolations of the coronal field at each time step.
   Aim: Model the continuous evolution of a small scale non-linear force-
  free coronal field based on surface motions.

 Features
           - 2 component model: photosphere and corona.
           - Extended area of Sun ~10x10 supergranules (140x140Mm²)
                                                (Radius of Sun = 696Mm).
           - Include flux emergence, cancellation etc. and supergranulation.
           - Continuous evolution of non-linear force-free coronal field
             with photospheric component as lower boundary.
 Consider
         - Magnetic topology and connectivity of complex non-potential small
             scale coronal field.
           - Locations of free magnetic energy and electric current.
                  Photospheric Model
Evolves the normal component of the magnetic field (Bz) at the
photosphere through an analytical boundary condition (van Ballegooijen).

 Evolution is in terms of the vector potential, A:

                   automatically in corona.
 Why analytically specified boundary condition?
       - avoid numerical overshoot (Gibb’s phenomenon)
       - avoid numerical diffusion
       - allows for full cancellation:
                            - diffusion is slow
                            - forcing is difficult numerically & causes pile-up
                            - our model overlaps the fragments
 Other methods exist: - may be computationally intensive.
                         - may work in terms of B rather than A.
Advection term only:
Advection and diffusion:
Flux Transport Movies
             Hexagonal Grid:
             Represent supergranular cells.

             Refine supergranule effect to be
             more realistic.

             Magnetic concentrations emerge
             as pairs within cells:
                 1- Fragments initially move in
                    opposite directions.
                 2- Move towards cell edges.
                 3- Move along cell edges,
                    interact with other magnetic

             (More detailed model: Parnell 2001)
                          Coronal Model
 Non-linear force-free fields:

Momentum equation:                                        Force-free condition

Ampère’s law

• α is current helicity
     - function of position
     - constant along field lines:
• 3 cases:          (potential field)
                        (nlfff)                                                  (Image: Yeates 2008)

• Describes local distribution of twist & shear in the magnetic field.

• Non-linear force-free field - contains free magnetic energy & electric currents.
                              - allows for highly twisted regions as well as relatively
                                untwisted regions (α ~ 0).
        Magneto-frictional Relaxation
 The Lorentz force acts against an artificial friction to cause the system to relax to
  a non-linear force-free equilibrium.

Momentum equation:

Coronal field
induction equation:

                                                          Photosphere is lower boundary.
                      Hyperdiffusion: a higher form
                       of diffusion that conserves
                      magnetic helicity. Only arises
                      when there is a gradient of α.
                        Basic Interactions
 Basic interactions are studied in order to gain a better understanding before
  moving onto more complex cases.
 27 simulations in total:
      - each case is simulated with 3 different strengths of
        overlying field (B₀=1G, 5G, 10G, the features have an
        absolute peak strength of ~ 88G).
     - 3 different orientations of bipole: same as, opposite to
       and perpendicular to overlying field.                       Emergence

 Consider:
     - locations of nulls
     - connectivity and interaction with overlying field
     - location of free magnetic energy and electric current
     - height of connections
     - comparison with potential field
Example: Cancellation
                   Example: Cancellation
 The bipole has opposite orientation to the overlying field.
    Example: Flyby
 Bipole is at 90° to the overlying field.

                           •Reconnection occurs as magnetic
                           elements pass one another.

                           •Magnetic energy:
            Example: Flyby

Potential field         Non-Potential field
Decayed Active Region:

•6 days full disc MDI magnetograms
•96 minute cadence
•180x125 pixel region
 (1 pixel = 1.96 arcsec)

•Centered in a 256x256 grid point box
•Corrected for flux balance
         Summary and Future Work
 Summary
  - Photospheric analytical boundary condition avoids undesirable numerical
    effects and allows for full cancellation.
  - A non-linear force-free field allows for free magnetic energy and electric
    currents, as well as twisting and shearing of the magnetic field.
  - We model a continuous evolution of the coronal field rather than single
    independent extrapolations at each time step; connections are maintained
    from one step to the next.

 Future Work
  - Refine the supergranule effect in the photospheric component.
  - Study more complex interactions with the two-component model.
  - Apply the magneto-frictional method to real magnetogram data. Compare the
    simulated coronal field to structures in observations.

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