Karen Meyer, Duncan Mackay and Aad van Ballegooijen
Outline of talk
Summary and future work
~ 120x120 arcsec²
Provide dominant flow pattern on solar surface at
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)
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
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.
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.
- 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.
- Magnetic topology and connectivity of complex non-potential small
scale coronal field.
- Locations of free magnetic energy and electric current.
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
Represent supergranular cells.
Refine supergranule effect to be
Magnetic concentrations emerge
as pairs within cells:
1- Fragments initially move in
2- Move towards cell edges.
3- Move along cell edges,
interact with other magnetic
(More detailed model: Parnell 2001)
Non-linear force-free fields:
Momentum equation: Force-free condition
• 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).
The Lorentz force acts against an artificial friction to cause the system to relax to
a non-linear force-free equilibrium.
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 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
- locations of nulls
- connectivity and interaction with overlying field
- location of free magnetic energy and electric current
- height of connections
- comparison with potential field
The bipole has opposite orientation to the overlying field.
Bipole is at 90° to the overlying field.
•Reconnection occurs as magnetic
elements pass one another.
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
- 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.
- 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.