Understanding the Photospheric and Near-Photospheric Magnetic Field

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Understanding the Photospheric and
Near-Photospheric Magnetic Field

• A View of the Past, Present, and Future of
  First-Principles Magnetic Field Modeling at the
  Photosphere and Below
• George Fisher, SSL UC Berkeley

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Goal of talk - Stimulate thinking and
discussion of how the HMI/AIA instruments can
best improve our knowledge of magnetic field
structure and evolution near the photosphere.
Three main topics

• A review of what Thin Flux Tube models have
  taught us about the origins of active region
  magnetic fields and the observed properties of
  active regions
• A biased and incomplete survey of what 3-D MHD
  simulations of the solar interior have taught us
  about photospheric magnetic fields
• A glimpse into the future of magnetic field
  modeling in the solar atmosphere and interior

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During the 1990s, most of the theoretical
understanding about the origin of active regions
as observed as sunspot groups and as bipolar
regions in full disk magnetograms came from thin
flux tube models.What motivates this approach,
and what is it exactly?
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Why we think of active regions as flux
tubes.(this shape of flux tube is known as an
O-loop)
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Thin flux tube models assume that the dynamics of
a flux tube can be described by the forces acting
on a thin 1-D tube imbedded in a 3-D model of the
solar interior
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The action of the Coriolis Force in thin flux
tube models provides one possible explanation for
the observed equatorial paucity of active regions

• The Coriolis force deflects tube toward the poles
  as it tries to rise radially

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The limits of the active region latitude belts
provide strong constraints on the initial
magnetic configuration and magnetic field
strength, according to thin flux tube calculations

• If B gt 105 G, no low-latitude equilibria possible
• If B lt 3x104 G, poleward motion too great for
  observed latitude distributions
• If active region flux tubes are initially
  toroidal, field strength is constrained.

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What are active region tilts and what is Joys
Law?
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Thin flux tube models do an excellent job of
explaining Joys law. The tilt comes from the
Coriolis force acting on the rising, expanding
plasma in an emerging, initially toroidal flux
loop.
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Torque balance between magnetic tension and
Coriolis forces determines the amount of tilt in
an emerging active region
(for northern hemisphere)
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Fisher, Fan Howard (1995) found from an
analysis of spot group data that the predicted
flux dependence of the tilt angle was consistent
with the data.
(Since this work was done, however, Tian Liu
(2003) have updated these results with magnetic
fluxes instead of polarity separations, allowing
for a more direct comparison with the theory.
Those results show a less clear agreement
between theory and observation.)
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Not only are there tilts, but quite significant
fluctuations of tilt
Analysis of 24,000 spot groups shows tilt
dispersion is not a function of latitude, but is
a function of d, with Da d-3/4.
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The observed variation of ?a with d suggests
convective turbulence as a possible mechanism for
tilt fluctuation
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Tilt fluctuations computed using tube dynamics
perturbed by convective turbulence can explain
the observed variation of ?a with d.
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Asymmetric spot motions can be explained by
asymmetric shapes in emerging O-loops

• Asymmetric shapes in the emerging loops originate
  from Coriolis forces that act to preserve angular
  momentum
• Panels (a), (b), (c) correspond to field
  strengths at the base of the convection zone of
  30, 60, and 100 kG respectively (Fan Fisher
  Sol. Phys. 166, 17)
• Caligari Moreno-Insertis Schüssler (1995)
  suggested that the emergence of these asymmetric
  loops will result in faster apparent motion of
  the leading spot group polarity c.f. the
  following polarity, a well known observational
  phenomenon.

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The Coriolis force is a possible explanation for
asymmetries in the morphology of active regions
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Field strength asymmetries could lead to
morphological differences between leading and
following polarity
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How twisted are typical active regions?
Where does active region twist come from?
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where
(Longcope Klapper 1997 Longcope, Fisher
Pevtsov 1998)
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What is the physical meaning of the source term
S? S depends only on the motion of tube axis.
For a thin flux tube H F2 (TwWr). (
Conservation of magnetic helicity H,where Tw is
twist, and Wr is writhe.)
Therefore, S exchanges writhe (Wr) with twist
(Tw).
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Could flux tube writhing account for observed
levels of active region twist? Possible sources
of writhing

• Joys Law tilts of active regions during
  emergence is one possibility, but is too small

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Writhing by convective turbulence is another
possibility

• Develop a tractable model of convective
  turbulence that includes kinetic helicity
• Solve equations of motion and twist evolution for
  a flux tube rising through such a turbulent
  medium
• Such a model was explored by Longcope et al.
  (1998)

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The writhing of initially untwisted flux tubes by
convective motions containing expected levels of
kinetic helicity leads to a twist distribution
with latitude that is consistent with observations
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The life cycle of an active region must somehow
transition between an emerging O-loop and active
region decay, as described by passive flux
transport models. A new idea for this transition
has been proposed by Schüssler Rempel.
1.
3.

1. Active region below the surface is an emerging
   O-loop
2. As the magnetic flux breaks through the
   photosphere, sunspots form and the initial
   coronal magnetic field is established
3. As the plasma in the spots cools and sinks, and
   the buoyant plasma from below emerges, the upper
   parts of these flux tubes are blown apart and are
   then controlled by convective motions. Passive
   flux transport models then describe the surface
   evolution of the active region field

2.
Images courtesy of Loraine Lundquist
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What sub-surface thin flux-tube models have told
us about the origin of active regions

• Active regions originate from a toroidal field at
  the base of the convection zone, whose sign
  changes across the equator, with a field strength
  in the range of 2x104 105G.
• The paucity of active regions near the equator
  could result from deflection toward the poles by
  the Coriolis force (low B), or from a lack of
  stable solutions (high B)
• Hales law and Joys law (active region
  orientation) can be reproduced with thin flux
  tube models, in which Coriolis forces balance
  magnetic tension
• The dependence of active region tilt on AR size
  might be explained by a balance between Coriolis
  forces and magnetic tension
• The dispersion of tilt versus active region size
  can be understood by the forces acting on a flux
  tube perturbed by convective motions
• Asymmetric spot motions (leading vs following)
  can be explained by the asymmetric shapes of
  O-loops. The asymmetric shapes result from the
  Coriolis force.
• The morphological asymmetry (the leading side
  being more compact than following side) of active
  regions may be explained by a field strength
  asymmetry in emerging O-loops, driven indirectly
  by Coriolis forces
• The observed helicity distribution with latitude
  of active regions can be explained by expected
  levels of kinetic helicity in convective motions
  that act to writhe magnetic flux tubes during
  their emergence toward the surface.

Future global MHD models of magnetic fields in
the solar interior need to include spherical
geometry and rotational effects, including
Coriolis forces.
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3-D MHD simulations of magnetic fields in the
solar interior can describe physics not that
cannot be addressed with thin flux tube models.
In the late 1990s, computers and numerical
techniques became powerful enough to address
problems of real significance for the Sun, rather
than highly idealized toy problems.

• MHD simulations can show us how active region
  scale flux tubes evolve in a model convection
  zone that is actually convecting
• MHD simulations have shown how kink unstable
  magnetic flux tubes may be able to explain many
  observed properties of island d-spot active
  regions
• MHD simulations have shown directly how a
  small-scale disordered magnetic dynamo can be
  driven by convective motions

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3D-MHD models of flux emergence confirm the
asymmetric shape of the W loop predicted by thin
flux tube models(Fig. 2 from Abbett Fisher Fan
2001)
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Active Region Fields in a Convectively Unstable
Background State
From Abbett et al. 2004

• Q What happens to an active region flux tube in
  a convection zone?

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Active Region Fields in a Convectively Unstable
Background State

• Q What are the conditions for the tube to retain
  its cohesion?
• Fieldline twist is relatively unimportant what
  matters is the axial field strength relative to
  the kinetic energy density of strong downdrafts

From Fan et al. 2003
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Island d-spot active regions can be understood as
twisted, kinking flux tubes (Linton, Fan)

• Properties of d-spot regions
• Sunspot umbrae of opposite polarity in a common
  penumbra
• Strong shear along neutral line
• Active region rotates as it emerges
• Large and frequent flares and CMEs
• Kinked geometry explains rotation, shear along
  neutral line
• Flares/CMEs might be explained by reconnection
  between the 2 legs of the intertwined loop
  structure

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On small scales, the solar magnetic field
appears, evolves, and disappears over very short
time scales
Is the small scale magnetic field on the Sun and
other stars the lint from the clothes in the
solar washing machine, or is it generated by its
own dynamo mechanism?
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We have performed our own simulations of
small-scale magnetic fields driven by convective
turbulence in a stratified model convection zone
without rotation, starting from a small seed
field. The magnetic energy grows by 12 orders of
magnitude, and saturates at a level of roughly 7
of the kinetic energy in convective motions.
This simulation took about 6 CPU months of
computing time.
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What does the generated magnetic field look like?
Here is a movie showing magnetograms of the
vertical component of the field in 2 slices of
the atmosphere, near the bottom and near the top
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Here is a snapshot showing volume renderings of
the entropy and the magnetic field strength in
the convective dynamo simulation at a time after
saturation
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This movie shows the time evolution of a volume
rendering of the magnetic field strength in the
convective dynamo after saturation has occurred
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How do we connect our simulation results to real
data for the Sun and stars?

• We must first convert the dimensionless units of
  the anelastic MHD code to real (cgs) units
  corresponding to the convective envelopes of real
  stars (1) demand that stellar surface
  temperature and density match those of model
  stellar envelopes, (2) Demand that the convective
  energy flux in the simulation match the stellar
  luminosity divided by the stellar surface area.
  We use mixing length theory to connect energy
  flux to the unit of velocity in the simulation.
  After applying these assumptions, we can scale a
  single simulation to the convective envelopes of
  main-sequence stars from spectral types F to M.
• To convert magnetic quantities from the
  simulations to observable signatures, we use the
  empirical relationship between magnetic flux and
  X-ray radiance (from Pevtsov et al) to predict
  surface X-ray fluxes for main-sequence stars

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Quantitative studies of magnetic dynamos on other
stars requires a quantitative knowledge of the
relationship between magnetic fields and
activity indicators such as X-ray
flux(Pevtsov et al. 2003, ApJ 598, 1387)
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So how does the convective dynamo model compare
to observed X-ray fluxes in main-sequence stars?
The convective dynamo model does an excellent job
of predicting the lower limit of X-ray emission
for slowly rotating stars, and for predicting the
amount of magnetic flux observed on the Quiet
Sun during solar minimum.
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Many Simulations of the Plasma just below the
solar photosphere now Include a great deal of
physical realism, including 3-D radiation
transfer and a realistic equation of state. This
allows for the self-consistent Formation of cool
micropores at magnetic flux concentrations, as
seen in this simulation from Dave Berciks (2002)
thesis.
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Simulation of a magnetic plage region using the
MURaM code by Vögler et al (2005). This code
solves the 3D MHD equations and non-grey LTE RT
equations in 3D for the convection zone and
photosphere.
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What is the future of first-principles magnetic
field modeling near the solar photosphere?

• A correct description of the physics of magnetic
  field evolution of the solar atmosphere must
  self-consistently couple very different regions
  of the solar atmosphere. Presently, there are 2
  approaches to this problem
• (1) Develop coupled models of the different
  regions, which communicate across a code-code
  interface
• (2) Implement numerical techniques that can
  accommodate greatly different physical conditions

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This figure shows Simulations of the Quiet Sun
using Abbetts new code, AMPS. These
simulations extend from the upper convection
zone, through the photosphere, a simplified
chromosphere, transition region, and a corona.
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AMPS (the Adaptive MHD Parallel Solver) was
designed from the outset to use the Paramesh
domain-decomposition libraries, allowing for an
efficient MPI/AMR environment. The code uses a
semi-implicit technique -- Newton-Krylov
formalism is used to evolve the troublesome
energy equation source terms implicitly, and the
semi-discrete formalism of Kurganov Levy 2000
(with 3rd order CWENO interpolation) is employed
as the shock-capture scheme and is used to
explicitly advance the continuity, induction,
and momentum equations. Lower left illustrates
how Paramesh divides the computational domain
into sub-regions, each handled by a separate
processor. Lower right shows Orszag-Tang vortex
and MHD blast tests with AMR.
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Questions

• What triggers the eruption of active regions from
  the base of the convection zone? Instability,
  secular heating, convective overshoot, or
  something else?
• Most active regions exhibit only small amounts of
  twist, and are consistent with a tube lying
  initially at the base of the convection zone with
  no twist. How then do the island d-spot regions
  acquire so much twist?
• How is the free energy from sub-surface fields
  transported into the corona?
• What is the magnetic connection between different
  active regions? Are active regions magnetically
  connected to each other in the dynamo region, or
  are they all separate?
• How do we relate active longitudes and active
  nests to a magnetic picture of the large-scale
  dynamo region at the base of the convection zone?
• Is the magnetic flux that gives birth to active
  regions in a smooth, slab-like geometry, or is
  the flux already pre-existing in the form of
  tubes?
• What happens when active region flux tubes
  collide in the solar interior? What happens when
  a new active region emerges into an old one?
• How do active region flux tubes interact with the
  small scale field in the Quiet Sun?
• What is the 3D analogue of the surface flux
  transport models? How does the following
  polarity from decaying, emerged active regions
  return to the dynamo regions?
• What happens to the magnetic roots of an emerged
  active region once the active region begins
  decaying?
• Can we infer the sub-surface structure of an
  active region by studying its surface evolution?
  (Try this for AR 8210!)
• Can we better predict the emergence of new active
  regions before it happens, either from
  helioseismic observation, or from a better
  knowledge of the physics of magnetic evolution
  below the photosphere?