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
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
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?
Why we think of active regions as flux tubes.
(this shape of flux tube is known as an Ω-loop)
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
Dv
i
 Fb  Ft  Fc  Fd
Dt

Fb  g ( e  i )r
2
B
Ft 

4
Fc  2 i   v
CD
Fd    e
| v | v
1/ 2
( / B)
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
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 > 105 G, no low-latitude equilibria possible
• If B < 3x104 G, poleward motion too great for
observed latitude distributions
• If active region flux tubes are initially toroidal,
field strength is constrained.
What are active region tilts and
what is Joy’s Law?
Thin flux tube models do an excellent job of explaining Joy’s
law. The tilt comes from the Coriolis force acting on the rising,
expanding plasma in an emerging, initially toroidal flux loop.
Torque balance between magnetic
tension and Coriolis forces determines the
amount of tilt in an emerging active region
(for northern hemisphere)
The result is that   sin   .
1/ 4
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.)
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 D ~ d-3/4.
The observed variation of Δ with d
suggests convective turbulence as a
possible mechanism for tilt fluctuation
Tilt fluctuations computed using tube dynamics
perturbed by convective turbulence can explain
the observed variation of Δα with d.
Asymmetric spot motions can be explained
by asymmetric shapes in emerging Ω-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.
The Coriolis force is a possible explanation for
asymmetries in the morphology of active regions
Field strength asymmetries could lead
to morphological differences between
leading and following polarity
How twisted are typical active regions?
Where does active region twist come from?
d
2 da
dq

2 q

  vA
;
 q 
 ( s, t ) ,
dt
a dt
s
dt
s
where

d ln  S  v
v
 
 s
, and   (s  )  .
dt
s
s
(Longcope & Klapper 1997; Longcope, Fisher & Pevtsov 1998)
What is the physical meaning of the source term ?
 depends only on the motion of tube axis.
For a thin flux tube: H = 2 (Tw+Wr). ( Conservation of
magnetic helicity H,where Tw is “twist”, and Wr is “writhe”.)
 
1
1
s 's"(r"r' )
Tw 
q( s)ds ; Wr 
ds'  ds"
,
3


2
4
| r"r ' |
d Tw
d Wr

  ( s)ds.
dt
dt
Therefore,  exchanges writhe (Wr) with twist (Tw).
Could flux tube writhing account for
observed levels of active region twist?
Possible sources of writhing:
• “Joy’s Law” tilts of active regions during
emergence is one possibility, but is too small…
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)
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
The life cycle of an active region must somehow transition between an emerging Ω-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.
2.
2.
3.
Images courtesy of Loraine Lundquist
Active region below the surface is an
emerging Ω-loop
As the magnetic flux breaks through the
photosphere, sunspots form and the initial
coronal magnetic field is established
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
What sub-surface thin flux-tube models have
told us about the origin of active regions
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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)
Hale’s law and Joy’s 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 Ω-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
Ω-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.
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 δ-spot active regions
• MHD simulations have shown directly how a small-scale
disordered magnetic dynamo can be driven by
convective motions
3D-MHD models of flux emergence confirm the asymmetric shape of
the  loop predicted by thin flux tube models
(Fig. 2 from Abbett Fisher & Fan 2001)
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?
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
B
Hp
a
Beq
Island δ-spot active regions can be understood as twisted, kinking
flux tubes (Linton, Fan)
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Properties of δ-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
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?
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.
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:
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:
This movie shows the time evolution of a volume rendering
of the magnetic field strength in the convective dynamo
after saturation has occurred
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
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)
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.
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 Bercik’s
(2002) thesis.
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.
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 selfconsistently 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 codecode interface
• (2) Implement numerical techniques that can
accommodate greatly different physical
conditions
This figure shows Simulations of the Quiet Sun using Abbett’s new code,
AMPS. These simulations extend from the upper convection zone,
through the photosphere, a simplified chromosphere, transition region, and
a corona.
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.
Questions
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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 δ-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?