SOI Team Science: Coronal Magnetic Field

*Coordinator: Xuepu Zhao

*Technical Summary

1. To compare with and understand  the long-term, large-scale coronal
plasma structures observed by SOHO coronal instruments high up to
10 (UVCS) and even 30 (LASCO) solar radii, we plan to develop a
helicity-free model more realistic than the potential field--source
surface and the potential field--current sheet models to extrapolate
the complete synoptic photospheric magnetic field measured by SOI/MDI
into the corona and heliosphere. This model mimics the interaction
between the accelerating solar wind and the helicity-free magnetic
field using volume and sheet currents flowing in the corona and
heliosphere.

2. To examine and understand the mechanism that works for the explosive
release of free magnetic energy during explosive events in the solar
atmosphere we plan to develop a helicity field model with both field-aligned
and gravity-perpendicular currents to extrapolate the high resolution
photospheric magnetic field measured by SOI/MDI into the corona.
The unprecedented frequency of the MDI magnetic field measurement
during a campaign program will ensure to detect the photospheric magnetic
states nearest to the starting and ending states of a explosive event.
Comparison of the predicted coronal magnetic field configuration with
observed images by other SOHO instruments may be used to test and
determine the explosive release model(s) of the free magnetic energy
that works in explosive events in the solar atmosphere.

*Team Members

Todd Hoeksema
Sasha Kosovichev
Phil Scherrer
Xuepu Zhao

*Detailed Specifications

1. Overview of the coronal magnetic field modeling

The difficult-to-observe coronal magnetic fields play important parts
in structuring the coronal plasma and in generating explosive events
such as flares, erupting prominences and coronal mass ejections.
It is desired to develop rather realistic models to extrapolate the
high quality photospheric field measured by SOI/MDI into the corona
in supporting other SOHO experiments such as EIT, UVCS, LASCO and SWAN.

Limitations of spatial and temporal resolution and nonuniform quality
of the input data are important factors that make it difficult to
investigate the response of the corona to drastic changing photospheric
conditions. The MDI will produce a series of full-disk photospheric
magnetic field observations free of the distortions from the Earth's
atmosphere and with 4" resolution about every 96 minutes and a
sensitivity better than 20 Gauss. During 8-hour campaigns, MDI
magnetograms can be taken in higher resolution (0.625" pixels)
as often as every minute to closely follow the development of
the photospheric field. These data should provide a basis for
predicting the coronal and heliospheric field and their changes
with unprecedented accuracy.

Improved understanding of the structure and dynamics of the magnetized
corona requires more realistic coronal magnetic field model to extrapolate
the high quality MDI photospheric field measurements into the corona.
The 3-D full MHD model is the most realistic. This approach uses a
time-dependent, numerical, magnetohydrodynamic model. The problem is
treated as an initial-boundary value problem in which the steady
state is found by holding the boundary conditions constant and allowing
the solution to relax in time from an essentially arbitrary initial
state. Thus its implementation must be time consuming even using
CRAY computers. In addition to the photospheric magnetic field as an
input to the model, the plasma properties  is also necessary to be
specified at the photosphere, which is difficult to be realistic
at the present. Besides, it involves the use of an artificial viscous
coefficient and the computation boundary conditions. Generally speaking,
such complex numerical calculations are rarely without uncertainties
and problems.

The corona is in a dichotomy of two distinct states, typified by
coronal holes and coronal helmet streamers. The magnetic field
in coronal holes is opened up by the solar wind to the heliosphere
with Alfvenic fluctuations along open field lines, that would be the
propagated twist in the original closed magnetic field (Low, 1994).
The fields within helmet-streamers are sometimes twisted, making up
one of major energy sources of explosive events. We plan to develop
two models that use, respectively, the routine and campaign MDI
observations of the magnetic field. One models the long-term and
large-scale structures by mimicking the interaction between the solar
wind and twist-free magnetic field in various heights. The model
includes effects of low-altitude large-scale horizontal currents
below cusp points of streamers, the heliospheric current sheet
above cusp points, and heliospheric currents above the Alfven critical
point. The other model builds on a magnetostatic atmosphere with
field-aligned currents as well as gravity-perpendicular currents
(Neukirch, 1995) to produce quasi-stable closed field geometry
and compare it with the observed quasi-stable plasma structures
just before and after a explosive event. By adjusting the field-aligned
and/or gravity-perpendicular currents in the corona to reproduce the
emission or white-light structures in the solar images observed just
before and after a explosive event, the predicted changes of magnetic
configuration may be used to detect which is the working model
among the explosive release models of the free magnetic energy
presented to understand explosive events.

2. The helicity-free field model

The interaction between the solar wind and magnetic field may be
characterized into three regions. In the inner region lower than
cusp points of coronal streamers the magnetic field dominates
over the plasma. In the outer region higher than the Alfven critical
point the solar wind totally controls the magnetic field,
solar wind velocity. In the middle region between the inner and
outer regions, the field is opened up but the plasma is stil
partly guided by the field.

Distortion of the field by the solar wind requires electric currents
to couple the hydrodynamic and magnetic forces. Observations of coronal
helmet streamers, coronal holes, and the solar wind imply the existence
of both volume and sheet currents in the corona and heliosphere.
Volume currents flow everywhere, but sheet currents flow mainly at
the boundaries between oppositely directed field lines (i.e. within
coronal streamers and near IMF sector boundaries) and between open
(holes) and closed (helmets) field regions. Pneuman and Kopp's
classic MHD model [1971] showed the presence of just such volume and
sheet currents.

The helicity-free model developed by the team (Zhao and Hoeksema, 1995a)
divides the solar atmosphere into three parts separated by two spherical
surfaces. The inner sphere, called the cusp surface, is located
approximately at the height of the cusp points of coronal streamers.
Above the cusp surface the coronal magnetic field is open everywhere.
The outer sphere, called the source surface, is located near the
field becomes purely radial. The model builds on a magnetostatic atmosphere
with large-scale horizontal currents (Bogdan and Low, 1986) and  uses
Schatten's techniques for calculating the effects of heliospheric
volume and sheet currents (Schatten et al, 1969; Schatten, 1971).

3. The helicity field model

Explosive events in the solar atmosphere, such as flares, erupting
prominences and coronal mass ejections, are believed to originate
in closed regions with enhanced and highly sheared coronal magnetic
fields, and the nature of the explosions is an explosive release of
energy, partly or totally, stored in the coronal magnetic field
(Sturrock, 1980). The magnetic energy to be released must be in excess of
the energy of a potential field. The energy stored in the magnetic field
may have a variety of sources. Shearing of field lines by photospheric
motion of the foot points, shearing of field lines by the expanding
(open) corona, and generation of currents by emergence of new magnetic
flux ropes in the vicinity of existing loops are specific possibilities
(Stix, 1989; Rust and Kumar, 1994)) though it is extremely difficult,
if not impossible,
for a force-free magnetic field in the form of an anchored bipolar field
to possess energy in excess of that in its associted open configuration
(Aly, 1984; Low, 1994).

Changes of magnetic configurations just before and after a event
may be different for the field-aligned current
and current sheet sources. For the former, it is
relaxation of sheared fields. For the latter, it is topologically
change of the field configuration. If both sources are functioned, both
magnetic helicity density and field topology would be changed.
Comparison the calculated field geometry changes with observed emission
structure variation before and after a event may determine
what kind of physical process works in the event.

The shear of the magnetic field in helmet streamers has been calculated
using force-free magnetic field models. A point often overlooked is
that such large-scale structure as the helmet streamer cannot be
modeled as a force-free magnetic field. The force-free assumption
in popular use is valid only at the base of the corona where the
the thermal pressure gradient and gravity may be neglected
in comparison to the magnetic force.

Neukirch (1994) recently presents a mathematical procedure to
calculate special self-consistent three-dimensional analytic
solutions of the magnetohydrostatic (MHS) equations. By prescribing
a special type of current that consists of a field-aligned part
and a gravity-perpendicular part, the non-linear MHS equations
can be reduced to a solvable Schrodinger type equation. The solution
both for constant and 1/r^2 gravitational field has been
derived with two free parameters, $\alpha$ and $a$. The parameter
$\alpha$ is similar to the constant in the linear force-free field model,
representing the magnetic helicity density. The parameter $a$
denotes the scale height of the horizontal component of the current.
The unknown coefficients in the solution can be determined using the measured
line-of-sight component of the photospheric magnetic field.

The MDI campaign data closely following the development of the
photospheric field is the best data to be used to obtain the photospheric
field very close the starting and ending states of the explosive events
observed by SOHO coronal instruments, and thus provide accurate input
to the helicity field model.

*Development Plan and Status

1. The helicity-free field model

The helicity-free field model has been developed and tested using WSO
data. The free parameters in the model has been preliminarily determined
using solar eclipse images and the heliospheric magnetic field
data measured by various in-ecliptic spacecraft and Ulysses.
Comparison of model calculations with observations showed that
the agreement is improved if the photospheric field measured by
$\lambda$5250 \AA line is assumed to be radial and is scaled up by a
latitude-dependent correction factor. In addition to successful
prediction of the location of coronal holes near the solar surface
and the location of the heliospheric current sheet, the model allows
to approximately reproduce the non-radial nature of coronal streamers
and the solar cycle and latitude dependence of the radial component
of the heliospheric magnetic field (Zhao and Hoeksema, 1995b;
Zhao, Hoeksema and Scherrer, 1995; Hoeksema, Zhao and Scherrer, 1995).

We have developed the mathematical model including effect of
the heliospheric current sheet on the coronal field below the cusp
points. It may improve the prediction for the structure in the inner
corona.

After the improvement, a rotating hairy Sun will be presented.

All tasks here are planed to be accomplished by the end of October, 1995
(before the SOHO launch).

2. The helicity field model

Develop and test the computer program to set up a composite synoptic
chart of the radial photospheric magnetic field using the WSO, KPNO and
GONG data (see discussion in the next section of Outstanding Problems).

Using KPNO and GONG data and the helicity-free field model to examine
the direction distribution of the photospheric field measured by
$\lambda$6768 lin and its possible saturation correction factor
(see discussion in the next section of Outstanding problems).

Develop the computer program for the helicity field model using
Neukirch's formulae.

Test the program using KPNO and GONG data of comparable spatial resolution,
and available MDI calibration data.

All work for the helicity field model will be finished by the
end of May, 1996.

*Outstanding Problems

1. The direction of the photospheric magnetic field

The field direction at the height where the spectral line
$\lambda$ 6768 $\AA$ emitted is an important factor that would
significantly affact the coronal magnetic field predicted.
can be used. Otherwise the line-of-sight field boundary condition
should be adopted (e.g. zhao and Hoeksema, 1993).

The photospheric magnetic field may be divided into strong fields
and weak fields, with a gap separating them in all but
transient conditions. The strong fields are found in sunspots,
faculae and in patches in the magnetic network. These field
elements are believed to be buoyant and consequently nearly vertical,
and to be created by flux emergence in $\Omega$-shaped loops.
Thus the strong field is supposed to be nearly radial.
The weak (intranetwork) field is found everywhere'' between the patches
of strong field and is suggested to be created by slowly upward floating
bottoms of {\bf U}-loop. It is also suggested that the weak field
intersects the photosphere at all angles. Because of its weakness and
mixed polarities, the weak field does not penetrate into the outer
atmosphere (Zwaan and Harvey, 1994 and References therein).
In the preliminary calculation
we will assume that the photospheric field measured by MDI
is radial everywhere at the photosphere. This assumption makes it
possible to use a large part of a magnetogram (not only the central strip of
the disk) as the input to a model. It is especially helpful
in calculation using the helicity field model.

2. The latitude-dependent correction factor

It is found recently (Wang and Sheeley, 1995; Zhao and Hoeksema, 1995b)
that in quantitatively extrapolating the photospheric field measured
by $\lambda$5250 $\AA$ line magnetograph it is necessary to correct
the saturation effect'' due to the magnetic and temperature
sensitivities of the spectral line.
The coronal magnetic field calculated using the WSO data
scaled up by a latitude-dependent correction factor is better than
by a constant factor in reproducing various observed structures.

What is correction factor for MDI data to revert the measured
field strength into true'' field strength.

3. Composite synoptic charts of the radial photospheric field.

In investigating the dynamic variation of the coronal magnetic field
due to an explosive event, the Cartension (XYZ) coordinate system
is usually adopted to map a limited photospheric field region into
the corona. A difficulty in the use of this approach is that boundary
conditions must be specified at the edges of the limited photospheric
region, and these boundary conditions are equivalent to assumptions
about how neighboring photospheric regions contribute to the coronal
magnetic field being calculated. In addtion, for such large-scale explosive
events as erupting prominences and coronal mass ejections, the effect of
spatial curvature must be accounted for in calculation. We plan to use
the spherical coordinate system with a higher angular resolution than that
used in the helicity-free field model.

As the input to the helicity field model, the synoptic chart of the
photospheric field is different from usual one. First, it is a synoptic
chart of the *radial* photospheric field. Secondly, the central
53.2-degree-width (or 79.8-degree-width) part of the synoptic chart
is replaced by the appropriate part of the magnetogram representing the
starting or ending state of the event in question. With the assumption of
radial photospheric magnetic field, the line-of-sight component in
the magnetogram (Level 1) is divided by \cos(\theta)cos(\phi) to obtain

4. Uniqueness of the solution

Both the helicity-free field and the helicity field models build on
special self-consistent three-dimensional analytic solutions of
the magnetohydrostatic (MHS) equations. General speaking, the solution
of the MHS equations is notoriously difficult due to their intrinsic
nonlinearity. Moreover, the problem can general only be reduced to a
system two coupled nonlinear partial differential equations (if e.g.,
the magnetic field is represented by Euler potentials) which has no
definite type in terms of elliptic, hyperbolic or parabolic. Thus it
is very difficult, if not impossible, to mathematically prove the
uniqueness of a special MHS solution from its general solutions.

The principle idea used to derive the analytic solutions on which
the models here build is to prescribe a special type of current flow
which allows then the reduction of the mathematical problem to a
single partial differential equation. The method depends strongly
on the inclusion of an outer gravitational field and on the trick
that the equation of state relating the plasma pressure to the
plasma density is not fixed a priori. These two features give
rise to an additional degree of freedom for the problem. That is
why there is free parameter $a$ in the solution.

When the current flows perpendicular to the gravity everywhere
three sets of special analytic solutions are obtained (Bogdan and Low,
1986). The set of solutions that used in the helicity-free field
model has been mathematically proved to be unique when the boundary
condition used is either radial field or line-of-sight (Zhao and
Hoeksema, 1993). The uniqueness of the special solution used
in the helicity field model appears possible to be proved.

*Comparable Projects / Datasets:

1. Bagenal and Gibson's coronal modeling starting from density observations.

2. Sheeley and Wang's modeling of solar wind velocity, field strength.

3. Linker and Mikic's 3-D MHD modeling of corona.

4. Other SOHO instruments - SUMER, CDS, UVCS, EIT, LACSO, and SWAN for
determining the structure and dynamics of the coronal plasma.

5. Projected availability of GONG magnetic time series.

*Bibliography

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Bogdan, T. J. and B. C. Low, Astrophys. J., 306, 271, 1986.
Hoeksema, J. T., X. P. Zhao and P. H. Scherrer, Solar Wind 8, in press, 1995.
Low, B. C., The Proceedings of the Third SOHO Workshop, 123, 1994.
Neukirch, Astron & Astrophys, in press, 1995.
Rust, D. M. and A. Kumar, The Proceedings of the Third SOHO Workshop,
39, 1994.
Schatten, K. H., Cosmic Electrodyn., 2, 232, 1971.
Schatten, K. H., J. W. Wilcox, and N. F. Ness, Solar Phys., 6, 442, 1969.
Stix, M., The Sun, 1989.
Sturrock, P. A., (ed.), Solar Flares, 1980.
Wang, Y.-M. and N. R. Sheeley, Jr., Astrophys. J., 447, L143, 1995.
Zwaan, C. and K. L. Harvey, Solar Magnetic Fields (Proceedings of
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Zhao X. P. and J. T. Hoeksema, Solar Phys., 143, 41, 1993.
Zhao X. P. and J. T. Hoeksema, J. Geophys. Res., 19, 1995a
Zhao X. P. and J. T. Hoeksema, Solar Wind 8, in press, 1995b.
Zhao X. P., J. T. Hoeksema and P. H. Scherrer, Proceedings of
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