SOI Team Science: Coronal Magnetic Field *Team Leader: J. Todd Hoeksema *Coordinator: Xuepu Zhao *Lead Programmer: Katie Scott *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, the magnetic field becomes radial everywhere due to the radial super-Alfvenic 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 Alfven critical radii. Above the source surface, the alread-opened 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. If the field direction is radial the radial field boundary condition 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 the radial field. 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 Aly, J. J., Astrophys. J., 283, 349, 1984. 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 the international conference held in Freiburg, 1993), p. 27, 1994. 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 the Fourth SOHO Workshop, in press, 1995.

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