A. N. McClymont

PROBLEMS AND PROGRESS IN COMPUTING THREE-DIMENSIONAL CORONAL ACTIVE REGION MAGNETIC FIELDS FROM BOUNDARY DATA

An overview of the whole process of reconstructing the coronal magnetic field from boundary data measured at the photosphere is presented. We discuss the errors and uncertainties in the data and in the data reduction process. The problems include noise in the magnetograph measurements, uncertainties in the interpretation of polarization signals, the 180° ambiguity in the transverse field, and the fact that the photosphere is not force-free. Methods for computing the three-dimensional structure of coronal active region magnetic fields, under the force-free assumption, from these boundary data, are then discussed. The methods fall into three classes: the ‘extrapolation’ technique, which seeks to integrate upwards from the photosphere using only local values at the boundary; the ‘current-field iteration’ technique, which propagates currents measured at the boundary along field lines, then iteratively recomputes the magnetic field due to this current distribution; and the ‘evolutionary’ technique, which simulates the evolution of the coronal field, under quasi-physical resistive magnetohydrodynamic equations, as currents injected at the boundary are driven towards the observed values. The extrapolation method is mathematically ill-posed, and must be heavily smoothed to avoid exponential divergence. It may be useful for tracing low-lying field lines, but appears incapable of reconstructing the magnetic field higher in the corona. The original formulation of the current-field iteration method had problems achieving convergence, but a recent reformulation appears promising. Evolutionary methods have been applied to several real datasets, with apparent success.

Rapid sunspot motion during a major solar flare

A major solar flare on 15 November, 1991 produced a striking perturbation in the position and shape of the sunspot related most closely to the flare. We have studied these perturbations by use of the aspect-sensor images from the Soft X-ray Telescope on board YOHKOH, and with ground-based data from the Mees Solar Observatory. The perturbation occurred during the impulsive phase of the flare, with a total displacement on the order of 1 arc sec. The apparent velocity of approximately 2 km s−1 exceeds that typically reported for sunspot proper motions even in flare events. We estimate that the magnetic energy involved in displacing the sunspot amounted to less than 4 × 1030 ergs, comparable to the radiant energy from the perturbed region. Examination of the Mees Observatory data shows that the spot continued moving at lower speed for a half-hour after the impulsive phase. The spot perturbation appears to have been a result of the coronal restructuring and flare energy release, rather than its cause.

A Test for Coronal Magnetic Field Extrapolations

As models for the physical properties of the corona above solar active regions grow more sophisticated, we will require better means for testing them. In this paper we discuss and apply such a test to a magnetic field model for an active region. This test is based on the expectation that the temperatures at different points on a given magnetic field line should be well correlated because of the rapid transport of heat along field lines in the corona. We use radio observations of an active region to measure the temperatures on field lines as they cross two isogauss surfaces (at 430 and 750 G) in the corona. The field lines and isogauss surfaces are derived from a coronal magnetic field model obtained via a nonlinear force-free field extrapolation of a photospheric vector magnetogram; for comparison, we also investigate a potential-field extrapolation of the same magnetogram. In a region in which strongly sheared fields are present, the nonlinear force-free field model does indeed show a good correlation between the temperatures in the two surfaces at points on the same field line, while the potential-field model does not. This diagnostic acts both as a test of the magnetic field model as well as of the interpretation of the radio data, and we show how this test can also aid in understanding the radio data.

Coronal Currents, Magnetic Fields, and Heating in a Solar Active Region

We compare microwave images of a solar active region with state-of-the-art fully nonlinear force-free extrapolations of the photospheric fields in order to study the link between coronal currents and heating of the corona. This extrapolation fully takes into account the nonuniform distribution of electric currents observed in the photosphere and its role in the coronal magnetic structure. We carry out the comparison for AR 6615, a complex region observed with the VLA on 1991 May 7. Under the assumption that the microwave emission is dominated by optically thick gyroresonance radiation, we may use the radio images to infer the temperature of the corona at different heights and locations. This is then compared with heating models based on the observed current distribution. We are able to reproduce the radio images remarkably well with a model in which temperature is structured along magnetic field lines, depends on the current on the field line, and increases with height in a manner similar to that inferred from static heated loop models. This result implies a direct link between electric currents and coronal heating.

RECONSTRUCTION OF THE THREE-DIMENSIONAL CORONAL MAGNETIC FIELD

Studies of solar flares indicate that the mechanism of flares is magnetic in character and that the coronal magnetic field is a key to understanding solar high-energy phenomena. In our ongoing research we are conducting a systematic study of a large database of observations which includes both coronal structure (from the Soft X-ray Telescope on the Yohkoh spacecraft) and photospheric vector magnetic fields (from the Haleakala Stokes Polarimeter at Mees Solar Observatory). We compare the three-dimensional nonlinear force-free coronal magnetic field, computed from photospheric boundary data, to images of coronal structure. In this paper we outline our techniques and present results for active region AR 7220/7222. We show that the computed force-free coronal magnetic field agrees well with Yohkoh X-ray coronal loops, and we discuss the properties of the coronal magnetic field and the soft X-ray loops.

The resistances of the photosphere and of a flaring coronal loop

We consider two aspects of solar flares from the point of view of circuit theory. First, we show that the so-called “dynamo models”, which invoke an analogy between the Earths magnetosphere-ionosphere circuit and the solar corona-photosphere circuit, are illfounded. Second, we consider the rate of coronal energy release in the impulsive phase of a modest flare, and show that, if the energy going into mass motion can be neglected, the corona must present a resistance of about 10−3 μ. Classical resistivity, even in a highly filamented circuit, cannot provide so high a resistance. Anomalous resistivity due to ion sound turbulence can provide the required resistance in this case, but is insufficient to explain the very high power levels inferred in some fast spikes.

Microwave Mode Coupling Above Active Regions as a Coronal Density Diagnostic

It is well recognized that the phenomenon of depolarization (the conversion of polarized radio emission into unpolarized emission) of microwaves over solar active regions can be used to infer the coronal electron density once the coronal magnetic field is known. In this paper we explore this technique using an active region for which we have excellent radio data showing depolarization at two frequencies, and for which we have an excellent magnetic field model which has been tested against observations. We show that this technique for obtaining coronal densities is very sensitive to a number of factors. When Cohens (1960) theory where depolarization is due to magnetic field rotation alone is used, the result is particularly sensitive to the location of the surface on which the magnetic field is orthogonal to the line of sight. Depending on whether we take into account the presence of electric currents in the photosphere or not, their extrapolation into the corona can result in very different heights being deduced for the location of the depolarization strip, and this changes the density which is then deduced from the depolarization condition. Such extreme sensitivity to the magnetic field model requires that field extrapolations be able to accurately predict the polarity of magnetic fields up to coronal heights as high as ∼ 105 km in order to exploit depolarization as a density diagnostic.

The structure and stability of coronal magnetic fields

The high degree of symmetry often assumed in studies of the structure and stability of coronal magnetic field configurations is restrictive and can yield misleading results. We have therefore developed fully three-dimensional numerical methods for constructing force-free equilibria and for examining their stability properties, which make no assumptions about symmetry. A test of the stability analysis has been performed by applying it to the Gold-Hoyle twisted flux tube, which is known to be kink-unstable if the helical field makes more than about one turn between the line-tying end-plates. Our preliminary result is that the critical number of turns is about 1.1, in good agreement with the previous best estimate. However, we find that the growth rate, which has not been discussed previously, is orders of magnitude smaller than expected, even when the flux tube is twisted far beyond the stability limit.