Linear Force-Free Field Modeling: Method
A method for the reconstruction of the linear force-free magnetic field in a bounded domain was elaborated. The methods allows to specify boundary conditions not only on the "photospheric" level but also on lateral parts of the volume. Dirichlet boundary value problem for the Helmholtz equation is solved for B_z component specified at the \Omega boundary. Chebyshev's iteration method with the optimal rearrangement of the iteration parameters sequence was used. The solution is obtained as for the positive-definite, so for the non-sign-definite difference analogue of the differential operator \nabla^2 u+\alpha^2 u. Specifying two scalar functions B_x and B_y on the intersection of the lateral part of the \Omega boundary with one selected plane z=const and using B_z inside the \Omega, we have found B_x and B_y throughout the \Omega. The algorithm was tested with the numerical procedure, which gives the analytic solution B of the linear force-free field (LFFF) equations for the dipole in a half-space. The r.m.s. deviation of the analytic solution B from the calculated B' does not exceed 1.0% (Figure 1). Comparison of B' with B'', which was calculated by the potential non-photospheric boundary conditions, shown that they differ significantly (Figure 2, 3, 4). Thus, specification of boundary conditions at non-photospheric boundaries of the volume is of particular importance when modeling the LFFF above an active region.

Figure 1. Average over the entire horizontal plane of the integration domain relative error of the numerical solution B' relative to the analytic solution B as a function of z.

Figure 2. Contours of B_z component of magnetic field at different heights. B' - linear force-free field with analytic force-free boundary conditions and for B''- linear force-free field with potential non-photospheric boundary conditions

Figure 3. Azimuth of the calculated magnetic field. See explanations on Figure 2

Figure 4. Topview (top panel) and sideview (bottom panel ) of the lines of force of fields B' and B''

Linear Force-Free Field Modeling: Application to AR 7216
The new numerical code for the LFFF modeling in a bounded domain above an active region was applied here to the AR NOAA 7216 (Figure 5). The code uses the B_z - component of the magnetic field at the photosphere as main data and the information about coronal fields as supplementary data. In the considered case of AR 7216 the potentiality of external X-ray coronal loops was used. The potential B_z substitution at the non-photospheric sides of the considered volume as boundary conditions secured a good modeling of the coronal field of the AR 7216. It has been shown that coronal magnetic field of the AR 7216 4 July,1992 at about 00:29 UT was non-linear force-free field with decreasing with high positive \alpha, slowly turning into potential one at the height about 10^5 km (differential shear) (Figure 6). Electric currents in coronal loops were anti-parallel to the magnetic field, their density decreased with height. The total coronal twisting of this large, quiet, vortexless active region was weak (\alpha = 0.001,...,0.007 arcsec^-1), but the sense of twisting - right-hand - was nontypical for active regions of the northern hemisphere.

Figure 5. Vector magnetogram of AR NOAA 7216. Solid lines - N polarity.

Figure 6. Comparison of observed SXT/YOHKOH X-ray loops (dots) and calculated field lines with different alpha

Magnetic Structure of an AR Filament
Investigations of filaments both quite and active region is point of major interest because they are connected with the CMEs and solar flares occurrence. Central question of these studies is the inferring of the filament magnetic field structure. It is not practical to measure the magnetic field vector of an active region filament because most of them are located too low in the corona for direct spectroscopic observations. That is way any indirect inferences concerning the magnetic field structure of an active region filament are interesting and useful for theoretical considerations.
We report here on observations of the active region NOAA 7597 in which the SN/C1.3 two-ribbon flare occurred on Oct 16, 1993. After the two-ribbon flare started (0919 UT), evaporated chromospheric plasma filled numerous magnetic loops (unvisible in H$_{\alpha}$ before the flare) that span the photospheric inversion line and the filament itself. Figure F1. shows longitudinal magnetic field (northern polarity - solid contour), orientation of the transverse magnetic field (short line segments) and the location and the orientation of the fine structure of the fileamnt (heavy short lines). By comparing the orientation of the photospheric transverse magnetic field with that of chromospheric fibrils (Figure F1) and soft X-Ray structure (Figure F4) one can see that they significantly differ. Thus, the orientation of transverse magnetic field are changed when we go from the photospheric to the coronal level. It could be nothing surprising but such changes take place in the volume occupied by the active region filament. The observed discrepancy can help us to make a conclusion about possible magnetic configuration of the observed filament. A possibile explanation is presented in Figure F2 and F3 and is based on the inverse configuration model. Let us suppose that converging and shearing flows bring together the magnetic loops footpoints and twist the loops themseves at both sides of the photospheric inversion line. Then the orientation of the transverse magnetic field above and underneath the filament will be different as it follows from Figure F3. The only thing one should keep in mind is, that in this case, any method for resolving of the 180 degree ambiguity will give us normal direction of the transverse field (Figure F2, dashed arrow) while the real thansverse (to the neutral line) component of the magnetic field of the filament at photospheric level pointed in S-N direction (Figure F2, solid line). This can be important when study the the flare activity (so called "bald patches"), the variation of the magnetic shear (Robert Cameroon, private communication) and also when processing the raw data of vector magnetic field.

Figure F1. The map of vector magnetic field (image plane, N polarity -- solid line, short lines - denote transverse field) is overlapped by the sketch of H\alpha loops (heavy lines)

Figure F2. Cartoon of the possible magnetic configuration of the active region filament: side view

Figure F3. Cartoon of the possible magnetic configuration of the active region filament: side view

Figure F4. The soft X-ray loops above the filament. The orientation of the loops is similar to that of the fine strcuture of the H\alpha filament

Current Helicity Imbalance
It is widely accepted that the energy source for non-stationary processes in the solar corona is the magnetic field energy stored in the form of electric currents. Since flares seemingly can occur in any active region, the energy build up mechanism must be easy accessible for all solar active regions. New conceptions of the magnetic energy build-up based on helical nature of solar magnetic fields were proposed recently. It has been shown that such non-stationary processes as filament eruptions, coronal mass ejections and active region loop expansions follow from the build-up of magnetic helicity in emerged flux ropes and the current helicity is of particular importans in the magnetic energy build-up. If we suggest that coronal fields evolve through sequences of force-free states, the energy of small-scale fluctuations of magnetic field and velocity, propagating upward from the photosphere along the field, can be transferred to the equilibrium magnetic field energy. This is equivalent to the generation of an electromotive force parallel to the equilibrium field (the alpha-effect). If the alpha-effect acts in some volume V, then there is a predominant (over the volume V) sign of the current helicity (see Figure 8) . The main purpose of the present study is to check whether there is a predominant sign of the current helicity at the photosphere of various active regions. The most appropriate parameter for this should be the imbalance of the current helicity over an active region. Is it has been inferred from calculations in 90\% of cases the imbalance \rho_h of current helicity h_c over an active region was quite significant: only in 4 cases out of 40 studied the values \rho_h were close to zero. Further, in 82.5\% of cases the predominant current helicity was negative (\rho_h<0) in the northern hemisphere and positive (\rho_h>0) in the southern hemisphere. The distribution of imbalance of current helicity in an AR versus latitude, is shown in Figure 7. The points are located predominantly in the 2-nd and 4-th quadrants. Unipolar spots (denoted by open circles) tend to have a stronger imbalance of current helicity than other magnetic structures at the same latitude; three of the four \delta-configurations (denoted by star) have extremal locations on the diagram, showing high values of \rho_h at high latitudes. Three \delta-configurations obey sign rule, but one doesn't. In the northern hemisphere two ARs do not obey sign rule.

Figure 7. Dependence of the current helicity imbalance (vertical) on solar latitude (horizontal)

Sign-Singularity of Current Helicity
Sign-singularity of 2D structures of current helicity can be studied by introducing a signed measure and by calculating some power-law exponent: cancellation exponent \kappa. It has been recently shown that current helicity (Figure 8) calculated using the photospheric magnetic field vector measurements possesses a well pronounced scaling behavior. In Figure 9 we show the results relative to the large bipolar active region NOAA 7316. A nice sign--singularity is present, with a well defined value of \kappa, in the region between 10^4 Km up to the resolution limit of the measurements, say about $10^3$ Km. There is a systematic tendency for the cancelation exponent in the leading part of an AR to be smaller than that in the following part. We think that such a difference between \kappa in the leading and following part is caused by the fact that the magnetic field of the leading spot, as a rule, is more compact than that in the following part. This fact must be reflected in the structure of h_c.
Well pronounced scaling laws take place for 2D structures of current helicity in ARs, indicating that the signed measure obtained with h_c is sign--singular, say the cancellations follow a nontrivial anomalous scaling.

Figure 8. Top: Greyscale map of vertical magnetic field.Middle: Greyscale map of vertical electric currents.Bottom: Greyscale map of the photospheric current helicity

Figure 9. An example of sign-singularuty spectra. The slope of spectrum defines the cancellation exponent

Variations of Scaling Properties with Flaring
The time variations of the cancellation exponent seem to be related to flare activity of an active region: the periods of enhanced flaring are accompanied by a drop and subsequent rise of the cancellation exponent. Here we show the sesults of study the flare-related changes of cancellation exponent on the basis of magnetic field observations of five active regions. We have found that in two cases the significant decrease of \kappa begins tens minutes (AR 7590 - 90 min; AR 6891 - 30 min) before strong flare (X-ray class C and greater). In AR 7315 decreasing took place at the time flare was in progress while in AR 7585 it turned out impossible to outline time interval due to significant gap in observations (about 3 hours between successive magnetograms). The decrease of the cancellation exponent in flaring active regions are greater then 30% while this decrease in case of quite active region NOAA 7216 doesn't exceed 11%.

Figure 10. Top: An example of sign-singularity spectra. Bottom: Variation of the cancellation exponent vs time in flareless active region AR NOAA 7216

Figure 11. Variation of the cancellation exponent vs time in four active regions (dashed line, asterisks). Vertical picks represent location in time and X-ray of observed flares

We suggest that the reduce of the \kappa indicates changes in the transverse magnetic field structure linked with solar flare. Variation of \kappa also assumes re-structuring of the current helicity or/and reorganization of the current system. The wide range of vortexes should dissipate over whole AR as flare is in progress (at least, \kappa defines the vortex behaveour at scales 1500 - 15000 km). It has also been found that at flaring periods a significant increasing of the current helicity imbalance took place which might imply a total reinforcement of twisting of a whole magnetic configuration.

Figure 12. AR NOAA 7315.Azimuth of transverse magnetic field before flare (thin line segments) and after (thin line segments)

Figure 13. AR NOAA 7590. Azimuth of transverse magnetic field before flare (thin line segments) and after (thin line segments)

Figure 14. AR NOAA 6891. Azimuth of transverse magnetic field before flare (thin line segments) and after (thin line segments)

Figure 15. AR NOAA 7585. Azimuth of transverse magnetic field before flare (thin line segments) and after (thin line segments)