Figure 5: Active regions have a complex magnetic structure consisting of multiple manetic flux systems separated by quasi-separatrix layers, or QSLs (Démoulin et al. 1996). This figure shows the quasi-separatrix layers for the 12 December 2006 event; on the left is a QSL computed from a NLFFF, on the right is a QSL computed from a potential field. The figure shows the plane z = 4 of the computational volume divided into three areas. Magnetic field lines in the blue area have both footpoints in the computational volume. The brightness in the blue area is proportional to the function where (x, y) is the starting point of a field line, (x′, y′) and (x′′, y′′) are the two end points, and {X 1,X 2 } define a vector {x′′ − x′, y′′ − y′}. Field lines in the brown area intersect the sides of the computational volume. Field lines in the green area intersect the top of the computational volume. The QSL calculation was performed only in the blue area. Modeling Non-Potential Magnetic Fields in Solar Active Regions M. G. Bobra 1, A. A. van Ballegooijen 1, E. E. DeLuca 1, J. L. Sandell 2 Harvard-Smithsonian Center for Astrophysics Columbia University Abstract: Many models aim to reproduce the non-linear force free fields in the solar corona; in this particular study, the magnetofrictional relaxation method is tested. This method produces non-linear force free fields in and around an active region filament. The method is based on line-of-sight magnetograms and ground- based H  images to define the location of a filament, which is represented by a flux rope. In this study, an ensemble of such models are created to reproduce Hinode X-Ray Telescope observations of highly sheared, non-potential loop structures in NOAA Active Region 10930, which occurred on 12 December We find that flux ropes with axial fluxes with 2 to.1 × Maxwells are needed to reproduce the observed loops. For each event, we also derive estimates of the current, torsion parameter  and quasi-separatrix layer distributions, as well as the relative magnetic helicity and magnetic free energy. Introduction The model in this study was developed by van Ballegooijen (2004) and involves inserting a weakly twisted flux rope into a potential field model (also see van Ballegooijen et al. 2004). After the flux rope is inserted, the coronal field is evolved using the ideal-MHD induction equation, and the plasma velocity is assumed to be proportional to the Lorentz force (magnetofrictional relaxation), so that the field evolves towards a non-linear force-free field. This method was used to model observations of non-potential coronal loops in NOAA active region 10930, which was observed by XRT on 12 December Three sets of observational data were used in this study: The line-of-sight component from a magnetogram taken by the Kitt Peak SOLIS Vector-Spectromagneotograph, full- disk H  image from Kanzelhöhe Observatory, and soft X-Ray images taken by the Hinode X-Ray Telescope with the Aluminum-Poly filter. These data represent the solar magnetic field configuration at the photosphere, chromosphere, and corona, respectively. See Figure 1. We find that flux ropes with axial fluxes in the range of 2 to.1 × Maxwells are needed to reproduce the observed loops. Cases with too little axial flux produced a potential field. Cases with too much axial flux produced a configuration with most field lines leaving the computational domain. Thus the upper and lower limits of the axial flux values are well constrained. Table 2 shows values for the helicity and free energy per model. As the amount of axial flux in the flux rope decreases, the amount of free energy decreases. Since the poloidal flux value is held constant, decreasing values of axial flux also causes larger values of . Free energy also decreases as axial flux decreases. Figure 3 shows the current distribution from both the x-y plane and from a perpendicular view, which shows the center of the flux rope. The current peaks exactly where the quasi- separatrix layers (see Figure 5) are located, which suggests that quasi-separatrix layers could be a source of coronal heating. Hinode is a Japanese mission developed and launched by ISAS/JAXA, with NAOJ as domestic partner and NASA and STFC (UK) as international partners. It is operated by these agencies in co-operation with ESA and the NSC (Norway). Figure 4, left: Many model-generated magnetic field lines at different heights in the solar atmosphere from the perspectives of an observer looking at the limb and disk (left and right, respectively). References: Acton, L., et al. 1992, Science, 258, 591 Berger, M. A., & Field, G. B. 1984, Journal of Fluid Mechanics, 147, 133 Birn, J., Forbes, T. G., & Hesse, M. 2006, ApJ, 645, 732 Bleybel, A., Amari, T., van Driel-Gesztelyi, L., & Leka, K. D. 2002, A&A, 395, 685 Bobra, M. G., van Ballegooijen, A. A., DeLuca, E. E. 2007, ApJ, in press Canfield, R. C., Hudson, H. S., & McKenzie, D. E. 1999, Geophys. Res. Lett., 26, 627 Chae, J., Wang, H., Qiu, J., Goode, P. R., Strous, L., & Yun, H. S. 2001, ApJ, 560, 476 Démoulin, P., Hénoux, J. C., Priest, E. R., & Mandrini, C. H. 1996, A&A, 308, 643 Foukal, P. 1971, Sol. Phys., 19, 59 Gary, G. A., Moore, R. L., Hagyard, M. J., & Haisch, B. M. 1987, ApJ, 314, 782 Data Results Figure 1: Three sets of observational data are used in this study. Left: An XRT soft X-Ray image. Middel: with Positive (red) and negative (green) contours of the LOS component of the photospheric field overlay an H  image. Right: LOS components from a SOLIS magnetogram overlay a XRT soft X-Ray image. Table 1, left: A table of parameters that are consistent through all four flux rope models. Table 2, bottom: A table of axial flux, free energy and helicity values per model. a (L, CMD) refers to the (Latitude, Central Meridian Distance). b The computational domain is in spherical coordinates; above, the coordinates have been translated into average x and y values. c θ = |2π(F pol L)(Φ axi ) −1 |, where L is the flux rope length. Figure 6, left and upper left: We have developed a technique to automatically recognize and extract loop-like structures within Hinode images.This involves applying image contrast enhancement processes and set theoretic morphological collation algorithms to reduce the features in an image into a locus of points that trace out a loop. It should be noted that this is not a formal loop detection algorithm; the process identifies structures of interest, which are then selected for further study. The process is automated and repeatable, to such an extent that human interaction is limited to selecting gross structures for analysis (Step 1), and thus represents a major improvement over having to manually select pixels in an image that correspond to a loop. Monte Carlo testing on TRACE images has shown this to be a stable and robust process. Figure 7, right: Currently, the NLFFF model incorporates a quantitative loop-matching technique that requires the user to draw a loop of interest (red); then, the model selects the closest matching field line (pink). A source of error in this method is that the user must draw a field line; incorporating a loop recognition and extraction algorithm would reduce such error. Hagenaar, H. J., & Shine, R. A. 2005, ApJ, 635, 659 Hagyard, M. J. 1990, Memorie della Societa Astronomica Italiana, 61, 337 Handy, B. N., et al. 1999, Sol. Phys., 187, 229 Hood, A. W., & Priest, E. R. 1981, Sol. Phys., 73, 289 Karpen, J. T., Tanner, S. E. M., Antiochos, S. K., & DeVore, C. R. 2005, ApJ, 635, 1319 Kuperus, M., & Raadu, M. A. 1974, A&A, 31, 189 Low, B. C., & Hundhausen, J. R. 1995, ApJ, 443, 818 Mandrini, C. H., Démoulin, P., van Driel-Gesztelyi, L., & López Fuentes, M. C. 2004, Ap&SS, 290, 319 Martin, S. F. 1998, Sol. Phys., 182, 107 Martin, S. F., Bilimoria, R., & Tracadas, P. 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A., & Martens, P. C. H. 1989, ApJ, 343, 971 van Ballegooijen, A. A., Priest, E. R., & Mackay, D. H. 2000, ApJ, 539, 983 Yang, W. H., Sturrock, P. A., & Antiochos, S. K. 1986, ApJ, 309, 383 Figure 3: Left: A plot of the current distribution at a height of z = 4, which corresponds to the center of the flux rope. The yellow line represents the x-axis of the image above, which shows the current distribution along a cross-section of the flux rope. White lines represent selected magnetic field lines from the model’s result. It is worth noting that the right-most magnetic field line, which looks like it is looping over itself, is actually a sheared line traveling along the axis of the flux rope. 12 December :12:57 UT Loop Recognition and Extraction Algorithm Quasi-Separatrix Layer Maps 1. Read in line-of-slight elements of SOLIS magnetogram (2D input). 2. Construct a potential field vector at every grid element of 3D computational domain. 3. Modify the vectors within a user- defined flux rope volume based on the location of a filament observed in an H-α image. 4. Smooth the sharp current sheets at the flux rope boundary with diffusion and relax into a non-linear force-free configuration by solving the ideal MHD induction equation. A solution is achieved when JxB per pixel is as small as possible. Model A non-linear force-free field (NLFFF) model of AR was constructed on the basis of a longitudinal magnetogram obtained with SOLIS on 2006 December 12 at 16:13 UT. The method involves computing a potential field model and inserting a (non-force-free) magnetic flux rope at the location of an observed H  filament. Finally magneto-frictional relaxation is applied by solving the ideal MHD induction equation, with the assumption that the plasma velocity is proportional to J x B, to produce a non-linear force free field (van Ballegooijen 2004; van Ballegooijen et al. 2007). The latest version of the code takes into account deviations from force-free conditions in the photosphere (see Metcalf et al. 2007), and includeshyper-diffusion in the corona to suppress artifacts in the coronal current distribution (for details see Bobra et al. 2007). The steps of the model are graphically illustrated in Figure 2. By overlaying an XRT image with the model's result, one can qualitatively determine how the model compares to observed structures. This process was repeated to create an ensemble of models with different flux rope parameters. Each model was compared to observation to determine which produced the best agreement. Figure 4 shows a model constructed with a flux rope that has an axial flux of Mx and a poloidal flux of Mx/cm (left-helical twist). Figure 2: The steps of the NLFFF model (van Ballegooijen, 2004, van Ballegooijen et al. 2007) have been graphically represented above. Step 1 Step 2 Soft X-Ray 16:12:57 UT Magnetogram 16:13:00 UT H  7:49:45 UT 12 December 2006 RMS distance of shortest leg: solar radii a b c