Nonlinear Force-Free Modeling of Coronal Magnetic Fields Part I: A Quantitative Comparison of Methods

Abstract

We compare six algorithms for the computation of nonlinear force-free (NLFF) magnetic fields (including optimization, magnetofrictional, Grad-Rubin based, and Green's function-based methods) by evaluating their performance in blind tests on analytical force-free-field models for which boundary conditions are specified either for the entire surface area of a cubic volume or for an extended lower boundary only. Figures of merit are used to compare the input vector field to the resulting model fields. Based on these merit functions, we argue that all algorithms yield NLFF fields that agree best with the input field in the lower central region of the volume, where the field and electrical currents are strongest and the effects of boundary conditions weakest. The NLFF vector fields in the outer domains of the volume depend sensitively on the details of the specified boundary conditions; best agreement is found if the field outside of the model volume is incorporated as part of the model boundary, either as potential field boundaries on the side and top surfaces, or as a potential field in a skirt around the main volume of interest. For input field (B) and modeled field (b), the best method included in our study yields an average relative vector error E n = |B − b| / |B| of only 0.02 when all sides are specified and 0.14 for the case where only the lower boundary is specified, while 162 CAROLUS J. SCHRIJVER ET AL. the total energy in the magnetic field is approximated to within 2%. The models converge towards the central, strong input field at speeds that differ by a factor of one million per iteration step. The fastest-converging, best-performing model for these analytical test cases is the Wheatland, Sturrock, and Roumeliotis (2000) optimization algorithm as implemented by Wiegelmann (2004).

Three-Dimensional Nonlinear Force-Free Field Reconstruction of Solar Active Region 11158 by Direct Boundary Integral Equation

“…Therefore BIE would be slow when extrapolating NLFFF from boundary data with computing grids compatible with current observations. DBIE is expected to make an improvement (Schrijver et al, 2006;Wiegelmann, 2008). Hence we will adopt a suitable parallel computing technique for DBIE.…”

“…As no iteration was performed to determine the set of factors by the BIE method in the comparison against the analytical force-free-field models of Low and Lou (1990). It was expected that completion of the iteration by DBIE will greatly reduce the computation time and should be feasible (Schrijver et al, 2006). Here we just adopt the Case II in Schrijver et al (2006), i.e., only bottom boundary data are used because this type of the boundary condition is close to the case of the Sun.…”

“…It was expected that completion of the iteration by DBIE will greatly reduce the computation time and should be feasible (Schrijver et al, 2006). Here we just adopt the Case II in Schrijver et al (2006), i.e., only bottom boundary data are used because this type of the boundary condition is close to the case of the Sun. The boundary size and the five evaluation metrics C vec , C cs , E n , E m and are the same as in Schrijver et al (2006).…”

“…Therefore, it is necessary to apply the DBIE method to real data by using high resolution boundary data as validation. The previous BIE method was thought to be slow when carried out on entire 3D domain (Schrijver et al, 2006;Wiegelmann, 2008) as the parallel algorithm was not implemented though the BIE technique itself should be suitable for parallel computation. In order to solve this problem, we implemented a Graphics Processing Unit (GPU) technique into our program to accelerate the computing processes.…”

Nonlinear force-free field extrapolation of the coronal magnetic field using the data obtained by the Hinode satellite

“…Currently, several different numerical NLFFF extrapolation methods have been developed to calculate the 3-D coronal magnetic fields from the photospheric vector magnetic field measurements. Descriptions, comparisons, and reviews of these methods can be found in the papers by Schrijver et al [2006], Metcalf et al [2008], Schrijver et al [2008], DeRosa et al [2009], Régnier [2007], and Wiegelmann [2008] and in section 5.3 of the book by Aschwanden [2005, and references therein].…”