Nonlinear force‐free coronal magnetic field extrapolation scheme based on the direct boundary integral formulation

Abstract

[1] The boundary integral equation (BIE) method was first proposed by Yan and Sakurai (2000) and used to extrapolate the nonlinear force-free magnetic field in the solar atmosphere. Recently, Yan and Li (2006) improved the BIE method and proposed the direct boundary integral equation (DBIE) formulation, which represents the nonlinear force-free magnetic field by direct integration of the magnetic field on the bottom boundary surface. On the basis of this new method, we devised a practical calculation scheme for the nonlinear force-free field extrapolation above solar active regions. The code of the scheme was tested by the analytical solutions of Low and Lou (1990) and was applied to the observed vector magnetogram of solar active region NOAA 9077. The results of the calculations show that the improvement of the new computational scheme to the scheme of Yan and Li (2006) is significant, and the force-free and divergence-free constraints are well satisfied in the extrapolated fields. The calculated field lines for NOAA 9077 present the X-shaped structure and can be helpful for understanding the magnetic configuration of the filament channel as well as the magnetic reconnection process during the Bastille Day flare on 14 July 2000.Citation: He, H., and H. Wang (2008), Nonlinear force-free coronal magnetic field extrapolation scheme based on the direct boundary integral formulation,

Magnetic Field Extrapolations into the Corona: Success and Future Improvements

“…-Boundary integral: the nonlinear force-free model can be describe as an exterior problem and a boundary integral equation written in which the magnetic field component within a volume can be determined by the magnetic field components on the boundary. The formulation of the boundary integrals can be found by Courant and Hilbert (1963) and recently applied to solar cases (D. Wang, Wei, and Yan, 1995;Yan andSakurai, 1997, 2000;Li, Yan, and Song, 2004;He and Wang, 2006;Yan and Li, 2006;He and Wang, 2008). -Force-free electrodynamics: the theory of force-free electrodynamics is applied to the modelling of coronal magnetic fields which has been applied successfully to pulsar magnetospheres (Contopoulos, Kalapotharakos, and Georgoulis, 2011;Contopoulos, 2013).…”

Variations of the 3‐D coronal magnetic field associated with the X3.4‐class solar flare event of AR 10930

“…The NLFFF model is a reasonable approximation to the quasi‐equilibrium corona [ Wiegelmann and Sakurai , ] and is commonly used for 3‐D coronal magnetic field reconstruction above active regions [e.g., Schrijver et al , ; He and Wang , ; Jing et al , ; DeRosa et al , ; He et al , ; Fuhrmann et al , ; Liu et al , ; Georgoulis et al , ; Jiang et al , ; Inoue et al , ]. The 2‐D photospheric vector magnetic field data (vector magnetograms), which can be measured sophisticatedly, act as the bottom boundary conditions in the NLFFF model and pose strong constraints to the upper coronal magnetic field distributions.…”

Coronal Mass Ejections: Models and Their Observational Basis

“…It is noted that the number of the twist is sensitive to the treatment of the 180°ambiguity of the horizontal magnetogram. For example, with the same vector magnetogram as in Yan et al (2001), He and Wang (2008) obtained a less-twisted flux rope. With the state-of-the-art extrapolation technique (Schrijver et al, 2008;Guo et al, 2010) and temperature tomography (Tripathi et al, 2009), the existence of a flux rope prior to some eruptions was further confirmed.…”