Nonlinear force-free field extrapolation in spherical geometry: improved boundary data treatment applied to a SOLIS/VSM vector magnetogram

Abstract

Context. Understanding the 3D structure of coronal magnetic field is important to understanding: the onset of flares and coronal mass ejections, and the stability of active regions, and to monitoring the magnetic helicity and free magnetic energy and other phenomena in the solar atmosphere. Routine measurements of the solar magnetic field are mainly carried out in the photosphere. Therefore, one has to infer the field strength in the upper layers of the solar atmosphere from the measured photospheric field based on the assumption that the corona is force-free. Meanwhile, those measured data are inconsistent with the above force-free assumption. Therefore, one has to apply some transformations to these data before nonlinear force-free extrapolation codes can be applied. Aims. Extrapolation codes in Cartesian geometry for modelling the magnetic field in the corona do not take the curvature of the Sun's surface into account and can only be applied to relatively small areas, e.g., a single active region. Here we apply a method for nonlinear force-free coronal magnetic field modelling and preprocessing of photospheric vector magnetograms in spherical geometry using the optimization procedure. Methods. We solve the nonlinear force-free field equations by minimizing a functional in spherical coordinates over a restricted area of the Sun. We extend the functional by an additional term, which allows us to incorporate measurement errors and treat regions lacking observational data. We use vector magnetograph data from the Synoptic Optical Long-term Investigations of the Sun survey (SOLIS) to model the coronal magnetic field. We study two neighbouring magnetically connected active regions observed on May 15 2009. Results. For vector magnetograms with variable measurement precision and randomly scattered data gaps (e.g., SOLIS/VSM), the new code yields field models that satisfy the solenoidal and force-free condition significantly better as it allows deviations between the extrapolated boundary field and observed boundary data within the measurement errors. Data gaps are assigned an infinite error. We extend this new scheme to spherical geometry and apply it for the first time to real data.

Magneto-Frictional Modeling of Coronal Nonlinear Force-Free Fields. Ii. Application to Observations

“…Most of them are modeled in Cartesian coordinates, although some models are reconstructed in spherical geometry (Su et al 2009;Tadesse et al 2011;Guo et al 2012) or with tetrahedral meshes (Amari et al 2014b). In particular, Amari et al (2014a) has developed another code to fulfill the need for reconstructions on the scale of local active regions within a global extrapolation, using an iterative GradRubin scheme adapted to spherical coordinates.…”

A Comparison Between Nonlinear Force-Free Field and Potential Field Models Using Full-Disk SDO/HMI Magnetogram

“…(??) to work with boundary data of different noise levels and qualities (Wiegelmann and Inhester, 2010;Tadesse et al, 2011). For those pixels, for which B obs was successfully inverted, we allow deviations between the model field B and the input fields observed B obs surface field using Eq.…”

“…In order to control the speed with which the lower boundary is injected during the NLFFF extrapolation, we have used the Langrangian multiplier of ν = 0.001 as suggested by Tadesse et al (2012c). For more details of the method used in this work we direct the readers to the study by Tadesse et al (2011).…”

“…DeRosa, M. L. and Schrijver, C. J. and has studied that different NLFFF models have markedly different field line configurations and provided widely varying estimates of the magnetic free energy in the coronal volume. The main reasons for that problem are (1) the forces acting on the field within the photosphere, (2) the uncertainties on vectorfield measurements, particularly on the transverse component, and (3) the large domain that needs to be modeled to capture the connections of an active region to its surroundings (Wiegelmann, 2007;Tadesse et al, 2011;Tadesse et al, 2012b;Tadesse et al, 2012c). In this study, we have considered those three points explicitly into account for modeling coronal magnetic field in a full-disk.…”

Coronal magnetic field modeling using stereoscopy constraints

“…The variational method has originally been proposed by Wheatland et al (2000) and was extended by Wiegelmann (2004) and Tadesse et al (2011). The essential approach is to minimize a scalar cost function L tot that consists of a number of terms L n quantifying constraints that the final solution should satisfy.…”