A Nonlinear Force-Free Magnetic Field Approximation Suitable for Fast Forward-Fitting to Coronal Loops. III. The Free Energy

Abstract

An analytical approximation of a nonlinear force-free magnetic field (NLFFF) solution was developed in Paper I, while a numerical code that performs fast forward-fitting of this NLFFF approximation to a line-of-sight magnetogram and coronal 3D loops has been described and tested in Paper II. Here we calculate the free magnetic energy E free = E N − E P , i.e., the difference of the magnetic energies between the nonpotential field and the potential field. A second method to estimate the free energy is obtained from the mean misalignment angle change ∆µ = µ P − µ N between the potential and nonpotential field, which scales as E free /E P ≈ tan 2 (∆µ). For four active regions observed with STEREO in 2007 we find free energies in the range of q free = (E free /E P ) ≈ 1% − 10%, with an uncertainty of less than ±2% between the two methods, while the free energies obtained from 11 other NLFFF codes exhibit a larger scatter of order ≈ ±10%. We find also a correlation between the free magnetic energy and the GOES flux of the largest flare that occurred during the observing period, which can be quantified by an exponential relationship, F GOES ∝ exp (q free /0.015), implying an exponentiation of the dissipated currents.

Effect of Magnetic Twist on Nonlinear Transverse Kink Oscillations of Line-tied Magnetic Flux Tubes

“…Therefore, we are not too far from this critical value. The range of values for the twist considered in this work are in agreement with the twist angles that are inferred from the estimations performed by Aschwanden (2013); Aschwanden et al (2014Aschwanden et al ( , 2016. These estimations are based on nonlinear force-free magnetic fields calculated using line-of-sight magnetograms and the geometry of observed coronal loops in 3D.…”

Magnetic Field Extrapolations into the Corona: Success and Future Improvements

“…For instance, the method has been developed and used for theorising the topology of magnetic field in different geometries (Titov, Priest, and Demoulin, 1993;Bungey, Titov, and Priest, 1996;Brown and Priest, 1999;Longcope and Klapper, 2002;Brown, 2002, 2004;Barnes, Longcope, and Leka, 2005;. The point-charge method has also been used to fit the observed 3D coronal loops following a nonlinear force-free assumption (Aschwanden, 2012a(Aschwanden, , 2012b(Aschwanden, , 2013Aschwanden and Malanushenko, 2012). Another successful model to determine the properties of the coronal magnetic field is the flux rope insertion model developed by van Ballegooijen (2004).…”

Determining the 3D Structure of the Corona Using Vertical Height Constraints on Observed Active Region Loops

“…This study uses an effective pattern recognition code that extracts the curvilinear loops from 2D imagery. The code, called Oriented Coronal CUrved Loop Tracing-2 (Occult-2), has been recently developed by Aschwanden and associates (Aschwanden, 2013b;Aschwanden, 2010). The code is extremely fast and we employed it, rather than the semi-manual method of Gary, Hu, and Lee (2013), for its speed and and its availability under the SolarSoftWare [SSW] library.…”

“…The important points of the first paper on using cubic Bézier splines included a rapid and flexible manual method to map on-disk coronal loops of a 2D EUV image into a 3D coronal loop. In this second paper, we employ the automated Occult-2 loop recognition routine (Aschwanden, 2013b). The method of Paper I employed 2D cubic Bézier splines to approximate a traced coronal loop trajectory in the plane-of-sky projection, using four control points of functions of x and y; the third coordinate, z, perpendicular to the plane of the sky, is initially zero, as does this paper.…”