Nonlinear Force-free Field Code Research Articles

Global Energetics of Solar Flares. X. Petschek Reconnection Rate and Alfvén Mach Number of Magnetic Reconnection Outflows

Abstract We investigate physical scaling laws for magnetic energy dissipation in solar flares, in the framework of the Sweet–Parker model and the Petschek model. We find that the total dissipated magnetic energy E diss in a flare depends on the mean magnetic field component B f associated with the free energy E f , the length scale L of the magnetic area, the hydrostatic density scale height λ of the solar corona, the Alfvén Mach number M A = v 1/v A (the ratio of the inflow speed v 1 to the Alfvénic outflow speed v A), and the flare duration τ f , i.e., , where the Alfvén speed depends on the nonpotential field strength B np and the mean electron density n e in the reconnection outflow. Using MDI/Solar Dynamics Observatory (SDO) and AIA/SDO observations and 3D magnetic field solutions obtained with the vertical-current approximation non-linear force-free field code we measure all physical parameters necessary to test scaling laws, which represents a new method to measure Alfvén Mach numbers M A, the reconnection rate, and the total free energy dissipated in solar flares.

Torsional Alfvénic Oscillations Discovered in the Magnetic Free Energy during Solar Flares

Abstract We report the discovery of torsional Alfvénic oscillations in solar flares, which modulate the time evolution of the magnetic free energy E f (t), while the magnetic potential energy E p (t) is uncorrelated, and the nonpotential energy varies as E np (t) = E p + E f (t). The mean observed time period of the torsional oscillations is P obs = 15.1 ± 3.9 minutes, the mean field line length is L = 135 ± 35 Mm, and the mean phase speed is v phase = 315 ± 120 km s−1, which we interpret as torsional Alfvénic waves in flare loops with enhanced electron densities. Most of the torsional oscillations are found to be decay-less, but exhibit a positive or negative trend in the evolution of the free energy, indicating new emerging flux (if positive), magnetic cancellation, or flare energy dissipation (if negative). The time evolution of the free energy has been calculated in this study with the Vertical-current Approximation (Version 4) Non-linear Force-free Field code, which incorporates automatically detected coronal loops in the solution and bypasses the non-force-freeness of the photospheric boundary condition, in contrast to traditional NLFFF codes.

The Formation of CME from Coupling Fan-spine Magnetic System: A Difficult Journey

Abstract We present the eruption of a mini-filament that caused a large-scale complicated coronal mass ejection (CME) on 2014 March 28, using the high-resolution observations taken by the Solar Dynamic Observatory. Three-dimensional coronal magnetic field extrapolated from the nonlinear force-free field code reveals that the magnetic environment of the eruption source region was a large fan-spine magnetic system that hosted another small fan-spine system under its fan, and the mini-filament was located under the fan structure of the small fan-spine system. Our analysis results suggest that the eruption of the mini-filament underwent three reconnection processes before the formation of the CME. First, the erupting filament triggered the null point reconnection in the small fan-spine system. During this stage, the sudden expansion of the spine field lines also excited a large-scale extreme-ultraviolet wave. Second, the spine field lines of the small fan-spine system were pushed up by the erupting filament and therefore further triggered the null point reconnection in the large fan-spine magnetic system. Third, during the second stage, magnetic reconnection also occurred between the two legs of the stretched confining field lines of the mini-filament. The present study suggests that the formation of the observed CME from the coupling fan-spine magnetic system was more complicated than previously thought, needing to undergo multistage magnetic reconnection processes in the low corona.

Comparison of Two Coronal Magnetic Field Models to Reconstruct a Sigmoidal Solar Active Region with Coronal Loops

Abstract Magnetic field extrapolation is an important tool to study the three-dimensional (3D) solar coronal magnetic field, which is difficult to directly measure. Various analytic models and numerical codes exist, but their results often drastically differ. Thus, a critical comparison of the modeled magnetic field lines with the observed coronal loops is strongly required to establish the credibility of the model. Here we compare two different non-potential extrapolation codes, a nonlinear force-free field code (CESE–MHD–NLFFF) and a non-force-free field (NFFF) code, in modeling a solar active region (AR) that has a sigmoidal configuration just before a major flare erupted from the region. A 2D coronal-loop tracing and fitting method is employed to study the 3D misalignment angles between the extrapolated magnetic field lines and the EUV loops as imaged by SDO/AIA. It is found that the CESE–MHD–NLFFF code with preprocessed magnetogram performs the best, outputting a field that matches the coronal loops in the AR core imaged in AIA 94 Å with a misalignment angle of ∼10°. This suggests that the CESE–MHD–NLFFF code, even without using the information of the coronal loops in constraining the magnetic field, performs as good as some coronal-loop forward-fitting models. For the loops as imaged by AIA 171 Å in the outskirts of the AR, all the codes including the potential field give comparable results of the mean misalignment angle (∼30°). Thus, further improvement of the codes is needed for a better reconstruction of the long loops enveloping the core region.

TRACING THE CHROMOSPHERIC AND CORONAL MAGNETIC FIELD WITH AIA, IRIS, IBIS, AND ROSA DATA

ABSTRACT The aim of this study is to explore the suitability of chromospheric images for magnetic modeling of active regions. We use high-resolution images ( ), from the Interferometric Bidimensional Spectrometer in the Ca ii 8542 Å line, the Rapid Oscillations in the Solar Atmosphere instrument in the Hα 6563 Å line, the Interface Region Imaging Spectrograph in the 2796 Å line, and compare non-potential magnetic field models obtained from those chromospheric images with those obtained from images of the Atmospheric Imaging Assembly in coronal (171 Å, etc.) and in chromospheric (304 Å) wavelengths. Curvi-linear structures are automatically traced in those images with the OCCULT-2 code, to which we forward-fitted magnetic field lines computed with the Vertical-current Approximation Nonlinear Force Free Field code. We find that the chromospheric images: (1) reveal crisp curvi-linear structures (fibrils, loop segments, spicules) that are extremely well-suited for constraining magnetic modeling; (2) that these curvi-linear structures are field-aligned with the best-fit solution by a median misalignment angle of –7°; (3) the free energy computed from coronal data may underestimate that obtained from cromospheric data by a factor of –4, (4) the height range of chromospheric features is confined to km, while coronal features are detected up to h = 35,000 km; and (5) the plasma-β parameter is for all traced features. We conclude that chromospheric images reveal important magnetic structures that are complementary to coronal images and need to be included in comprehensive magnetic field models, something that is currently not accomodated in standard NLFFF codes.

NONLINEAR FORCE-FREE FIELD EXTRAPOLATION OF A CORONAL MAGNETIC FLUX ROPE SUPPORTING A LARGE-SCALE SOLAR FILAMENT FROM A PHOTOSPHERIC VECTOR MAGNETOGRAM

Solar filament are commonly thought to be supported in magnetic dips, in particular, of magnetic flux ropes (FRs). In this Letter, from the observed photospheric vector magnetogram, we implement a nonlinear force-free field (NLFFF) extrapolation of a coronal magnetic FR that supports a large-scale intermediate filament between an active region and a weak polarity region. This result is the first in that current NLFFF extrapolations with presence of FRs are limited to relatively small-scale filaments that are close to sunspots and along main polarity inversion line (PIL) with strong transverse field and magnetic shear, and the existence of a FR is usually predictable. In contrast, the present filament lies along the weak-field region (photospheric field strength $\lesssim 100$ G), where the PIL is very fragmented due to small parasitic polarities on both side of the PIL and the transverse field has a low value of signal-to-noise ratio. Thus it represents a far more difficult challenge to extrapolate a large-scale FR in such case. We demonstrate that our CESE--MHD--NLFFF code is competent for the challenge. The numerically reproduced magnetic dips of the extrapolated FR match observations of the filament and its barbs very well, which supports strongly the FR-dip model for filaments. The filament is stably sustained because the FR is weakly twisted and strongly confined by the overlying closed arcades.

THE MAGNETIC FIELD OF ACTIVE REGION 11158 DURING THE 2011 FEBRUARY 12-17 FLARES: DIFFERENCES BETWEEN PHOTOSPHERIC EXTRAPOLATION AND CORONAL FORWARD-FITTING METHODS

We developed a {\sl coronal non-linear force-free field (COR-NLFFF)} forward-fitting code that fits an approximate {\sl non-linear force-free field (NLFFF)} solution to the observed geometry of automatically traced coronal loops. In contrast to photospheric NLFFF codes, which calculate a magnetic field solution from the constraints of the transverse photospheric field, this new code uses coronal constraints instead, and this way provides important information on systematic errors of each magnetic field calculation method, as well as on the non-forcefreeness in the lower chromosphere. In this study we applied the COR-NLFFF code to active region NOAA 11158, during the time interval of 2011 Feb 12 to 17, which includes an X2.2 GOES-class flare plus 35 M and C-class flares. We calcuated the free magnetic energy with a 6-minute cadence over 5 days. We find good agreement between the two types of codes for the total nonpotential $E_N$ and potential energy $E_P$, but find up to a factor of 4 discrepancy in the free energy $E_{free}=E_N-E_P$, and up to a factor of 10 discrepancy in the decrease of the free energy $\Delta E_{free}$ during flares. The coronal NLFFF code exhibits a larger time variability, and yields a decrease of free energy during the flare that is sufficient to satisfy the flare energy budget, while the photospheric NLFFF code shows much less time variability and an order of magnitude less free energy decrease during flares. The discrepancy may partly be due to the pre-processing of photospheric vector data, but more likely due to the non-forcefreeness in the lower chromosphere. We conclude that the coronal field cannot be correctly calculated based on photospheric data alone, but requires additional information on coronal loop geometries.

EXTRAPOLATION OF THE SOLAR CORONAL MAGNETIC FIELD FROMSDO/HMI MAGNETOGRAM BY A CESE-MHD-NLFFF CODE

Due to the absence of direct measurement, the magnetic field in the solar corona is usually extrapolated from the photosphere in numerical way. At the moment, the nonlinear force-free field (NLFFF) model dominates the physical models for field extrapolation in the low corona. Recently we have developed a new NLFFF model with MHD relaxation to reconstruct the coronal magnetic field. This method is based on CESE--MHD model with the conservation-element/solution-element (CESE) spacetime scheme. In this paper, we report the application of the CESE--MHD--NLFFF code to \SDO/HMI data with magnetograms sampled for two active regions (ARs), NOAA AR 11158 and 11283, both of which were very non-potential, producing X-class flares and eruptions. The raw magnetograms are preprocessed to remove the force and then inputted into the extrapolation code. Qualitative comparison of the results with the \SDO/AIA images shows that our code can reconstruct magnetic field lines resembling the EUV-observed coronal loops. Most important structures of the active regions are reproduced excellently, like the highly-sheared field lines that suspend filaments in AR 11158 and twisted flux rope which corresponds to a sigmoid in AR 11283. Quantitative assess of the results shows that the force-free constraint is fulfilled very well in the strong-field regions but apparently not that well in the weak-field regions because of data noise and numerical errors in the small currents.

MODELING MAGNETIC FIELD STRUCTURE OF A SOLAR ACTIVE REGION CORONA USING NONLINEAR FORCE-FREE FIELDS IN SPHERICAL GEOMETRY

We test a nonlinear force-free field (NLFFF) optimization code in spherical geometry using an analytical solution from Low and Lou. Several tests are run, ranging from idealized cases where exact vector field data are provided on all boundaries, to cases where noisy vector data are provided on only the lower boundary (approximating the solar problem). Analytical tests also show that the NLFFF code in the spherical geometry performs better than that in the Cartesian one when the field of view of the bottom boundary is large, say, $20^\circ \times 20^\circ$. Additionally, We apply the NLFFF model to an active region observed by the Helioseismic and Magnetic Imager (HMI) on board the Solar Dynamics Observatory (SDO) both before and after an M8.7 flare. For each observation time, we initialize the models using potential field source surface (PFSS) extrapolations based on either a synoptic chart or a flux-dispersal model, and compare the resulting NLFFF models. The results show that NLFFF extrapolations using the flux-dispersal model as the boundary condition have slightly lower, therefore better, force-free and divergence-free metrics, and contain larger free magnetic energy. By comparing the extrapolated magnetic field lines with the extreme ultraviolet (EUV) observations by the Atmospheric Imaging Assembly (AIA) on board SDO, we find that the NLFFF performs better than the PFSS not only for the core field of the flare productive region, but also for large EUV loops higher than 50 Mm.

A NEW CODE FOR NONLINEAR FORCE-FREE FIELD EXTRAPOLATION OF THE GLOBAL CORONA

Reliable measurements of the solar magnetic field are still restricted to the photosphere, and our present knowledge of the three-dimensional coronal magnetic field is largely based on extrapolation from photospheric magnetogram using physical models, e.g., the nonlinear force-free field (NLFFF) model as usually adopted. Most of the currently available NLFFF codes have been developed with computational volume like Cartesian box or spherical wedge while a global full-sphere extrapolation is still under developing. A high-performance global extrapolation code is in particular urgently needed considering that Solar Dynamics Observatory (SDO) can provide full-disk magnetogram with resolution up to $4096\times 4096$. In this work, we present a new parallelized code for global NLFFF extrapolation with the photosphere magnetogram as input. The method is based on magnetohydrodynamics relaxation approach, the CESE-MHD numerical scheme and a Yin-Yang spherical grid that is used to overcome the polar problems of the standard spherical grid. The code is validated by two full-sphere force-free solutions from Low & Lou's semi-analytic force-free field model. The code shows high accuracy and fast convergence, and can be ready for future practical application if combined with an adaptive mesh refinement technique.

Nonlinear Force-Free Modeling of Coronal Magnetic Fields. II. Modeling a Filament Arcade and Simulated Chromospheric and Photospheric Vector Fields

We compare a variety of nonlinear force-free field (NLFFF) extrapolation algorithms, including optimization, magneto-frictional, and Grad – Rubin-like codes, applied to a solar-like reference model. The model used to test the algorithms includes realistic photospheric Lorentz forces and a complex field including a weakly twisted, right helical flux bundle. The codes were applied to both forced “photospheric” and more force-free “chromospheric” vector magnetic field boundary data derived from the model. When applied to the chromospheric boundary data, the codes are able to recover the presence of the flux bundle and the field’s free energy, though some details of the field connectivity are lost. When the codes are applied to the forced photospheric boundary data, the reference model field is not well recovered, indicating that the combination of Lorentz forces and small spatial scale structure at the photosphere severely impact the extrapolation of the field. Preprocessing of the forced photospheric boundary does improve the extrapolations considerably for the layers above the chromosphere, but the extrapolations are sensitive to the details of the numerical codes and neither the field connectivity nor the free magnetic energy in the full volume are well recovered. The magnetic virial theorem gives a rapid measure of the total magnetic energy without extrapolation though, like the NLFFF codes, it is sensitive to the Lorentz forces in the coronal volume. Both the magnetic virial theorem and the Wiegelmann extrapolation, when applied to the preprocessed photospheric boundary, give a magnetic energy which is nearly equivalent to the value derived from the chromospheric boundary, but both underestimate the free energy above the photosphere by at least a factor of two. We discuss the interpretation of the preprocessed field in this context. When applying the NLFFF codes to solar data, the problems associated with Lorentz forces present in the low solar atmosphere must be recognized: the various codes will not necessarily converge to the correct, or even the same, solution.