Force-free magnetic fields in the solar atmosphere

Reliable measurements of the solar magnetic field are restricted to the level of the photosphere. For about half a century attempts have been made to calculate the field in the layers above the photosphere, i.e. in the chromosphere and in the corona, from the measured photospheric field. The procedure is known as magnetic field extrapolation. In the superphotospheric parts of active regions the magnetic field is approximately force-free, i.e. electric currents are aligned with the magnetic field. The practical application to solar active regions has been largely confined to constant-α or linear force-free fields, with a spatially constant ratio, α, between the electric current and the magnetic field. We review results obtained from extrapolations with constant-α force-free fields, in particular on magnetic topologies favourable for flares and on magnetic and current helicities. Presently, different methods are being developed to calculate non-constant-α or nonlinear force-free fields from photospheric vector magnetograms. We also briefly discuss these methods and present a comparison of a linear and a nonlinear force-free magnetic field extrapolation applied to the same photospheric boundary data. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Preprocessing of Vector Magnetograph Data for a Nonlinear Force-Free Magnetic Field Reconstruction

Knowledge regarding the coronal magnetic field is important for the understanding of many phenomena, like flares and coronal mass ejections. Because of the low plasma beta in the solar corona, the coronal magnetic field is often assumed to be force-free and we use photospheric vector magnetograph data to extrapolate the magnetic field into the corona with the help of a nonlinear force-free optimization code. Unfortunately, the measurements of the photospheric magnetic field contain inconsistencies and noise. In particular, the transversal components (say B _x and B _y) of current vector magnetographs have their uncertainties. Furthermore, the magnetic field in the photosphere is not necessarily force free and often not consistent with the assumption of a force-free field above the magnetogram. We develop a preprocessing procedure to drive the observed non–force-free data towards suitable boundary conditions for a force-free extrapolation. As a result, we get a data set which is as close as possible to the measured data and consistent with the force-free assumption.     

Nonlinear Force-Free Reconstruction of the Global Solar Magnetic Field: Methodology

We present a novel numerical method that allows the calculation of nonlinear force-free magnetostatic solutions above a boundary surface on which only the distribution of the normal magnetic field component is given. The method relies on the theory of force-free electrodynamics and applies directly to the reconstruction of the solar coronal magnetic field for a given distribution of the photospheric radial field component. The method works as follows: we start with any initial magnetostatic global field configuration (e.g. zero, dipole), and along the boundary surface we create an evolving distribution of tangential (horizontal) electric fields that, via Faraday’s equation, give rise to a respective normal-field distribution approaching asymptotically the target distribution. At the same time, these electric fields are used as boundary condition to numerically evolve the resulting electromagnetic field above the boundary surface, modeled as a thin ideal plasma with non-reflecting, perfectly absorbing outer boundaries. The simulation relaxes to a nonlinear force-free configuration that satisfies the given normal-field distribution on the boundary. This is different from existing methods relying on a fixed boundary condition – the boundary evolves toward the a priori given one, at the same time evolving the three-dimensional field solution above it. Moreover, this is the first time that a nonlinear force-free solution is reached by using only the normal field component on the boundary. This solution is not unique, but it depends on the initial magnetic field configuration and on the evolutionary course along the boundary surface. To our knowledge, this is the first time that the formalism of force-free electrodynamics, used very successfully in other astrophysical contexts, is applied to the global solar magnetic field.     

Optimization code with weighting function for the reconstruction of coronal magnetic fields

We developed a code for the reconstruction of nonlinear force-free and non-force-free coronal magnetic fields. The 3D magnetic field is computed numerically with the help of an optimization principle. The force-free and non-force-free codes are compiled in one program. The force-free approach needs photospheric vector magnetograms as input. The non-force-free code additionally requires the line-of-sight integrated coronal density distribution in combination with a tomographic inversion code. Previously the optimization approach has been used to compute magnetic fields using all six boundaries of a computational box. Here we extend this method and show how the coronal magnetic field can be reconstructed only from the bottom boundary, where the boundary conditions are measured with vector magnetographs. The program is planed for use within the Stereo mission.     

Longitude variations of solar magnetic fields of different intensity in cycle 23 as inferred from the SOHO/MDI data

SOHO/MDI magnetograms have been used to analyze the longitude distribution of the squared solar magnetic field 〈B ²〉 in the activity cycle no. 23. The energy of the magnetic field (〈B ²〉) is shown to change with longitude. However, these variations hardly fit the concept of active longitudes. In the epochs of high solar activity, one can readily see a relationship between longitude variations of the medium-strong ((|B| > 50 G or |B| > 100 G) and relatively weak (|B| ≤ 50 G or |B| ≤ 100 G) fields at all latitudes. In other periods, this relationship is revealed mainly at the latitudes not higher than 30°. The background fields (|B| ≤ 25 G) also display longitude variations, which are, however, not related to those of the strong fields. This makes us think that the fields of solar activity are rather inclusions to the general field than the source of the latter.     

The Force-Free Electrodynamics Method for the Extrapolation of Coronal Magnetic Fields from Vector Magnetograms

We present a new improved version of our force-free electrodynamics (FFE) numerical code in spherical coordinates that extrapolates the magnetic field in the inner solar corona from a photospheric vector magnetogram. The code satisfies the photospheric boundary condition and the condition ??B=0 to machine accuracy. The performance of our method is evaluated with standard convergence parameters, and is found to be comparable to that of other nonlinear force-free extrapolations.     

Magnetic Energy Storage and Current Density Distributions for Different Force-Free Models

In the last decades, force-free-field modelling has been used extensively to describe the coronal magnetic field and to better understand the physics of solar eruptions at different scales. Especially the evolution of active regions has been studied by successive equilibria in which each computed magnetic configuration is subject to an evolving photospheric distribution of magnetic field and/or electric-current density. This technique of successive equilibria has been successful in describing the rate of change of the energetics for observed active regions. Nevertheless the change in magnetic configuration due to the increase/decrease of electric current for different force-free models (potential, linear and nonlinear force-free fields) has never been studied in detail before. Here we focus especially on the evolution of the free magnetic energy, the location of the excess of energy, and the distribution of electric currents in the corona. For this purpose, we use an idealised active region characterised by four main polarities and a satellite polarity, allowing us to specify a complex topology and sheared arcades to the coronal magnetic field but no twisted flux bundles. We investigate the changes in the geometry and connectivity of field lines, the magnetic energy and current-density content as well as the evolution of null points. Increasing the photospheric current density in the magnetic configuration does not dramatically change the energy-storage processes within the active region even if the magnetic topology is slightly modified. We conclude that for reasonable values of the photospheric current density (the force-free parameter α<0.25 Mm⁻¹), the magnetic configurations studied do change but not dramatically: i) the original null point stays nearly at the same location, ii) the field-line geometry and connectivity are slightly modified, iii) even if the free magnetic energy is significantly increased, the energy storage happens at the same location. This extensive study of different force-free models for a simple magnetic configuration shows that some topological elements of an observed active region, such as null points, can be reproduced with confidence only by considering the potential-field approximation. This study is a preliminary work aiming at understanding the effects of electric currents generated by characteristic photospheric motions on the structure and evolution of the coronal magnetic field.     

An Improved Approach to Non-Force-Free Coronal Magnetic Field Extrapolation

We develop an approach to deriving the three-dimensional non-force-free coronal magnetic field from vector magnetograms. Based on the principle of minimum dissipation rate, a general non-force-free magnetic field is expressed as the superposition of one potential field and two constant-α (linear) force-free fields. Each is extrapolated from its bottom boundary data, providing the normal component only. The constant-α parameters are distinct and determined by minimizing the deviations between the numerically computed and measured transverse magnetic field at the bottom boundary. The boundary conditions required are at least two layers of vector magnetograms, one at the photospheric level and the other at the chromospheric level, presumably. We apply our approach to a few analytic test cases, especially to two nonlinear force-free cases examined by Schrijver et al. (Solar Phys. 235, 161, 2006). We find that for one case with small α parameters, the quantitative measures of the quality of our result are better than the median values of those from a set of nonlinear force-free methods. The reconstructed magnetic-field configuration is valid up to a vertical height of the transverse scale. For the other cases, the results remain valid to a lower vertical height owing to the limitations of the linear force-free-field solver. Because our method is based on the fast-Fourier-transform algorithm, it is much faster and easy to implement. We discuss the potential usefulness of our method and its limitations.     

Magnetic Field Elements at High Latitude: Lifetime and Rotation Rate

Using one-minute cadence time-series full disk magnetograms taken by the SOHO/MDI, we have studied the magnetic field elements at high latitude (poleward of 65° in latitude). It is found that an average lifetime of the magnetic field elements is 16.5 h during solar minimum, much longer than that during solar maximum (7.3 h). During solar minimum, number of the magnetic field elements with the dominant polarity is about 3 times as that of the opposite polarity elements. Their lifetime is 21.0 h on average, longer than that of the opposite polarity elements (2.3 h). It is also found that the lifetime of the magnetic field elements is related with their size, consistent with the magnetic field elements in the quiet sun at low latitude found by Hagenaar et al. (Astrophys. J. 511:932, 1999). During solar maximum, the polar regions are equally occupied by magnetic field elements with both polarities, and their lifetimes are roughly the same on average. No evidence shows there is a correlation between the lifetime and size of the magnetic field elements. Using an image cross-correlation method, we also measure the solar rotation rate at high latitude, up to 85° in latitude. The rate is ω=2.914−0.342sin ² φ−0.482sin ⁴ φ μrad s⁻¹ sidereal. It agrees with previous studies using the spectroscopic and image cross-correlation methods, and also agrees with the results using the element tracking method when the sample of the tracked magnetic field elements is large. The consistency of those results strongly suggests that this rate at high latitude is reliable.     

Modeling of Local Magnetic Field Enhancements within Solar Flux Ropes

To model and study local magnetic-field enhancements in a solar flux rope we consider the magnetic field in its interior as a superposition of two linear (constant α) force-free magnetic-field distributions, viz. a global one, which is locally similar to a part of the cylinder, and a local torus-shaped magnetic distribution. The newly derived solution for a toroid with an aspect ratio close to unity is applied. The symmetry axis of the toroid and that of the cylinder may or may not coincide. Both the large and small radii of the toroid are set equal to the cylinder’s radius. The total magnetic field distribution yields a flux tube which has a variable diameter with local minima and maxima. In principle, this approach can be used for the interpretation and analysis of solar-limb observations of coronal loops.     

A Fluid Dynamics Approach for the Computation of Non—linear Force—Free Magnetic Field

Inspired by the analogy between the magnetic field and velocity field of incompressible fluid flow, we propose a fluid dynamics approach for computing nonlinear force-free magnetic fields. This method has the advantage that the divergence-free condition is automatically satisfied, which is a sticky issue for many other algorithms, and we can take advantage of modern high resolution algorithms to process the force-free magnetic field. Several tests have been made based on the well-known analytic solution proposed by Low & Lou. The numerical results are in satisfactory agreement with the analytic ones. It is suggested that the newly proposed method is promising in extrapolating the active region or the whole sun magnetic fields in the solar atmosphere based on the observed vector magnetic field on the photosphere.     

The Rotation Rates of Solar Magnetic Fields During Solar Cycles 21 – 23

Employing the synoptic maps of the photospheric magnetic fields from the beginning of solar cycle 21 to the end of 23, we first build up a time – longitude stackplot at each latitude between ±35°. On each stackplot there are many tilted magnetic structures clearly reflecting the rotation rates, and we adopt a cross-correlation technique to explore the rotation rates from these tilted structures. Our new method avoids artificially choosing magnetic tracers, and it is convenient for investigating the rotation rates of the positive and negative fields by omitting one kind of field on the stackplots. We have obtained the following results. i) The rotation rates of the positive and negative fields (or the leader and follower polarities, depending on the hemispheres and solar cycles) between latitudes ±35° during solar cycles 21–23 are derived. The reversal times of the leader and follower polarities are usually not consistent with the years of the solar minimum, nevertheless, at latitudes ±16°, the reversal times are almost simultaneous with them. ii) The rotation rates of the three solar cycles averaged over each cycle are calculated separately for the positive, negative and total fields. The latitude profiles of rotation of the positive and negative fields exhibit equatorial symmetries with each other, and those of the total fields lie between them. iii) The differences in rotation rates between the leader and follower polarities are obtained. They are very small near the equator, and increase as latitude increases. In the latitude range of 5° – 20°, these differences reach 0.05 deg day⁻¹, and the mean difference for solar cycle 22 is somewhat smaller than cycles 21 and 23 in these latitude regions. Then, the differences reduce again at latitudes higher than 20°.     

Magnetic Stereoscopy of Coronal Loops in NOAA 8891

The Solar TErrestrial RElations Observatory (STEREO) requires powerful tools for the three-dimensional (3D) reconstruction of the solar corona. Here we test such a program with data from SOHO and TRACE. By taking advantage of solar rotation, a newly developed stereoscopy tool for the reconstruction of coronal loops is applied to the solar active region NOAA 8891 observed from 1 March to 2 March 2000. The stereoscopic reconstruction is composed of three steps. First, we identify loop structures in two TRACE images observed from two vantage viewpoints approximately 17 degrees apart, which corresponds to observations made about 30 hours apart. In the second step, we extrapolate the magnetic field in the corona with the linear force-free field model from the photospheric line-of-sight SOHO/MDI data. Finally, combining the extrapolated field lines and one-dimensional loop curves from two different viewpoints, we obtain the 3D loop structures with the magnetic stereoscopy tool. We demonstrate that by including the magnetic modeling this tool is more powerful than pure geometrical stereoscopy, especially in resolving the ambiguities generated by classical stereoscopy. This work will be applied to the STEREO mission in the near future.     

New Boundary Integral Equation Representation for Finite Energy Force-Free Magnetic Fields in Open Space above the Sun

A boundary integral equation to describe a force-free magnetic field with finite energy content in the open space above the solar surface is found. This is a new representation for a 3-D nonlinear force-free field in terms of the boundary field and its normal gradient at the boundary. Therefore the magnetic field observed on the solar surface can be incorporated into the formulation directly and a standard numerical technique, the boundary element method, can be applied to solve the field. A numerical test case demonstrates the power of the method by recovering the analytical solution to the desired accuracy and its application to practical solar magnetic field problems is straightforward and promising.     

A systematic cross-search for radio/infrared counterparts of XMM-<Emphasis Type="Italic">Newton</Emphasis> sources

We present a catalog of cross-correlated radio, infrared and X-ray sources using a very restrictive selection criteria with an IDL-based code developed by us. The significance of the observed coincidences was evaluated through Monte Carlo simulations of synthetic sources following a well-tested protocol. We found 3320 coincident radio/X-ray sources with a high statistical significance characterized by the sum of error-weighted coordinate differences. For 997 of them, 2MASS counterparts were found. The percentage of chance coincidences is less than 1%. X-ray hardness ratios of well-known populations of objects were used to provide a crude representation of their X-ray spectrum and to make a preliminary diagnosis of the possible nature of unidentified X-ray sources. The results support the fact that the X-ray sky is largely dominated by Active Galactic Nuclei at high galactic latitudes (|b|≥10°). At low galactic latitudes (|b|≤10°) most of unidentified X-ray sources (∼94%) lie at |b|≤2°. This result suggests that most of the unidentified sources found toward the Milky Way plane are galactic objects. Well-known and unidentified sources were classified in different tables with their corresponding radio/infrared and X-ray properties. These tables are intended as a useful tool for researchers interested in particular identifications.     

How Should One Optimize Nonlinear Force-Free Coronal Magnetic Field Extrapolations from SDO/HMI Vector Magnetograms?

The Helioseismic and Magnetic Imager (HMI) on board the Solar Dynamics Observatory (SDO) provides photospheric vector magnetograms with a high spatial and temporal resolution. Our intention is to model the coronal magnetic field above active regions with the help of a nonlinear force-free extrapolation code. Our code is based on an optimization principle and has been tested extensively with semianalytic and numeric equilibria and applied to vector magnetograms from Hinode and ground-based observations. Recently we implemented a new version which takes into account measurement errors in photospheric vector magnetograms. Photospheric field measurements are often affected by measurement errors and finite nonmagnetic forces inconsistent for use as a boundary for a force-free field in the corona. To deal with these uncertainties, we developed two improvements: i) preprocessing of the surface measurements to make them compatible with a force-free field, and ii) new code which keeps a balance between the force-free constraint and deviation from the photospheric field measurements. Both methods contain free parameters, which must be optimized for use with data from SDO/HMI. In this work we describe the corresponding analysis method and evaluate the force-free equilibria by how well force-freeness and solenoidal conditions are fulfilled, by the angle between magnetic field and electric current, and by comparing projections of magnetic field lines with coronal images from the Atmospheric Imaging Assembly (SDO/AIA). We also compute the available free magnetic energy and discuss the potential influence of control parameters.     

A model of solar flares triggered by interactions between force-free current loops and plasmoids

We present a model of solar flares triggered by collisions between current loops and plasmoids. We investigate a collision process between a force-free current loop and a plasmoid, by using 3-D resistive MHD code. It is shown that a current system can be induced in the front of a plasmoid, when it approaches a force-free current loop. This secondary current produced in the front of the plasmoid separates from the plasmoid and coalesces to the force-free current loop associated with the magnetic reconnection. The core of the plasmoid stays outside the reconnection region, maintaining high density. The core can be confined by the current system produced around the plasmoid. This collison process between a current loop and a plasmoid may explain the triggering of solar flares observed byYohkoh.     

Nonlinear Force-Free Modeling of the Corona in Spherical Coordinates

We present a code for solving the nonlinear force-free equations in spherical polar geometry, with the motivation of modeling the magnetic field in the corona. The code is an implementation of the Grad–Rubin method. Our method is applicable to a spherical domain of arbitrary angular size. The implementation is based on a global spectral representation for the magnetic field that makes no explicit assumptions about the form of the magnetic field at the transverse boundaries of the domain. We apply the code to a bipolar test case with analytic boundary conditions, and demonstrate the convergence of the Grad–Rubin method and the self-consistency of the resulting numerical solution.     

A Fast Current-Field Iteration Method for Calculating Nonlinear Force-Free Fields

Existing methods for calculating nonlinear force-free magnetic fields are slow, and are likely to be inadequate for reconstructing coronal magnetic fields based on high-resolution vector magnetic field data from a new generation of spectro-polarimetric instruments. In this paper a new implementation of the current-field iteration method is presented, which is simple, fast, and accurate. The time taken by the method scales as N ⁴, for a three-dimensional grid with N ³ points. The method solves the field-updating part of the iteration by exploiting a three-dimensional Fast Fourier Transform solution of Ampere’s law with a current density field constructed to satisfy the required boundary conditions, and uses field line tracing to solve the current-updating part of the iteration. The method is demonstrated in application to a known nonlinear force-free field and to a bipolar test case.