Computation of Relative Magnetic Helicity in Spherical Coordinates

Magnetic helicity is a quantity of great importance in solar studies because it is conserved in ideal magnetohydrodynamics. While many methods for computing magnetic helicity in Cartesian finite volumes exist, in spherical coordinates, the natural coordinate system for solar applications, helicity is only treated approximately. We present here a method for properly computing the relative magnetic helicity in spherical geometry. The volumes considered are finite, of shell or wedge shape, and the three-dimensional magnetic field is considered to be fully known throughout the studied domain. Testing of the method with well-known, semi-analytic, force-free magnetic-field models reveals that it has excellent accuracy. Further application to a set of nonlinear force-free reconstructions of the magnetic field of solar active regions and comparison with an approximate method used in the past indicates that the proposed method can be significantly more accurate, thus making our method a promising tool in helicity studies that employ spherical geometry. Additionally, we determine and discuss the applicability range of the approximate method.     

Validation and Benchmarking of a Practical Free Magnetic Energy and Relative Magnetic Helicity Budget Calculation in Solar Magnetic Structures

In earlier works we introduced and tested a nonlinear force-free (NLFF) method designed to self-consistently calculate the coronal free magnetic energy and the relative magnetic helicity budgets of observed solar magnetic structures. In principle, the method requires only a single, photospheric or low-chromospheric, vector magnetogram of a quiet-Sun patch or an active region and performs calculations without three-dimensional magnetic and velocity-field information. In this work we strictly validate this method using three-dimensional coronal magnetic fields. Benchmarking employs both synthetic, three-dimensional magnetohydrodynamic simulations and nonlinear force-free field extrapolations of the active-region solar corona. Our time-efficient NLFF method provides budgets that differ from those of more demanding semi-analytical methods by a factor of approximately three, at most. This difference is expected to come from the physical concept and the construction of the method. Temporal correlations show more discrepancies that are, however, soundly improved for more complex, massive active regions, reaching correlation coefficients on the order of, or exceeding, 0.9. In conclusion, we argue that our NLFF method can be reliably used for a routine and fast calculation of the free magnetic energy and relative magnetic helicity budgets in targeted parts of the solar magnetized corona. As explained in this article and in previous works, this is an asset that can lead to valuable insight into the physics and triggering of solar eruptions.     

Topology of Magnetic Field and Coronal Heating in Solar Active Regions

Force-free magnetic fields can be computed by making use of a new numerical technique, in which the fields are represented by a boundary integral equation based on a specific Green's function. Vector magnetic fields observed on the photospheric surface can be taken as the boundary conditions of this equation. In this numerical computation, the following two points are emphasized: (1) A new method for data reduction is proposed, for removing uncertainties in boundary data and determining the parameter in this Green's function, which is important for solving the boundary integral equation. In this method, the transverse components of the observed boundary field are calibrated with a linear force-free field model without changing their azimuth. (2) The computed 3-D fields satisfy the divergence-free and force-free conditions with high precision. The alignment of these field lines is mostly in agreement with structures in Hα and Yohkoh soft X-ray images. Since the boundary data are calibrated with a linear force-free field model, the computed 3-D magnetic field can be regarded as a quasi-linear force-free field approximation. The reconstruction of 3-D magnetic field in active region NOAA 7321 was taken as an example to quantitatively exhibit the capability of our new numerical technique.     

Non-ideal evolution of non-axisymmetric, force-free magnetic fields in a magnetar

Recent numerical magnetohydrodynamic calculations by Braithwaite and collaborators support the 'fossil field' hypothesis regarding the origin of magnetic fields in compact stars and suggest that the resistive evolution of the fossil field can explain the reorganization and decay of magnetar magnetic fields. Here, these findings are modelled analytically by allowing the stellar magnetic field to relax through a quasi-static sequence of non-axisymmetric, force-free states, by analogy with spheromak relaxation experiments, starting from a random field. Under the hypothesis that the force-free modes approach energy equipartition in the absence of resistivity, the output of the numerical calculations is semiquantitatively recovered: the field settles down to a linked poloidal–toroidal configuration, which inflates and becomes more toroidal as time passes. A qualitatively similar (but not identical) end state is reached if the magnetic field evolves by exchanging helicity between small and large scales according to an α-dynamo-like, mean-field mechanism, arising from the fluctuating electromotive force produced by the initial random field. The impossibility of matching a force-free internal field to a potential exterior field is discussed in the magnetar context.     

Basic Properties of Mutual Magnetic Helicity

We derive the magnetic helicity for configurations formed by flux tubes contained fully or only partially in the spatial domain considered (called closed and open configurations, respectively). In both cases, magnetic helicity is computed as the sum of mutual helicity over all possible pairs of magnetic flux tubes weighted by their magnetic fluxes. We emphasize that these mutual helicities have properties which are not those of mutual inductances in classical circuit theory. For closed configurations, the mutual helicity of two closed flux tubes is their relative winding around each other (known as the Gauss linkage number). For open configurations, the magnetic helicity is derived directly from the geometry of the interlaced flux tubes so it can be computed without reference to a ground state (such as a potential field). We derive the explicit expression in the case of a planar and spherical boundary. The magnetic helicity has two parts. The first one is given only by the relative positions of the flux tubes on the boundary. It is the only part if all flux tubes are arch-shaped. The second part counts the integer number of turns each pair of flux tubes wind about each other. This provides a general method to compute the magnetic helicity with discrete or continuous distributions of magnetic field. The method sets closed and open configurations on an equal level within the same theoretical framework.     

Topological differences and similarities between force-free and potential models of coronal magnetic fields

Potential fields and linear force-free fields are often used as models for the magnetic field of the Sun's corona. They can be written as analytical expressions in terms of boundary values at the photosphere. Because of their relative simplicity compared with nonlinear force-free fields, these two models are of particular importance in topological analysis of solar phenomena. However, it has been suggested by Hudson and Wheatland (1999) that the topologies of potential and force-free models are in general not even qualitatively equivalent. In this paper, their example is re-examined and it is found that the opposite conclusions hold. In general, potential and force-free fields are topologically similar sufficiently close to localized sources. The exception to this are structurally unstable states, such as bifurcation states, where a small change of current can produce a significant change of topology.     

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.     

Thin-Shell Magnetohydrodynamic Equations for the Solar Tachocline

We present a new system of equations designed to study global-scale dynamics in the stably-stratified portion of the solar tachocline. This system is derived from the 3D equations of magnetohydrodynamics in a rotating spherical shell under the assumption that the shell is thin and stably-stratified (subadiabatic). The resulting thin-shell model can be regarded as a magnetic generalization of the hydrostatic primitive equations often used in meteorology. It is simpler in form than the more general anelastic or Boussinesq equations, making it more amenable to analysis and interpretation and more computationally efficient. However, the thin-shell system is still three-dimensional and as such represents an important extension to previous 2D and shallow-water approaches. In this paper we derive the governing equations for our thin-shell model and discuss its underlying assumptions, its context relative to other models, and its application to the solar tachocline. We also demonstrate that the dissipationless thin-shell system conserves energy, angular momentum and magnetic helicity.     

Including stereoscopic information in the reconstruction of coronal magnetic fields

We present a method to include stereoscopic information about the three-dimensional structure of flux tubes into the reconstruction of the coronal magnetic field. Due to the low plasma beta in the corona we can assume a force-free magnetic field, with the current density parallel to the magnetic field lines. Here we use linear force-free fields for simplicity. The method uses the line-of-sight magnetic field on the photosphere as observational input. The value of is determined iteratively by comparing the reconstructed magnetic field with the observed structures. The final configuration is the optimal linear force-free solution constrained by both the photospheric magnetogram and the observed plasma structures. As an example we apply our method to SOHO MDI/EIT data of an active region. In the future it is planned to apply the method to analyse data from the SECCHI instrument aboard the STEREO mission.     

Comments on solar linear force-free field and application of FFT analysis

We begin with some comments on the methods of computation of the magnetic fields of solar active regions in terms of force-free field, and point out some mistakes in the solutions. Then we propose a method of quasi-linear force-free field, and probe the application of fast Fourier transform (FFT) in its computation. The magnetic field of the active region NOAA 9165 on 2000-09-15 is taken as an example, and three different methods are used in the determination of the configuration of magnetic lines. The results are then compared with the fine Hα filtergrams of Ganyu Station of Purple Mountain Observatory.     

Photospheric Injection of Magnetic Helicity: Connectivity-Based Flux Density Method

Magnetic helicity quantifies the degree to which the magnetic field in a volume is globally sheared and/or twisted. This quantity is believed to play a key role in solar activity due to its conservation property. Helicity is continuously injected into the corona during the evolution of active regions (ARs). To better understand and quantify the role of magnetic helicity in solar activity, the distribution of magnetic helicity flux in ARs needs to be studied. The helicity distribution can be computed from the temporal evolution of photospheric magnetograms of ARs such as the ones provided by SDO/HMI and Hinode/SOT. Most recent analyses of photospheric helicity flux derived a proxy to the helicity-flux density based on the relative rotation rate of photospheric magnetic footpoints. Although this proxy allows a good estimate of the photospheric helicity flux, it is still not a true helicity flux density because it does not take into account the connectivity of the magnetic field lines. For the first time, we implement a helicity density that takes this connectivity into account. To use it for future observational studies, we tested the method and its precision on several types of models involving different patterns of helicity injection. We also tested it on more complex configurations – from magnetohydrodynamics (MHD) simulations – containing quasi-separatrix layers. We demonstrate that this connectivity-based proxy is best-suited to map the true distribution of photospheric helicity injection.     

Magnetic Helicity Budget of Solar-Active Regions from the Photosphere to Magnetic Clouds

We have analyzed the long-term evolution of two active regions (ARs) from their emergence through their decay using observations from several instruments on board SoHO (MDI, EIT and LASCO) and Yohkoh/SXT. We have computed the evolution of the relative coronal magnetic helicity combining data from MDI and SXT with a linear force-free model of the coronal magnetic field. Next, we have computed the injection of helicity by surface differential rotation using MDI magnetic maps. To estimate the depletion of helicity we have counted all the CMEs of which these ARs have been the source, and we have evaluated their magnetic helicity assuming a one to one correspondence with magnetic clouds with an average helicity contain. When these three values (variation of coronal magnetic helicity, injection by differential rotation and ejection via CMEs) are compared, we find that surface differential rotation is a minor contributor to the helicity budget since CMEs carry away at least 10 times more helicity than the one differential rotation can provide. Therefore, the magnetic helicity flux needed in the global balance should come from localized photospheric motions that, at least partially, reflect the emergence of twisted flux tubes. We estimate that the total helicity carried away in CMEs can be provided by the end-to-end helicity of the flux tubes forming these ARs. Therefore, we conclude that most of the helicity ejected in CMEs is generated below the photosphere and emerges with the magnetic flux.     

EVOLUTION OF MAGNETIC HELICITY IN MAGNETIC RECONNECTION

This paper presents a definition of magnetic helicity specifically for two-dimensional magnetic fields and derives the associated helicity equation. The newly defined helicity is closely related to its three-dimensional counterpart and serves as a measure of the shear of magnetic field. Based on this, a numerical simulation is carried out on magnetic reconnection occurring in the lower solar atmosphere. It is found that the helicity dissipation due to magnetic reconnection is very small. A large amount of helicity is transferred upward and escapes from the domain of the solution, and the total helicity is approximately conserved during the magnetic reconnection and helicity transfer. This is in support of the applicability of a postulate, which was proposed by Taylor (1974, 1986) concerning the approximate conservation of magnetic helicity in the presence of resistive dissipation and magnetic reconnection in a highly conductive laboratory plasma, to the solar atmosphere.     

Estimating the Relative Helicity of Coronal Magnetic Fields

To quantify changes of the solar coronal field connectivity during eruptive events, one can use magnetic helicity, which is a measure of the shear or twist of a current-carrying (non-potential) field. To find a physically meaningful quantity, a relative measure, giving the helicity of a current-carrying field with respect to a reference (potential) field, is often evaluated. This requires a knowledge of the three-dimensional vector potential. We present a method to calculate the vector potential for a solenoidal magnetic field as the sum of a Laplacian part and a current-carrying part. The only requirements are the divergence freeness of the Laplacian and current-carrying magnetic field and the sameness of their normal field component on the bounding surface of the considered volume.     

RECONSTRUCTION OF THE THREE-DIMENSIONAL CORONAL MAGNETIC FIELD

Studies of solar flares indicate that the mechanism of flares is magnetic in character and that the coronal magnetic field is a key to understanding solar high-energy phenomena. In our ongoing research we are conducting a systematic study of a large database of observations which includes both coronal structure (from the Soft X-ray Telescope on the Yohkoh spacecraft) and photospheric vector magnetic fields (from the Haleakala Stokes Polarimeter at Mees Solar Observatory). We compare the three-dimensional nonlinear force-free coronal magnetic field, computed from photospheric boundary data, to images of coronal structure. In this paper we outline our techniques and present results for active region AR 7220/7222. We show that the computed force-free coronal magnetic field agrees well with Yohkoh X-ray coronal loops, and we discuss the properties of the coronal magnetic field and the soft X-ray loops.     

Modelling the Global Solar Corona II: Coronal Evolution and Filament Chirality Comparison

This paper considers the hemispheric pattern of solar filaments using newly developed simulations of the real photospheric and 3D coronal magnetic fields over a six-month period, on a global scale. The magnetic field direction in the simulation is compared directly with the chirality of observed filaments, at their observed locations. In our model the coronal field evolves through a continuous sequence of nonlinear force-free equilibria, in response to the changing photospheric boundary conditions and the emergence of new magnetic flux. In total 119 magnetic bipoles with properties matching observed active regions are inserted. These bipoles emerge twisted and inject magnetic helicity into the solar atmosphere. When we choose the sign of this active-region helicity to match that observed in each hemisphere, the model produces the correct chirality for up to 96% of filaments, including exceptions to the hemispheric pattern. If the emerging bipoles have zero helicity, or helicity of the opposite sign, then this percentage is much reduced. In addition, the simulation produces a higher proportion of filaments with the correct chirality after longer times. This indicates that a key element in the evolution of the coronal field is its long-term memory, and the build-up and transport of helicity from low to high latitudes over many months. It highlights the importance of continuous evolution of the coronal field, rather than independent extrapolations at different times. This has significant consequences for future modelling such as that related to the origin and development of coronal mass ejections.     

Computing Nonlinear Force-Free Coronal Magnetic Fields in Spherical Geometry

We describe a newly developed code for the extrapolation of nonlinear force-free coronal magnetic fields in spherical coordinates. The program uses measured vector magnetograms on the solar photosphere as input and solves the force-free equations in the solar corona. The method is based on an optimization principle and the heritage of the newly developed code is a corresponding method in Cartesian geometry. We test the newly developed code with the help of a semi-analytic solution and rate the quality of our reconstruction qualitatively by magnetic field line plots and quantitatively with a number of comparison metrics. We find that we can reconstruct the original test field with high accuracy. The method is fast if the computation is limited to low co-latitudes (say 30°≤θ≤150°), but it becomes significantly slower if the polar regions are included.     

Magnetic Helicity Injection in Solar Active Regions

We present the evolution of magnetic field and its relationship with mag- netic(current)helicity in solar active regions from a series of photospheric vector magnetograms obtained by Huairou Solar Observing Station,longitudinal magne- tograms by MDI of SOHO and white light images of TRACE.The photospheric current helicity density is a quantity reflecting the local twisted magnetic field and is related to the remaining magnetic helicity in the photosphere,even if the mean current helicity density brings the general chiral property in a layer of solar active regions.As new magnetic flux emerges in active regions,changes of photospheric cur- rent helicity density with the injection of magnetic helicity into the corona from the subatmosphere can be detected,including changes in sign caused by the injection of magnetic helicity of opposite sign.Because the injection rate of magnetic helicity and photospheric current helicity density have different means in the solar atmosphere, the injected magnetic helicity is probably not proportional to the current helicity den- sity remaining in the photosphere.The evidence is that rotation of sunspots does not synchronize exactly with the twist of photospheric transverse magnetic field in some active regions(such as,delta active regions).They represent different aspects of mag- netic chirality.A combined analysis of the observational magnetic helicity parameters actually provides a relative complete picture of magnetic helicity and its transfer in the solar atmosphere.     

A Nonlinear Force-Free Magnetic Field Approximation Suitable for Fast Forward-Fitting to Coronal Loops. I. Theory

We derive an analytical approximation of nonlinear force-free magnetic field solutions (NLFFF) that can efficiently be used for fast forward-fitting to solar magnetic data, constrained either by observed line-of-sight magnetograms and stereoscopically triangulated coronal loops, or by 3D vector-magnetograph data. The derived NLFFF solutions provide the magnetic field components B _x (x), B _y (x), B _z (x), the force-free parameter α(x), the electric current density j(x), and are accurate to second-order (of the nonlinear force-free α-parameter). The explicit expressions of a force-free field can easily be applied to modeling or forward-fitting of many coronal phenomena.     

A note on the evolution of magnetic helicity in active regions

The evolution of magnetic helicity in a rapidly growing active region is governed mainly by the advection of magnetic fields into the photosphere. Dissipation at the photosphere makes only a minor contribution to helicity evolution. Current helicity dissipation in the three-dimensional domain around the active region can be ignored.