Linear Force-free Magnetic Fields Research Articles

Magnetohydrodynamics simulation of magnetic flux rope formation in a quadrupolar magnetic field configuration

Magnetic flux ropes (MFRs) play an important role in high-energetic events like solar flares and coronal mass ejections in the solar atmosphere. Importantly, solar observations suggest an association of some flaring events with quadrupolar magnetic configurations. However, the formation and subsequent evolution of MFRs in such magnetic configurations still need to be fully understood. In this paper, we present idealized magnetohydrodynamics (MHD) simulations of MFR formation in a quadrupolar magnetic configuration. A suitable initial magnetic field having a quadrupolar configuration is constructed by modifying a three-dimensional linear force-free magnetic field. The initial magnetic field contains neutral lines, which consist of X-type null points. The simulated dynamics initially demonstrate the oppositely directed magnetic field lines located across the polarity inversion lines (PILs) moving towards each other, resulting in magnetic reconnections. Due to these reconnections, four highly twisted MFRs form over the PILs. With time, the foot points of the MFRs move towards the X-type neutral lines and reconnect, generating complex magnetic structures around the neutral lines, thus making the MFR topology more complex in the quadrupolar configuration than those formed in bipolar loop systems. Further evolution reveals the non-uniform rise of the MFRs. Importantly, the simulations indicate that the pre-existing X-type null points in magnetic configurations can be crucial to the evolution of the MFRs and may lead to the observed brightenings during the onset of some flaring events in the quadrupolar configurations.

Observation-based modelling of the energetic storm particle event of 14 July 2012

Aims. We model the energetic storm particle (ESP) event of 14 July 2012 using the energetic particle acceleration and transport model named ‘PArticle Radiation Asset Directed at Interplanetary Space Exploration’ (PARADISE), together with the solar wind and coronal mass ejection (CME) model named ‘EUropean Heliospheric FORcasting Information Asset’ (EUHFORIA). The simulation results illustrate both the capabilities and limitations of the utilised models. We show that the models capture some essential structural features of the ESP event; however, for some aspects the simulations and observations diverge. We describe and, to some extent, assess the sources of errors in the modelling chain of EUHFORIA and PARADISE and discuss how they may be mitigated in the future. Methods. The PARADISE model computes energetic particle distributions in the heliosphere by solving the focused transport equation in a stochastic manner. This is done using a background solar wind configuration generated by the ideal magnetohydrodynamic module of EUHFORIA. The CME generating the ESP event is simulated by using the spheromak model of EUHFORIA, which approximates the CME’s flux rope as a linear force-free spheroidal magnetic field. In addition, a tool was developed to trace CME-driven shock waves in the EUHFORIA simulation domain. This tool is used in PARADISE to (i) inject 50 keV protons continuously at the CME-driven shock and (ii) include a foreshock and a sheath region, in which the energetic particle parallel mean free path, λ∥, decreases towards the shock wave. The value of λ∥ at the shock wave is estimated from in situ observations of the ESP event. Results. For energies below ∼1 MeV, the simulation results agree well with both the upstream and downstream components of the ESP event observed by the Advanced Composition Explorer. This suggests that these low-energy protons are mainly the result of interplanetary particle acceleration. In the downstream region, the sharp drop in the energetic particle intensities is reproduced at the entry into the following magnetic cloud, illustrating the importance of a magnetised CME model.

Coronal Rain in Randomly Heated Arcades

Abstract Adopting the MPI-AMRVAC code, we present a 2.5-dimensional magnetohydrodynamic simulation, which includes thermal conduction and radiative cooling, to investigate the formation and evolution of the coronal rain phenomenon. We perform the simulation in initially linear force-free magnetic fields that host chromospheric, transition-region, and coronal plasma, with turbulent heating localized on their footpoints. Due to thermal instability, condensations start to occur at the loop top, and rebound shocks are generated by the siphon inflows. Condensations fragment into smaller blobs moving downwards, and as they hit the lower atmosphere, concurrent upflows are triggered. Larger clumps show us clear coronal rain showers as dark structures in synthetic EUV hot channels and as bright blobs with cool cores in the 304 Å channel, well resembling real observations. Following coronal rain dynamics for more than 10 hr, we carry out a statistical study of all coronal rain blobs to quantify their widths, lengths, areas, velocity distributions, and other properties. The coronal rain shows us continuous heating–condensation cycles, as well as cycles in EUV emissions. Compared to the previous studies adopting steady heating, the rain happens faster and in more erratic cycles. Although most blobs are falling downward, upward-moving blobs exist at basically every moment. We also track the movement of individual blobs to study their dynamics and the forces driving their movements. The blobs have a prominence-corona transition-region-like structure surrounding them, and their movements are dominated by the pressure evolution in the very dynamic loop system.

Analytical Three-dimensional Magnetohydrostatic Equilibrium Solutions for Magnetic Field Extrapolation Allowing a Transition from Non-force-free to Force-free Magnetic Fields

For the extrapolation of magnetic fields into the solar corona from measurements taken in the photosphere (or chromosphere) force-free magnetic fields are typically used. This does not take into account that the lower layers of the solar atmosphere are not force-free. While some numerical extrapolation methods using magnetohydrostatic magnetic fields have been suggested, a complementary and numerically comparatively cheap method is to use analytical magnetohydrostatic equilibria to extrapolate the magnetic field. In this paper, we present a new family of solutions for a special class of analytical three-dimensional magnetohydrostatic equilibria, which can be of use for such magnetic field extrapolation. The new solutions allow for the more flexible modeling of a transition from non-force-free to (linear) force-free magnetic fields. In particular, the height and width of the region where this transition takes place can be specified by choosing appropriate model parameters.

Solar and Galactic Magnetic Halo Structure: Force-Free Dynamos?

Magnetic fields may relax dissipatively to the minimum energy force-free condition whenever they are not constantly created or distorted. We review the axially symmetric solutions for force-free magnetic fields, especially for the non-linear field. A new formulation for the scale invariant state is given. Illustrative examples are shown. Applications to both stellar coronas and galactic halos are possible. Subsequently we study whether such force-free fields may be sustained by classical magnetic dynamo action. Although the answer is `not indefinitely’, there may be an evolutionary cycle wherein the magnetic field repeatedly relaxes to the minimum energy condition after a period of substantial growth and distortion. Different force-free dynamos may coexist at different locations. Helicity transfer between scales is studied briefly. A dynamo solution is given for the temporal evolution away from an initial linear force-free magnetic field due to both α 2 and ω terms. This can be used at the sub scale level to create a `delayed’ α effect.

Determining the parameter for the linear force-free magnetic field model with multi-dipolar configurations using deep neural networks

Abstract Recent advances in the field of neural networks have made convolutional neural networks (CNNs) a conventional algorithm for many computer vision tasks including image recognition and object detection. Because modeling the coronal magnetic field of the Sun is an important objective in heliophysics, this study extends the use of CNNs to the application of coronal magnetic field modeling. We employ a simple one-parameter model of linear force-free magnetic fields (LFFFs) to model active regions of multiple dipolar configurations including the Active Region (AR) 11117. We use state-of-the-art architectures such as ResNet and Inception networks, and develop our customized network “SolarNet” to determine the associated LFFF parameter alpha from a set of pseudo-coronal loop images, which are generated using the modeled active regions. Our results show very high accuracy of determining the LFFF parameter alpha, thereby demonstrating the effectiveness of the generic and customized deep CNN architectures to understand the coronal magnetic field.

Reconstructing nonlinear force-free fields by a constrained optimization

It seems that the potential and linear force-free magnetic fields are inadequate to represent the observed magnetic events occurring in different regions of the solar corona. To reconstruct the nonlinear force-free fields from the solar surface magnetograms, various analytical and numerical methods have already been examined by different authors. Here, using the Lagrange multiplier technique, a constrained optimization approach for reconstructing force-free magnetic fields is proposed. In the optimization procedure the solenoidal property is considered as a constraint on the initial non-force-free field. In the Wheatland et al. (2000) method as an unconstrained optimization, both solenoidal and force-free conditions are fulfilled approximately. In contrast, the constrained optimization method, up to numerical precision, leads us to a nearly force-free magnetic field with exactly zero divergence. The solutions are obtained and tested by the Low and Lou (1990) semi-analytic solution.

The Plasma Parameters and Geometry of Cool and Warm Active Region Loops

Abstract How the solar corona is heated to high temperatures remains an unsolved mystery in solar physics. In the present study we analyze observations of 50 whole active region loops taken with the Extreme-ultraviolet Imaging Spectrometer on board the Hinode satellite. Eleven loops were classified as cool loops (<1 MK) and 39 as warm loops (1–2 MK). We study their plasma parameters, such as densities, temperatures, filling factors, nonthermal velocities, and Doppler velocities. We combine spectroscopic analysis with linear force-free magnetic field extrapolation to derive the 3D structure and positioning of the loops, their lengths and heights, and the magnetic field strength along the loops. We use density-sensitive line pairs from Fe xii, Fe xiii, Si x, and Mg vii ions to obtain electron densities by taking special care of intensity background subtraction. The emission measure loci method is used to obtain the loop temperatures. We find that the loops are nearly isothermal along the line of sight. Their filling factors are between 8% and 89%. We also compare the observed parameters with the theoretical Rosner–Tucker–Vaiana (RTV) scaling law. We find that most of the loops are in an overpressure state relative to the RTV predictions. In a follow-up study, we will report a heating model of a parallel-cascade-based mechanism and will compare the model parameters with the loop plasma and structural parameters derived here.

Formation and Initiation of Erupting Flux Rope and Embedded Filament Driven by Photospheric Converging Motion

Abstract In this paper, we study how a flux rope (FR) is formed and evolves into the corresponding structure of a coronal mass ejection (CME) numerically driven by photospheric converging motion. A two-and-a-half-dimensional magnetohydrodynamics simulation is conducted in a chromosphere-transition-corona setup. The initial arcade-like linear force-free magnetic field is driven by an imposed slow motion converging toward the magnetic inversion line at the bottom boundary. The convergence brings opposite-polarity magnetic flux to the polarity inversion, giving rise to the formation of an FR by magnetic reconnection and eventually to the eruption of a CME. During the FR formation, an embedded prominence gets formed by the levitation of chromospheric material. We confirm that the converging flow is a potential mechanism for the formation of FRs and a possible triggering mechanism for CMEs. We investigate the thermal, dynamical, and magnetic properties of the FR and its embedded prominence by tracking their thermal evolution, analyzing their force balance, and measuring their kinematic quantities. The phase transition from the initiation phase to the acceleration phase of the kinematic evolution of the FR was observed in our simulation. The FR undergoes a series of quasi-static equilibrium states in the initiation phase; while in the acceleration phase the FR is driven by Lorentz force and the impulsive acceleration occurs. The underlying physical reason for the phase transition is the change of the reconnection mechanism from the Sweet–Parker to the unsteady bursty regime of reconnection in the evolving current sheet underneath the FR.

Toroidal linear force-free magnetic fields with axial symmetry

Aims. Interplanetary magnetic flux ropes are often described as linear force-free fields. To account for their curvature, toroidal configurations must be used. The aim is to find an analytic description of a linear force-free magnetic field of the toroidal geometry in which the cross section of flux ropes can be controlled. Methods. The solution is found as a superposition of fields given by linear force-free cylinders tangential to a generating toroid. The cylindrical field is expressed in a series of terms that are not all cylindrically symmetric. Results. We found the general form of a toroidal linear force-free magnetic field. The field is azimuthally symmetric with respect to the torus axis. It depends on a set of coefficients that enables controlling the flux rope shape (cross section) to some extent. By varying the coefficients, flux ropes with circular and elliptic cross sections were constructed. Numerical comparison suggests that the simple analytic formula for calculating the helicity in toroidal flux ropes of the circular cross section can be used for flux ropes with elliptic cross sections if the minor radius in the formula is set to the geometric mean of the semi-axes of the elliptic cross section.

A Minimum Energy Fit Method to Reconstruct Photospheric Velocity and Magnetic Diffusivity in Active Regions from Observed Magnetograms and Dopplergrams

We introduce MEF-R, a generalization of the minimum energy fit (MEF; Longcope, Astrophys. J. 612, 1181, 2004) to a non-ideal (resistive) gas. The new technique requires both vector magnetograms and Doppler velocities as input fields. However, in the case of active regions observed only with the Michelson–Doppler Imager (MDI) onboard the Solar and Heliospheric Observatory (SOHO) such as AR 9077, we have only access to line-of-sight magnetograms. We reconstruct two-dimensional maps of the magnetic diffusivity η(x,y) together with velocity components v x (x,y), v y (x,y), and v z (x,y) under the linear force-free magnetic field approximation. Computed maps for v z (x,y) very well match the Doppler velocities v r (x,y). We find the average value 〈η〉≈108 m2 s−1 with a standard deviation of ≈ 1010 m2 s−1. Such high values of η(x,y) are to be expected at some places since our magnetic diffusivity is actually eddy-diffusivity. Inside AR 9077, the maps of η(x,y) do not resemble closely the maps from classical models of the magnetic diffusivity, but they are closer to η as a function of temperature than to η as a function of electric current density.

THREE-DIMENSIONAL PROMINENCE-HOSTING MAGNETIC CONFIGURATIONS: CREATING A HELICAL MAGNETIC FLUX ROPE

The magnetic configuration hosting prominences and their surrounding coro- nal structure is a key research topic in solar physics. Recent theoretical and observational studies strongly suggest that a helical magnetic flux rope is an es- sential ingredient to fulfill most of the theoretical and observational requirements for hosting prominences. To understand flux rope formation details and obtain magnetic configurations suitable for future prominence formation studies, we here report on three-dimensional isothermal magnetohydrodynamic simulations including finite gas pressure and gravity. Starting from a magnetohydrostatic corona with a linear force-free bipolar magnetic field, we follow its evolution when introducing vortex flows around the main polarities and converging flows towards the polarity inversion line near the bottom of the corona. The con- verging flows bring feet of different loops together at the polarity inversion line and magnetic reconnection and flux cancellation happens. Inflow and outflow signatures of the magnetic reconnection process are identified, and the thereby newly formed helical loops wind around pre-existing ones so that a complete flux rope grows and ascends. When a macroscopic flux rope is formed, we switch off the driving flows and find that the system relaxes to a stable state containing a helical magnetic flux rope embedded in an overlying arcade structure. A major part of the formed flux rope is threaded by dipped field lines which can stably support prominence matter, while the total mass of the flux rope is in the order of 4-5.e14 g.

Modeling Magnetic Flux Ropes

AbstractThe magnetic configuration hosting prominences can be a large-scale helical magnetic flux rope. As a necessary step towards future prominence formation studies, we report on a stepwise approach to study flux rope formation. We start with summarizing our recent three-dimensional (3D) isothermal magnetohydrodynamic (MHD) simulation where a flux rope is formed, including gas pressure and gravity. This starts from a static corona with a linear force-free bipolar magnetic field, altered by lower boundary vortex flows around the main polarities and converging flows towards the polarity inversion. The latter flows induce magnetic reconnection and this forms successive new helical loops so that a complete flux rope grows and ascends. After stopping the driving flows, the system relaxes to a stable helical magnetic flux rope configuration embedded in an overlying arcade. Starting from this relaxed isothermal endstate, we next perform a thermodynamic MHD simulation with a chromospheric layer inserted at the bottom. As a result of a properly parametrized coronal heating, and due to radiative cooling and anisotropic thermal conduction, the system further relaxes to an equilibrium where the flux rope and the arcade develop a fully realistic thermal structure. This paves the way to future simulations for 3D prominence formation.

MAGNETIC ENERGY AND HELICITY BUDGETS IN THE ACTIVE-REGION SOLAR CORONA. II. NONLINEAR FORCE-FREE APPROXIMATION

Expanding on an earlier work that relied on linear force-free magnetic fields, we self-consistently derive the instantaneous free magnetic energy and relative magnetic helicity budgets of an unknown three-dimensional nonlinear force-free magnetic structure extending above a single, known lower-boundary magnetic field vector. The proposed method does not rely on the detailed knowledge of the three-dimensional field configuration but is general enough to employ only a magnetic connectivity matrix on the lower boundary. The calculation yields a minimum free magnetic energy and a relative magnetic helicity consistent with this free magnetic energy. The method is directly applicable to photospheric or chromospheric vector magnetograms of solar active regions. Upon validation, it basically reproduces magnetic energies and helicities obtained by well-known, but computationally more intensive and non-unique, methods relying on the extrapolated three-dimensional magnetic field vector. We apply the method to three active regions, calculating the photospheric connectivity matrices by means of simulated annealing, rather than a model-dependent nonlinear force-free extrapolation. For two of these regions we correct for the inherent linear force-free overestimation in free energy and relative helicity that is larger for larger, more eruptive, active regions. In the third studied region, our calculation can lead to a physical interpretation of observed eruptive manifestations. We conclude that the proposed method, including the proposed inference of the magnetic connectivity matrix, is practical enough to contribute to a physical interpretation of the dynamical evolution of solar active regions.

On the Shape of Force-Free Field Lines in the Solar Corona

This paper studies the shape parameters of looped field lines in a linear force-free magnetic field. Loop structures with a sufficient amount of kinking are generally seen to form S or inverse S (Z) shapes in the corona (as viewed in projection). For a single field line, we can ask how much the field line is kinked (as measured by the writhe), and how much neighbouring flux twists about the line (as measured by the twist number). The magnetic helicity of a flux element surrounding the field line can be decomposed into these two quantities. We find that the twist helicity contribution dominates the writhe helicity contribution, for field lines of significant aspect ratio, even when their structure is highly kinked. These calculations shed light on some popular assumptions of the field. First, we show that the writhe of field lines of significant aspect ratio (the apex height divided by the footpoint width) can sometimes be of opposite sign to the helicity. Secondly, we demonstrate the possibility of field line structures which could be interpreted as Z-shaped, but which have a helicity value sign expected of an S-shaped structure. These results suggest that caution should be exercised in using two-dimensional images to draw conclusions on the helicity value of field lines and flux tubes.

A Test on Two Codes for Extrapolating Solar Linear Force-free Magnetic Fields

We evaluate the performances of two newly-implemented codes for extrapolating the solar linear force-free magnetic fields, by measuring their quantified responses to the lower boundary vector field data on a finite region using analytical models. The codes are based on two boundary integral formulas with different mechanisms in utilizing the transverse boundary field: the first one only utilizes the transverse boundary field to derive the value of the force-free parameter, while the other one explicitly utilizes the whole transverse boundary field, in addition to the vertical field component. Studies on the test cases show that both of the codes could reproduce the analytical model fields with reasonable accuracy within the valid domain, provided a sufficient amount of data is available at the lower boundary. The code utilizing explicitly all three components of the boundary field shows generally better performances and requires a smaller boundary-data area in order to achieve the same degree of accuracy in the metric quantities such as the normalized vector error, vector correlation, vector angular difference, and magnetic energy; however, the accuracy in the divergence-free metric and especially the force-free metric are less ideal.

Linear force-free field of a toroidal symmetry

Context. Interplanetary flux ropes are often described by linear force-free fields. To account for their curvature, toroidal configurations valid for large aspect ratios are used. However, in some cases, flux ropes need to be approximated by a toroid with a low aspect ratio. Aims. The aim is to find an analytical description of a linear force-free magnetic field of toroidal geometry, not restricted to large aspect ratios. Methods. The solution is found as a superposition of fields given by linear force-free cylinders tangential to a generating toroid. Results. The obtained solution describes toroidal flux ropes with reasonable shapes and magnetic field values. The field is exactly linear force-free for arbitrary aspect ratios. The new solution can be applied for solar and interplanetary flux ropes, astrophysical objects, and laboratory plasma.

Saturation of tearing modes in reversed field pinches with locally linear force-free magnetic fields and its application to quasi-single-helicity states

A simple formula for predicting the width of a saturated island, formed as a consequence of tearing perturbation of linear force-free fields in cylindrical geometry, is derived. The formula makes it possible to calculate the saturated island width in terms of the values of parameters characterizing the initial force-free equilibrium and can be applied to equilibria of interest for reversed field pinches. In particular it is applied, in this paper, to force-free equilibria with piecewise constant radial profile of the pinch parameter, which have been recently suggested to be relevant for the formation of quasi-single-helicity states. The main result is that the island width becomes larger as a parameter, that quantifies the departure from a relaxed Taylor state, increases.

Magnetic Energy and Helicity Budgets in the Active Region Solar Corona. I. Linear Force‐Free Approximation

Expanding on an earlier work that relied on linear force-free magnetic fields, we self-consistently derive the instantaneous free magnetic energy and relative magnetic helicity budgets of an unknown three-dimensional nonlinear force-free magnetic structure extending above a single, known lower-boundary magnetic field vector. The proposed method does not rely on the detailed knowledge of the three-dimensional field configuration but is general enough to employ only a magnetic connectivity matrix on the lower boundary. The calculation yields a minimum free magnetic energy and a relative magnetic helicity consistent with this free magnetic energy. The method is directly applicable to photospheric or chromospheric vector magnetograms of solar active regions. Upon validation, it basically reproduces magnetic energies and helicities obtained by well-known, but computationally more intensive and non-unique, methods relying on the extrapolated three-dimensional magnetic field vector. We apply the method to three active regions, calculating the photospheric connectivity matrices by means of simulated annealing, rather than a model-dependent nonlinear force-free extrapolation. For two of these regions we correct for the inherent linear force-free overestimation in free energy and relative helicity that is larger for larger, more eruptive, active regions. In the third studied region, our calculation can lead to a physical interpretation of observed eruptive manifestations. We conclude that the proposed method, including the proposed inference of the magnetic connectivity matrix, is practical enough to contribute to a physical interpretation of the dynamical evolution of solar active regions.

The GMRES method applied to the BEM extrapolation of solar force-free magnetic fields

Context.Since the 1990's, Yan and colleagues have formulated a kind of boundary integral formulation for the linear or non-linear solar force-free magnetic fields with finite energy in semi-infinite space, and developed a computational procedure by virtue of the boundary element method (BEM) to extrapolate the magnetic fields above the photosphere.