Force-free Field Research Articles

Electromagnetic effects on charged particles in a near-horizon-extreme-Kerr geometry

We investigate the motions of charged particles in the near horizon region of an extreme Kerr black hole with weak electromagnetic fields. There is an enhanced symmetry in the near-horizon-extreme-Kerr geometry. We find that when the electromagnetic field respects this enhanced symmetry, which we refer to as the maximally symmetric electromagnetic field, the equations of motion of charged particles get simplified into a set of decoupled first-order differential equations. We discuss the motions of charged particles in two maximally symmetric electromagnetic fields, one being the force-free field and the other being the vacuum fields. Even though the radial motions are similar to the geodesics in near-horizon-extreme-Kerr geometry, the angular motions could be affected by the electromagnetic field significantly. In particular, for the vacuum solution that is produced by a weakly charged black hole, there exist stable vortical motions if the electromagnetic parameter is above the critical value ${\mathfrak{B}}_{c}=\sqrt{3}$. These vortical motions do not cross the equatorial planes, and the charged particles in them radiate nonthermally. We discuss the corresponding astrophysical implications.

Circular ribbon flare triggered from an incomplete fan-spine configuration

Context. Circular ribbon flares are characterised by circular, semi-circular, or elliptical ribbon brightenings. As the physics of such solar events involves a true 3D magnetic topology, they have been extensively studied in contemporary solar research. Aims. In order to understand the triggering processes and the complex magnetic topology involved in circular ribbon flares, we carried out a thorough investigation of an M-class circular ribbon flare that originated within close proximity of a quasi-separatrix layer (QSL). Methods. We combined multi-wavelength Atmospheric Imaging Assembly (AIA) and Reuven Ramaty High Energy Solar Spectroscopic Imager (RHESSI) observations with photospheric Helioseismic and Magnetic Imager (HMI) observations and coronal magnetic field modelling analysis using the non-linear force free field (NLFFF) model. Results. The circular ribbon flare occurred from a complex magnetic configuration characterised by negative magnetic patches surrounded by positive-polarity regions on three sides. As the negative polarity patches were not surrounded by positive-polarity regions on all four sides, the corresponding coronal field was devoid of any null points. This led to the formation of an incomplete fan-spine-like configuration that deviated from classical fan-spine configurations in null-point topology. Further, an observationally identified QSL structure was situated within the active region, very close to the flaring region. The presence of the QSL was verified by the NLFFF modelling. The far end of the spine-like lines terminated very close to one footpoint location of the QSL lines. Our analysis suggests that activities at this location led to the activation of a flux rope situated within the fan-like lines and triggering of the circular ribbon flare via perturbation of the overall fan-spine-like structure. Further, we identified RHESSI X-ray sources from the footpoints of the QSL structure, which suggests that slipping reconnections can also lead to discernible signatures of particle acceleration.

Beltrami fields in partially ionized magnetized dusty plasma

The relaxed state of a magnetized four-component partially ionized dusty (PID) plasma is explored. PID plasma is composed of inertial electrons, positive ions, neutrals, and static negatively charged dust particles. When the neutrals are dragged along with the ions, the steady-state solution of vortex dynamic equations for inertial electron and ion species yields a triple Beltrami relaxed state, which is a linear combination of three single force-free fields and is characterized by three self-organized structures. The impact of plasma parameters such as Beltrami parameters and neutral (degree of ionization) and dust species densities on the nature of the relaxed state is investigated. Furthermore, the role of scale separation and its implication in space plasma are also presented. The present research will aid in elucidating the role of neutral and dust species in space, astrophysical, and laboratory plasmas.

Quiet Sun flux rope formation via incomplete Taylor relaxation

Context. Low-altitude nanoflares are among the candidates for atmospheric heating in the quiet Sun’s corona. Low-altitude twisted magnetic fields may be involved in such events, as they are in larger flares. But for nanoflares, the exact role, topology, and formation mechanisms of these twisted fields remain to be studied. Aims. In this paper, we investigate the formation and evolution of a preflare flux rope in a fully stratified, 3D magnetohydrodynamics simulation of the quiet Sun using the Bifrost code. This study focuses on the time period before the rope eventually reconnects with an overlying field, resulting in a nanoflare-scale energy on the order of 1017 J. One puzzle is that this modeled flux rope does not form by any of the mechanisms usually at work in larger flares, such as flux emergence, flux cancellation, or tether-cutting reconnection. Methods. Using Lagrangian markers to trace representative field lines, we follow the spatiotemporal evolution of the flux rope. By focusing on current volumes (which we call current sheets) between these lines, we identify flux bundles and associated reconnecting field line pairs. We also analyze the time-varying distribution function for the force-free parameter as the flux rope relaxes. Lastly, we compare different seeding methods for tracing magnetic field lines, and discuss their relevance to the analysis. Results. We show that the modeled flux rope is gradually built from the coalescence of numerous current-carrying flux tubes. This occurs through a series of component reconnections that are continuously driven by the complex flows in the underlying convection zone. These reconnections lead to an inverse cascade of helicity from small scales to larger scales. We also find that the system attempts to relax toward a linear force-free field, but that the convective drivers and the nanoflare event prevent full Taylor relaxation. Conclusions. Using a self-consistently driven simulation of a nanoflare event, we show for the first time an inverse helicity cascade tending toward a Taylor relaxation in the Sun’s corona, resulting in a well-ordered flux rope that later reconnects with surrounding fields. This provides context clues toward understanding the buildup of nanoflare events in the quiet Sun through incomplete Taylor relaxations, when no relevant flux emergence or cancellation is observed.

A force-free magnetic field solution in toroidal coordinates

While there are no known analytic solutions for force-free fields in toroidal coordinates, with a reasonable boundary condition, it is possible to find a solution for the surface field and, with a restriction on the form of the field, the interior of the torus as well.

A Large-Scale Dataset of Three-Dimensional Solar Magnetic Fields Extrapolated by Nonlinear Force-Free Method

It has been widely accepted that solar magnetic field manipulates all solar activities, especially violent solar bursts in solar corona. Thus, it is extremely important to reconstruct three-dimentional (3D) magnetic field of solar corona from really observed photospheric magnetogram. In this paper, a large-scale dataset of 3D solar magnetic fields of active regions is built by using the nonlinear force-free magnetic field (NLFFF) extrapolation from vector magnetograms of Helioseismic and Magnetic Imager (HMI) on Solar Dynamics Observatory (SDO). In this dataset, all space-weather HMI active region patches (SHARPs) with the corresponding serial numbers of national oceanic and atmospheric administration (NOAA) are included. They are downloaded from the SHARP 720 s series of JSOC every 96 minutes. In addition, each sample is labelled with a finer grained label for solar flare forecast. This paper is with the purpose of open availability of data resource and source code to the peers for refraining from repeated labor of data preparation. Meanwhile, with such a large-scale, high spatio-temporal resolution and high quality scientific data, we anticipate a wide attention and interest from artificial intelligence (AI) and computer vision communities, for exploring AI for astronomy over such a large-scale dataset.

The Initiation and Back-reaction of the X5.4 Flare on 2012 March 7

In this paper, we study the evolution of the X5.4 flare (SOL2012-03-07T00:02) in NOAA Active Region 11429, focusing on its initiation mechanisms and back-reaction effects. To help our study, three-dimensional (3D) coronal magnetic field models are extrapolated from the photospheric magnetograms of the Helioseismic and Magnetic Imager on board the Solar Dynamics Observatory under the assumptions of nonlinear force-free field (NLFFF) and non-force-free field (non-FFF). We investigate the 3D magnetic structure and MHD kink instability, torus instability, and double-arc instability (DAI), and find that this flare is most likely triggered by the tether-cutting reconnection and the subsequent DAI. For the back-reactions of the flare, both NLFFF and non-FFF models clearly show an increase in horizontal magnetic field (B h ) and a decrease in inclination angle (ϕ) of the magnetic field near the polarity inversion line, from the photosphere up to a certain height (5 Mm and 8 Mm for non-FFF and NLFFF, respectively). In addition, the non-FFF model shows an enhancement of the downward Lorentz force acting on the photosphere, and the location of the enhancement spatially coincides with the location of the flare onset. The observed back-reaction is likely a consequence of magnetic reconnection.

Evolution of coronal magnetic field parameters during X5.4 solar flare

The coronal magnetic field over NOAA Active Region 11,429 during a X5.4 solar flare on 7 March 2012 is modeled using optimization based Non-Linear Force-Free Field extrapolation. Specifically, 3D magnetic fields were modeled for 11 timesteps using the 12-min cadence Solar Dynamics Observatory (SDO) Helioseismic and Magnetic Imager photospheric vector magnetic field data, spanning a time period of 1 hour before through 1 hour after the start of the flare. Using the modeled coronal magnetic field data, seven different magnetic field parameters were calculated for 3 separate regions: areas with surface |Bz|≥ 300 G, areas of flare brightening seen in SDO Atmospheric Imaging Assembly imagery, and areas with surface |B| ≥ 1000 G and high twist. Time series of the magnetic field parameters were analyzed to investigate the evolution of the coronal field during the solar flare event and discern pre-eruptive signatures. The data shows that areas with |B| ≥ 1000 G and |Tw|≥ 1.5 align well with areas of initial flare brightening during the pre-flare phase and at the beginning of the eruptive phase of the flare, suggesting that measurements of the photospheric magnetic field strength and twist can be used to predict the flare location within an active region if triggered. Additionally, the evolution of seven investigated magnetic field parameters indicated a destabilizing magnetic field structure that could likely erupt.

A Comparative Study of Solar Active Region 12371 with Data-constrained and Data-driven Magnetohydrodynamic Simulations

We performed two data-based magnetohydrodynamic (MHD) simulations for solar active region 12371, which produced an M6.5 flare. The first simulation is a full data-driven simulation where the initial condition is given by a nonlinear force-free field (NLFFF). This NLFFF was extrapolated from photospheric magnetograms approximately 1 hr prior to the flare, and then a time-varying photospheric magnetic field is imposed at the bottom surface. The second simulation is also a data-driven simulation, but it stops driving at the bottom before the time of flare onset and then switches to the data-constrained simulation, where the horizontal component of the magnetic field varies according to an induction equation, while the normal component is fixed with time. Both simulations lead to an eruption, with both simulations producing highly twisted field lines before the eruption, which were not found in the NLFFF alone. After the eruption, the first simulation based on the time-varying photospheric magnetic field continues to produce sheared field lines after the flare without reproducing phenomena such as postflare loops. The second simulation reproduces the phenomena associated with flares well. However, in this case, the evolution of the bottom magnetic field is inconsistent with the evolution of the observed magnetic field. In this Letter, we report potential advantages and disadvantages in data-constrained and data-driven MHD simulations that need to be taken into consideration in future studies.

Data-constrained MHD Simulation of a Multi-ribbon Flare Corresponding to a Successful and a Confined Eruption

The formation and eruption mechanisms of multi-ribbon flares are extremely complicated, especially when the flare is associated with homologous eruptions in the same region. In this paper, we investigate such an event, corresponding to a successful eruption and a confined eruption. This is an M7.1 flare, starting at 12:33 UT on 2011 September 24 in active region NOAA 11302. We obtain the coronal magnetic configuration for this region, using a nonlinear force-free field extrapolation based on the photospheric magnetogram at 12:00 UT. Taking this as the initial condition, we perform a data-constrained MHD simulation to study the evolution of the magnetic topology for this region. We analyze the magnetic null points and the 3D squashing degree for this region, indicating the existence of three flux ropes and two spine–fan structures. The model reproduces the rising processes of the two flux ropes, which form two homologous eruptions consistent with the observations as shown in 94 Å: a large-scale successful eruption that is followed by a small-scale confined eruption. By analyzing the magnetic configuration, the Lorentz force, and the decay index, we find that the torus instability plays an important role in driving the successful eruption of the large flux rope. The magnetic reconnection above the medium flux rope changes the direction of the overlying magnetic field, which provides a downward component of the Lorentz force to confine the eruption of the medium flux rope.

Topological nature of the Parker magnetostatic theorem

The two-plate initial boundary-value problem of Parker is reviewed, treating the relaxation of a 3D magnetic field prescribed with an arbitrary topology to a terminal force-free field in a cold, viscous, electrically perfect fluid conductor. Anchored by their foot-points at the perfectly conducting rigid plates, the relaxing field preserves its topology. The Parker magnetostatic theorem states that for most prescribed field topologies, the terminal field must embed current sheets. The elements of this theorem are reviewed and analyzed to relate this initial boundary-value problem to (i) the variational problem for a force-free field of a given topology and (ii) the direct construction of a force-free field in terms of its pair of Euler flux functions. New insights and understanding are presented on the theorem as the compelling basis of the Parker theory of solar coronal heating.

Tracking magnetic flux and helicity from the Sun to Earth

Aims.We analyze the complete chain of effects – from the Sun to Earth – caused by a solar eruptive event in order to better understand the dynamic evolution of magnetic-field-related quantities in interplanetary space, in particular that of magnetic flux and helicity.Methods.We study a series of connected events – a confined C4.5 flare, a flare-less filament eruption, and a double-peak M-class flare – that originated in NOAA active region (AR) 12891 on late 2021 November 1 and early 2021 November 2. We deduce the magnetic structure of AR 12891 using stereoscopy and nonlinear force-free (NLFF) magnetic field modeling, allowing us to identify a coronal flux rope and to estimate its axial flux and helicity. Additionally, we compute reconnection fluxes based on flare ribbon and coronal dimming signatures from remote sensing imagery. Comparison to corresponding quantities for the associated magnetic cloud (MC) deduced from in situ measurements from Solar Orbiter and near-Earth spacecraft allows us to draw conclusions on the evolution of the associated interplanetary coronal mass ejection (CME). The latter analysis is aided by the application of geometric fitting techniques (graduated cylindrical shell modeling; GCS) and interplanetary propagation models (drag-based ensemble modeling; DBEM) to the interplanetary CME.Results.NLFF modeling suggests the magnetic structure of the host AR was in the form of a left-handed (negative-helicity) flux rope reaching altitudes of 8−10 Mm above photospheric levels, which is in close agreement with the corresponding stereoscopic estimate. GCS and DBEM modeling suggest that the ejected flux rope propagated in a self-similar expanding manner through interplanetary space. Comparison of magnetic fluxes and helicities processed by magnetic reconnection in the solar source region and the respective budgets of the MC indicate a considerable contribution from the eruptive process, though the pre-eruptive budgets also appear to be relevant.

Changes of Magnetic Energy and Helicity in Solar Active Regions from Major Flares

Magnetic free energy powers solar flares and coronal mass ejections, and the buildup of magnetic helicity might play a role in the development of unstable structures that subsequently erupt. To better understand the roles of energy and helicity in large flares and eruptions, we have characterized the evolution of magnetic energy and helicity associated with 21 X-class flares from 2010 to 2017. Our sample includes both confined and eruptive events, with 6 and 15 in each category, respectively. Using the Helioseismic and Magnetic Imager vector magnetic field observations from several hours before to several hours after each event, we employ (a) the Differential Affine Velocity Estimator for Vector Magnetograms to determine the photospheric fluxes of energy and helicity, and (b) nonlinear force-free field extrapolations to estimate the coronal content of energy and helicity in source-region fields. Using superposed epoch analysis, we find, on average, the following: (1) decreases in both magnetic energy and helicity, in both photospheric fluxes and coronal content, that persist for a few hours after eruptions, but no clear changes, notably in relative helicity, for confined events; (2) significant increases in the twist of photospheric fields in eruptive events, with twist uncertainties too large in confined events to constrain twist changes (and lower overall twist in confined events); and (3) on longer timescales (event time +12 hr), replenishment of free magnetic energy and helicity content to near preevent levels for eruptive events. For eruptive events, magnetic helicity and free energy in coronal models clearly decrease after flares, with the amounts of decrease proportional to each region’s pre-flare content.

Sympathetic Partial Filament Eruptions Caused by the Interaction between Two Nearby Filaments

To better understand the physical connections in sympathetic solar eruptions, we investigated the interaction between two nearby filaments and their successive partial eruptions in the active region (AR) NOAA 12866 on 2021 September 9 by using data from the Solar Dynamics Observatory and the New Vacuum Solar Telescope. Based on Hα and extreme ultraviolet observations, we found that the right part of one filament (F1) became active first and experienced an obvious rolling motion. Then the whole body of the filament became wider and expanded toward another filament (F2). They collided with each other, and the interaction between them was accompanied by the brightening and bidirectional flows that appeared between them. This implies that magnetic reconnection occurred between the threads of two filaments. The interaction resulted in a rightward motion of F2 at first, and then its activation, and finally part of it erupted. Furthermore, when the erupted F2 deflected rapidly toward the middle part of F1, the left part of F1 erupted with its overlying magnetic fields pushed by F2. These observational results imply that these successive eruptions within a short time are physically linked, and this was caused by the interaction of the filaments. Nonlinear force-free field extrapolation reveals that the magnetic structure of the filament F1 was composed of several magnetic flux ropes with different twists. These results further advance our understanding of partial filament eruptions and sympathetic solar eruptions.

Formation and Eruption of Hot Channels during an M6.5 Class Solar Flare

We investigate the formation and eruption of hot channels associated with the M6.5 class flare (SOL 2015-06-22T18:23) that occurred in NOAA AR 12371 on 2015 June 22. Two flare precursors are observed before the flare’s main phase. Observations in 94 and 131 Å by SDO/AIA revealed the early morphology of the first hot channel as a group of hot loops, termed the seed hot channel. A few seed hot channels are formed above the polarity inversion line (PIL) and the formation is associated with the parallel motion of the footpoint brightenings along the PIL, which proceeds into the early stage of the flare main phase. During this process, seed hot channels build up and rise slowly, accelerating at the peak of the second precursor. They merge during the process of acceleration, forming a larger hot channel, which then forms an inverted-γ-shape kinking structure. Before the flare peak, the second kinking hot channel with negative crossing appears near the first kinking hot channel that erupted. The eruption of these two hot channels produces two peaks on the main flare's GOES light curve. The propagation of the footpoint brightenings along the PIL indicates that the first kinking hot channel may be formed due to zipper reconnection. The occurrence of merging between seed hot channels observed by AIA is supported by the extrapolated nonlinear force-free field models. The observed writhing motion of the first kinking hot channel may be driven by the Lorentz force.

A Data-constrained Magnetohydrodynamic Simulation of the X1.0 Solar Flare of 2021 October 28

The solar active region NOAA 12887 produced a strong X1.0 flare on 2021 October 28, which exhibits X-shaped flare ribbons and a circle-shaped erupting filament. To understand the eruption process with these characteristics, we conducted a data-constrained magnetohydrodynamics simulation using a nonlinear force-free field of the active region about an hour before the flare as the initial condition. Our simulation reproduces the filament eruption observed in the Hα images of GONG and the 304 Å images of SDO/AIA, and suggests that two mechanisms can possibly contribute to the magnetic eruption. One is the torus instability of the preexisting magnetic flux rope (MFR) and the other is upward pushing by magnetic loops newly formed below the MFR via continuous magnetic reconnection between two sheared magnetic arcades. The presence of this reconnection is evidenced by the SDO/AIA observations of the 1600 Å brightening in the footpoints of the sheared arcades at the flare onset. To clarify which process is more essential for the eruption, we performed an experimental simulation in which the reconnection between the sheared field lines is suppressed. In this case too, the MFR could erupt, but at a much reduced rising speed. We interpret this result as indicating that the eruption is not only driven by the torus instability, but additionally accelerated by newly formed and rising magnetic loops under continuous reconnection.

Modelling the magnetic structure of a large-scale horse-shoe-like filament in a decaying and diffuse active region

ABSTRACTA large-scale, horse-shoe-like filament was investigated and the magnetic field around it was reconstructed. This is an intermediate filament (IF) that appeared on the solar disc for the first time at 02:00 ut on 2015 November 7, and took 8 d to move to the central median on the solar disc. The active region AR 12452 around which the filament occurred was diffuse so that the magnetic field nearby was weak, the average field strength is 106 G. Therefore, the existing approaches to extrapolating the coronal magnetic field and to constructing the filament configuration in the region with strong background field do not work well here. On the basis of the regularized Biot–Savart laws method, we successfully constructed a data-constrained, non-linear force-free field configuration for this IF observed on 2015 November 14. The overall IF configuration obtained in this way matches well the morphology suggested by a 304 Å image taken by the Atmospheric Imaging Assembly on board Solar Dynamics Observatory. Magnetic dips in the configuration were coincident in space with the H α features of the filament, which is lower in altitude than the features seen in 304 Å. This suggests that the cold plasma fills the lower part of the filament, and hot plasma is situated in the higher region. A quasi-separatrix layer wraps the filament, and both the magnetic field and the electric current are stronger near the inner edge of the filament.

Topological Evolution of an Unwinding Blowout Jet

We investigate the topological evolution of coronal jet containing a sigmoid-like flux rope using a nonlinear force-free field model obtained with the flux-rope insertion method and magnetofrictional relaxation. We examine the topological evolution predicted by the unstable model with respect to observations from Solar Dynamic Observatory’s Atmospheric Imaging Array. We also calculate the squeezing factor, an approximation for sharp discontinuities in the magnetic field, and the coiling rate, an approximation for the amount of twist in the field. We identify at least two topological features where magnetic reconnection is likely taking place: an internal anemone-like region, near the filament, and an external region between the closed dome of the coronal jet and the ambient field. We also find evidence of reconnection below the filament, but it is not clear if the two inner regions are the same. We find that the internal region inflates the jet dome into the external region, which in turn initiates the fast eruption, allowing the inner region to unwind and the filament to escape. Finally, we examine the thermal evolution of the jet and trace the regions of enhanced emission-measure-weighted temperature (T EM) to the location of the expected reconnection regions. We find that magnetic field lines associated with the internal reconnection region are tied to increased T EM and emission in extreme-UV observations, indicative of heating. We identify this eruption as an untwisting jet, where unwinding magnetic field lines impart energy along the magnetic field forming the observed features of the coronal jet.

Sigmoid Formation through Slippage of a Single J-shaped Coronal Loop

A well-known precursor of an imminent solar eruption is the appearance of a hot S-shaped loop, also known as a sigmoid, in an active region (AR). Classically, the formation of such an S-shaped loop is envisaged to be implemented by magnetic reconnection of two oppositely oriented J-shaped loops. However, the details of reconnection are elusive due to weak emission and subtle evolution during the preeruptive phase. In this paper, we investigate how a single J-shaped loop transforms into an S-shaped one through the slippage of one of its footpoints in NOAA AR 11719 on 2013 April 11. During an interval of about 16 minutes, the J-shaped loop slips through a low-corona region of strong electric current density in a bursty fashion, reaching a peak apparent speed of the slipping footpoint as fast as 1000 km s−1 and over. The enhancement of electric current density, as suggested by nonlinear force-free field modeling, indicates that the “nonidealness” of coronal plasma becomes locally important, which may facilitate magnetic reconnection. The loop segment undergoing slipping motions is heated; meanwhile, above the fixed footpoint coronal emission dims due to a combination effect of the lengthening and heating of the loop; the latter of which is manifested in the temporal variation of dimming slope and of emission measure. These features together support an asymmetric scenario of sigmoid formation through slipping reconnection of a single J-shaped loop, which differs from the standard tether-cutting scenario involving a double J-shaped loop system.

Magnetic Helicity Evolution and Eruptive Activity in NOAA Active Region 11158

Coronal mass ejections are among the Sun’s most energetic activity events yet the physical mechanisms that lead to their occurrence are not yet fully understood. They can drive major space weather impacts at Earth, so knowing why and when these ejections will occur is required for accurate space weather forecasts. In this study we use a 4 day time series of a quantity known as the helicity ratio, ∣H J ∣/∣H V ∣ (helicity of the current-carrying part of the active region field to the total relative magnetic helicity within the volume), which has been computed from nonlinear force-free field extrapolations of NOAA active region 11158. We compare the evolution of ∣H J ∣/∣H V ∣ with the activity produced in the corona of the active region and show this ratio can be used to indicate when the active region is prone to eruption. This occurs when ∣H J ∣/∣H V ∣ exceeds a value of 0.1, as suggested by previous studies. We find the helicity ratio variations to be more pronounced during times of strong flux emergence, collision and reconnection between fields of different bipoles, shearing motions, and reconfiguration of the corona through failed and successful eruptions. When flux emergence, collision, and shearing motions have lessened, the changes in helicity ratio are somewhat subtle despite the occurrence of significant eruptive activity during this time.