Solar eruptions are the most powerful drivers of space weather. To understand their cause and nature, it is crucial to know how the coronal magnetic field evolves before eruption. Here we study the formation process of a relatively large-scale magnetic flux rope (MFR) in active region NOAA~12371 that erupts with a major flare and coronal mass ejection on 2015 June 21. A data-driven numerical magnetohydrodynamic model is employed to simulate three-dimensional coronal magnetic field evolution of one-day duration before the eruption. Comparison between the observed features and our modeled magnetic field discloses how the pre-eruption MFR forms. Initially, the magnetic field lines were weakly twisted as being simple sheared arcades. Then a long MFR was formed along the polarity inversion line due to the complex photospheric motion, which is mainly shearing rather than twisting. The presence of the MFR is evidenced by a coherent set of magnetic field lines with twist number above unity. Below the MFR a current sheet is shown in the model, suggesting that tether-cutting reconnection plays a key role in the MFR formation. The MFR's flux grows as more and more field lines are twisted due to continuous injection of magnetic helicity by the photospheric motions. Meanwhile, the height of the MFR's axis increases monotonely from its formation. By an analysis of the decay index of its overlying field, we suggest that it is because the MFR runs into the torus instability regime and becomes unstable that finally triggers the eruption.
We present results from 3D magnetohydrodynamic (MHD) simulations of the emergence of a twisted convection zone flux tube into a pre-existing coronal dipole field. As in previous simulations, following the partial emergence of the sub-surface flux into the corona, a combination of vortical motions and internal magnetic reconnection forms a coronal flux rope. Then, in the simulations presented here, external reconnection between the emerging field and the pre-existing dipole coronal field allows further expansion of the coronal flux rope into the corona. After sufficient expansion, internal reco…
We carry out a three-dimensional magneto-hydrodynamic (MHD) simulation to model the initiation of the coronal mass ejection (CME) on 13 December 2006 in the emerging δ-sunspot active region NOAA 10930. The setup of the simulation is similar to a previous simulation by Fan (2011), but with a significantly widened simulation domain to accommodate the wide CME. The simulation shows that the CME can result from the emergence of a east-west oriented twisted flux rope whose positive, following emerging pole corresponds to the observed positive rotating sunspot emerging against the southern edge of t…
Loss of equilibrium of magnetic flux ropes is a leading candidate for the origin of solar coronal mass ejections (CMEs). The aim of this paper is to explore to what extent this mechanism can account for the initiation of CMEs in the global context. A simplified MHD model for the global coronal magnetic field evolution in response to flux emergence and shearing by large-scale surface motions is described and motivated. Using automated algorithms for detecting flux ropes and ejections in the global magnetic model, the effects of key simulation parameters on the formation of flux ropes and the nu…
We performed magnetohydrodynamic simulation of a formation process of coronal mass ejections (CMEs), focusing on interaction (reconnection) between an ejecting flux rope and its ambient field. We examined three cases with different ambient fields: no ambient field, and cases with dipole field of two opposite directions which are parallel and anti-parallel to that of the flux rope surface. As a result, while the flux rope disappears in the anti-parallel case, in other cases the flux ropes can evolve to CMEs and show different amounts of rotation of the flux rope. The results imply that the inte…
Using three-dimensional magnetohydrodynamic (MHD) simulations, we investigate the eruption of coronal flux ropes underlying coronal streamers and the development of a prominence eruption. We initialize a quasi-steady solution of a coronal helmet streamer, into which we impose at the lower boundary the slow emergence of a part of a twisted magnetic torus. As a result a quasi-equilibrium flux rope is built up under the streamer. With varying sizes of the streamer and the different length and total twist of the emerged flux rope, we found different scenarios for the evolution from quasi-equilibri…
An eruption event launched from solar active region (AR) NOAA 11719 is investigated based on coronal EUV observations and photospheric magnetic field measurements obtained from Solar Dynamic Observatory. The AR consists of a filament channel originating from major sunspot and its south section is associated with inverse-S sigmoidal system as observed in AIA passbands. We regard the sigmoid as the main body of the flux rope (FR). There also exists a twisted flux bundle crossing over this FR. This overlying flux bundle transforms in shape similar to kink-rise evolution which has correspondence w…
The temporal evolution of a solar prominence inserted in a three-dimensional magnetic flux rope is investigated numerically. Using the model of Titov Demoulin (1999) under the regime of weak twist, the cold and dense prominence counteracts gravity by modifying the initially force-free magnetic configuration. In some cases a quasi-stationary situation is achieved after the relaxation phase, characterized by the excitation of standing vertical oscillations. These oscillations show a strong attenuation with time produced by the mechanism of continuum damping due to the inhomogeneous transition be…
Current magnetohydrodynamic (MHD) simulations of the initiation of solar eruptions are still commonly carried out with idealized magnetic field models, whereas the realistic coronal field prior to eruptions can possibly be reconstructed from the observable photospheric field. Using a nonlinear force-free field extrapolation prior to a sigmoid eruption in AR 11283 as the initial condition in a MHD model, we successfully simulate the realistic initiation process of the eruption event, as is confirmed by a remarkable resemblance to the SDO/AIA observations. Analysis of the pre-eruption field reve…
Flux rope formation through flux cancellation of sheared coronal arcades in a 3D convectively-driven MHD simulation
University of Oslo（奥斯陆大学）；Sorbonne Université, Observatoire de Paris-PSL, École Polytechnique, IP Paris, CNRS, Laboratoire de Physique des Plasmas（索邦大学，巴黎天文台-PSL，巴黎综合理工学院，IP巴黎，法国国家科学研究中心，等离子体物理实验室）
Large-scale solar eruptive activities have a close relationship with coronal magnetic flux ropes. Previous numerical studies have found that the equilibrium of a coronal flux rope system could be disrupted if the axial magnetic flux of the rope exceeds a critical value, so that the catastrophe occurs, initiating the flux rope to erupt. Further studies discovered that the catastrophe does not necessarily exist: the flux rope system with certain photospheric flux distributions could be non-catastrophic. It is noteworthy that most previous numerical studies are under the ideal magnetohydrodynamic…
We study a filament eruption, two-ribbon flare, and coronal mass ejection (CME) that occurred in Active Region NOAA 10898 on 6 July 2006. The filament was located South of a strong sunspot that dominated the region. In the evolution leading up to the eruption, and for some time after it, a counter-clockwise rotation of the sunspot of about 30 degrees was observed. We suggest that the rotation triggered the eruption by progressively expanding the magnetic field above the filament. To test this scenario, we study the effect of twisting the initially potential field overlying a pre-existing flux-…
We carry out magnetohydrodynamic (MHD) simulations of the quasi-static evolution and eruption of a twisted coronal flux rope under a coronal streamer built up by an imposed flux emergence at the lower boundary. The MHD model incorporates a simple empirical coronal heating, optically thin radiative cooling, and field aligned thermal conduction, and thus allows the formation of prominence condensations. We find that during the quasi-static evolution, prominence/filament condensations of an elongated, sigmoid morphology form in the dips of the significantly twisted field lines of the emerged flux…
Solar prominences are subject to all kinds of perturbations during their lifetime, and frequently demonstrate oscillations. The study of prominence oscillations provides an alternative way to investigate their internal magnetic and thermal structures as the oscillation characteristics depend on their interplay with the solar corona. Prominence oscillations can be classified into longitudinal and transverse types. We perform three-dimensional ideal magnetohydrodynamic simulations of prominence oscillations along a magnetic flux rope, with the aim to compare the oscillation periods with those pr…
Large-scale solar eruptions are believed to have a magnetic flux rope as the core structure. However, it remains elusive as to how the flux rope builds up and what triggers its eruption. Recent observations found that a prominence erupted following multiple episodes of "flux feeding". During each episode, a chromospheric fibril rose and merged with the prominence lying above. In this letter, we carried out 2.5-dimensional magnetohydrodynamic (MHD) numerical simulations to investigate whether the flux-feeding mechanism can explain such an eruption. The simulations demonstrate that the discrete …
The aim of this work is to investigate and characterise non-thermal particle behaviour in a three-dimensional (3D) magnetohydrodynamical (MHD) model of unstable multi-threaded flaring coronal loops. We have used a numerical scheme which solves the relativistic guiding centre approximation to study the motion of electrons and protons. The scheme uses snapshots from high resolution numerical MHD simulations of coronal loops containing two threads, where a single thread becomes unstable and (in one case) destabilises and merges with an additional thread. The particle responses to the reconnection…
We present a 3D MHD simulation that qualitatively models the coronal mag- netic field evolution associated with the eruptive flare that occurred on December 13, 2006 in the emerging δ-sunspot region NOAA 10930 observed by the Hinode satellite. The simulation is set up where we drive the emergence of an east-west oriented magnetic flux rope at the lower boundary into a pre-existing coronal field constructed from the SOHO/MDI full-disk magnetogram at 20:51:01 UT on December 12, 2006. The resulting coronal flux rope embedded in the ambi- ent coronal magnetic field first settles into a stage of qu…
Multi-instrument observations of two filament eruptions on 24 February and 11 May 2011 suggest the following updated scenario for eruptive flare, CME and shock wave evolution. An initial destabilization of a filament results in stretching out of magnetic threads belonging to its body and rooted in the photosphere along the inversion line. Their reconnection leads to i) heating of parts of the filament or its environment, ii) initial development of the flare arcade cusp and ribbons, and iii) increasing similarity of the filament to a curved flux rope and its acceleration. Then the pre-eruption …
We present a comprehensive analysis of the three-dimensional magnetic flux rope structure generated during the Lynch et al. (2019) magnetohydrodynamic (MHD) simulation of a global-scale, 360 degree-wide streamer blowout coronal mass ejection (CME) eruption. We create both fixed and moving synthetic spacecraft to generate time series of the MHD variables through different regions of the flux rope CME. Our moving spacecraft trajectories are derived from the spatial coordinates of Parker Solar Probe's past encounters 7 and 9 and future encounter 23. Each synthetic time series through the simulati…
In this paper, we analyze the full evolution, from a few days prior to the eruption to the initiation, and the final acceleration and propagation, of the CME that occurred on 2008 April 26 using the unprecedented high cadence and multi-wavelength observations by STEREO. There existed frequent filament activities and EUV jets prior to the CME eruption for a few days. These activities were probably caused by the magnetic reconnection in the lower atmosphere driven by photospheric convergence motions, which were evident in the sequence of magnetogram images from MDI (Michelson Doppler Imager) onb…
In the solar corona, magnetic flux rope is believed to be a fundamental structure accounts for magnetic free energy storage and solar eruptions. Up to the present, the extrapolation of magnetic field from boundary data is the primary way to obtain fully three-dimensional magnetic information of the corona. As a result, the ability of reliable recovering coronal magnetic flux rope is important for coronal field extrapolation. In this paper, our coronal field extrapolation code (CESE-MHD-NLFFF, Jiang & Feng 2012) is examined with an analytical magnetic flux rope model proposed by Titov & Demouli…
Data-driven Magnetohydrodynamic Simulation of the Initiation of a Coronal Mass Ejection with Multiple Stages
KU Leuven（荷语鲁汶大学）；Nanjing University（南京大学）；University of Maria Curie-Skłodowska（玛丽亚·居里-斯克沃多夫斯卡大学）；Observatoire de Paris（巴黎天文台）；CNRS（法国国家科学研究中心）；UPMC（皮埃尔和玛丽·居里大学）；Université Paris Diderot（巴黎第七大学）；LUNEX EMMESI Institut（卢森堡EMMESI研究所）
In this paper, we produce synthetic radio views on simulated flux ropes in the solar corona, where finite-beta magnetohydrodynamic (MHD) simulations serve to mimic the flux rope formation stages, as well as their stable endstates. These endstates represent twisted flux ropes where balancing Lorentz forces, gravity and pressure gradients determine the full thermodynamic variation throughout the flux rope. The obtained models are needed to quantify radiative transfer in radio bands, and allow us to contrast weak to strong magnetic field conditions. Field strengths of up to 100 G in the flux rope…
In recent years, space weather research has focused on developing modelling techniques to predict the arrival time and properties of coronal mass ejections (CMEs) at the Earth. The aim of this paper is to propose a new modelling technique suitable for the next generation of Space Weather predictive tools that is both efficient and accurate. The aim of the new approach is to provide interplanetary space weather forecasting models with accurate time dependent boundary conditions of erupting magnetic flux ropes in the upper solar corona. To produce boundary conditions, we couple two different mod…
We present magnetohydrodynamic simulation of the evolution from quasi-equilibrium to onset of eruption of a twisted, prominence-forming coronal magnetic flux rope underlying a corona streamer. The flux rope is built up by an imposed flux emergence at the lower boundary. During the quasi-static phase of the evolution, we find the formation of a prominence-cavity system with qualitative features resembling observations, as shown by the synthetic SDO/AIA EUV images with the flux rope observed above the limb viewed nearly along its axis. The cavity contains substructures including ``U''-shaped or …
Solar eruptions are spectacular magnetic explosions in the Sun's corona, and how they are initiated remains unclear. Prevailing theories often rely on special magnetic topologies that may not generally exist in the pre-eruption source region of corona. Here, using fully three-dimensional magnetohydrodynamic simulations with high accuracy, we show that solar eruptions can be initiated in a single bipolar configuration with no additional special topology. Through photospheric shearing motion alone, an electric current sheet forms in the highly sheared core field of the magnetic arcade during its…
In this paper, we present multiwavelength observations of helical kink instability as a trigger of a CME which occurred in AR NOAA 11163 on 24 February 2011. The CME was associated with a M3.5 limb flare. High resolution observations from SDO/AIA suggest the development of helical kink instability in the erupting prominence, which implies a flux rope structure of the magnetic field. A brightening starts below the apex of the prominence with its slow rising motion (~100 km/s) during the activation phase. A bright structure, indicative of a helix with ~3-4 turns, was transiently formed at this p…
The combination of magnetohydrodynamic (MHD) simulation and multi-wavelength observations is an effective way to study mechanisms of magnetic flux rope eruption. We develop a data-driven MHD model using the zero-$β$ approximation. The initial condition is provided by nonlinear force-free field derived by the magneto-frictional method based on vector magnetic field observed by the Helioseismic and Magnetic Imager (HMI) aboard the Solar Dynamics Observatory (SDO). The bottom boundary uses observed time series of the vector magnetic field and the vector velocity derived by the Differential Affine…
Current coronal mass ejection (CME) models set their lower boundary to be in the lower corona. They do not calculate accurately the transfer of free magnetic energy from the convection zone to the magnetically dominated corona because they model the effects of flux emergence using kinematic boundary conditions or simply assume the appearance of flux at these heights. We test the importance of including dynamical flux emergence in CME modeling by simulating, in 2.5D, the emergence of sub-surface flux tubes into different coronal magnetic field configurations. We investigate how much free magnet…
We review and discuss insights on ideal magnetohydrodynamic (MHD) instabilities that can play a role in destabilizing solar coronal flux rope structures. For single flux ropes, failed or actual eruptions may result from internal or external kink evolutions, or from torus unstable configurations. We highlight recent findings from 3D magnetic field reconstructions and simulations where kink and torus instabilities play a prominent role. For interacting current systems, we critically discuss different routes to coronal dynamics and global eruptions, due to current channel coalescence or to tilt-k…
Streamer blowout (SBO) coronal mass ejections (CMEs) represent a particular class of CMEs that are characterized by a gradual swelling of the overlying streamer and a slow CME containing a flux-rope structure. SBO CMEs arising from the streamer flank fall into a special category of SBO CMEs involving three lower arches under the higher streamer arcade. However, the initiation mechanism for this special category of SBO CMEs remains elusive, due to the observational limitations. Here we report critical observations of a SBO CME associated with the eruption of a polar crown filament that originat…
On SOL2017-09-06 solar active region 12673 produced an X9.3 flare which is regarded as largest to occur in solar cycle 24. In this work we have preformed a magnetohydrodynamic (MHD) simulation in order to reveal the three-dimensional (3D) dynamics of the magnetic fields associated with the X9.3 solar flare. We first performed an extrapolation of the 3D magnetic field based on the observed photospheric magnetic field prior to the flare and then used it as the initial condition for an MHD simulation. Consequently, the simulation showed a dramatic eruption. In particular, we found that a large co…
We intend to provide a comprehensive answer to the question on whether all Coronal Mass Ejections (CMEs) have flux rope structure. To achieve this, we present a synthesis of the LASCO CME observations over the last sixteen years, assisted by 3D MHD simulations of the breakout model, EUV and coronagraphic observations from STEREO and SDO, and statistics from a revised LASCO CME database. We argue that the bright loop often seen as the CME leading edge is the result of pileup at the boundary of the erupting flux rope irrespective of whether a cavity or, more generally, a 3-part CME can be identi…
Coronal mass ejections (CMEs) are large-scale eruptions of plasma from the coronae of stars. Understanding the plasma processes involved in CME initiation has applications to space weather forecasting and laboratory plasma experiments. James et al. (Sol. Phys. 292, 71, 2017) used EUV observations to conclude that a magnetic flux rope formed in the solar corona above NOAA Active Region 11504 before it erupted on 14 June 2012 (SOL2012-06-14). In this work, we use data from the Solar Dynamics Observatory to model the coronal magnetic field of the active region one hour prior to eruption using a n…
Using a 3D MHD simulation, we model the quasi-static evolution and the onset of eruption of a coronal flux rope. The simulation begins with a twisted flux rope emerging at the lower boundary and pushing into a pre-existing coronal potential arcade field. At a chosen time the emergence is stopped with the lower boundary taken to be rigid. Then the coronal flux rope settles into a quasi-static rise phase during which an underlying, central sigmoid-shaped current layer forms along the so called hyperbolic flux tube (HFT), a generalization of the X-line configuration. Reconnections in the dissipat…
Solar flares and coronal mass ejections (CMEs) are the most powerful explosions in the Sun. They are major sources of potentially destructive space weather conditions. However, the possible causes of their initiation remain controversial. By using high resolution data observed by NST of BBSO, supplemented by Solar Dynamics Observatory (SDO) observations, we present unusual observations of a small-scale emerging flux rope near a large sunspot, whose eruption produced an M-class flare and a coronal mass ejection. The presence of the small-scale flux rope was indicated by static nonlinear force f…
The separation of a filament and sigmoid is observed during an X1.4 flare on July 12, 2012 in solar active region 11520, but the corresponding magnetic field change is not clear. We construct a data-constrained magnetohydrodynamic simulation of the filament-sigmoid system with the flux rope insertion method and magnetic flux eruption code, which produces the magnetic field evolution that may explain the separation of the low-lying filament and high-lying hot channel (sigmoid). The initial state of the magnetic model contains a magnetic flux rope with a hyperbolic flux tube, a null point struct…
Solar active region 12673 produced two successive X-class flares (X2.2 and X9.3) approximately 3 hours apart in September 2017. The X9.3 was the recorded largest solar flare in Solar Cycle 24. In this study we perform a data-constrained magnetohydrodynamic simulation taking into account the observed photospheric magnetic field to reveal the initiation and dynamics of the X2.2 and X9.3 flares. According to our simulation, the X2.2 flare is first triggered by magnetic reconnection at a local site where at the photosphere the negative polarity intrudes into the opposite-polarity region. This magn…
From a magnetohydrodynamic (MHD) simulation of the eruption of a prominence hosting coronal flux rope, we carry out forward synthesis of the circular polarization signal (Stokes V signal) of the FeXIII emission line at 1074.7 nm produced by the MHD model as measured by the proposed COronal Solar Magnetism Observatory (COSMO) Large Coronagraph (LC) and infer the line-of-sight magnetic field BLOS above the limb. With an aperture of 150 cm, integration time of 12 min, and a resolution of 12 arcsec, the LC can measure a significant BLOS with sufficient signal to noise level, from the simulated flu…
We present a magnetohydrodynamic (MHD) simulation of the coronal mass ejection (CME) on 13 December 2006 in the emerging delta-sunspot active region 10930, improving upon a previous simulation by Fan (2016) as follows. (1) Incorporate an ambient solar wind instead of using a static potential magnetic field extrapolation as the initial state. (2) In addition to imposing the emergence of a twisted flux rope, also impose at the lower boundary a random electric field that represents the effect of turbulent convection, which drives field-line braiding and produces resistive and viscous heating in t…
A filament eruption was observed on October 31, 2010 in the images recorded by the Atmospheric Imaging Assembly (AIA) on board the Solar Dynamic Observatory (SDO) in its Extreme Ultra-Violet (EUV) channels. The filament showed a slow rise phase followed by a fast rise and was classified to be an asymmetric eruption. In addition, multiple localized brightening which was spatially and temporally associated with the slow rise phase were identified leading us to believe that the tether-cutting mechanism to be the cause of the initiation of the eruption. An associated flux rope was detected in high…
We investigate the formation times of eruptive magnetic flux ropes relative to the onset of solar eruptions, which is important for constraining models of coronal mass ejection (CME) initiation. We inspected uninterrupted sequences of 131 Å images that spanned more than eight hours and were obtained by the Atmospheric Imaging Assembly (AIA) on board the Solar Dynamics Observatory (SDO) to identify the formation times of hot flux ropes that erupted in CMEs from locations close to the limb. The appearance of the flux ropes as well as their evolution toward eruptions were determined using morphol…
Electron acceleration in the solar corona is often associated with flares and the eruption of twisted magnetic structures known as flux ropes. However, the locations and mechanisms of such particle acceleration during the flare and eruption are still subject to much investigation. Observing the exact sites of particle acceleration can help confirm how the flare and eruption are initiated and how they evolve. Here we use the Atmospheric Imaging Assembly to analyse a flare and erupting flux rope on 2014-April-18, while observations from the Nancay Radio Astronomy Facility allows us to diagnose t…
The chapter "Solar Active Region Electric Currents Before and During Eruptive Flares" is a discussion on electric currents in the pre-eruption state and in the course of eruptions of solar magnetic structures, using information from solar observations, nonlinear force-free field extrapolations relying on these observations, and three-dimensional magnetohydrodynamic (MHD) models. The discussion addresses the issue of neutralized vs. non-neutralized currents in active regions and concludes that MHD models are able to explain non-neutralized currents in active regions by the existence of strong m…
2.5D time-dependent ideal magnetohydrodynamic (MHD) models in Cartesian coordinates were used in previous studies to seek MHD equilibria involving a magnetic flux rope embedded in a bipolar, partially open background field. As demonstrated by these studies, the equilibrium solutions of the system are separated into two branches: the flux rope sticks to the photosphere for solutions at the lower branch but is suspended in the corona for those at the upper branch. Moreover, a solution originally at the lower branch jumps to the upper, as the related control parameter increases and reaches a crit…
We present the multi-wavelength observations of a flux rope that was trying to erupt from NOAA AR 11045 and the associated M-class solar flare on 12 February 2010 using space and ground based observations from TRACE, STEREO, SOHO/MDI, Hinode/XRT and BBSO. While the flux rope was rising from the active region, an M1.1/2F class flare was triggered nearby one of its footpoints. We suggest that the flare triggering was due to the reconnection of a rising flux rope with the surrounding low-lying magnetic loops. The flux rope reached a projected height of ~0.15 Rs with a speed of ~90 km/s while the …
Magnetohydrodynamics simulation of active region NOAA 11515 is performed to examine the initiation of the M5.6 flaring event that starts around 10:43 UT on 2012 July 2. The simulation is conducted using an extrapolated non-force-free magnetic field generated from the photospheric vector magnetogram of the active region as the initial magnetic field. The magnetic field shows the presence of a three-dimensional (3D) magnetic null with the corresponding dome overlying a filament and a low-lying magnetic flux rope, observed in 304~Å~ and 131~Å~ respectively. The simulated dynamics, triggered by th…
We present results from 3D visco-resistive magnetohydrodynamic (MHD) simulations of the emergence of a convection zone magnetic flux tube into a solar atmosphere containing a pre-existing dipole coronal field, which is orientated to minimize reconnection with the emerging field. We observe that the emergence process is capable of producing a coronal flux rope by the transfer of twist from the convection zone as found in previous simulations. We find that this flux rope is stable, with no evidence of a fast rise, and that its ultimate height in the corona is determined by the strength of the pr…
It is widely accepted that coronal magnetic flux ropes are the core structures of large-scale solar eruptive activities, which inflict dramatic impacts on the solar-terrestrial system. Previous studies have demonstrated that varying magnetic properties of a coronal flux rope system could result in a catastrophe of the rope, which may trigger solar eruptive activities. Since the total mass of a flux rope also plays an important role in stabilizing the rope, we use 2.5-dimensional magnetohydrodynamic (MHD) numerical simulations in this letter to investigate how a flux rope evolves as its total m…
Coronal mass ejections (CMEs) are closely related to eruptive filaments and usually are the continuation of the same eruptive process into the upper corona. There are failed filament eruptions when a filament decelerates and stops at some greater height in the corona. Sometimes the filament after several hours starts to rise again and develops into the successful eruption with a CME formation. We propose a simple model for the interpretation of such two-step eruptions in terms of equilibrium of a flux rope in a two-scale ambient magnetic field. The eruption is caused by a slow decrease of the …
