Formation of an observed eruptive flux rope above the torus instability threshold through tether-cutting magnetic reconnection

Eruptive events such as solar flares and coronal mass ejections (CMEs) are a manifestation of an abrupt and explosive release of energy stored in twisted magnetic field lines in the solar corona (Antiochos et al., 1999; Fleishman et al., 2020). In a magnetically dominated corona, these transient events generally relax the coronal magnetic field by releasing the stored magnetic energy (Aschwanden, 2004). Observations suggest that before the eruption, since the coronal magnetic field lines (MFLs) are rooted in the photosphere, they become stressed and twisted by the photospheric (rotational and shear) flows. Consequently, they end up storing the magnetic energy (Priest & Forbes, 2002; Shibata & Magara, 2011; Priest, 2014). However, the physical process triggering the release of the stored energy still needs to be fully understood (Shibata & Magara, 2011; Priest, 2014).

The magnetic structure central to the eruptive events is thought to be a magnetic flux rope (MFR), which is a bundle of twisted magnetic field lines winding about a common axis (Chen, 2011; Priest, 2014; He et al., 2022). Under favourable conditions, the eruptions are believed to be initiated by the sudden rise of the flux ropes. In general, the rise can be attributed to either ideal magnetohydrodynamic (MHD) instabilities of the MFR, such as the kink or torus instability (Fan & Gibson, 2007; Kliem & Török, 2006; Démoulin & Aulanier, 2010), or it may be due to magnetic reconnection (MR); this is a process that changes the magnetic topology of MFLs and converts magnetic energy into heat, kinetic energy, and acceleration of the charged particles below the flux rope (Amari et al., 2003; Aulanier et al., 2012; Kumar et al., 2016). Therefore, it becomes imperative to understand the creation of the MFRs to understand solar eruptions. Typically, there are two standard scenarios proposed for MFR formation. In the first scenario, the flux rope is considered to be pre-existing below the photosphere and emerges into the corona during the flux emergence process (Chen & Shibata, 2000; Fan, 2001, 2010). In contrast, the second scenario suggests that the flux rope can be generated in the solar atmosphere through reconnections inside a sheared coronal arcade (van Ballegooijen & Martens, 1989; Moore et al., 2001; Amari et al., 2003; Aulanier et al., 2010; Jiang et al., 2021). These reconnection-based studies often rely on force-free fields and utilise prescribed flows at the bottom boundary to initiate MR. This paper also focuses on the process of flux rope formation through MR in a coronal arcade by simulating the dynamical evolution of a complex active region. However, in contrast to many previous works, here, the evolution is triggered by the Lorentz force (instead of the boundary flow) at the lower heights of the solar atmosphere.

The instability of an MFR plays an important role in the eruptive nature of flares. For explaining eruptions in bipolar coronal arcades, the “tether-cutting” model (Moore et al., 2001) attributed the formation and subsequent eruption of the flux rope to runaway tether-cutting reconnection that begins when shearing and converging photospheric flows have brought oppositely directed field lines into close-enough proximity. In this situation, ideal magnetohydrodynamic instabilities such as the “kink instability” (Török et al., 2004; Török & Kliem, 2005) and the “torus instability” (Kliem & Török, 2006; Aulanier et al., 2010) are also proposed to be causing the eruption. Recently, the modelling of eruptions has been further advanced using fully 3D MHD simulations (Aulanier et al., 2012; Janvier et al., 2013, 2014, 2015). Their model shows the eruption of a flux rope through torus instability and reconnection at the current sheet that develops under the flux rope. Relevant to this, “slip-running reconnection” has also been identified as a possible mechanism to trigger eruptive events (Aulanier et al., 2006; Janvier et al., 2013).

To construct an appropriate coronal magnetic field, the non-force-free-fields (NFFFs) extrapolation technique (Hu & Dasgupta, 2008; Hu et al., 2008, 2010) is utilised. As the name suggests, the extrapolated field supports a non-zero Lorentz force, which is crucial for the spontaneous onset of the eruption. Consequently, the NFFF extrapolations differ from the widely used nonlinear-force-free-fields (NLFFFs) extrapolations which correspond to the vanishing Lorentz force equilibrium state (e.g. Wiegelmann, 2008; Wiegelmann & Sakurai, 2012; Duan et al., 2017). Utilising the NFFF extrapolations, recent MHD simulations successfully replicated the required coronal dynamics that lead to solar flares, coronal jets, and coronal dimmings (Prasad et al., 2018; Nayak et al., 2019; Prasad et al., 2020; Kumar et al., 2022). In exploring the flux rope formation, we chose to numerically simulate the onset of the eruption of NOAA AR 12241, as the corresponding multi-wavelength observations suggest the origin of a flux rope over the polarity inversion line (PIL) (Joshi et al., 2017). In this paper, we perform an MHD simulation initiated with NFFF extrapolation of AR 12241 to study the magnetic flux rope formation and the progress of its eruption leading to the flare. Interestingly, we find that the rope is formed during the time of pre-flaring activity of the M6.9 flare on 2014 December 18, and the subsequent rise of the flux rope coincides with the development of the parallel ribbons of the flare (Joshi et al., 2017). We find that our simulation results attribute the flux rope formation to tether-cutting reconnection that occurs at a low-lying hyperbolic flux tube (HFT) by the suitable converging Lorentz force. Moreover, a slip-running reconnection (Aulanier et al., 2005, 2010) contributes to the extension of the flux rope along the PIL. The eruptive nature of the flux rope is attributed to two factors: the development of the flux rope in a torus unstable zone and the runaway tether-cutting reconnection.

The rest of the paper is arranged as follows: Section 2 briefly mentions the important aspects of the flaring event, while Section 3 provides details of the initial non-force-free extrapolated field. Section 4 presents the simulation results and their relation to the multi-wavelength observations. The key findings of the paper are summarised in Section 5.

The M6.9 flare occurred in NOAA AR 12241 on 2014 December 18. A detailed study of the various observational aspects of the flare was already published by Joshi et al. (2017), where these authors divided the flaring process into two components. The first component corresponds to the development of a flux rope over the PIL and its rise, leading to parallel flare ribbons. In the second component, the erupting flux rope reaches the three-dimensional (3D) magnetic null located at a much higher height and initiates MR at the null, forming the larger quasi-circular ribbon. This paper aims to simulate the dynamical evolution leading to the first component. For completeness, we first highlight the corresponding observations in Figure 1. The photospheric vector magnetogram of AR 12241 is from the Helioseismic Magnetic Imager (HMI; Schou et al., 2012) on board the Solar Dynamic Observatory (SDO; Pesnell et al., 2012) at 21:24 UT on 2014 December 18 (see Fig. 1(a)). The magnetogram is taken from the ‘hmi.sharp_cea_720s’ data series (Bobra et al., 2014) that provides vector magnetograms of the Sun with a temporal cadence of 12 minutes and a spatial resolution of $0^{\prime\prime}.5$. To obtain the magnetic field on a Cartesian grid, the magnetogram is initially remapped onto a Lambert cylindrical equal-area (CEA) projection and then transformed into heliographic coordinates (Gary & Hagyard, 1990). We further cropped the field of view to 680$\times$340 pixels to focus on the region of interest. The dark green line shows the PIL, along which the flux rope is observed to develop.

The flare starts at around 21:41 UT and peaks at $\approx$ 21:58 UT. Figures 1(b)–(d) illustrate SDO/AIA 131 Å images. To adequately compare the simulation results with observations, SDO/AIA filtergrams are also CEA projected and remapped to the same spatial resolution as the magnetic field data, while the same field of view is used. Panel (b) shows the formation of the flux rope (marked by red arrow) at around 21:35 UT, and its subsequent rise and expansion can be identified in panels (c) and (d). Similar flux rope dynamics are plotted in Figure 5 of Joshi et al. (2017). Notably, there are the brightenings moving in the eastward direction (marked by yellow arrows in panels (c) and (d)). Panels (e) and (f) present SDO/AIA 304 Å images, which depict the creation of parallel ribbons during the flare. In addition to the parallel ribbons, the plots also document the development of isolated brightenings (marked by green arrows) on the eastward and westward sides of the parallel ribbons. SDO/AIA 1600 Å images further confirm the development of the parallel ribbons and the isolated brightenings (panels (g) and (h)). Also notably, at low heights, a cool-material filament also exists along the PIL (Wang et al., 2017). However, its eruption during the flux rope formation and subsequent rise is not observationally established (Joshi et al., 2017). This suggests that its presence does not affect the flux rope dynamics.

We utilise the non-force free extrapolation model developed by Hu & Dasgupta (2008); Hu et al. (2008, 2010) to extrapolate the coronal magnetic field of AR 12241 at 21:24 UT corresponding to the vector magnetogram shown in Figure 1(a). For the details of the model, readers are referred to Hu et al. (2010) and references therein. The model is based on the principle of minimum energy dissipation rate, according to which a plasma system prefers to relax toward a state that has minimum dissipation rate (Bhattacharyya et al., 2007). The magnetic field ${\bf{B}}$ is then shown to satisfy the double-curl Beltrami equation (Bhattacharyya et al., 2007). The field ${\bf{B}}$ then can be written as the superposition of three linear-force-free fields (one of which is selected to be potential in the model) (Hu & Dasgupta, 2008). Consequently, the Lorentz force associated with the field is non-zero. An iterative approach based on the minimisation of the average deviation between the observed and the calculated transverse field on the photospheric boundary is then employed to get the extrapolated NFFF. Recently, the NFFF model has been successfully utilised to explain various transient events in active regions such as flares, coronal jets, and coronal dimmings (Prasad et al., 2018; Nayak et al., 2019; Prasad et al., 2020).

Notably, the magnetogram displayed in Figure 1(a) has a spatial extent of $680\times 340$ pixels along the $x$ and $y$ axes of a Cartesian coordinate system. To minimise the computational cost, the original domain is re-scaled to a $320\times 160$ pixel grid in $x$ and $y$ directions. The vertical extension of the domain is taken to be $160$ pixels. In such a re-scaling, the inherent magnetic structures remain preserved (Prasad et al., 2020). In physical units, the size of the computational domain is roughly $245$ Mm $\times$ $122.5$ Mm $\times$ $122.5$ Mm along $x$, $y,$ and $z$.

The magnetic field lines of the extrapolated initial field are plotted in Figure 2. In this and subsequent figure, the $x$, $y$, and $z$ axes of the computational domain are represented by red, green, and blue axes, respectively. The red line in the $z=0$ plane denotes the PIL. The yellow magnetic loops mark the magnetic arches of the arcade over the PIL, with the side and top views shown in panels (a) and (b). Large-scale magnetic loops are plotted in cyan-coloured MFLs. Notably, there is the bifurcation in the field lines towards the southeast direction, more visible in the top view in panel (b), showing a sharp change in the field line connectivity. Noticeably, the cyan-coloured MFLs leave the computational domain through the lateral boundary ($y=0$). We performed an additional extrapolation with an extended computational box along the lateral boundary (not shown) and found that these MFLs connect to the appropriate opposite polarity regions, validating the accuracy of the extrapolated field. Panel (c) of Figure 2 shows the direct volume rendering of the Lorentz force density, which reveals the existence of the force at the lower heights and its sharp decay with height. Moreover, the panel also identifies the presence of a strong Lorentz force in the vicinity of the PIL. Notably, the Lorentz force plays a crucial role in triggering the arcade dynamics, eventually leading to the MFR formation and the onset of the flare. Figure 2(d) plots the variations of horizontally averaged strength for the magnetic field, current density, and Lorentz force density with pixel height, $z$. The plots further confirm a faster decay of the Lorentz force density compared to the current density and the field strength. Consequently, the extrapolated field can be considered reasonably force-free at coronal heights, favouring the typical description of the solar corona.

This paper presents an MHD simulation of magnetic field evolution early in an M6.9 flare in AR 12241. The prime focus of the paper is to understand the formation of the magnetic flux rope during the initiation phase of the eruption that produced the flare. The initial magnetic field is generated by extrapolating the photospheric vector magnetogram of the active region obtained from HMI/SDO at 21:24 UT on 2014 December 18, using the non-force-free extrapolation technique. The extrapolated field has a Lorentz force at lower heights that becomes negligible in the corona, matching the standard picture of the coronal magnetic field. The Lorentz force, however, plays a crucial role in generating the self-consistent dynamical evolution from the initial static state.

The initial non-force-free extrapolated field shows the presence of a sheared coronal arcade enveloping the PIL. During the MHD evolution, under-resolved scales would evolve in the simulation as non-parallel MFLs come into close proximity with each other. The employed numerics then regularise these scales with simulated magnetic reconnections by producing locally adaptive residual dissipation. The simulated MFL evolution shows the origin of twisted field lines inside the initially sheared arcade at around 21:35 UT. The field lines represent a magnetic flux rope upon achieving a twist greater than one. The simulation demonstrates that magnetic reconnection at a low height in the solar atmosphere is responsible for the flux-rope formation and the M-class flare. The onset of that reconnection is attributed to developing favourable field line geometry in the form of a hyperbolic flux tube due to the suitable converging Lorentz force. Hence, the development of the flux rope is caused by tether-cutting reconnection at a low-lying HFT. In addition, the flux rope is formed in a region having a decay index $>1.5$. This suggests that the newly formed flux rope is unstable via the torus instability and, therefore, has an eruptive nature. After its formation, the flux rope exhibits a slow rising motion while staying in the torus-unstable region.

SDO/AIA multi-wavelength observations (in particular, the AIA 131 Å channel) of the M6.9 flare show the signature of the formation of a flux rope during the pre-flare stage. The initial formation location of the flux rope and its subsequent rise away from the photosphere matches the observations in 131 Å, indicating that magnetic reconnection plays a vital role in the pre-flare activity. After its formation, the flux rope shows a rising motion and an extension with time of its east end. In comparing our simulations with the SDO/AIA 304 Å and 1600 Å observations of the actual eruption, we find that our simulations closely fit the observed development of parallel flare ribbons and movement of brightenings with time towards the east. As noted above, the eastern feet of the MFLs of the flux rope show an eastward movement. This simulated movement coincides with the concurrent eastward expansion of brightenings in the 131 Å images. Interestingly, the simulation shows that the MFLs follow the high-value ln $Q$-contours; this suggests that slip-running reconnection is responsible for the motion and, hence, the brightenings. The generation of the post-reconnection arcade fits the parallel flare ribbons seen in the AIA 304 and 1600 Å images. Moreover, observable brightening also occurs near the eastern leg of the simulated flux rope, suggesting a causal connection between the flux rope and the brightenings.

Overall, the presented simulation successfully captures the important observational features of the flaring event onset in the form of the flux rope development, the rise of the flux rope, and the early-flare arcade. This alignment of our simulation’s results with observational data validates the initial assumption of our method, where we start with an unbalanced plasma state and allow the non-force-free field-induced Lorentz force to initiate the dynamical evolution of the active region. Moreover, the well-matched flare ribbons and post-flare loops, as depicted in our simulations, underscore the critical role of the Lorentz force in driving the reconnection processes leading to flare and eruption. On the flip side, the current simulation was not able to demonstrate the fast-rise dynamics of the erupting flux rope, as the rising motion of the flux rope appears to cease towards the end of the simulation. This indicates that the runaway tether-cutting reconnection is not efficient enough by itself to lift the flux rope further. In the absence of prescribed photospheric boundary flows, the inefficiency of the reconnection can be attributed to the constantly depleting magnetic energy through the magnetic reconnection and the continuously operating viscous dissipation. Moreover, the incompressibility condition assumed in the simulation may also play a role in restricting the faster rise of the flux rope. Relevantly, compressible MHD simulations driven by photospheric shearing flows or flux emergence have ascribed the eruption of flux ropes to the torus instability instead of the run-away tether-cutting reconnection (Aulanier et al., 2010; Fan, 2010). Therefore, the present simulation can be advanced further to simulate the eruption more realistically by relaxing the incompressibility and including the appropriate boundary flows, which are kept as future projects.