Solar eruptions are spectacular manifestation of explosive release of magnetic energy in the Sun’s atmosphere, i.e., the solar corona, and therefore, unveiling the relevant magnetic field structures and their evolution holds a central position in the study of solar eruptions. Magnetic flux rope (MFR), a bundle of twisted magnetic field lines winding around a common axis with the same sign, is believed to be a fundamental structure in solar eruptions [1–3], especially in those which successfully produce coronal mass ejections (CMEs). By reconstruction of the cross section of ICME (i.e., CMEs that evolves into the inter-planetary space) from the in-situ data obtained by satellites passing through the ICME, it has been well established that typical ICMEs have structure of highly twisted MFR (e.g., [4,5]).

Although there is little doubt that MFR constitutes the core structure of CMEs, whether MFR exists in the solar corona before CME initiation is still in intense debates [6,7]. Currently, there are two different opinions; one is that MFR does not exist before solar eruption, and it is the latter that creates MFR through magnetic reconnection; the other is that MFR should exist prior to eruption and it is the ideal magnetohydrodynamic (MHD) instability of the MFR that initiates the eruption. The typical scenarios for the first opinion include the runaway tether-cutting reconnection model [8–10] and the magnetic breakout model [11–14]. In these models, the coronal magnetic field before eruption is strongly sheared and eruption is triggered by magnetic reconnection, internally within the sheared arcade (i.e., tether-cutting), or above it (i.e., breakout), while MFR is built up during the eruption through reconnection which transforms the sheared arcade into the rope. For the models assuming the pre-existence of MFR, such as the catastrophe model [15,16], the torus instability and kink instability models [17–21], an MFR is proposed to either emerge from below the photosphere (i.e., in the convective zone, where the turbulent convection can create thin, twisted magnetic tubes [22,23]), or forms slowly by reconnection in the lower atmosphere [24] through the so-called flux cancellation process [25]. Observations seem to indicate that both opinions are possible. For example, on one hand, Song et al [26] presented a good observation that an MFR can formed during a CME. On the other hand, an MFR characterized by a hot sigmoid structure may pre-exist before eruption, as manifested by precursor oscillation [27] or precursor external magnetic reconnection between the top of the MFR and ambient magnetic field [28].

Regardless of which model is relevant to the real case in the corona, it is commonly agreed that flare reconnection (i.e., the main reconnection that occurs below the erupting MFR) can shape substantially the on-the-fly MFR. In the purely two-dimensional (2D) standard flare model, a plasmoid (corresponding to the cross section of MFR in 3D) rises from the top of the flare current sheet, and reconnection in the current sheet continuously adds poloidal flux to the MFR, which thus grows and expands during the eruption. As a result, the observed double flare ribbons, which indicate the locations of the reconnecting field-line footpoints in the opposite magnetic polarities, are continuously separated with each other. However, in a fully 3D case, it is not that straightforward how the reconnection shapes the MFR. Numerical simulations of the simplest magnetic configuration (i.e., a bipolar magnetic field), aided with accurate analysis of magnetic topology evolution, have been developed to study how an MFR evolves with reconnection during eruption [20,29], and the findings are becoming known as the standard flare model in 3D [29–31], although it is still an over-simplified version of the realistic cases as demonstrated in recent data-constrained and data-driven simulations (e.g., [32–34]).

In the standard 3D model, the erupting MFR is separated from the ambient field by a quasi-separatrix layer (QSL [35]), and in more details, this QSL intersects with itself below the MFR, forming a hyperbolic flux tube (HFT, [36]), and the flare reconnection occurs mainly in the HFT. The footprints of this QSL at the bottom surface, i.e., the photosphere, forms two thin strips of J shape on each side of the main polarity inversion line (PIL), and the legs of the MFR are anchored within the hooked parts of the J-shaped strips. Thus, the observed flare ribbons usually exhibit double-J pattern, and the transient coronal holes [37], i.e., post-eruptive twin coronal dimmings, are naturally suggested to map the feet of erupting MFRs, along which mass leakage into interplanetary space could take place [38–40], and the boundaries of such twin coronal dimmings are outlined by the hooks of flare ribbons. With the ongoing of reconnection, the arms of the double-J ribbons separate, and their hooks gradually extends outwards. In such process, the flare reconnection, which occurs between the pre-reconnection sheared arcade (as shown in the classic cartoon of tether-cutting model, i.e., Figure 1 of [10]), should increase the toroidal (axial) flux by increasing the number of field lines within the MFR.

However, such a “standard” type of flare reconnection in 3D still cannot explain fully the observations of “standard” two-ribbon flares. A well-known, unexplained fact is that the feet of the erupting flux rope, as manifested by twin coronal dimmings and also by the hook ends of double-J flare ribbons, are found to be drifting progressively away from the main PIL during eruption [37], even though the photosphere can be regarded as motionless during the short time scale of eruption. To this end, Aulanier and Dudík [41] analyzed in more details the reconnection process in their simulation of flux rope eruption and showed that the flare reconnection actually occurs in three different types of events according to their different effect in building up the flux rope. The first one is named as aa-rf reconnection, which is the standard 3D flare reconnection that occurs between two arcades and results in a long field line joining the flux rope and a short one as a flare loop. The second one is the so-called rr-rf reconnection, which occurs within the flux rope by reconnecting two flux-rope field lines with each other and generates a new multi-turn flux-rope field line and a flare loop. The third one is ar-rf reconnection, in which an inclined arcade reconnects with the leg of a flux-rope field line, and it generates new flux-rope field line rooted far away from the PIL and a flare loop. Thus, it is the ar-rf reconnection that actually leads to gradual drifting of the MFR footpoints.

Observations show even more features not explained (or not mentioned) in the “advanced” standard model of [41]. For instance, using high-resolution observations, [42] found two closed-ring-shaped flare ribbons in the case of a buildup of highly twisted MFRs with the development of a flare reconnection. During the separation of the main flare ribbons, the flare rings expand significantly, starting from almost point-like brightening. Note that such closed circular shape flare ribbons have different nature from those formed by null-point topology which also produce circular ribbon due to reconnection in the null’s spine-fan separatrix [32]. It is predicted by theoretical models [35] that if the MFR grows to sufficiently twisted, the hooks of the double-J shaped footprints of the QSL can indeed close onto themselves, becoming two closed rings, although this is not reproduced by the numerical model of [20], possibly because their simulation run is stopped before the MFR grows to such a high degree of twist.

Very recently, a few papers reported that there is a systematic decrease of the toroidal flux of erupting MFR after its fast increase (e.g., [40,42,43]). In particular, Xing et al [40] developed a practical method for estimation of the toroidal flux of MFR during eruption by combining twin coronal dimmings and the hooks of flare ribbons. They found that the toroidal flux of the CME flux rope for all four studied events shows a two-phase evolution: a rapid increasing phase followed by a decreasing phase, and moreover, the evolution is well synchronous in time with that of the flare soft X-ray flux. The increase of MFR’s toroidal flux can be easily understood by the aa-rf reconnection while the subsequent decrease remains unclear. Although Xing et al [40] invoked the rr-rf reconnection as the mechanism responsible for such decrease, it is still unknown whether the increase-to-decrease evolution of toroidal flux can self-consistently be reproduced in any MHD simulation.

This series of papers are devoted to a comprehensive analysis of a new MHD simulation of eruption [44], focusing on the formation of MFR during eruption. That simulation demonstrated in fully 3D that solar eruption can be initiated from a single magnetic arcade without the formation of MFR before the triggering of eruption. This is different from the aforementioned simulations [20,41], in which the eruption is initiated by torus instability (which is a kind of ideal MHD instability [17]) of an MFR formed well before the eruption. With the new MHD simulation, one can follow whole life, i.e., the birth and subsequent evolution, of an MFR during the eruption. As the first paper of this series, here we show, for the first time, that both the closing of the hook ends of the QSLs at the MFR’s feet and the increase-to-decrease evolution of the toroidal magnetic flux can be self-consistently reproduced by the simulation, suggesting that they are genuine features of erupting MFRs. We further quantified the evolution of reconnection flux during the eruption, and found that the evolution of QSLs is rather complex within the MFR. In the second article of this series, we will illustrate the 3D configuration of the different types of magnetic reconnection in building up the MFR and disclose why the QSLs evolve in such a complex way, following the pioneering work by [41].

Recently, Jiang et al [44] performed an ultra-high accuracy, fully 3D MHD simulation and demonstrated that solar eruptions can be initiated in a single sheared arcade with no additional special topology. Their simulation shows that, “through photospheric shearing motion alone, an electric current sheet forms in the highly sheared core field of the magnetic arcade during its quasi-static evolution. Once magnetic reconnection sets in, the whole arcade is expelled impulsively, forming a fast-expanding twisted flux rope with a highly turbulent reconnecting region underneath”. They further found that the high-speed reconnection jet plays the key role in driving the eruption. The simplicity and efficacy of this scenario, in the theoretical point of view, argue strongly for its fundamental importance in the initiation of solar eruptions. Since the model do not need a pre-existing MFR, the MFR itself comes into being after the eruption initiation.

Here we focus on the formation and evolution of MFR during the eruption by using a simulation run like the one in [44], but with a lower resolution than the original ones. Such simulation solves the full set of MHD equations and starts from a bipolar potential magnetic field and a hydrostatic plasma stratified by solar gravity with typical coronal temperature. Then shearing flows along the PIL, which are implemented by rotating the two magnetic polarities at the photosphere in the same count-clockwise direction, are applied on the bottom boundary to energize the coronal field until an eruption is triggered, and after then the surface flow is stopped. The whole computational box extends as (−32, −32, 0) < (x, y, z) < (32, 32, 64) with length unit of 11.5 Mm. We solve a full MHD equation with both solar gravity and plasma pressure included, but with the energy equation simplified as an isothermal process. The time unit of the model is τ = 105 s, and the shearing motion is applied by approximately 120τ before the onset of the eruption, during which a current sheet is gradually built up. Since no explicit resistivity is used in the MHD model, magnetic reconnection is triggered when the current sheet is sufficiently thin such that its width is close to the grid resolution. For more details of the simulation settings, the readers are referred to [44]. In that paper, the simulation is managed to be of very high resolutions with Lundquist number achieving ∼ 10⁵ for a length unit. Therefore, the secondary tearing instability (or plasmoid instability) is triggered in the current sheet and the magnetic topology becomes extremely complicated in small scales along with formation of the large-scale MFR. Such a complexity substantially complicates our analysis of the large-scale magnetic topology evolution associated with the erupting MFR, thus in this paper we used a lower-resolution run (corresponding to a Lundquist number of ∼ 10³). In the lower-resolution run, the basic evolution of the MFR during the eruption is not changed as compared to the high-resolution run, except that the small-scale structure will not arise, and thus the QSLs are computed in a cleaner pattern. Moreover, with the lower resolution, we can run the simulation longer and thus follow a longer evolution of MFR.

To help revealing the variation of the magnetic topology, we study the distribution and evolution of two parameters, the magnetic squashing degree and the magnetic twist number, which are commonly used for the study of 3D magnetic fields and their dynamics [2,29,30,45–47]. The magnetic squashing degree Q quantifies the gradient of magnetic field-line mapping with respect to their footpoints, and it is helpful for searching QSLs (and true separatries) of magnetic fields [36], which can have extremely large values of Q (e.g., ≥ 10⁵) and are preferential sites of magnetic reconnection. By locating the QSLs from the high values of Q we can see how the magnetic topology is evolved by the magnetic reconnection. Specifically, for a field line starting at one footpoint (x, y) and ending at the other footpoint (X, Y) where X and Y are both functions of x and y, the squashing degree Q associated with this field line is given by [36] Q = a 2 + b 2 + c 2 + d 2 | a d − b c | (1) where a = ∂ X ∂ x , b = ∂ X ∂ y , c = ∂ Y ∂ x , d = ∂ Y ∂ y . (2)

The magnetic twist number T _w [48] is defined for a given (closed) field line by taking integration of T _w = ∫ _L J⋅B/B ² dl/(4π) along the length L of the field line between two conjugated footpoints on the photosphere. Note that T _w is not identical to the classic winding number of field lines about a common axis, but an approximation of the number of turns that two infinitesimally close field lines wind about each other [47].

In this paper, we have studied the magnetic evolution of an MFR formed during the eruption in an MHD simulation. The MFR is generated absolutely by tether-cutting reconnection of the pre-eruption, strongly sheared arcade. In the early phase, the MFR is partially separated from its ambient field by a QSL that has a double-J shaped footprint on the bottom surface. With the ongoing of the reconnection, the arms (i.e., the straight parts) of the two J-shaped footprints continually separate from each other, which eventually pass through the centers of each polarity. Meanwhile, the hooks of the J shaped footprints expand and eventually become closed almost at the eruption peak time, and thereafter the MFR is fully separated with the un-reconnected field by a QSL. The reconnection substantially shapes the MFR by first increasing quickly and then decreasing gradually its total toroidal flux, which explains a recent observation of magnetic flux variation in erupting MFR’s foot. In the whole eruption, nearly half of the AR’s flux is reconnected, and the reconnection rate, as measured by the increasing rate of the reconnection flux, synchronizes well with the energy conversion rate (i.e., magnetic energy releasing rate and the kinetic energy increasing rate). In the early stage, i.e., before the reconnection rate reaches its peak, the reconnected flux almost equals to the toroidal flux in the MFR, whereas after the peak time the toroidal flux in the MFR decreases despite that the reconnected flux continues to increase. The increase of toroidal flux is owing to the flare reconnection in the early phase that transforms the sheared arcade to twisted field lines, while its decrease should be a result of reconnection between field lines in the interior of the MFR in the later phase, as first disclosed in [41].

Our simulation shows that the QSLs associated with the MFR in the later phase become more complex than expected, since there are new QSLs formed within the MFR, while the flux associated with these new QSLs becomes extremely highly twisted. This is due to a fast expansion of the MFR as well as its complex 3D nature, and thus at certain locations field lines reconnect with others in the MFR or themselves. Such reconnection may happen multiple times for field lines rooted at the same locations, even making some of field lines self-closed in the corona, which might be an important way for a CME flux to be totally detached from the Sun. The details of such complexity and the involved reconnection, and whether such complexity is hinted in observation, are to be elaborated in future works.