# Evolution of Magnetic Helicity During Eruptive Flares and Coronal Mass Ejections

## Abstract

During eruptive solar flares and coronal mass ejections, a non-potential magnetic arcade with much excess magnetic energy goes unstable and reconnects. It produces a twisted erupting flux rope and leaves behind a sheared arcade of hot coronal loops. We suggest that the twist of the erupting flux rope can be determined from conservation of magnetic flux and magnetic helicity and equipartition of magnetic helicity. It depends on the geometry of the initial pre-eruptive structure. Two cases are considered, in the first of which a flux rope is not present initially but is created during the eruption by the reconnection. In the second case, a flux rope is present under the arcade in the pre-eruptive state, and the effect of the eruption and reconnection is to add an amount of magnetic helicity that depends on the fluxes of the rope and arcade and the geometry.

### Keywords

Sun: flares Sun: magnetic topology Magnetic reconnection Helicity## 1 Introduction

The standard understanding of eruptive solar flares (*e.g.*, Schmieder and Aulanier, 2012; Priest, 2014; Aulanier, 2014; Janvier, Aulanier, and Démoulin, 2015) is that excess magnetic energy and magnetic helicity build up until a threshold is reached at which point the magnetic configuration either goes unstable or loses equilibrium, either by breakout (Antiochos, DeVore, and Klimchuk, 1999; DeVore and Antiochos, 2008) or magnetic catastrophe (Démoulin and Priest, 1988; Priest and Forbes, 1990; Forbes and Isenberg, 1991; Lin and Forbes, 2000; Wang, Shen, and Lin, 2009) or kink instability (Hood and Priest, 1979) or by torus instability (Lin *et al.*, 1998; Kliem and Török, 2006; Démoulin and Aulanier, 2010; Aulanier *et al.*, 2010; Aulanier, Janvier, and Schmieder, 2012).

Some solar flares (known as *eruptive flares*) are associated with the eruption of a magnetic structure containing a prominence (observed as a coronal mass ejection) and typically produce a two-ribbon flare, with two separating \(\mbox{H}\upalpha\) ribbons joined by a rising arcade of flare loops. Others are contained and exhibit no eruptive behaviour. While some coronal mass ejections are associated with eruptive solar flares, others occur outside active regions and are associated with the eruption of a quiescent prominence. Coronal mass ejections outside active regions do not produce high-energy products, because their magnetic and electric fields are much smaller than in eruptive solar flares, but their magnetic origin and evolution may well be qualitatively the same.

Magnetic helicity is a measure of the twist and linkage of magnetic fields, and its basic properties were developed by Woltjer (1958), Taylor (1974), Moffatt (1978), Berger and Field (1984), Berger and Ruzmaikin (2000) and Demoulin, Pariat, and Berger (2006). It was first suggested to be important in coronal heating, solar flares and coronal mass ejections by Heyvaerts and Priest (1984), who proposed that, when the stored magnetic helicity is too great, it may be ejected from the Sun in an erupting flux rope (see also Rust and Kumar, 1994; Low and Berger, 2003; Kusano *et al.*, 2002). The flux of magnetic helicity through the photosphere, its buildup in active regions and its relation to sigmoids has been studied by Pevtsov, Canfield, and Metcalf (1995), Canfield and Pevtsov (1998), Canfield, Hudson, and McKenzie (1999), Canfield, Hudson, and Pevtsov (2000), Pevtsov and Latushko (2000), Pevtsov (2002), Green *et al.* (2002), Pariat *et al.* (2006), and Poisson *et al.* (2015).

Indeed, the measurement of magnetic helicity in the corona is now a key topic with regard to general coronal evolution (Chae, 2001; Démoulin *et al.*, 2002; Mandrini *et al.*, 2004; Zhang, Flyer, and Low, 2006; Zhang and Flyer, 2008; Mackay, Green, and van Ballegooijen, 2011; Mackay, DeVore, and Antiochos, 2014; Gibb *et al.*, 2014). Also, since magnetic helicity is well conserved on timescales smaller than the global diffusion time, measuring it in interplanetary structures such as flux ropes and magnetic clouds (Gulisano *et al.*, 2005; Qiu *et al.*, 2007; Hu *et al.*, 2014; Hu, Qiu, and Krucker, 2015) allows us to link the evolution of CMEs in the solar wind with their source at the Sun (Nindos, Zhang, and Zhang, 2003; Luoni *et al.*, 2005).

The common scenario described above for an eruptive solar flare or coronal mass ejection is that, after the slow buildup of magnetic helicity in a magnetic structure, the eruption is triggered and drives three-dimensional reconnection which adds energy to post-flare loops. Originally, it was thought that all of the magnetic energy stored in excess of potential would be released during a flare, and therefore that the final post-flare state would be a potential magnetic field. However, the modern realisation is that magnetic helicity conservation provides an extra constraint, which produces a different final state. It is also observed that flare loops in an eruptive flare do not relax to a potential state, since the low-lying loops remain quite sheared (Asai *et al.*, 2004; Warren, O’Brien, and Sheeley, 2011; Aulanier, Janvier, and Schmieder, 2012). Our aim in this paper is to determine two key observational consequences of the reconnection process, namely, the amount of magnetic helicity, and therefore twist, in the erupting flux rope, and in the shear of the underlying flare loops.

- i)
conservation of magnetic flux;

- ii)
conservation of magnetic helicity;

- iii)
and equipartition of magnetic helicity.

The third assumption implies that the same amount of magnetic helicity is transferred by reconnection to the erupting flux rope and underlying arcade (Wright and Berger, 1989). We have considered an alternative possibility, namely, preferential transfer of magnetic helicity to the flux rope during the reconnection, which would imply the flaring loops have vanishing self-helicity. However, this seems less likely, because, although the flare loops are generally seen as non-twisted structures, they are also observed to be non-potential, since the low-lying loops appearing early in the flare possess more shear than the high-lying loops (Aulanier, Janvier, and Schmieder, 2012).

The reconnection of twisted tubes has been studied in landmark papers by Linton, Dahlburg, and Antiochos (2001), Linton and Antiochos (2002, 2005), Linton and Priest (2003), in which they consider also the extra constraint of energy both numerically and analytically. Straight tubes of a variety of twists and inclinations are brought together by an initial stagnation-point flow and they are allowed to reconnect self-consistently by either bouncing, slingshot, merging or tunnelling reconnection. Although energy is not considered in detail here, we make initial comments about energy considerations in Section 3.3 and hope to develop them further in future numerical treatments.

The structure of the paper is as follows. In Section 2 we summarise the basic properties of magnetic helicity that are needed for our analysis. Then in Section 3 we present our simple model for a sheared magnetic arcade in which no flux rope is present initially but is created during the eruption by the reconnection. This is followed in Section 4 by a different model in which the initial arcade contains a flux rope, which erupts and leaves behind a less-sheared arcade. Finally, suggestions for follow-up are given in Section 5.

## 2 Magnetic Helicity Preliminaries

Magnetic helicity is a topological quantity that comprises two parts: the *self-helicity* measures the twisting and kinking of a flux tube, whereas the *mutual helicity* refers to the linkage between different flux tubes (Berger, 1986, 1999). Their sum, the relative helicity, is a global invariant that is conserved during ideal evolution and that decays extremely slowly (over the global magnetic diffusion time, \(\tau_{d}\)) in a weakly resistive medium – *i.e.*, one for which the global magnetic Reynolds number is large (\(R_{m}\gg1\)). Thus, during magnetic reconnection in the solar atmosphere over a timescale \(t\ll\tau_{d}\), the total magnetic helicity is approximately conserved, but it may be converted from one form to another, say, from mutual to self.

## 3 Modelling the Eruption of a Simple Magnetic Arcade with no Flux Rope

Consider a simple sheared arcade (Figure 1(a)), which reconnects and produces an erupting flux rope together with an underlying arcade (Figure 1(b)). Our aim is to deduce the helicity in the flux rope (and so its twist) in terms of the shear and helicity of the initial arcade. In addition, we shall deduce the distribution of twist within the flux rope.

### 3.1 Overall Process

We first consider the overall process and model the configuration in the simplest way by treating the initial arcade as two untwisted flux tubes side by side. We compare the initial state in Figure 3(a) having two untwisted flux tubes (of equal flux \(F_{A}=F_{B}=F_{a}\)) side by side with the reconnected state in Figure 3(b) having a twisted erupting flux rope (of mean twist \({\bar{\Phi}}\), say) overlying flare loops.

*flux conservation*, so that the total final flux (\(F_{R}+F_{U}\)) equals the total initial flux (\(2F_{a}\)). We also assume that reconnection feeds flux simultaneously and equally into the rope and underlying loops, so that \(F_{R}=F_{U}\). The net result is that in the reconnected configuration (Figure 3(b)) the fluxes of the erupting flux rope and underlying loops are both equal to half the initial total flux (\(F_{R}=F_{U}=F_{a}\)). We also assume

*helicity equipartition*, so that the released mutual helicity is added equally to the erupting flux rope and underlying loops. Then the magnetic helicity in the reconnected state, namely, the sum of the self-helicities and mutual helicity of the flux rope and the underlying sheared loops, may be written for simplicity (assuming \(\theta_{4}=\theta _{3}\)) as

### 3.2 Evolution of the Process

#### 3.2.1 Using Helicity Conservation to Deduce the Mean Flux Rope Twist

Next, we consider the evolution of the process by again supposing the initial configuration consists of two untwisted flux tubes stretching from fixed photospheric flux sources to sinks (Figure 5(a)). The reconnected state again consists of an erupting twisted flux rope (R) and underlying arcade of loops (U) (Figure 5(c)).

During reconnection, in going from Figure 5(a) to Figure 5(c) via intermediate states of the form Figure 5(b), we assume *conservation of magnetic flux*, so that, if the new flux rope (R) joining \(\mbox{A}_{+}\) to \(\mbox{B}_{-}\) gains flux \(F\), then so does the underlying arcade (U) joining \(\mbox{B}_{+}\) to \(\mbox{A}_{-}\), while both flux tubes (A) and (B) of flux \(F_{a}\) lose flux \(F\) so that their fluxes both become \(F_{a}-F\). We suppose the footpoints form a parallelogram, so that \(\theta_{3}=\theta_{4}\). Initially, the arcade is modelled as consisting of two flux tubes (A and B) side by side, the centres of whose footpoints are indicated by large dots separated by a distance \(L\) (Figure 5(a)). During the course of the reconnection (Figure 5(b)), when a fraction \(\bar{F}=F/F_{a}\) of the flux has been reconnected, there is an overlying flux rope (R) of flux \(F\), the centres of whose footpoints are located at \(\mbox{R}_{+}\) and \(\mbox{R}_{-}\), together with a underlying flux loop (U) with footpoints at \(\mbox{U}_{+}\) and \(\mbox{U}_{-}\) (Figure 5(d)). Also, the remaining unreconnected flux consists of two flux tubes (A and B) whose footpoints are located at \(\mbox{A}_{+}\), \(\mbox{A}_{-}\) and \(\mbox{B}_{+}\), \(\mbox{B}_{-}\), respectively (Figure 5(d)). The centres of the upper ends of the four flux tubes are located at distances \(\frac{1}{2}L(1-\bar{F})\), \(L(1-\frac{1}{2}\bar {F})\), \(L(1+\frac{1}{2}\bar{F})\) and \(\frac{1}{2}L(3+\bar{F})\), respectively, from the right-hand end of the arcade (Figure 5(c)), and so they are separated by distances \(\frac{1}{2}L\), \(L\bar{F}\) and \(\frac{1}{2}L\), as shown in Figure 5(d). The edges of the upper ends of the tubes are located at distances \(L(1-\bar{F})\), \(L\), \(L(1+\bar{F})\) and \(2L\) from the right-hand end of the arcade.

*magnetic helicity conservation*, so that the total magnetic helicity (

*i.e.*, the sum of self and mutual helicity) is conserved during the reconnection process, namely,

*magnetic helicity equipartition*, so that the effect of reconnection of A and B is to give self-helicity equally to the flux rope (R) and underlying loops (U),

*i.e.*,

The expressions for the mutual helicities are given in the Appendix. By substituting them into the expressions for initial and post-reconnection helicity, helicity conservation determines the flux rope self-helicity (and therefore its mean twist, \({\bar{\Phi}}\)). It transpires that, although all of the mutual helicities change during the course of the reconnection, the sums \(H_{{AR}}+H_{{AU}}\) and \(H_{{BR}}+H_{{BU}}\) remain the same. In other words, the unreconnected parts of flux tubes A and B are spectators, since they have not taken part in the reconnection, and we just need to take account of the other parts and the way they reconnect to give flux tubes R and U in Figure 5(d). But this process is exactly what we have already considered in Section 3.1 with one difference, namely, that the length \(\bar{L}\) is replaced by \(\bar{L}\bar{F}\) and the angles \(\theta_{1}\) and \(\theta_{2}\) are replaced by \(\theta_{1}^{{RU}}\) and \(\theta _{2}^{{RU}}\), respectively.

#### 3.2.2 Deducing the Distribution of Twist Within the Flux Rope

### 3.3 Energy Considerations

If the initial state is driven to reconnect by footpoint motions, then there is no need for the initial magnetic energy to exceed the final energy, since any change in energy could come from the work done by the footpoints. If, however, the reconnection arises from some kind of instability, then an extra constraint, which we have not considered so far, arises from the condition that the magnetic energy of the initial state must exceed that of the final state. This will then rule in some of the changes we have considered and rule out others. We do not here give a definitive answer to which changes are allowed, but only undertake a preliminary investigation. In Section 3.3.1 we first consider a general expression for the free energy and demonstrate that there is indeed energy release during reconnection to start with, which implies that the magnetic helicity transfers we have been considering so far provide an upper limit on what will happen in reality. Then in Section 3.3.2 we describe a simple model for calculating the free energy and again demonstrate that in some cases the free energy is indeed positive. In future, we look to a full numerical solution to provide a more in-depth account of the constraints produced by energy considerations.

Two minor points are worth making first. It may naturally be thought that two parallel untwisted tubes side by side (with flux going from footpoints \(\mbox{A}_{+}\) and \(\mbox{B}_{+}\) to \(\mbox{A}_{-}\) and \(\mbox{B}_{-}\), respectively) will not reconnect. This is indeed true if the initial state is potential and the footpoints form a rectangle, with reflectional symmetry, since the initial field is in a state of minimum possible energy (*e.g.*, Longcope, 1996). In our case, however, the footpoints form a non-rectangular parallelogram lacking this symmetry and the initial state is force-free (Figure 3(a)), so there exists a lower-energy (potential) state with the same footpoints but with flux going from \(\mbox{A}_{+}\) to \(\mbox{B}_{-}\) as well as to \(\mbox{B}_{+}\); some reconnection is therefore likely to occur. However, the minimum-energy state that has given footpoint fluxes and given total magnetic helicity is not the potential state but a piecewise linear force-free field (*e.g.*, Longcope and Malanushenko, 2008). So, provided the initial state is a nonlinear force-free field, there will certainly exist a lower-energy state with the latter connections that preserves magnetic helicity.

The second minor point is that, since the initial states are untwisted and the final states are twisted, it may be thought at first that the final state must have a higher energy than the initial state. But that conclusion is valid only for tubes of a given constant cross-sectional radius, and it will not necessarily hold for tubes that are allowed to expand as they arch up from their footpoints.

#### 3.3.1 General Aspects

*i.e.*, the energy above potential) can be written as a sum over terms involving the total currents, \(I_{i}\), flowing along their paths (Jackson, 1999, §5.17),

#### 3.3.2 Simple Model

We shall suppose for simplicity that our flux tubes expand rapidly up to form relatively uniform tubes in the corona, for which the flux (\(F_{0}\)) and twist (\(\Phi\)) are given. If these are held constant, then, as the radius \(a\) increases and the tube expands more into the corona, so the central axial field \(B_{0}\) decreases and the energy \(W\) decreases.

## 4 Modelling the Eruption of a Magnetic Arcade Containing a Flux Rope

Consider next a twisted magnetic flux rope of initial twist \(\Phi_{Ri}\) and flux \(F_{I}\) situated under a coronal arcade of flux \(2F_{a}\), and suppose that it reconnects with the arcade to produce an erupting flux rope whose core is the original flux rope, but which is now enveloped by a sheath of extra flux \(F_{a}\) and twist \(\Phi_{R}\), as sketched in Figure 2. We here develop a simple model to determine \(\Phi_{R}\) in terms of the geometry of the initial state.

The initial self-helicity of the flux rope is \({\Phi}_{Ri}F_{13}^{2}/(2\pi)\) and is preserved as the self-helicity of the core of the erupting flux rope after reconnection. The final self-helicity of the new part \(\mbox{P}_{2}\mbox{N}_{2}\) of the erupting flux rope is \({\Phi}_{R}F_{22}^{2}/(2\pi)\), while the final self-helicity of the underlying loops \(\mbox{P}_{3}\mbox{N}_{1}\) is \({\Phi}_{U}F_{31}^{2}/(2\pi)\).

The initial mutual helicity has three parts, namely: \([1-\frac{1}{2}(\theta _{6}+\theta_{7})/\pi]F_{13}F_{21}\) due to \(\mbox{P}_{1}\mbox{N}_{3}\) lying under \(\mbox{P}_{2}\mbox{N}_{1}\), where \(\theta_{6}\) is the angle \(\mbox{N}_{1}\mbox{P}_{1}\mbox{P}_{2}\) and \(\theta _{7}\) is the angle \(\mbox{P}_{2}\mbox{N}_{3}\mbox{N}_{1}\); \([1- \frac{1}{2}(\theta_{5}+ \theta_{8})/\pi ]F_{13}F_{32}\) due to \(\mbox{P}_{1}\mbox{N}_{3}\) lying under \(\mbox{P}_{3}\mbox{N}_{2}\), where \(\theta _{5}\) is the angle \(\mbox{N}_{2}\mbox{P}_{1}\mbox{P}_{3}\) and \(\theta_{8}\) is the angle \(\mbox{P}_{3}\mbox{N}_{3}\mbox{N}_{2}\); and \([(\theta_{2}-\theta_{1})/\pi] F_{21}F_{32}\) due to \(\mbox{P}_{3}\mbox{N}_{2}\) lying alongside \(\mbox{P}_{2}\mbox{N}_{1}\), where \(\theta_{1}\) is the angle \(\mbox{P}_{3}\mbox{P}_{2}\mbox{N}_{2}\) and \(\theta_{2}\) is the angle \(\mbox{P}_{3}\mbox{N}_{1}\mbox{N}_{2}\).

After reconnection, the mutual helicity decreases to the sum of three parts, namely: \(-[\frac{1}{2}(\theta_{6}+\theta_{7})/\pi]F_{13}F_{31}\) due to the initial flux rope \(\mbox{P}_{1}\mbox{N}_{3}\) now lying over the underlying arcade of loops \(\mbox{P}_{3}\mbox{N}_{1}\); \(-[\frac{1}{2}(\theta_{5}+\theta_{8})/\pi]F_{13}F_{22}\) due to the initial flux rope \(\mbox{P}_{1}\mbox{N}_{3}\) lying under the new part \(\mbox{P}_{2}\mbox{N}_{2}\) of the erupting flux rope; and \(-(\theta_{3}/\pi)F_{31}F_{22}\) due to the underlying arcade of loops \(\mbox{P}_{3}\mbox{N}_{1}\) lying under the new part \(\mbox{P}_{2}\mbox{N}_{2}\) of the erupting flux rope.

*magnetic helicity equipartition*, so that the self-helicities added to the underlying arcade and flux rope are equal and the released mutual helicity is shared equally between the two flux tubes,

*i.e.*,

*magnetic helicity conservation*so that the initial and final helicities (self plus mutual) are the same, namely

## 5 Discussion

We have set up a simple model for estimating the twist in erupting prominences, in association with eruptive two-ribbon flares and/or with coronal mass ejections. It is based on three simple assumptions, namely, conservation of magnetic flux, conservation of magnetic helicity and equipartition of magnetic helicity. While the first and second are well established, the third is more of a reasonable conjecture. In future, it would be interesting to test the model and the conjecture with both observations and computational experiments.

During the main phase of a flare, the shear of the flare loops is observed to decrease in time, so that they become oriented more perpendicular to the polarity inversion line. This is a natural consequence of our model, where flux is added to the flux rope first from the innermost parts of the arcade (*i.e.*, closest to the polarity inversion line), so that, as can be seen in Figures 1(b), 2(b) and 9(b), the final shear of the arcade is smaller than the initial shear. In other words, the change in shear is a consequence of the geometry of the three-dimensional reconnection process.

The cause of the eruption is a separate topic that has been discussed extensively elsewhere (*e.g.*, Priest, 2014), and it includes either nonequilibrium, kink instability, torus instability or breakout. One puzzle is what happens with confined flares, where the flare loops and \(\mbox{H}\upalpha\) ribbons form but there is no eruption. A distinct possibility is that the overlying magnetic field and flux are too strong to allow the eruption, but this needs to be tested by comparing nonlinear force-free extrapolations with observations (*e.g.*, Wiegelmann, 2008; Mackay, Green, and van Ballegooijen, 2011; Mackay and Yeates, 2012). Another puzzle is the cause of preflare heating. One possibility is the slow initiation of reconnection before a fast phase (Yuhong Fan, private communication), but another is that the flux rope goes unstable to kink instability which spreads the heating nonlinearly throughout the flux rope in a multitude of secondary current sheets (Hood, Browning, and van der Linden, 2009); if the surrounding field is stable enough, such an instability can possibly occur without an accompanying eruption.

The present simple model can be developed in several ways, which we hope to pursue in the future. One is to conduct computational experiments, in which the energies before and after reconnection will be calculated in order to check which states are energetically accessible. A second way is to extend the model to more realistic initial configurations with more elements, in which the reconfiguration is by quasi-separator or separator reconnection (Priest and Titov, 1996; Longcope, 1996; Longcope *et al.*, 2005; Longcope and Malanushenko, 2008; Parnell, Maclean, and Haynes, 2010) and the internal structure of the flux rope and arcade are taken into account. In particular, the distribution of magnetic flux within the arcade will be included, both normal to and parallel to the polarity inversion line.

## Notes

### Acknowledgements

We are grateful to Mitch Berger, Pascal Démoulin, Alan Hood and Clare Parnell for helpful comments and suggestions and to the UK STFC, High Altitude Observatory and Montana State University for financial support.

### Disclosure of Potential Conflicts of Interest

The authors declare that they have no conflicts of interest.

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