The observational fact that solar flares generally occur tens of minutes after the CME is initiated implies that magnetic reconnection often takes place soon after the CME progenitor is stretched up in the low corona (well before it is stretched up to be open field), and the “Aly–Sturrock” constraint is not applicable to these resistive MHD processes. Actually most resistive MHD numerical simulations have been performed to reproduce CMEs like this, without the necessity to consider the constraint.
Even in the ideal MHD framework, there are still several ways to circumvent the Aly–Sturrock constraint, allowing the initial magnetic field to have the energy higher than that of the open field, which means that the transition from the initial closed field to the open field is feasible from the energy point of view, and an eruption driven by ideal MHD processes is feasible:
(1) The initial magnetic field is not simply linked to the solar surface, e.g., it contains detached magnetic field lines (e.g., Aly, 1991; Hu et al., 2003). However, it is reminded here that whereas it is hard to construct a 3D force-free field with detached field lines interwound with simply connected field lines, the detached field in most analytical solutions is a self-closed flux rope in 2D. In the 3D reality, a twisted flux rope should be anchored to the solar surface, and the twisted field lines are simply connected again. It keeps an open question whether a really detached magnetic island in 3D, as that in the time-dependent MHD solutions of Gibson and Low (1998), is stable even if an analytical solution is available.
(2) The initial magnetic field is not force-free, i.e., the Lorenz force is balanced with gravity and pressure gradient. For example, it is estimated that the initial total energy would be increased by 10% if gravity is considered (Forbes, 2000), and the magnetic field with cross-field electric current may contain magnetic energy in excess of that in an open field (Low and Smith, 1993; Wolfson and Dlamini, 1997).
(3) Only a part of the initial magnetic field becomes open. Wolfson and Low (1992) demonstrated that the closed magnetic field in 2D may contain more energy than a partially open field with the same flux distribution at the solar surface, meaning that partial opening of the closed field through an ideal MHD process is allowable energetically, although the authors did not show how this proceeds. It was further pointed out by Low (1997) that the situation in 3D becomes much easier since an anchored flux rope in 3D can erupt by simply pushing its way amidst neighboring magnetic fields which can remain closed. As illustrated by Figure 35, Sturrock et al. (2001) quantified such a process by demonstrating that the initial magnetic energy of the closed field is larger than that of a partially open field, where only a part of the line-tied twisted field lines ruptures to the infinity. This indicates that such a rupture process is at least permissible from the energy point of view. With MHD numerical simulations, Fan (2005) reproduced such a rupture process in the early phase, while numerical resistivity led to magnetic reconnection in the current sheet formed below the erupting flux rope later. In a similar simulation, Inoue and Kusano (2006) inhibited magnetic reconnection by forcing the plasma velocity to be zero on and around the current sheet, and found that the flux rope can still erupt. With the zero-beta MHD simulations, Török and Kliem (2007) also claims that only torus instability enables the final eruption. More recently, Rachmeler et al. (2009) used a quasi-Lagrangian simulation technique to simulate the end state of the coronal field with different twists. Without magnetic reconnection in their simulations, they demonstrated that a twist flux rope, which is anchored to the solar surface at two ends, can rupture into a CME through an ideal MHD process. The key point is that some parts of the overlying arcade are pushed aside, leaving space for the erupting flux rope. It is noted that their current simulations are not based on a full MHD code, where mass and plasma dynamics are not included. Future efforts on MHD simulations without numerical resistivity are strongly encouraged to demonstrate that CMEs can erupt via ideal MHD processes.
The significance of the above discussions is that the ideal MHD process is possible to release magnetic energy powering CMEs from a theoretical point of view. The observations, on the other hand, indicate that the majority of CMEs, especially those fast events, are strongly associated with solar flares, which result from magnetic reconnection (McAllister and Martin, 2000, see also Section 2 of this paper). In a unified model, Shibata (2003) emphasized that magnetic reconnection plays a crucial role in eruptions from small-scale jets to large-scale CMEs. Magnetic reconnection is also a key ingredient in the standard model for CME/flares, i.e., the so-called CSHKP model. The analytical modeling of this model in 2D indicates that the reconnection rate should be larger than a threshold in order for the flux rope to erupt into infinity, although the threshold is as small as 5 × 10–3 (Lin and Forbes, 2000; Lin, 2002). This conclusion is different from that in Rachmeler et al. (2009), and the possible reason is that in the 2D case, all the overlying field lines above the flux rope would be stretched up as the flux rope erupts, which imposes stronger and stronger magnetic tension force hindering the flux rope from further eruption; whereas in the 3D case, some of the overlying magnetic loops can slip away from the erupting flux rope, as seen from Figure 35.
It is worth pointing out that although the resistivity of the coronal plasma is extremely low, it is not zero anyway, i.e., ideal MHD is just a theoretical model. As the closed structure erupts due to loss of equilibrium or instabilities via ideal MHD processes, current sheet formation is inevitable, where microscopic instabilities may excite anomalous resistivity or other non-ideal terms, and further launch magnetic reconnection. Therefore, the essence of the debate on the necessity of magnetic reconnection is to clarify whether reconnection plays a crucial role or a minor role in CME eruptions. It might be safe to say that magnetic reconnection plays an important role in most CMEs, especially the fast events which are of interest to space weather research. This is why the CME acceleration was found to be temporally correlated with the magnetic reconnection rate (Qiu et al., 2004). On the other hand, some other effects can also facilitate the eruption of the CMEs, e.g., ideal MHD instabilities, mass drainage, gas pressure, solar wind, etc. Especially, the higher corona, say, at 105 km above the surface, is often not force-free (Gary, 2001), making these factors more effective. This explains why there is only a weak correlation between CME velocities and the peak SXR flux of the associated flare (Hundhausen et al., 1994; Yashiro et al., 2002; Vršnak et al., 2005; Yeh et al., 2005). As indirect evidence, there are some CMEs with proton flares, which were associated with a relatively weak impulsive phase of the flare (Cliver et al., 1983), and the filament eruption on 1981 December 5 generated an intense solar energetic particle (SEP) event and interplanetary type II radio bursts, but lacking an impulsive microwave burst (Kahler et al., 1986). All these indicate that some CMEs may be mainly accelerated by processes other than magnetic reconnection, which requires further confirmation.
When judging whether magnetic reconnection is involved in CME eruptions, the mostly used evidence is the SXR or EUV brightenings, especially solar flares, in the low corona. A ray-like structure under the some CMEs was also considered to be the direct evidence of a current sheet (Ciaravella et al., 2002, and followers). The width of such an assumed current sheet was estimated to be 104 km, and the corresponding resistivity was estimated to be 7 orders of magnitude larger than the anomalous resistivity (or 12 orders larger than the classical one) (Lin et al., 2007). It still awaits to be assured whether such a ray-like structure under some CMEs is really the current sheet where magnetic reconnection happens or it is actually the outward reconnection jet.
Living Rev. Solar Phys. 8, (2011), 1
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