We here try to understand the nature of such magnetic structure by referring to filament/prominence and sigmoid, both of which are observed before the onset of a flare (precursor), where an emphasis is put on their formation processes. We also explain how to reconstruct invisible magnetic structure using surface magnetic field which is observed. Force-free-field modeling is one of the possible methods of reconstruction. We also explain a famous conjecture on the energy state of force-free field, which is known as Aly–Sturrock conjecture.
Observations have revealed various structural features of filament (Martin, 1990, 1998; Schmieder et al., 2006; Rust and Kumar, 1994). A filament tends to form around the polarity inversion line separating opposite polarity regions (main polarity regions), forming a filament channel. Along the inversion line is observed the main body of a filament, which is called ‘spine’. There are also small weak-flux regions distributed in a filament channel (Martin, 1998; Chae et al., 2001), and these regions, which are called parasitic or satellite polarity regions compared to the main polarity regions, contribute to forming secondary structure such as filament feet called ‘barbs’. An important result about parasitic polarity regions is that these regions have the opposite polarity to the nearby main polarity region (Martin et al., 1994), and field lines connecting to parasitic polarity regions are suggested to have dipped structure (Aulanier and Démoulin, 1998; Aulanier et al., 1998; López Ariste et al., 2006).
There is also known a hemispheric chirality rule of filaments: ‘dextral’ filaments tend to appear in the northern hemisphere where the magnetic field with left-handed twist is preferentially observed, while ‘sinistral’ filaments are frequently observed in the southern hemisphere where the right-handed twist is dominant (Rust and Kumar, 1994; Martin, 1998; Pevtsov et al., 2003).
Theoretical studies of the magnetic structure containing a filament has been done extensively. Low and Hundhausen (1995) and Low (1996) suggest that a twisted flux tube emerges via magnetic buoyancy to form a flux rope above the surface, inside which a filament is formed (Figure 17, top-left panel). Antiochos et al. (1994) and DeVore and Antiochos (2000) demonstrate that a sheared arcade contains a filament, which is formed by shear flows around polarity inversion line (Figure 17 top-right panel). van Ballegooijen and Martens (1989) presents a result showing that shear flows followed by converging flows toward polarity inversion line causes reconnection at the surface (called ‘flux cancellation’), which creates a twisted flux rope inside which a filament is formed (Figure 17 bottom panel). The last one has been developed later to explain the global hemispheric pattern of filaments (van Ballegooijen et al., 2000; Mackay and van Ballegooijen, 2005). The origins of these surface motions and possible relation between twist and chirality mentioned above have been studied from the viewpoint of flux emergence. Recently, Magara (2007) shows that the emergence of a twisted flux tube undulating along tube axis naturally reproduces those observed structural features (Figure 18). Field lines close to tube axis (inner field lines) form the main body of a filament. On the other hand, field lines composing the twisted flux tube which are located away from tube axis (outer field lines) form a coronal arcade that overlies the main body of a filament, while they underlie the main body by forming barbs. Regarding the chirality of filaments, this work suggests that a flux tube of left-handed (right-handed) twist tends to form a dextral (sinistral) filament (see the middle panels in Figure 18). Also, it was shown that the emergence of U-loops distributed below the axis causes apparent flux cancellation on the surface (Magara, 2011). In this respect, the evolution of U-loops has been observed and analyzed by van Driel-Gesztelyi et al. (2000). They show the peculiar motions of magnetic polarities on the surface and suggest that flux cancelation proceeds without significant energy release.
Although there are some observations suggesting that a filament is formed via the emergence of a twisted flux tube (Lites, 2005; Okamoto et al., 2008), it should be mentioned that a number of filaments are formed away from emerging active regions. These filaments form along the polarity inversion line of decaying active regions, in between active regions, and even in the polar regions (polar crown filaments).
Recently, the dynamic nature of filament/prominence has well been captured with advanced observing tools, which provides the detailed information on plasma motions in a filament/prominence (Berger et al., 2008; Okamoto et al., 2007). The modeling focused on the dynamic nature of filament/prominence has also been reported (Antiochos et al., 1999b; Karpen and Antiochos, 2008).
Sigmoid is observed as either S or inverse S-shaped structure with soft X-ray enhancement in the corona, and it has been known as the precursor of a big cusp-shaped flare (Tsuneta et al., 1992a) or coronal mass ejection (CME) (Canfield et al., 1999; Sterling and Hudson, 1997). Recently, using the soft X-ray observations by Hinode, McKenzie and Canfield (2008) found that sigmoid is not a single loop but consists of many loops.
Gibson et al. (2002) use linear force-free field modeling to analyze an observed sigmoid. Pevtsov (2002) shows an interesting result on the spatial relationship between a filament and a sigmoid. Régnier and Amari (2004) use nonlinear force-free field modeling to study a global magnetic structure containing a filament, sigmoid, and a large -loop overlying the filament and sigmoid. They explain that the filament and sigmoid are composed of the loops that have a smaller aspect ratio of height to footpoint separation than the overlying -loop. This result suggests that a filament and sigmoid are located at inner central part of a magnetic structure formed through the emergence of a twisted flux tube, as we discussed in Section 3.1.2.
Regarding the geometrical feature of sigmoid, Titov and Démoulin (1999) suggests that sigmoid is composed of double J-shaped structure which is formed at the interface between an emerging flux tube and the overlying potential-like field. On the other hand, an emerging flux tube itself also produces similar double J-shaped structure inside the flux tube when it is twisted (see Figure 19, Magara, 2004). The twist of field lines is important in forming sigmoidal structure (Matsumoto et al., 1998); a potential field-like structure is formed by a weakly twisted flux tube (Figure 20, Magara, 2006). Here the illuminated field lines have a relatively high current density distributed at their chromospheric footpoints.
The formation of sigmoid may be related to the emergence of U-loops distributed below the axis of a twisted flux tube. Magara and Longcope (2001) show that these U-loops can explain the observational chirality rule of sigmoid (Canfield et al., 1999), mentioning that inverse S-shaped (foward S-shaped) sigmoid tends to have left-handed (right-handed) twist. They also found that high current density is preferentially distributed at the chromospheric footpoints of these U-loops (Figure 21).
Figure 22 schematically explains the sigmoid formation displayed in Figure 21 (here the cross section of an emerging flux tube is shown). The upper half of the flux tube contains -loops which continue to expand via active magnetic buoyancy. The lower half is occupied by U-loops which tend to remain below the surface because the mass accumulates at dipped part of U-loops. However, those U-loops distributed close to the axis of the flux tube can emerge because they have a shallow dip (Magara, 2011), and they form sigmoidal structure (in this respect, since the axis becomes bent after emerging, such a shallow dip might disappear to make U-loops take an -shape; see Magara, 2004). Below these emerged U-loops, the magnetic field is vertically stretched and forms a current sheet in the corona, which in fact plays a key role in producing a flare, as explained in the succeeding sections. Generally, it is not easy to directly observe a magnetic structure formed in the corona, although the appearance of sigmoid might be an indicator that the axis of the flux tube already reaches the corona and a current sheet is being developed below the axis. This may explain why sigmoid is often observed as a precursor of a flare.
The three-dimensional (3D) distribution of current density inside an emerging flux tube was investigated by Manchester IV et al. (2004). Recently, the temporal development of 3D distribution of current density has been reported in Archontis et al. (2009), where a comparison between 3D distribution of current density and soft X-ray images of a sigmoid is presented. They show that the double J-shaped structure are merged into a single sigmoidal structure, just as observations show (Figure 23).
It is not clear whether sigmoid is just a thin current layer formed at the interface between two magnetic flux domains such as overlying and emerging fields, or it is more like a volumetric structure occupied by field lines with strong field-aligned current flowing on them. Three-dimensional modeling focused on the magnetic structure of an observed sigmoid such as force-free field modeling (see the next section) could provide useful information to clarify this issue.
As we have explained in the previous section, the field-aligned electric current is a key factor in producing a flare, so we need the modeling that describes how the field-aligned current is distributed in a magnetic structure. In this sense, the potential-field modeling is insufficient because it cannot tell the distribution of electric current in a magnetic structure. This is provided by the force-free-field modeling where the magnetic field keeps force-balance by itself inside a magnetic structure, satisfying the following equation except at boundaries of the structure:linear force-free field, otherwise it is called nonlinear force-free field. In any force-free field, the electric current always flows along field lines (volumetric field-aligned current).
Force-free field provides a method to reconstruct coronal magnetic field from surface magnetic field, the latter of which is easier to observe than the former. By using this method, Nakagawa et al. (1971) studied the magnetic structure of an isolated sunspot. Similarly, Sheeley Jr and Harvey (1975) construct a magnetic configuration formed by discrete flux sources at the surface. The helical nature of force-free field has been investigated by Sakurai (1979). By considering a series of force-free states, Barnes and Sturrock (1972), Low and Nakagawa (1975), and Low (1977) investigated the characteristics of evolving magnetic structure.
Recent developments are found in the attempt to combine force-free field modeling and the observation of surface magnetic field. Wheatland et al. (2000) introduce an optimization method to calculate force-free field, and according to this method Wiegelmann et al. (2000) calculate a force-free field based on observed surface magnetic field. Wiegelmann and Neukirch (2006) extended this method to calculate a magnetohydrostatic state. Régnier and Priest (2007) estimated difference among energy states of nonlinear force-free, linear force-free, and potential field in several active regions. A good review on various methods to calculate force-free field is given by Wiegelmann (2008). Very recently, Valori et al. (2010) have done two important steps in coronal field modeling. First, they show that it is possible to compute the coronal field even when it is significantly twisted (more than one turn), which was not obvious from previous studies. Second, they relate the specific behavior of the extrapolated field to its MHD evolution when the test field is out of equilibrium.
Let us mention a famous conjecture on the energy state of force-free field. Barnes and Sturrock (1972) calculated a series of force-free fields in cartesian coordinates, showing that the energy state of these force-free fields could be greater than that of the open field with the same vertical magnetic flux at the surface. However, Aly (1984) presents a conjecture that the energy of any force-free fields might be smaller than that of the open field. Yang et al. (1986) reconsidered this problem, showing that the Aly’s conjecture seems to be right, which later has also been supported by Sturrock (1991). The conjecture is now known as the Aly–Sturrock conjecture on the energy state of force-free field, mentioning that the fully open-field state associated with a current sheet of infinite length and infinitesimal thickness can be reached only asymptotically and the energy of this fully open field cannot be exceeded by any other force-free fields. This in fact caused difficulty in creating a force-free state during a preflare phase, which should be reduced to the fully open-field state in a postflare phase. There have, however, been proposed various ways to avoid this difficulty, as listed below:
Regarding the energetics of force-free fields, Kusano et al. (1995) demonstrate a difference between two energy states of linear force-free fields that have the same magnetic helicity (see Section 3.3). They present an energy-bifurcation theory in which a vertically elongated magnetic arcade tends to be reduced to a state in which a twisted flux rope exists.
Living Rev. Solar Phys. 8, (2011), 6
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