3.2 Magnetic structure

Flux emergence is an essential process by which a magnetic structure containing free energy is formed on the Sun. It is still difficult to grasp the whole magnetic structure formed on the Sun observationally, while parts of the structure can be deduced from observed objects such as filament (or equivalently prominence which is observed on the limb of the Sun) and sigmoid that shows ‘S’ or ‘inverse-S’ shape in soft X-rays (Pevtsov et al., 1995; Rust and Kumar, 1996; Canfield et al., 1999Jump To The Next Citation Point). Modeling of these objects is therefore a key to the understanding of the magnetic structure related to flares.

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.

3.2.1 Filament (Prominence)

Observations have revealed various structural features of filament (Martin, 1990, 1998Jump To The Next Citation Point; Schmieder et al., 2006; Rust and Kumar, 1994Jump To The Next Citation Point). 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, 1998Jump To The Next Citation Point; 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).

View Image

Figure 17: Magnetic structure of a filament (from Low and Hundhausen, 1995Jump To The Next Citation Point) (top-left panel), DeVore and Antiochos (2000Jump To The Next Citation Point) (top-right panel) and van Ballegooijen and Martens (1989Jump To The Next Citation Point) (bottom panel).

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, 1998Jump To The Next Citation Point; Pevtsov et al., 2003).

View Image

Figure 18: Possible magnetic structure of a filament formed via the emergence of a twisted flux tube of left handedness. The top panel shows pre- and post-emergence states. The flux tube undulates along the axis when it emerges. Field lines composing the twisted flux tube which are located away from tube axis (outer field lines) are displayed in red, while inner field lines close to tube axis are in white. Note that the inner field lines form the main body of a filament while the outer field lines overlie the main body by forming a coronal arcade and underlie the main body by forming barbs. The middle panels show top and side views of the main body formed by the inner field lines (blue) and barbs formed by the outer field lines (green and orange). The bottom panels schematically show the spatial relationship among the coronal arcade, main body and barbs of a dextral filament. The sinistral case is given by the mirror symmetry of the dextral case (flux tube has right handedness). From Magara (2007Jump To The Next Citation Point), except for the top panel, which is now given from a different viewing angle (same 3D plot), and the bottom-left panel, which is adapted from Martin (1998).

Theoretical studies of the magnetic structure containing a filament has been done extensively. Low and Hundhausen (1995Jump To The Next Citation Point) and Low (1996Jump To The Next Citation Point) 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 17View Image, top-left panel). Antiochos et al. (1994) and DeVore and Antiochos (2000Jump To The Next Citation Point) demonstrate that a sheared arcade contains a filament, which is formed by shear flows around polarity inversion line (Figure 17View Image 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 17View Image 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 18View Image). 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 18View Image). Also, it was shown that the emergence of U-loops distributed below the axis causes apparent flux cancellation on the surface (Magara, 2011Jump To The Next Citation Point). 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).

3.2.2 Sigmoid

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., 1992aJump To The Next Citation Point) or coronal mass ejection (CME) (Canfield et al., 1999Jump To The Next Citation Point; 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.

View Image

Figure 19: Double J-shaped structure formed inside an emerging flux tube. Colors of field lines represent the strength of current density measured at their chromospheric footpoints. Top, side and perspective views are presented at top-left, bottom-left, and bottom-right panels, respectively. The top-right panel shows the distribution of current density at a chromospheric plane (from Magara, 2004Jump To The Next Citation Point).

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 Ω-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 Ω-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 19View Image, Magara, 2004Jump To The Next Citation Point). 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 20View Image, Magara, 2006Jump To The Next Citation Point). Here the illuminated field lines have a relatively high current density distributed at their chromospheric footpoints.

View Image

Figure 20: Magnetic structure formed by a (a) highly and (b) weakly twisted flux tube. The bottom panels (c, d) show an observed example in each case (from Magara, 2006).

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 (2001Jump To The Next Citation Point) 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 21View Image).

View Image

Figure 21: (a) Sigmoid observed by the soft X-ray telescope on board Yohkoh (courtesy of the Yohkoh team members). (b) Two-dimensional map of gas density (color map) and the magnetic field (black arrows) projected onto the middle plane at y = 0, which is obtained by the emergence of a twisted flux tube (initially placed along the y-axis). The red, white, and orange dots represent the locations where red (above the axis), white (axis), and orange (below the axis) field lines cross this plane. (c) Three-dimensional viewgraph of these magnetic field lines. The bottom map at z = 0 shows horizontal velocity field (white arrows), vertical velocity (color map), and vertical magnetic field (contour lines). (d) Top view of (c), where a color map shows the absolute value of the current density measured at z = 3 (chromospheric level) (from Magara and Longcope, 2001Jump To The Next Citation Point).
View Image

Figure 22: Sigmoid formation by an emerging flux tube. Shown is the cross section of the flux tube.

Figure 22View Image schematically explains the sigmoid formation displayed in Figure 21View Image (here the cross section of an emerging flux tube is shown). The upper half of the flux tube contains Ω-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, 2011Jump To The Next Citation Point), 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 Ω-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.

View Image

Figure 23: Comparison between simulations (left and middle columns) and soft X-ray observations (right column). The left column shows the evolution of the isosurface of current density, the middle one shows the corresponding snapshots of the heating term, and the right column shows soft X-ray images at three different times during the evolution of the sigmoidal structure (from Archontis et al., 2009Jump To The Next Citation Point).

The three-dimensional (3D) distribution of current density inside an emerging flux tube was investigated by Manchester IV et al. (2004Jump To The Next Citation Point). 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 23View Image).

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.

3.2.3 Force-free field

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:

j × B = 0, (10 )
where c- j = 4π∇ × B is the current density. Equation (10View Equation) can be transformed to ∇ × B = αB where α is a scalar, and using ∇ ⋅ B = 0 we can obtain B ⋅ ∇ α = 0, so α is constant along each field line. When α is constant all over the magnetic structure, it is called 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 (1972Jump To The Next Citation Point), 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.

3.2.4 Aly–Sturrock conjecture

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 (1991Jump To The Next Citation Point). 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.

  Go to previous page Go up Go to next page