Morphologically speaking, the field-aligned current introduces distortion to a magnetic structure where magnetic field lines tend to be aligned with the so-called polarity inversion line defined as the boundary between positive and negative polarity regions on the surface. When there is no field-aligned current, and when the inversion line is nearly straight, field lines overlie the inversion line transversely, forming a potential field without any free energy. The configuration of magnetic field therefore indicates whether field-aligned current (or free energy) exist or not in a magnetic structure (it is not generally true that field lines overlie the inversion line transversely in a potential field; for example, if the inversion line is bent, the angle between field lines and the inversion line deviates significantly from 90°, while there are cases with field lines locally almost parallel to the inversion line in quadrupolar configurations).
The morphology of flux emergence has been studied by observing emerging flux regions (EFRs) on the Sun (Kurokawa, 1987; Tanaka, 1991; Leka et al., 1996; Strous et al., 1996; Ishii et al., 1998; Otsuji et al., 2007). Observations show that an arch filament system (AFS) appears in the early phase of flux emergence (Figure 12a). The top of an AFS rises at about 10 km s–1 in a chromospheric level (Bruzek, 1969; Chou and Zirin, 1988), while slower rising motions (about 0.1 – 1 km s–1) have been observed in the photosphere (Tarbell et al., 1988). In the late phase of flux emergence, a dark filament is sometimes observed above the polarity inversion line, suggesting that a sheared magnetic structure is formed (see Section 3.2.1). Figure 12 shows the evolution of an active region observed in white light and H. An important result from those observations is that the emerging magnetic field is less sheared in the early phase of flux emergence, while a sheared structure develops in the late phase.
The evolution of EFRs observed on the Sun provides the key information on the subsurface structure of emerging magnetic field. As we mentioned in the introduction, it has been suggested that the magnetic field is confined to form thin flux tubes in the convection zone. The swirling motions of convective plasma in helical turbulence might add some twist to these flux tubes (Longcope et al., 1998), and the twisted field lines naturally generate the field-aligned current. Also flux tubes might be twisted enough to keep their coherence when they rise through the convection zone (Emonet and Moreno-Insertis, 1998; Cheung et al., 2006; Fan, 2008). An idealized model of such a twisted flux tube is the so-called Gold–Hoyle flux tube (Gold and Hoyle, 1960), in which field lines are uniformly twisted while the current density takes the highest value at the axis of the flux tube and decreases toward the boundary of the flux tube. Assuming that such a twisted flux tube emerges into the surface, part of the flux tube with less field-aligned current first appears and forms a potential field-like structure, which is reminiscent of an AFS observed on the surface. As emergence proceeds, inner central part of the flux tube that contains strong field-aligned current appears, forming a sheared arcade. This thought experiment presents a scenario of forming a sheared magnetic structure with free energy in the corona. The dynamic process suggested by this scenario will be discussed in the succeeding sections.
Recently, much efforts have been made to clarify the dynamic nature of flux emergence using numerical simulations. These simulations had first been performed in two dimension (see Figure 13). Shibata et al. (1989a) performed simulations in a flux-sheet configuration to reproduce several key features of emerging magnetic field and associated flow. They derived a self-similar solution expressing the expansion of emerging magnetic field (Shibata et al., 1989b, 1990b; Tajima and Shibata, 1997), which is driven by the Parker (i.e., buoyancy) instability (Parker, 1955). The solution shows how the rise velocity () and gas density () of plasma, and the strength of horizontal magnetic field () depend on height, given by
Similar two-dimensional simulations have been performed by Nozawa et al. (1992) to study the effect of convection on flux emergence. Shibata et al. (1990a) studied the convective collapse (Parker, 1978; Spruit and Zweibel, 1979) that occurs at photospheric footpoints of emerged loops, showing that the field strength becomes a kilo Gauss at the footpoints. The interaction of emerging and preexisting coronal fields has also been investigated by Yokoyama and Shibata (1996), which is further developed by Miyagoshi and Yokoyama (2004) where thermal conduction is taken into account.
Another type of two-dimensional simulations of flux emergence has also been done in a flux-tube configuration (Krall et al., 1998; Magara, 2001). Magara (2001) demonstrates that a flux tube rising through the convection zone becomes flattened when it approaches the surface where the nature of the background gas layer changes from a convectively unstable state to a stable one. This is because when the top part of a flux tube comes close to the surface, it stops rising while the bottom part still continues to rise, making the flux tube extend horizontally to form a flux sheet-like structure just below the surface (see the middle panel of Figure 14a). At the same time, the mass contained in the flux sheet-like structure is squeezed out, locally reducing the density below the surface. By applying the concept of the Rayleigh–Taylor instability to this region (high density region lies on a flux sheet with low gas density), we show that the flux sheet can emerge when its horizontal extent becomes greater than the critical wavelength (see Figure 14b), which is given byet al., 2004, see Figure 14c). A more precise analysis of successful emergence has been obtained by taking the stratification of magnetic field into account, that is, et al. (2004) and Murray and Hood (2008) have made an extended survey of successful emergence.
The emergence in a flux-tube configuration is significantly different from the emergence in a flux-sheet configuration. In a flux-tube configuration field lines have different geometric shapes depending on their locations inside the flux tube, and this causes the difference in evolution among these field lines. To see how different it is, we should know the relation between the dynamic nature and geometrical shape of emerging field lines, which is demonstrated below.
The outer field lines composing a twisted flux tube, which are located near the boundary of the flux tube, have helical structure, while the inner field lines close to tube axis have a strong axial component. This geometrical difference between the outer and inner field lines causes different dynamic behavior of these field lines after they emerge (Magara and Longcope, 2003, see Figure 15). Figure 16 schematically explains this. The outer field lines form -loops with a large aspect ratio of height to footpoint separation on the surface, while the inner field lines form relatively flat -loops with a smaller aspect ratio than the outer field lines. In the outer field lines, a diverging downflow is strong because they have large curvature at the top so that the gravity works effectively, enhancing magnetic buoyancy and making these field lines continuously expand. On the other hand, a diverging flow is weak along the inner field lines in a flatter shape, where sometimes the mass even accumulates somewhere at the field line, forming dipped structure. Accordingly, while the outer field lines form a continuously expanding arcade, the inner field lines form a quasi-static structure below the overlying arcade.
A quantitative analysis on the dynamic behavior of emerging field lines in a flux-tube configuration has been made by Magara (2004). The rise velocity of a field line with the curvature at the top is given by
Continuously increasing computational power enables to investigate flux emergence in the three dimension. Fan (2001) simulated the pattern of surface flows driven by the emergence of a twisted flux tube and compared it with observations. Abbett and Fisher (2003) present an integrated simulation where a subsurface convection model and a coronal model are combined. They have confirmed that emerging magnetic field tends to be relaxed to a force-free field state in the chromosphere and corona. Nozawa (2005), Murray et al. (2006) and Murray and Hood (2007, 2008) have done an extended survey of flux emergence by changing the subsurface configuration of magnetic field.
One of the issues related to flux emergence is the behavior of the axis of an emerging flux tube. It can be expected that the emergence of the axis becomes easy when the axis is strongly bent and has an shape because the mass drains efficiently along the axis, thereby enhancing buoyancy. This conjecture has been confirmed by a series of works: in Magara (2001) which keeps a straight axis in the horizontal direction (2.5-dimensional simulation), the axis does not emerge (see Figure 14a), while when a flux tube is assumed to have a curved axis, the axis emerges. In fact, the emergence of the axis proceeds more efficiently when a curved (convex-up) flux tube is initially assumed (Hood et al., 2009; MacTaggart and Hood, 2009). Archontis et al. (2004, 2005, 2007), Isobe et al. (2005, 2006), and Galsgaard et al. (2005, 2007) have studied the interaction of emerging and preexisting fields in various three-dimensional configurations (see Section 4.3). A series of works done by Manchester (Manchester IV, 2001; Manchester IV et al., 2004; Manchester IV, 2007) have shown the origin of shear flows observed on the surface (see Section 4.3). Recently, studies taking realistic factors into account such as radiation, thermal conduction, viscosity and partial ionization, have enabled to make a detailed comparison between simulations and observations (Leake and Arber, 2006; Cheung et al., 2007, 2008; Abbett, 2007; Hansteen et al., 2007).
Living Rev. Solar Phys. 8, (2011), 6
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