3.1 Emergence of magnetic field (flux emergence)

In this section we start with the morphology of flux emergence, then explain the dynamic nature of flux emergence and the characteristics of magnetic structures formed via flux emergence.

3.1.1 Morphology

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 12View Imagea). 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 12View Image shows the evolution of an active region observed in white light and Hα. 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.

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Figure 12: Evolution of an active region, showing the development of magnetic shear. The left panels are Hα images and the right panels are white-light images (from Kurokawa, 1989).

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.

3.1.2 Dynamics

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 13View Image). 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., 1989bJump To The Next Citation Point, 1990b; Tajima and Shibata, 1997Jump To The Next Citation Point), which is driven by the Parker (i.e., buoyancy) instability (Parker, 1955). The solution shows how the rise velocity (vz) and gas density (ρ) of plasma, and the strength of horizontal magnetic field (Bh) depend on height, given by

vz ∼ ωnz, ρ ∝ z− 4, Bh ∝ z− 1. (5 )
Here, z is the height and ωn represents the growth rate in the nonlinear phase of the Parker instability, given by
1- −1∕2cs ωn ∼ 2 ωl ≃ 0.1 (1 + 2 β∗) Λ , (6 )
where ωl, β ∗, cs, and Λ are the linear growth rate of the instability, plasma beta, adiabatic sound velocity, and pressure scale height of the flux sheet initially assumed. The solution explains that the drain of mass along magnetic field evacuates magnetic loops where the gas density decreases with height while both the rise velocity and Alfvén velocity increases with height. A comparison of this analytic result to an MHD simulation is given in Figure 13View Image.
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Figure 13: Top: Two-dimensional simulations of flux emergence in a flux-sheet configuration. Contours represent magnetic field lines. Bottom: Distributions of rising velocity (vz), Alfvén velocity (v A), density (ρ), and horizontal magnetic field (B x) with height during the expansion of emerging magnetic field. The dashed curves in the graphs (c) and (d) at the bottom panel represent the analytic curves of the self-similar solution (from Shibata et al., 1989bJump To The Next Citation Point).

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 (1996Jump To The Next Citation Point), which is further developed by Miyagoshi and Yokoyama (2004) where thermal conduction is taken into account.

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Figure 14: (a) A rising flux tube becomes flattened when it approaches the surface. (b) A developed magnetic sheet and its horizontal extent (λ) (from Magara, 2001Jump To The Next Citation Point). (c) Histogram of the observed horizontal extent of emerging magnetic field (from Pariat et al., 2004Jump To The Next Citation Point).

Another type of two-dimensional simulations of flux emergence has also been done in a flux-tube configuration (Krall et al., 1998; Magara, 2001Jump To The Next Citation Point). Magara (2001Jump To The Next Citation Point) 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 14View Imagea). 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 14View Imageb), which is given by

c2s,i λ = 4π g--, (7 )
where cs,i and g are the photospheric isothermal sound velocity and gravitational acceleration. This length is about 2 Mm in the photosphere, and observations also suggested that there is a threshold of the horizontal extent of emerging magnetic field which is about 2 Mm (Pariat et al., 2004, see Figure 14View Imagec). A more precise analysis of successful emergence has been obtained by taking the stratification of magnetic field into account, that is,
( ) ∂ γ 2 k2⊥ − Λ---lnB > − --βδ + k∕∕ 1 + -2- , (8 ) ∂z 2 kz
where Λ, γ, β, and ( ) δ = ∂∂-lnln-TP − ∂∂-lnlnTP- ad are the photospheric pressure scale height, ratio of specific heat, plasma beta, and excess of superadiabaticity, while k ∕∕ and k ⊥ are the wavenumbers in two horizontal directions in unit of local scale height (the former is the field-aligned component and the latter the cross-field component) and kz is the wavenumber in the vertical direction (Newcomb, 1961; Acheson, 1979). Using this formula, Archontis et al. (2004Jump To The Next Citation Point) and Murray and Hood (2008Jump To The Next Citation Point) have made an extended survey of successful emergence.
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Figure 15: A magnetic structure formed by the emergence of a twisted flux tube. Two field lines (outer and inner) composing the flux tube are presented. The arrows on these field lines represent flow velocity. The outer field line forms an expanding loop, below which the inner field line forms a sheared loop which is less dynamic (from Magara and Longcope, 2003Jump To The Next Citation Point).

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.

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Figure 16: Difference in evolution between the outer and inner field lines composing a twisted flux tube (from Magara and Longcope, 2003Jump To The Next Citation Point).

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, 2003Jump To The Next Citation Point, see Figure 15View Image). Figure 16View Image schematically explains this. The outer field lines form Ω-loops with a large aspect ratio of height to footpoint separation on the surface, while the inner field lines form relatively flat Ω-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 (2004Jump To The Next Citation Point). The rise velocity of a field line with the curvature κ at the top is given by

Λ √ --- ⟨vrise⟩ = --m- gκ, (9 ) 2
where ⟨vrise⟩ means an average in time, Λm is the scale height of magnetic-field strength and g is the gravitational acceleration. This indicates that the rise velocity of an emerging field line depends on a geometrical property (curvature) as well as stratification of magnetic field.

3.1.3 Latest progress

Continuously increasing computational power enables to investigate flux emergence in the three dimension. Fan (2001Jump To The Next Citation Point) 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 Ω 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 14View Imagea), 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. (2004Jump To The Next Citation Point, 2005, 2007), Isobe et al. (2005Jump To The Next Citation Point, 2006Jump To The Next Citation Point), and Galsgaard et al. (2005, 2007Jump To The Next Citation Point) 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, 2001Jump To The Next Citation Point; Manchester IV et al., 2004Jump To The Next Citation Point; Manchester IV, 2007Jump To The Next Citation Point) 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).

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