### 3.3 Magnetic helicity

As we have seen, the topology of magnetic field is one of the key factors in determining the dynamical nature of a magnetic structure. A quantity describing how much a magnetic structure is twisted, sheared and braided is given by the magnetic helicity, which is originally defined as
where is the magnetic field and is the vector potential for . In a semi-infinite region above the surface where the cartesian coordinates is used (-axis is directed upward and is the surface), the magnetic helicity defined by Equation (11) becomes gauge-invariant and is called relative magnetic helicity (Berger and Field, 1984; Finn and Antonsen Jr, 1985) if the vector potential is given by DeVore (2000)
Here is the vector potential for the potential field that has the same vertical magnetic flux at the surface as . The exact form of is
where is the scalar potential for the potential field, given by
Here the potential field is also defined in the semi-infinite space (). In any numerical simulations, the domain of simulation is finite, surrounded by boundaries. To compromise this unavoidable discrepancy, simulations are terminated before the emerging magnetic field reaches the boundaries of the domain so that the emerging field feels that the domain is infinite (still, there are effects of waves and flows reflected by boundaries on emerging field).

The relative magnetic helicity quantitatively describes how much a magnetic structure is sheared compared to a (shearless) potential field. In a right-handed coordinate system the relative magnetic helicity takes a positive (negative) value when a magnetic structure is sheared in a right-handed (left-handed) way, while it take zero in the case of a potential field. Hereafter, we simply use the magnetic helicity to express the relative magnetic helicity.

The helicity flux across the surface is given by

where the first term on the right hand side of Equation (15) represents the shear/twist by horizontal motions and will be called the shear term. The second term, called the emergence term, would clearly vanish in the absence of vertical flows.

Figure 24 shows a result on the injection of magnetic flux, energy and helicity via the emergence of a twisted flux tube. Since the flux tube has left-handed twist, negative magnetic helicity is injected during the emergence of the flux tube. This figure indicates that the injection of magnetic energy and helicity is carried out mainly by the emergence term during the early phase of emergence, while it is done by the shear term during the late phase when the emergence of magnetic field almost terminates (see the time variation of emerged magnetic flux at the top right of the figure). In some case, the emergence term gives net loss of magnetic energy and helicity in the late phase of emergence when a strong downflow arises along emerged loops, dragging magnetic field downward below the surface and causing the net loss (Magara, 2008). Another important result from Figure 24 is that the injection of magnetic flux, energy, and helicity has already saturated before the inner central part of the flux tube expands into the corona and make a well developed magnetic structure. This suggests that the strong energy injection at the surface may not be directly related to the activation of a magnetic structure formed on the Sun; the activation may depend on the magnetic configuration developed in the corona.

We have discussed the injection of magnetic helicity via the emergence of an integrated flux tube, while the emergence of magnetic field sometimes proceeds in the form of a partially split flux tube, as illustrates in Figure 25a (Zwaan, 1985). The dynamic process and associated helicity injection caused by a partially split flux tube were numerically simulated in Magara (2008) where a flux tube composed of two splitting portions forms a quadrupolar-like distribution at the surface. In this simulation the mutual magnetic helicity (Priest and Forbes, 2000) arises when these two portions emerge (Figures 25b – d), and the sign of mutual helicity is opposite to the sign of self helicity contained by those individual portions. We cannot say that this is a general result although the opposite sign of helicity is also observed in real active regions where multiple flux tubes successively emerged (Lim and Chae, 2009; Park et al., 2010). This suggests that these multiple flux tubes might correspond to the splitting portions of a global flux tube, forming multiple flux domains on the surface and a flare is expected to occur at the interface between different flux domains (see Sections 4.2 and 4.3).

Deriving observationally how much magnetic helicity is injected into the atmosphere has widely been done, which is a key to the understanding of the relationship between helicity evolution and the occurrence of active phenomena including flares (Chae, 2001; Moon et al., 2002; Nindos and Zhang, 2002; Kusano et al., 2002; Démoulin and Berger, 2003; Yang et al., 2004; Pariat et al., 2005; Jeong and Chae, 2007; Magara and Tsuneta, 2008). These studies are important in that we can derive the characteristics of helicity evolution and use it to predict the occurrence of active phenomena on the Sun. A recent review by Démoulin and Pariat (2009) is worth reading for those who are interested in this subject.