5.1 Magnetic helicity

Magnetic helicity is a quantity closely related to a property of the force-free field (Woltjer, 1958), and is defined by
∫ Hm = A ⋅ BdV, (43 ) V
where B = ∇ × A and A is the vector potential. When B is given, A is not unique and a gradient of any scalar function can be added without changing B. Such gauge freedom does not affect the value of Hm if the volume V is bounded by a magnetic surface (i.e., no field lines go through the surface). Figure 8View Image shows simple torus configurations and their magnetic helicities. As can be guessed from the figures, magnetic helicity is a topological quantity describing how the field lines are twisted or mutually linked, and is conserved when resistive diffusion of magnetic field is negligible. In the case of the solar corona, the bottom boundary (the photosphere) is not a magnetic surface, and field lines go through it. Even under such conditions, an alternative form for the magnetic helicity which does not depend on the gauge of A can be defined (Berger and Field, 1984; Finn and Antonsen Jr, 1985). On the Sun one finds the hemispheric helicity sign rule (see, e.g., Pevtsov et al., 1995; Wang and Zhang, 2010, and references therein). For various features like active regions, filaments, coronal loops, and interplanetary magnetic clouds the helicity is negative in the northern and positive in the southern hemisphere.
View Image

Figure 8: Magnetic helicity of field lines in torus configuration: untwisted (left), twisted by T turns (middle), and two untwisted but intersecting tori (right). Φ stands for the total magnetic flux.

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