### 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
where and is the vector potential. When is given, is not unique and
a gradient of any scalar function can be added without changing . Such gauge freedom
does not affect the value of if the volume is bounded by a magnetic surface (i.e.,
no field lines go through the surface). Figure 8 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
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.