11 Magnetic helicity interpreted and Woltjer-Taylor relaxation
11.1 Introduction
The previous chapter introduced the concept of magnetic helicity by using arguments de-
riving from the energy principle and showed that global helicityK=
∫
d^3 rA·Bis a
conserved quantity in an ideal plasma. We show in this chapter that magnetic helicity has
an important topological interpretation and furthermore that the conservation property of
global helicity is more robust than magnetic energy conservation. By robust,it is meant
that when some dissipation exists and the plasma is no longer ideal, helicity conservation
holds up much better than magnetic energy conservation, i.e., helicity isnearly conserved
even though energy is not conserved. This difference in the relative loss rates of energy and
helicity provides the basis for the Woltjer-Taylor theory which predicts a minimum-energy,
conserved-helicity final state which any arbitrary initial state will always relax towards.
Relaxation theory has been very successful for predicting the general behavior of many
laboratory, space, and astrophysical plasmas. Finally, it will be shown that helicity can be
manifested in more than one way and in particular that the kink instabilitycan be consid-
ered as a conversion of helicity from one form (twist) to another form (writhe).
11.2 Topological interpretation of magnetic helicity
11.2.1Linkage helicity
Consider the two thin linked untwistedflux tubes labeled asflux tube #1 andflux tube #2
shown in Fig. 11.1. The respectivefluxes areΦ 1 andΦ 2 and the tube axes follow the
contoursC 1 andC 2 .and the tube volumes areV 1 andV 2 .The magnetic field is assumed to
vanish outside of theflux tubes.
The helicity of the volumeVcontaining the two linkedflux tubes is
K=
∫
VA·Bd^3 r. (11.1)