Introduction to magnetic helicity

This paper reviews several aspects of magnetic helicity, including its history from Gauss to the present, its relation to field structure, its role in Taylor relaxation, and how it is defined for sub-volumes of space. Also, its importance in solar physics will be discussed. Magnetic helicity quantifies various aspects of magnetic field structure. Examples of fields possessing helicity include twisted, kinked, knotted, or linked magnetic flux tubes, sheared layers of magnetic flux, and force-free fields. Helicity thus allows us to compare models of fields in different geometries, avoiding the use of parameters specific to one model. Magnetic helicity is conserved in ideal magnetohydrodynamics and approximately conserved during reconnection. Often, physical systems are described in terms of interacting parts: for example one might separate the solar magnetic field into an interior field and an atmospheric (coronal) field. We can obtain insight into how the parts of a magnetic system interact by describing how magnetic helicity is transferred from one part to another. This transfer is governed by an equation similar to Poynting's theorem for the transfer of energy through boundaries. In a confined volume, widespread reconnection may reduce the magnetic energy of a field while approximately conserving its magnetic helicity. As a result, the field relaxes to a minimum energy state, often called the Taylor state, where the current is parallel to the field. Such relaxation processes are important to both fusion and astrophysical plasmas. Recent observations show that structures in the northern hemisphere of the sun have predominantly negative helicity, and structures in the south have predominantly positive helicity. Helicity injection by differential rotation may explain this dependence.