Magnetic reconnection is a fundamental process in plasma physics that allows the transfer of energy from the magnetic field to the plasma in the form of kinetic energy, thermal energy, or particle acceleration (Yamada et al., 2010). The basic process of magnetic reconnection is the following: when two magnetic field lines of opposite directions are close enough, an intense current sheet is created between the two and a topological reorganization of the magnetic field lines occurs. Basically, this mechanism involves a violation of Alfvén's theorem whose origin is the magnetic diffusivity in standard MHD, or for example the Hall effect in collisionless plasmas. In this chapter, we present the elementary mechanism of magnetic reconnection whose main applications range from solar flares – the most violent events in the solar system – to magnetic substorms in planetary magnetospheres which produce spectacular aurorae (see Figure 7.1). Magnetic reconnection is also invoked for stellar coronae, accretion disks, dynamos, and tokamaks, and laboratory experiments have been designed specifically to study this phenomenon, such as the Magnetic Reconnection Experiment (MRX) built in 1995 at the Princeton Plasma Physics Laboratory. Reconnection is actually a fairly general term that is also used in fluid mechanics when the topology of the vorticity lines is modified. Nowadays, the reconnection process is even observed between quantized vorticies in superfluid helium (Bewley et al., 2008).

A Current Sheet in Ideal MHD

We will first consider the two-dimensional stationary magnetic configuration of Figure 7.2 where magnetic field lines of opposite directions are separated by a distance 2l. For example, we can think about two close solar magnetic loops (see Figure 3.4). This external configuration being imposed, one wants to know the properties of the inner region of thickness 2l. We will assume that the norm of

the magnetic field is constant in the external region. From Maxwell's equations, we have

hence for (we assume that the magnetic field varies linearly in the inner region; see Figure 7.2). In other words, a current sheet of thickness 2l appears between the two regions of different magnetic polarity. This current is even more intense given that the sheet is thin and therefore the regions of different magnetic polarity are close.