Under ideal conditions, plasmas have infinite conductivity and the field must be frozen to the fluid, such that its connections are preserved. This is, in general, the case in the solar corona, where the characteristic lengths of the plasma are so high that the diffusion term is negligible in the induction equation, (1.2.9). However, it is possible
slow shocks diffusion region diffusion region vi vi vo vo vi vi vo vo
(a) Sweet-Parker model: slow reconnection regime (b) Petschek model: fast reconnection regime
Figure 1.15: Comparison between (a) the Sweet-Parker model of slow reconnection, where energy conversion happens in a large diffusion region, and (b) the Petschek model, of fast reconnection, where most of the energy conversion takes place in four slow-mode shocks that come out of a small diffusion region.
that, in certain regions, the magnetic diffusivity becomes important locally, allowing non-ideal effects to occur. Two and three dimensional null points are potential locations for that non-idealness to happen.
Magnetic reconnection is the process in which field lines break and then merge with other field lines, allowing them to change their connectivity. The process is directly linked to the diffusion of the field and associated with the release of magnetic energy, which is partly converted into internal energy of the plasma. The characteristics of these processes in two-dimensions are well gathered and exposed in Priest and Forbes (2000). Reconnection may occur in presence of high electric fields and electric currents. Some effects of magnetic reconnection can be: 1) the partial conversion of magnetic energy into heat, a process known as ohmic dissipation, 2) acceleration of plasma by converting magnetic energy into bulk kinetic energy, 3) Generation of shock waves and current filamentation and 4) changes of global connections of the field lines, that allow the field to relax to a lower energy state, affecting the paths of fast particles and heat, which are generally directed along magnetic field lines.
Magnetic reconnection may be studied by either resistive (non-ideal) MHD models, with the classical ohmic dissipation, which can be mainly applied for highly collisional plasmas, or using particle models, involving multi- fluid theory, applicable in the higher corona, where collisionless effects dominate. Nevertheless, even in the latter case, an MHD approach can give a valid characterization and provides a macroscopical view of the general process. A very brief history of the study of magnetic reconnection starts with Dungey (1953), who showed that the collapse of a magnetic X-point would create a current sheet capable of accelerating particles and generating heat in solar flares (pointed out earlier by Cowling, 1953), and first stated that “lines of force can be broken and rejoined”. The first model came with Parker (1957) and Sweet (1958), who studied the process of two bipolar magnetic fields coming together. Parker was the first to use the term reconnection of field lines. They showed that the reconnection rate was equivalent to the inflow plasma speed, which turned out to be way too small for solar flares. This mechanism is now referred as to slow reconnection. Furth et al. (1963) showed that resistive instabilities occur in a one-dimensional current sheet. This is known as the tearing mode instability. Then, Petschek (1964) showed how conversion of magnetic energy into heat and kinetic energy was also possible in slow-mode shock waves, generated by a diffusion region much smaller than the one formed in the Sweet-Parker model. This was the first of many regimes of fast reconnection. Biskamp (1986) found a different solution to Petschek, which, finally
Priest and Forbes (1986) included into a whole family of solutions for both fast and slow reconnection, with the cases of Petschek and Biskamp as particular solutions. Figure 1.15 shows a comparison between the Sweet-Parker and the Petschek models.
In three dimensions, magnetic reconnection is very different from reconnection in 2D (Priest et al., 2003). Schindler et al. (1988) showed how, in contrast to the two-dimensional case, in three dimensions, reconnection can happen either at the location of magnetic null points or in absence of them. Instead, the condition for reconnection to occur is that, within a region of non-idealness, the integral along a field line of the electric field parallel to it is different from zero,
Z
Ekds6= 0. (1.5.5)
In fact, if the region of non-idealness is a single isolated region with a singly peaked form for this integral, then its maximum value gives the rate of three-dimensional reconnection. The different regimes of three-dimensional reconnection may be classified as follows. 1) torsional fan and torsional spine reconnection, where torsional motions concentrate the current along the spine or in the plane of the fan (Pontin et al., 2004; Priest and Pontin, 2009; Wyper and Jain, 2010), 2) spine-fan reconnection, where shearing motions concentrate the current along both (Pontin et al., 2005; Priest and Pontin, 2009), 3) separator reconnection, where current concentrates along the separator line that joins two nulls and represents the intersection of two separatrix surfaces (Parnell et al., 2010), and 4) QSL reconnection, where reconnection occurs at quasi-separatrix layers, where the mapping of magnetic field lines changes continuously but extremely rapidly (Priest and D´emoulin, 1995).