Figure 26 shows a current sheet where magnetic reconnection occurs. The speed of reconnection (reconnection rate) is measured as the magnetic flux reconnecting per unit time, . Here, is the magnetic flux per unit length in the direction perpendicular to the plane containing the current sheet. For a steady state, the Faraday’s equation and Ohm’s law give the reconnection rate as follows:

Here and are the electric field and current density in a diffusion region with finite magnetic diffusivity , and in a steady state we have , implying that outside the diffusion region (inflow region) with zero diffusivity is equal to in the diffusion region. Also is the inflow speed of plasma entering the diffusion region and is the magnetic field, both of which are measured in the inflow region. In deriving Equation (16), we assume that the diffusivity takes nonzero value only in the diffusion region.The reconnection rate given by Equation (16) can be non-dimensionalized using the Alfvén Mach number in the inflow region;

where is the Alfvén velocity measured in the inflow region. We then define the reconnection time in a current sheet of the length : where is the Alfvén transit time. When we take as the typical length of coronal magnetic structure (10In the Sweet–Parker model for magnetic reconnection (Sweet, 1958; Parker, 1957), the non-dimensionalized reconnection rate is given by

where is called the Lundquist number (magnetic Reynolds number with the typical velocity ). typically takes a value of 10

Petschek proposed another steady model for magnetic reconnection that enables faster energy release than the Sweet–Parker model. The reconnection rate in the Petschek model is given by

which is 0.01 – 0.1 in the corona (Petschek, 1964). This value gives a reconnection time comparable to the time scale of a flare. Forbes and Priest (1987) later extended the model proposed by Petschek. However, the evolution to the steady state assumed in the Petschek’s model has not been fully understood yet, and the controlling factor of the reconnection rate is still a controversial issue (Biskamp, 1993; Priest and Forbes, 2000). In this respect, Forbes and Priest (1987) did an extensive investigation into the system where two-dimensional steady reconnection is operated. They show that the speed of magnetic reconnection is controlled by the spatial pattern of flow in the inflow region. They have demonstrated that a diverging pattern of inflow produces a flux pile-up configuration around the diffusion region, while a converging pattern of inflow gives a magnetic configuration that is similar to the one suggested by the Petschek model. In order to derive the reconnection rate from observations (Isobe et al., 2002; Nagashima and Yokoyama,
2006), it is crucial to obtain the inflow speed of plasma around a current sheet (Yokoyama et al., 2001;
Narukage and Shibata, 2006; Hara and Ichimoto, 1997; Hara et al., 2006; Miklenic et al., 2009, and
references therein). Suppose that the reconnection rate is , which is suggested by the
Petschek model, the inflow speed would be 10 – 100 km s^{–1} because the Alfvén velocity in the corona is
estimated as

The evolution leading to the steady state assumed in the Petschek model has widely been investigated with the help of numerical simulations. Here we discuss two types of reconnection: (i) driven-type reconnection (Sato and Hayashi, 1979), and (ii) spontaneous-type reconnection (Ugai and Tsuda, 1977). In the driven-type reconnection, fast reconnection is achieved by an external object that drives an inflow toward a current sheet. Sato and Hayashi (1979) performed two-dimensional MHD simulations in which an inflow is driven as boundary condition, showing that the Petschek-type reconnection occurs when the resistivity is locally enhanced in a current sheet. Biskamp (1986) also followed the concept of the driven-type reconnection, but the result shows that when a uniform resistivity is assumed the Petschek-type reconnection does not arise, instead the Sweet–Parker-type reconnection occurs.

For the spontaneous-type reconnection, resistivity is locally enhanced inside a current sheet via some microscale instabilities attributed to the nature of a current sheet, then an inflow spontaneously arises without any external sources, which draws plasma toward a current sheet. In this case the nature of a current sheet is the primary factor causing fast reconnection. Ugai and Tsuda (1977) performed two-dimensional MHD simulations of the spontaneous-type reconnection, and successfully reproduced the Petschek-type reconnection by locally enhancing resistivity inside a current sheet. Their result was later confirmed by Scholer (1989) showing that fast reconnection is closely related to the local enhancement of resistivity. Yokoyama and Shibata (1994) presented a result on the role of locally enhanced resistivity in the Petschek-type reconnection.

In either driven or spontaneous case, the local enhancement of resistivity seems an essential process, by which the Petschek-type reconnection occurs (Kulsrud, 2001; Uzdensky and Kulsrud, 2000). Also there is an issue about the origin of the external source assumed in the driven-type reconnection. This is crucial when we apply the driven-type reconnection to space plasma because we should identify the object that drives an inflow in a free space occupied by space plasma. In the case of laboratory plasma, we may easily identify such external objects outside a current sheet.

Very recently, it was shown that the classical Petschek-type solution with fast stationary magnetic reconnection is possible with a spatially uniform resistivity (Baty et al., 2009a,b) when a non-uniform viscosity distribution is assumed, which gives a new insight into the relation between the distribution of resistivity and the speed of reconnection. Also the self-similarity aspect of Petschek-type reconnection has been investigated (Nitta, 2010, and references therein).

Following the discussion presented above, we then focus on how the resistivity is locally enhanced in a current sheet. Firstly, it should be noticed that even if the resistivity is uniformly distributed in a current sheet, the sheet is subject to the tearing instability (Furth et al., 1963; Steinolfson and van Hoven, 1984; Horton and Tajima, 1988), which eventually introduces nonuniformity into the current sheet where a series of magnetic islands are formed.

These magnetic islands tend to coalesce together to make a large magnetic island, and during this coalescencing process the magnetic energy is efficiently converted to kinetic energy (Bhattacharjee et al., 1983; Sakai and Ohsawa, 1987; Tajima et al., 1987). Nonsteady reconnection associated with multiple magnetic islands (Choe and Cheng, 2000) often causes impulsive bursty reconnection (Priest, 1985). Recently, Karlický and Bárta (2007) and Bárta et al. (2008) performed a series of simulations where tearing process is incorporated into the evolution of a flare.

In a developed phase of this process, the thickness of a current sheet is locally reduced so as to produce an extremely thin layer (Magara and Shibata, 1999). Tanuma et al. (2001) investigated this thinning process in detail by performing two-dimensional MHD simulations, which reproduces the long-term evolution of a current sheet subject to the tearing instability. They have shown that secondary tearing instability occurs in an extremely thin layer. It is suggested that successive tearing processes might produce a hierarchy of multi-scale magnetic islands coexisting in a current sheet, which becomes a fractal current sheet (Shibata and Tanuma, 2001; Nishizuka et al., 2009), as shown in Figures 27 and 28.

If the thickness of a current sheet is reduced to microscales such as the ion’s Larmor radius or ion’s inertial length, kinetic processes such as the decoupling of ions and electrons as well as the interaction of waves and charged particles become important. These kinetic processes might enhance the resistivity significantly over the Spitzer resistivity based on collisions among particles (Spitzer, 1962). The coupling between macroscopic (MHD) and microscopic (kinetic) processes is important during magnetic reconnection, which has intensively been investigated by theory, simulation, and laboratory (Treumann and Baumjohann, 1997; Biskamp et al., 1995; Horiuchi and Sato, 1999; Yamada et al., 2000; Birn et al., 2001; Shay et al., 2001; Bhattacharjee et al., 2003; Drake et al., 2003; Hanasz and Lesch, 2003; Heitsch and Zweibel, 2003; Rogers et al., 2003; Ji et al., 2004; Craig and Watson, 2005).

It has been speculated that the speed of magnetic reconnection is related to the behavior of magnetic island, i.e., plasmoid formed in a current sheet (Shibata, 1999). While staying in a current sheet, a plasmoid significantly reduces the speed of reconnection by inhibiting an inflow accompanied by magnetic flux from entering a current sheet. When a plasmoid moves out of a current sheet, substantial amount of magnetic flux can come into the sheet, thereby triggering magnetic reconnection. This facilitates the ejection of a plasmoid via strong reconnection outflow (reconnection jet), which in turn enables new magnetic flux to continuously enter the current sheet. The positive feedback between plasmoid ejection and inflow enhancement contributes to producing fast reconnection, and eventually a plasmoid is ejected at about the Alfvén velocity. The whole process is named as plasmoid-induced reconnection by Shibata (1999). Recently, a detailed relation between plasmoid velocity and reconnection rate has been investigated by performing a series of numerical simulations (Nishida et al., 2009).

Living Rev. Solar Phys. 8, (2011), 6
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