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:
The reconnection rate given by Equation (16) can be non-dimensionalized using the Alfvén Mach number in the inflow region;reconnection time in a current sheet of the length : 4 – 105 km), is something between 10 and 100 s.
In the Sweet–Parker model for magnetic reconnection (Sweet, 1958; Parker, 1957), the non-dimensionalized reconnection rate is given by14 in the corona, so . This gives from Equation (18), which is much longer than the typical time scale of a flare (102 – 104 s). Hence, this model cannot explain rapid energy release in a flare (Parker, 1963).
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
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 as28 erg/s is a typical value of energy-release rate in the impulsive phase of a flare (e.g., Masuda et al., 1994).
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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