Magnetic Reconnection

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Magnetic Reconnection

The solar corona has dynamic boundary conditions: (1) The solar dynamo in the interior of the Sun constantly generates new magnetic flux from the bottom of the con- vection zone (i.e., the tachocline) which rises by buoyancy and emerges through the photosphere into the corona, (2) the differential rotation as well as convective motion at the solar surface continuously wrap up the coronal field with every rotation, and (3) the connectivity to the interplanetary field has constantly to brake up to avoid excessive magnetic stress. These three dynamic boundary conditions are the essential reasons why the coronal magnetic field is constantly stressed and has to adjust by restructuring the large-scale magnetic field by topological changes, called magnetic reconnection processes. Of course, such magnetic restructuring processes occur wherever the magnetic stresses build up, e.g., in the canopy-like divergent field in the transition region, in highly tangled coronal regions in active regions, or at coronal hole boundaries. A classical example is a trans-equatorial coronal hole that sometimes is observed to rotate almost rigidly during several solar rotations, although the underlying photosphere dis- plays the omnipresent differential rotation (in latitude): The shape of the coronal hole can only be preserved quasi-statically, if the photospheric magnetic field constantly dis- connects and reconnects at the eastern and western boundaries. Topological changes in form of magnetic reconnection always liberate free non-potential energy, which is converted into heating of plasma, acceleration of particles, and kinematic motion of coronal plasma. Magnetic reconnection processes can occur in a slowly-changing quasi-steady way, which may contribute to coronal heating ( 9), but more often hap- pen as sudden violent processes that are manifested asflaresandcoronal mass ejections ( 11-17). These dynamic processes are the most fascinating plasma processes we can observe in the universe, displaying an extreme richness of highly dynamic phenomena observable in all wavelengths.

We concentrate here mainly on magnetic reconnection processes in the solar corona, including associated processes in the photospheric and chromospheric boundaries, but there is also a rich literature on magnetic reconnection processes in planetary mag- netospheres (e.g., Kivelson & Russell 1995, 9; Treumann & Baumjohann 1997, 7;

Scholer 2003), in laboratory tokomaks and spheromaks (e.g., Bellan 2002), as well as in other astrophysical objects such as comets, planets, stars, accretion disks, etc. In-

407

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408 CHAPTER 10. MAGNETIC RECONNECTION

troductions into magnetic reconnection in the solar corona can be found in textbooks (Priest 1982, 10.1; Sturrock 1994, 17; Priest & Forbes 2000; Somov 2000, 16-22;

Tajima & Shibata 2002, 3.3), or in the following recent reviews and encyclopedia ar- ticles (Forbes 2001; Schindler & Hornig 2001; Ugai 2001; Hood et al. 2002; Biskamp 2003; Kliem et al. 2003), while specific applications to solar flares can be found in the Proceedings of some Yohkoh conferences (Bentley & Mariska 1996; Watanabe et al. 1998; Martens & Cauffman 2002).

10.1 Steady 2D Magnetic Reconnection

Quasi-steady reconnection of magnetic fields enables the coronal plasma to dissipate magnetic energy, a process that has been proposed to yield direct plasma heating of the corona (e.g., Parker 1963a, 1972, 1979, 1983; Sturrock & Uchida 1981; Van Bal- legooijen 1986) or to supply direct plasma heating in flares (e.g., Sweet 1958; Parker 1963a; Petschek 1964; Carmichael 1964; Sturrock 1966). This concept represents one of the most fundamental building blocks that has been used in many theoretical models of coronal heating and solar flares, which we outline in the following.

When a new magnetic flux system is pushed towards a pre-existing old magnetic flux system, for instance as the solar wind runs into the magnetopause at the Earth’s bow shock, or as a new emerging flux region pushes through the chromosphere up- wards into a pre-existing coronal magnetic field, a new dynamic boundary is formed where the magnetic field can be directed in opposite directions at both sides of the boundary. The magnetic field has then necessarily to drop to zero at the boundary to allow for a continuous change from a positive to a negative magnetic field strength.

Thus the balance between the magnetic and thermal pressure (Eq. 6.2.4) across the neutral boundary layer,

yields a higher thermal pressure ( ) in the neutral layer (where

) than on both sides with finite field strengths and

. In a one-dimensional model we would have an infinite neutral boundary layer. In reality, however, the process of bringing two oppositely-directed magnetic flux systems together will always have a finite area of first contact, which limits the extent of the neutral boundary layer and channels outflows to both sides, so that the simplest scenario is a two-dimensional model as shown in Fig. 10.1, where the lateral inflows (driven by external forces) will create outflows along the neutral line in an equilibrium situation. The plasma- parameter

"!$#

%

#

becomes larger than unity in the central region (because ^'&(

), so that the plasma can flow across the magnetic field lines, which is calleddiffusion region, and is channeled into the outflow regions along the neutral boundary. Outside of the diffusion region the plasma- drops again below unity and the magnetic flux is frozen-in. The highly pointed magnetic field lines in the outflow region experience a high curvature force that tries to smooth out the cusps in the outflow region until a balance between the outward-directed magnetic tension force and the inward-directed magnetic pressure force plus thermal pressure is achieved (see 6.2.2 for Lorentz force

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Figure 10.1: Basic 2-dimensional model of a magnetic reconnection process, driven by two oppositely-directed inflows (in x-direction), which collide in the diffusion region and create oppositely-directed outflows (in y-direction). The central zone with a plasma- parameter of

is called diffusion region (grey box) (Schindler & Hornig 2001).

near X-point). This magnetic field line relaxation process is also calledsling shot effect, which is the basic conversion mechanism of magnetic into kinetic energy. The stationary outflows are sandwiched between twostanding slow shocks(which do not propa- gate). The end result is a thin diffusion region with width and length (Fig. 10.1).

The whole process can evolve into a steady-state equilibrium with continuous inflows and outflows, driven by external forces. Since the Lorentz force creates an electric field in perpendicular direction to the 2D-plane of the flows (i.e. perpendicular to the image plane of Fig. 10.1), a current in the neutral layer is associated with the electric field according to Ohm’s law (Eq. 5.1.10),

which coined the termcurrent sheetfor the diffusion region. The finite resistivity requires, strictly speaking, a treatment in the framework of resistive MHD ( 6.1.5), although the processes outside the diffusion region can be approximated with the ideal MHD equations ( 6.1.3).

10.1.1 Sweet-Parker Reconnection Model

There exists no full analytical solution for the steady-state situation of the reconnection geometry shown in Fig. 10.1 using the full set of resistive MHD equations ( 6.1.5), but separate analytical solutions for the external (ideal MHD) region and special solutions for the (resistive MHD) diffusion region are available that can be matched with some simplifications. One such solution is the Sweet-Parker model (Sweet 1958; Parker 1963a), where it is assumed that the diffusion region is much longer than wide, . For steady, compressible flows (

), it was found that the outflows roughly have

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Alfv´enic speeds,

and that the outflow speed

relates to the inflow speed

reciprocally to the cross- sections and (according to the continuity equation),

and that thereconnection rate , defined as the Mach number ratio of the external inflow speed to the (Alfv´enic) outflow speed , is (with the approximation

,

, and ),

TheLundquist number (ormagnetic Reynolds number) is defined by

#

in analogy to the Reynolds number

# defined for a general fluid velocity . From Eqs. (10.1.4-6) it follows the relation

So, for typical coronal conditions (with a large Lundquist number of

) the reconnection rate is typically

, which yields inflow speeds in the order of

!

km s

#"

$

km s

(with Eq. 10.1.5) and yields extremely thin current sheets with a thickness of

!

# "

%$

(with Eq. 10.1.4). So, a current sheet with a length of

km would have a thickness of only

m. In typical flares, ener- gies of^& ^('

)

erg (Table 9.4, Fig. 9.27) are dissipated over flare durations of^+*

-,

s, which imply much larger dissipation rates than obtained with the Sweet-Parker current sheet,

.

&/

. *

.10

. *

2

4365

2

87:9;5

2

87:9

#/<

5

so the Sweet-Parker reconnection rate is much too slow to explain the magnetic dissipation in solar flare events.

10.1.2 Petschek Reconnection Model

A much faster reconnection model was proposed by Petschek (1964), by reducing the size of the diffusion region to a very compact area

that is much shorter than the Sweet-Parker current sheet ( ), (Fig. 10.2). Summaries of the Petschek model can be found, e.g., in Priest (1982, p.351), Jardine (1991), Priest & Forbes (2000, p.130), Treumann & Baumjohann (1997, p.148), or Tajima & Shibata (2002,

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Sweet-Parker model

δ ∆

Petschek model

Slow shocks δ ∆

Figure 10.2: Geometry of the Sweet-Parker (top) and Petschek reconnection model (bottom).

The geometry of the diffusion region (grey box) is a long thin sheet ( ) in the Sweet- Parker model, but much more compact ( ) in the Petschek model. The Petschek model also considers slow mode MHD shocks in the outflow region.

p.225). Because the length of the current sheet is much shorter, the propagation time through the diffusion region is shorter and the reconnection process becomes faster.

However, in a given external area with size

comparable with the length of the Sweet-Parker current sheet, a much smaller fraction of the plasma flows through the Petschek diffusion region with size , where finite resistivity exists and field lines reconnect. Most of the inflowing plasma turns around outside of the small diffusion region andslow mode shocksarise where the abrupt flow speed changes from

to

in the outflow region (Fig. 10.2, bottom). The shock waves represent an obstacle in the flow and thus are the main sites where inflowing magnetic energy is converted into heat and kinetic energy. Simple energy considerations show that the inflowing kinetic energy is split up roughly in equal parts into kinetic and thermal energy in the outflowing plasma (Priest & Forbes 2000). Petschek (1964) estimated the maximum flow speed by assuming a magnetic potential field in the inflow region and found that the external field

at large distance scales logarithmically with the distance

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,

2

5

Petschek (1964) estimated the maximum reconnection rate at a distance

where the internal magnetic field is half of the external value, i.e.,

#

, which yields with Eq. (10.1.9),

#

so the reconnection rate

#

depends only logarithmically on the magnetic Reynolds number

# . So, for coronal conditions, where the magnetic Reynolds number is very high, i.e.,

, the Petschek reconnection rate is

+,

according to Eq. (10.1.10), yielding an inflow speed of

+,

km s

for typical coronal Alfv´en speeds of

km s

. Thus, the Petschek reconnection rate is about 3 orders of magnitude faster than the Sweet-Parker reconnection rate.

10.1.3 Generalizations of Steady 2D Reconnection Models

The semi-quantitative Petschek model has been generalized in a number of mathe- matically more rigorous formulations that are summarized in Priest (1982) and Priest

& Forbes (2000). The generalizations concern the magnetic field discontinuity in the slow mode shock regions, the matching of flow velocities between the diffusion region and the external region, and the compressibility of the plasma (Green & Sweet 1967;

Petschek & Thorne 1967; Sonnerup 1970; Yeh & Axford 1970; Cowley 1974a,b; Yang

& Sonnerup 1976; Roberts & Priest 1975; Priest & Soward 1976; Soward & Priest 1977; Mitchell & Kan 1978). The structure of the diffusion region was modeled in greater detail (Priest & Cowley 1975; Milne & Priest 1981; Priest 1972; Somov 1992).

A unification of fast, steady, almost-uniform reconnectionsolutions of the MHD equations was accomplished by Priest & Forbes (1986), who derived the following expression for the reconnection rate in the external diffusion region,

2

5 ,

,

which contains the Sweet-Parker, the Petschek (

), and Sonnerup solution (

) as special cases. Solutions with

produce aslow mode compression, solutions with

produceslow mode expansions, also calledflux pile-up regime(Litvinenko 1999b; Litvinenko & Craig 2000; Craig & Watson 2000b), while the intermediate range of

!

produces hybrid solutions of slow mode and fast mode expansions.

The unified solutions have been extended by including nonlinearity effects in the inflow, compressibility, energetics, and reverse currents (Jardine & Priest 1988a,b,c 1989, 1990; Jardine 1991). Thealmost-uniform reconnectionsolutions refer to the magnetic field in the inflow region (for which Petschek assumed a potential field). As an extension,non-uniform reconnectionsolutions for highly curved magnetic field lines in

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Figure 10.3: Numerical simulations of steady 2D reconnection, showing the flow trajectories (top) and magnetic field lines (bottom) in the top-left quadrant of the symmetric configuration shown in Figs. 10.1 and 10.2. The simulations are performed for an external reconnection rate

of and for three different values of the magnetic Reynolds number, (a)

, (b) , and (c) ! " . Note that the current sheet (feature at

bottom right of each frame) becomes more elongated with increasing magnetic Reynolds number

, asymptotically approaching the Sweet-Parker solution (Biskamp 1986).

the inflow region have also been calculated (Priest & Lee 1990; Strachan & Priest 1994). Solutions with no flows across the separatrices have also been found (Craig &

Rickard 1994; Craig & Henton 1994), which confirm theanti-reconnection theorem:

“Steady MHD reconnection in two dimensions with plasma flow across separatrices is impossible in an inviscid plasma with a highly sub-Alfv´enic flow and a uniform magnetic diffusivity” (Priest & Forbes 2000). To circumvent this problem, reconnection solutions have been sought by including viscosity in the central diffusion region and separatrix layers (Priest & Forbes 2000). Alternative 2D models explore also asymmetric geometries (Watson & Craig 1998; Ji et al. 2001), cylindrical geometries (Watson

& Craig 2002), strongly sheared configurations instead of the conventional stagnation point flow topology (Craig & Henton 1995; Craig & McClymont 1997), and solutions for partially ionized plasmas (Ji et al. 2001), applicable to the photosphere. The latter effect is interesting because partially ionized plasmas with ambi-polar diffusion can enhance the reconnection rate (Zweibel 1989; Brandenburg & Zweibel 1994).

10.1.4 Numerical Simulations of Steady 2D Reconnection

Numerical simulations of magnetic reconnection have been performed for a variety of boundary conditions by a number of groups (e.g., Ugai & Tsuda 1977; Sato 1979;

Biskamp 1986; Scholer 1989; Yan et al. 1992, 1993), and analytical solutions have been compared with the numerical results (Forbes & Priest 1987; Lee & Fu 1986; Jin

& Ip 1991). An example is shown in Fig. 10.3, showing the steady-state situation that occurred after formation of the standing shock. Note that numerical simulations

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generally have a much lower magnetic Reynolds number (

$

) than the coronal plasma (

). The simulations by Biskamp (1986) shown in Fig. 10.3 found a slow reconnection rate as predicted by the Sweet-Parker model, as well as an elongation of the current sheet with higher magnetic Reynolds numbers. This simulation could not reproduce the Petschek model, but a fast reconnection rate could be obtained by appropriate choice of the boundary conditions and using non-uniform (en- hanced) resistivity in the diffusion region (Yan et al. 1992), consistent with the Petschek model. A puzzle remains why simulations with uniform resistivity cannot reproduce the analytical solutions of the Petschek type in the limit of a high magnetic Reynolds number (Priest & Forbes 2000).

10.2 Unsteady/Bursty 2D-Reconnection

When the diffusion region becomes too long (such as in the Parker-Sweet model), it goes unstable tosecondary tearing(Furth et al. 1963) and animpulsive bursty regime of reconnection ensues (Priest 1986; Lee & Fu 1986; Kliem 1995; Priest & Forbes 2000, 6-7). Such unsteady reconnection modes are very likely to operate in solar flares, because bursty and intermittent pulses (on time scales of seconds to subsec- onds) have been observed in hard X-ray and radio signatures of particle acceleration during virtually all flares ( 13, 15). We describe in the following a few such unsteady reconnection modes, such as the tearing instability ( 10.2.1), the coalescence instability ( 10.2.2), and their combined dynamics (i.e. the regime of bursty reconnection, 10.2.3). There are also other unsteady reconnection types, such as X-type collapse (Dungey, 1953; Craig & McClymont 1991,1993; Craig & Watson 1992a; Mc- Clymont & Craig (1996), Priest & Forbes, 2000, p.205), resistive reconnection in 3D (e.g. Schumacher et al. 2000; Priest & Forbes, 2000, p.230), or collisionless reconnection (e.g. Drake et al. 1997; Haruki & Sakai, 2001a,b), which have not yet been applied to solar flares, but have been discovered in the Earth’s magnetotail (Øieroset et al. 2001).

10.2.1 Tearing Mode Instability and Magnetic Island Formation

In current sheet formations, resistive instabilities ( 6.3.8) can occur, where the magnetic field lines can move independently of the plasma due to the non-zero resistivity (opposed to thefrozen-flux theorem for zero resistivity). In magnetic reconnection regions with high magnetic Reynolds numbers ( ^# ), where the outward diffusion (on a time scale of

# , with

the width of the current sheet and

the magnetic diffusivity) is much larger than the Alfv´enic transit time

# v , i.e. , three different types of resistive instabilities can occur ( 6.8.3, Fig. 6.11):gravitational, rippling, andtearing mode(Furth et al. 1963). Essen- tially, an Alfv´enic disturbance can trigger an instability before it can be stabilized by magnetic diffusion, when (i.e., for large Reynolds numbers ^# ).

The tearing mode, which has a wavelength greater than the width of the sheet

, has a growth time ^! of

!

)

$

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X O X O X O Magnetic field B

Magnetic field B

Figure 10.4: Magnetic island formation by tearing mode instability in magnetic reconnection region. Magnetically neutral X and O points are formed at the boundary between regions of oppositely directed magnetic field, with plasma flow in the directions indicated by arrows (after Furth et al. 1963).

for wavenumbers in the approximate range

#

. (e.g. see derivations in Furth et al. 1963; Priest 1982, p.272; White 1983; or Sturrock 1994, p.272). Thus, the mode with the longest wavelength has the fastest growth rate,

!

The tearing mode produces magnetic islands in 2D (see Fig. 10.4), or magnetic flux ropes in 2.5D, respectively. These structures saturate in the nonlinear phase of the tearing mode (if coalescence is not permitted) and their subsequent diffusion at the diffusive timescale is extremely slow (since in the corona). The energy release of the tearing-mode instability occurs during the process of island formation.

Tearing modes have been applied to solar flares in a number of theoretical studies (e.g. Sturrock 1966; Heyvaerts et al. 1977; Spicer 1977a,b, 1981a; Somov & Verneta, 1989; Kliem 1990), and numerical MHD simulations have been performed (Biskamp &

Welter 1989). Kliem (1995) estimated the growth time of the tearing mode for coronal conditions (

cm

)

,

"

K, B=200 G, with smallest current sheets half widths of

"

)

cm), which yields ^!

s. This time scale is comparable with the duration of elementary time structures observed in form of hard X-ray pulses and radio type III bursts. Because the tearing mode has a threshold current density orders of magnitude below the threshold of kinetic current-driven instabilities, it will occur first. Continued shearing and tearing may reduce the width of the current sheet until the threshold of a kinetic instability is reached (Kliem 1995).

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t=0

-3 0 3

y

t=0

-3 0 3

t=13

-3 0 3

y

t=13

-3 0 3

t=41

-3 0 3

y

t=41

-3 0 3

t=90

-3 0 3

y

t=90

-3 0 3

t=298

0 5 10 15 20

x -3

0 3

y

t=298

0 5 10 15 20

-3 0 3

Figure 10.5: MHD simulation of the coalescence instability for a Lundquist number of

and a plasma- . The magnetic field is shown in the left panels, the velocity field

in the right panels. The initial resistivity perturbation is shown shaded (Schumacher & Kliem 1997a).

10.2.2 Coalescence Instability

While the tearing mode leads to filamentation of the current sheet, the resulting filaments are not stable in a dynamic environment. If two neighbored filaments approach each other and there is still non-zero resistivity, they enter another instability, thecoa- lescence instability, which merges the two magnetic islands into a single one (Pritchett

& Wu 1979; Longcope & Strauss 1994; Haruki & Sakai 2001a,b). An example of an MHD simulation is shown in Fig. 10.5 (Schumacher & Kliem 1997a). The coalescence

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instability completes the collapse in sections of the current sheet, initiated by the tearing mode instability, and thus releases the main part of the free energy in the current sheet (Leboef et al. 1982). There is no complete analytical description of the coalescence instability, but numerical MHD simulations (Pritchett & Wu, 1979; Biskamp &

Welter, 1979; 1989; Leboef et al. 1982; Tajima et al. 1982,1987; Schumacher & Kliem 1997a) show that the evolution consists of two phases: first the pairing of current filaments as an ideal MHD process, and then a resistive phase of pair-wise reconnection between the approaching filaments. The characteristic time scale of the ideal phase is essentially the Alfv´enic transit time through the distance^'

! between the approaching current filaments,

' !

' ! ' !

,

where ^'

! is the velocity of the approaching filaments. For coronal conditions (say

cm

%)

, B=200 G, ^'

!

km) we estimate coalescence times of

' !

,

s, which is again a typical time for the observed modulation of hard-X pulses and type III electron beams in flares.

10.2.3 Dynamic Current Sheet and Bursty Reconnection

In praxis, the two previously described processes of tearing instability and coalescence instability occur iteratively, leading to a scenario ofdynamic current sheet evolution, also known asimpulsive bursty reconnection(Leboef et al. 1982; Priest 1985a; Tajima et al. 1987; Kliem, 1988; 1995). A long current sheet is first subject to tearing that creates many filaments, while rapid coalescence clusters and combines then groups of closely-spaced filaments, these are then again unstable to secondary tearing, to secondary coalescence, and so forth. MHD simulations reproduce this iterative chain of successive tearing and coalescence events (Malara et al. 1992; Kliem et al. 2000). An example of such a numerical simulation from the study of Kliem et al. (2000) is shown in Fig. 10.8 (magnetic field evolution). Let us review three key studies (Tajima et al. 1987; Karpen et al. 1995; Kliem et al. 2000), where numerical MHD simulations of this process have been applied to solar flares.

Tajima et al. (1987) performed numerical MHD simulations of the nonlinear coalescence instability between current-carrying loops and derived also an analytical model of the temporal evolution of the electromagnetic fields (see also two comprehensive reviews on this subject by Sakai & Ohsawa 1987 and Sakai & De Jager 1996, and references therein). This nonlinear system evolves into an oscillatory relaxation dynam- ics, driven by the interplay of the j

"

B force and the hydrodynamic pressure response, which could be modeled analytically by Sakai & Ohsawa (1987). The oscillatory behavior is very appealing, because it provides a possible explanation for the numerous quasi-periodic time structures observed in radio and hard X-rays during flares. An oscillatory regime of fast reconnection has also been found from other theoretical work on current instabilities in current sheets (Smith 1977) and X-point relaxation (Craig &

McClymont 1991, 1993).

Karpen et al. (1995) performed 2.5-dimensional numerical MHD simulations of shear-driven magnetic reconnection in a double arcade with quadrupolar magnetic

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Figure 10.6:Magnetic field lines near the reconnection region at four different times (565, 575, 585, 595 s) during a strong-shear MHD simulation by Karpen et al. (1995). Note the tearing along the vertical current sheet (first frame), which forms two magnetic islands (second frame), which are ejected from the sheet and merge with the flux systems above or below the sheet (third frame), followed by another tearing plus magnetic island formation (forth frame). (Karpen et al. 1995).

topology. For strong shear, the initial X-point was found to lengthen upward into a current sheet, which reconnects gradually for a while but then began to undergo multiple tearing. Several magnetic islands develop in sequence, move toward the ends of the sheet, and disappear through reconnection with the overlying and underlying field (Fig. 10.6). A second study with similar quadrupolar configuration was performed, but with asymmetric shear in dipoles with markedly unequal field strengths (Karpen et al. 1998). Similar intermittency was found in the shear-driven magnetic reconnection process, and the simulations moreover show that each dissipated magnetic island leaves a footprint in form of fine filaments in the overlying separatrix layer (Fig. 10.7).

This dynamic behavior is essentially identical with the pattern of repeated tearing and coalescence, first investigated by Leboef et al. (1982) and dubbedimpulsive bursty reconnectionby Priest (1985b). In Fig. 10.7 there are also some other dynamic processes present: (a) a thin region along the slowly rising inner separatrix is compressed, (b) a downflow with v

km s

, (c) followed by an upflow along the same field lines. Although these simulations by Karpen et al. (1995; 1998) are carried out with parameters corresponding to chromospheric conditions, it demonstrates that magnetic reconnection in sheared flare arcades occurs in a bursty and intermittent mode, and not in a quasi-stationary Sweet-Parker or Petschek mode. The physical origin of this intermittent reconnection dynamics is most essential to understand the rapidly-varying time structures of accelerated particles.

A recent work on impulsive bursty reconnectionapplied to solar flares was carried out by Kliem et al. (2000). Fig. 10.8 shows the evolution of tearing, magnetic

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Figure 10.7: Mass density difference ratio (grey scale) and projection of magnetic field lines into the image plane (dashed lines) at 800 s and 1000 s in the vicinity of the reconnection region, during a MHD simulation of a sheared arcade. The location a corresponds to a thin compressed region along the slowly rising inner separatrix, b to a narrow downflow falling outside of the left outer separatrix, and c indicates a broader upflow that follows along the same field lines (Karpen et al. 1998).

island formations, magnetic islands coalescence, secondary tearing, and so forth. Tear- ing and coalescence in the bursty magnetic reconnection mode modulates also particle acceleration on time scales that are observed in radio and hard X-rays, and is more consistent with flare observations than steady reconnection scenarios. The iterative processes of tearing and colaescence may repeat down to microscopic scales (of the ion Larmor radius or the ion inertial length), producing afractal current sheet(Shibata

& Tanuma 2001). A similar concept is that of MHD turbulent cascading, which leads to a similar high fragmentation at the smallest spatial scales, calledturbulent reconnec-

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-2 0 2

y

t=80 B, η max(η)=0.0005

-2 0

2 j max(j)=2.8

-4 0 4

y

t=158 max(η)=0.0030

-4 0

4 max(j)=3.0

-6 0 6

y

t=210 max(η)=0.0024

-6 0

6 max(j)=2.8

-6 0 6

y

t=273 max(η)=0.0065

-6 0

6 max(j)=3.1

-8 80

y

t=409 max(η)=0.0069

-8 80 max(j)=2.9

-8 80

y

t=479 max(η)=0.0070

-8 80 max(j)=3.0

0 20 40 60

x -12 120

y

t=821 max(η)=0.0049

0 20 40 60

x

-12 120 max(j)=2.6

Figure 10.8: Two-dimensional MHD simulation of dynamic magnetic reconnection, showing the magnetic field (left panels) and current density (right panels). Regions with anomalous resistivity are shown shaded in the magnetic field plot (at y=0) (Kliem et al. 2000).

tion(e.g., Kim & Diamond 2001; Matthaeus 2001b; see also Fig. 9.8) and applied to flares (Moore et al. 1995; Somov & Oreshina 2000). The two concepts of fractal cur-

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rent sheets and turbulent reconnection could possibly be discriminated observationally from the frequency distribution of time scales, since fractal processes are scale-free and generally produce powerlaw distributions, while turbulent processes are controlled by incoherent random processes that generally produce exponential distributions ( 9.8.1).

10.3 3D Magnetic Reconnection

In the previous Sections we described magnetic reconnection in two dimensions, which can approximate the three-dimensional reality as long as the magnetic field configuration has a translational symmetry in the third dimension, the ignorable coordinate.

There are two types of nullpoints in 2D reconnection, X-points and O-points, but there is a much richer variety of magnetic 3D topologies, where 3D volumes with oppositely- directed magnetic fields are divided by 2D separatrix surfaces, the intersection line of two separatrix surfaces form 1D separator lines, and the intersections of separator lines form3D nullpoints. The field of 3D magnetic reconnection is still in a very exploratory phase, encompassing a complex variety of mathematical topologies (e.g., Priest & Forbes 2000, 8; Parnell 1996), while only a few of all possible mathematical topologies have been identified by observations. Here we focus mainly on 3D topologies that seem to be most relevant for solar flares.

10.3.1 3D X-Type Reconnection

There are different theoretical definitions of 3D reconnection, in terms of (1) changes in the magnetic connectivity, (2) the electric field component, (3) plasma flows across the separatrices, or (4) changes in magnetic helicity (e.g., Priest & Forbes 2000, 8.1).

A simple practical discrimination rule between 2D and 3D X-type reconnection is the criterion whether both the pre-reconnection and post-reconnection field lines can be mapped in the same 2D plane or not. We illustrate this in Fig. 10.9 for reconnection between open (having only one footpoint on the solar surface) and closed field lines (with both conjugate footpoints on the solar surface). For 2D reconnection we can have bipolar (open with open), tripolar (open with closed), and quadrupolar (closed with closed) reconnection (Fig. 10.9 top). For 2D reconnection, a necessary condition is that the sequence of magnetic polarities of the reconnecting field lines is alternating in a common 2D plane, e.g., ^#

,

# # ,

for a quadrupolar geometry (Fig. 10.9 top right). If the sequence is not alternating, e.g., ^# ^#

, # ,

(Fig.10.9 bottom right), the reconnection geometry cannot be represented in a 2D plane, because either the pre- reconnection or the post-reconnection field lines would intersect, so they can only be separated topologically in 3D space.

The 2D reconnection geometries (Fig. 10.9 top) can have an arbitrary extension in the third dimension (along the neutral line), which are calledloop arcades. The 3D reconnection geometries (Fig. 10.9 bottom), in contrast, have more complicated topological constraints in the third dimension, so that they generally involveinteract- ing loopsrather thanloop arcades. Sakai & De Jager (1991) classify the geometries of coalescing loops into 1D, 2D, and 3D cases, which correspond to the quadrupolar geometries shown in Fig. 10.9 (right panels). The angle between the magnetic fields

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BIPOLAR TRIPOLAR QUADRUPOLAR

(open-open) (open-closed) (closed-closed)

N N N N N N

N N N

2D

3D

Figure 10.9: Classification of X-type magnetic reconnection topologies: (1)bipolar models have reconnection between two open field lines, (2)tripolar models have reconnection between an open and a closed field line, and (3)quadrupolar models have reconnection between two closed field lines. The pre-reconnection field lines are rendered in light grey, at the time of reconnection with dotted linestyle, while the post-reconnection field lines, as they occur after relaxation into a near-potential field state, are rendered in dark grey color. 2D versions, invariant in the third dimension (forming arcades) are shown in the upper row, while 3D versions are captured in the lower row. The pre-reconnection field lines (light grey) are located behind each other in the 3D versions, but approach each other in the image plane during reconnection. Note that the number of neutral lines (marked with symbol N, perpendicular to the image plane) is different in the corresponding 2D and 3D cases (Aschwanden 2002b).

at the reconnection points, which is simply

(anti-parallel fields) is the 2D-case, can take any arbitrary value in 3D reconnection geometries. The reconnection rate depends very much on this angle, being most efficient for anti-parallel fields, but scales with ^<

#

for skewed angles

in the Petschek model (Soward 1982). Thus, reconnection can still occur for almost parallel magnetic fields, although with reduced efficiency, a phenomenon that is called component reconnectionin magnetospheric physics. Sakai & Koide (1992) classify magnetic reconnection between two coalescing loops into six types, depending on the parallelity or anti-parallelity of the magnetic field component , the current along the loop, and the azimuthal field component

.

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1+ 1-

2+ 2-

Separatrix surface

Separator

Nullpoint

1+ 2- 1+

Spine curve

Nullpoint

Fane surface

Figure 10.10:Topology of 3D reconnection features for a quadrupolar region (left) and a parasitic region (right). In the quadrupolar region (left), a new emerging dipole (2+,2-) joins a pre-existing older dipole region (1+,1-), which are separated by a separatrix surface. The intersection of the two separatrix surfaces embedding both the old and new dipole region intersects at the separator line, which intersects with the photospheric surface at magnetic nullpoints. In the parasitic region (right) a unipolar flux region (2-) emerges in the center of a pre-existing open field region with polarity (1+). The new regions is shielded from the pre-existing open field by a dome-likefan surface, where the symmetry axis is called spine, containing a nullpoint at the intersection with the fan dome.

10.3.2 Topology of 3D Nullpoints

Wherever multiple magnetic dipoles occur, each one defines a domain that contains a volume of magnetic field lines with the same connectivity of positive to negative footpoints, as illustrated in Plate 10. Different dipolar domains are separated by separatrix surfaces in 3D space, intersections of 2D separatrix surfaces form 1D separators, and intersections of 1D separators form 3D nullpoints. The most natural example is the emergence of a secondary bipole in the neighborhood of a pre-existing dipole, which form together a new quadrupolar configuration, as shown in Fig. 10.10 (left). The new dipole region (2+,2-) pushes the old pre-existing field lines (1+,1-) aside in the coronal volume and forms a new separatrix as dividing surface. Strictly speaking, there is also a third domain of magnetic field lines with connections (1-,2+) inbetween, and per- haps a forth with connections (1+,2-) above, depending on the relative strength of the magnetic fluxes.

If the new emerging region is unipolar (which is nothing else than a vertically ori- ented dipole) and emerges in an open-field region with opposite polarity, it forms a parasitic polarity (Fig. 10.10 right), a configuration also calledanemone regionin the quiet Sun, or -spotin flaring regions. In such a unipolar region, surrounded by opposite magnetic polarity, the new magnetic domain is separated by a dome-like separatrix surface, which is calledfan dome. The symmetry axis of the unipolar region is called

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424 CHAPTER 10. MAGNETIC RECONNECTION SPINE

Reconnection FAN

Reconnection ^SEPARATORReconnection

Fan surface

Spine

Nullpoint Fan surface

Spine

Nullpoint

Fan surface Fan surface

Separator

Nullpoint Nullpoint

Fan dome

Spine

Nullpoint Fan dome

Spine

Nullpoint

Fan dome

Separator

Nullpoint

Figure 10.11: Classification of 3D nullpoint reconnection topologies: (1)spine reconnection (left), (2)fan reconnection (middle), and (3) separator reconnection (right), shown for a cylindrical geometry (top row) and for a dome-like fan surface geometry on the solar surface (bottom row). Spine and separator curves are marked with thick lines, fan surfaces with hatched areas, and 3D nullpoints with black dots. The pre-reconnection line is rendered with light grey color and the post-reconnection field line with dark grey color.

spine curve, which intersects the fan dome at a 3D nullpoint, and continues above the fan dome as spine curve (Fig. 10.10 right). A field line that connects two nullpoints, is called aseparator. Nullpoints have been studied in multiple sources (with hexagonal geometry) in the photospheric network up to 12-cell configurations (Inverarity & Priest 1999).

10.3.3 3D Spine, Fan, and Separator Reconnection

There is an infinite variety of such 3D topologies, because the complexity increases with the emergence of every new dipole that pushes into the pre-existing maze of coronal magnetic topologies (see Fig. 5.23 for more examples). However, the topological complexity cannot grow to infinity. At some point magnetic stresses will break up highly sheared structures and the neighbored field lines will reconnect to a simpler topology corresponding to a lower energy state. Field lines that are stressed or pushed

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Figure 10.12: This numerical simulation shows the collapse of a separator line. Top left:

Initial magnetic field configuration with a pair of 3D nullpoints. Top right: Currents start to accumulate in the nullpoints (shown with isosurfaces). Bottom left: The separator surface is stretched in vertical direction (where the thin vectors indicate the driving velocity and the thick white vectors the high velocity outflow jets driven by the reconnection). Bottom right: The process of outflow jets continues until no more magnetic energy is left and the current sheet fades then away (Galsgaard et al. 2000).

towards a 3D separatrix layer, a fan surface, a spine, or a separator, will experience a high plasma- near the zero-magnetic field zone and can slip through the nullpoints and reconnect to new field lines on the opposite side. Three special types of 3D recon- nections are illustrated in Fig. 10.11, calledspine reconnection,fan reconnection, and separator reconnection. In the case ofspine reconnection, a field line penetrates the fan surface, swirls around the spine and reconnects at the opposite side of the fan surface and spine curve (Fig. 10.10 left). In the case offan reconnection, the field line merely swirls around the spine, rotates around the fan dome, and reconnects at the other side (Fig. 10.10 middle). The case ofseparator directionis a special case of fan reconnection, where the spine curve is replaced by a separator line (Fig. 10.10 right). Since 3D nullpoints are created in pairs, they are always (at least in the initial stage) connected with a separator field line. We will show related observations in Section 10.6, which seem to fit these theoretical reconnection modes.

The theory or reconnection in 3D is presented in greater detail in the textbook of Priest & Forbes (2000, 8), while shorter reviews and introductions can be found

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in Priest (1996), Priest & Schrijver (1999), Brown & Priest (1999), Forbes (2000a), Schindler & Hornig (2001), and Hood et al. (2002). Analytical studies quantify the magnetic field in 3D nullpoint topologies (Brown & Priest 2001), the current distributions near 3D nullpoints (Rickard & Titov 1996), and solutions for fan, spine, and separator reconnection (Craig & McClymont 1999; Craig et al. 1999; Craig & Wat- son 2000a; Ji & Song 2001). The classical Sweet-Parker reconnection rate is found to be the slowest possible one in present 3D reconnection models, but it not clear what type of 3D reconnection yields the fastest reconnection rate suitable for flares. Numer- ical simulations of 3D reconnection topologies have been performed by Galsgaard et al. (1997b, 2000), which showed that reconnection is not restricted to singular points (nullpoints) but can also occur along separators. The experiments show (Fig. 10.12) that current accumulation is generated by a shear flow across the fan plane of the two nulls, the spine axis through the null points becomes disrupted by the development of the separator current sheet, looses its identity as a singular line and becomes integrated into the separator surfaces. The experiments of Galsgaard et al. (2000) also suggest that separator reconnection is the most important type of null reconnection among the three possible types (spine, fan, separator).

10.4 Magnetic Reconnection in the Chromosphere

There are a number of small-scale phenomena in the photosphere, chromosphere, and transition region that involve magnetic reconnection processes and may produce signatures in the lower corona such as microflares and nanoflares. An overview of such small-scale phenomena is given in Table 9.3 and their role for coronal heating is dis- cussed in 9.6. Here we concentrate on the aspect of magnetic reconnection of these small-scale events.

10.4.1 Magnetic Flux Emergence

The magnetic dynamo in the solar interior constantly generates magnetic fluxtubes that emerge through the photosphere and add new magnetic flux systems to the corona, probably moored in deep subsurface structures for the lifetime of an active region (Schrijver & Title 1999). Fig. 10.13 shows a frequency distribution of the emergence rate of magnetic dipoles, which are calledephemeral regions(Harvey & Martin 1973;

Martin 1988) if they have an area smaller than

deg

, oractive regionsif they are larger. Thus, ephemeral regions are essentially “mini-active regions” in quiet Sun areas. The size distribution shown in Fig. 10.13 encompasses 4 orders of magnitude in magnetic flux (

"

Mx) and 8 orders of magnitude variation in the occurrence rate (Hagenaar et al. 2003). A comparison of the magnetic flux emergence rate with the network flux implies an overall mean replacement time of

,

hr in the quiet Sun (Hagenaar et al. 2003).

Emerging flux systems in active regions are alineated with overlying arch filament systems (Strous et al. 1996; Strous & Zwaan 1999), emerging in form of -, U-loops, or “sea-serpent-like” shapes (Zwaan 1987; Van Driel-Gesztelyi et al. 2000;

Van Driel-Gesztelyi 2002). Newly emerging magnetic dipoles appear at the edges of

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Figure 10.13: Frequency distribution of emerging magnetic bipoles per day, per flux unit of

Mx]. The distribution includesephemeral regions (ER: Mx) and active

regions (AR: Mx, area deg). The variation by a factor 8 is mainly caused by the solar cycle. The histograms include ephemeral regions studied with MDI, with a detection threshold of

Mx (Hagenaar et al. 2003).

supergranulation cells (Hagenaar 2001). During emergence, magnetic dipoles grow in size and their rate of divergence is of order

km s

, while the magnetic flux increases with a rate of ^. ^# ^. ^*

"

$

Mx s

(Hagenaar 2001). The emergence of growing new magnetic flux structures necessarily forces topological changes in the magnetic field of the overlying corona, which may involve magnetic reconnection processes. MHD simulations of such magnetic fluxtubes have been performed in subphotospheric zones (see 6.2.5 and references therein) and in coronal heights (Shibata et al. 1989b,1990). When the emerging field has the same orientation as the overlying coronal field, an approximately current-free field forms in the interaction region. When the emerging field is anti-parallel, however, a current sheet forms and could initiate magnetic reconnection (Shibata et al. 1989b), as shown in Fig. 10.14.

The rise velocity is relatively small, however, in the order of

km s

, driven by the Parker instability (undular mode of magnetic buoyancy instability), in the model of Shibata et al. (1989a,b). The emergence of a bipole does not necessarily trigger a

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Figure 10.14:Numerical 2D MHD simulation of an emerging dipole into an anti-parallel coronal field: magnetic field (top), velocity field (middle), and density (bottom). A current sheet forms (dashed line in top panel) that enables magnetic reconnection (Shibata et al. 1989b).

microflare or flare event, because the emerging flux region could be too small or could have the wrong orientation (Martin et al. 1984). Harvey (1996) found that only 8%

of soft X-ray bright points (which are the sites of microflares) were associated with an emerging bipole, while a much larger fraction was associated with magnetic cancellation features (Harvey et al. 1994). Karpen & Boris (1986) conclude that preflare brightenings and flares probably do not result directly from the emerging fluxtubes themselves, but from coupled energy release processes, such as current-driven plasma microinstabilities. Altogether it appears that magnetic flux emergence does not trigger flares directly, but increases the magnetic complexity locally, e.g., by adding a new separator surface and creating new nullpoints (Fig. 10.10), which after further increases in magnetic complexity ultimately may escalate into a flare-like event. However, flare models that are directly driven by flux emergence have also been proposed (Heyvaerts et al. 1977), see 10.5. Furthermore, 3D magnetic modeling of quasi-separatrix layers in a flare event and comparison with H and soft X-ray images was found to be

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consistent with the concept of magnetic reconnection driven by emergence of sheared magnetic field (Schmieder et al. 1997a,b).

10.4.2 Magnetic Flux Cancellation

The reverse process tomagnetic flux emergenceismagnetic flux cancellation, which can occur in at least three different manifestations (Martin et al. 1985; Livi et al. 1985;

Van Driel-Gesztelyi 2002; Parnell 2002b): (1) by submergence, subduction, or retrac- tion of fluxtubes, (2) by converging flows, or (3) by flux dispersion or diffusion. The first process essentially corresponds to a downward motion of a dipole, which sinks through the photosphere, but does not necessarily involve a magnetic reconnection process. Evidence for such a process was found from the timing of disappearance in different heights of chromospheric and photospheric magnetograms (Harvey et al. 1999).

The second process, however, corresponds to a collision between two conjugate magnetic polarity elements and thus involves a magnetic reconnection process (e.g., model of Litvinenko 1999a). The third one is a fragmentation process driven by surface random flows, which may or may not involve magnetic reconnection between individual fragments.

A theoretical model of magnetic reconnection for converging flows that produce magnetic cancellation was quantified by Litvinenko (1999a). From the equations of continuity, momentum, and Ohm’s law, Litvinenko (1999a) derives a reconnection inflow speed

of

where

$

# . # . $

km is the chromospheric density scale height. Us- ing a VAL-C standard atmospheric density model, a maximum of the inflow speed of

1

m s

is found at a height of

km above the photosphere (assuming a magnetic field of

G, see Fig. 10.15). This approaching speed corresponds to observed speeds of some cancelling features. The height of

km above the photosphere (in the temperature minimum region) is thus considered as the most favorable location for chromospheric reconnection processes. This process could explain mass upflows into an associated filament during a photospheric cancellation event (Litvinenko & Martin 1999).

A more general model of chromospheric reconnection driven by converging flows has been developed by Chae et al. (2003), including the adiabatic energy equation which contains the case of an isothermal current sheet used by Litvinenko (1999a) as special case (for a specific heat ratio of

). Moreover, Chae’s model (2003) is formulated only in terms of observable parameters (safe for the magnetic diffusivity

). The model of Chae et al. (2003) assumes a (vertical) Sweet-Parker current sheet in the chromosphere with length , width , inflow speed

, and outflow speed

, which obey the relations given in Eqs. (10.1.1-7). An observable is the magnetic flux loss^. ^# ^. ^*, which is defined as the magnetic flux that is processed through the current

sheet, ..