ELECTRON ACCELERATION BY MULTI-ISLAND COALESCENCE

Figure 6 shows the first example of the electron trajectories (hereafter particle 1). We divided the trajectory into four segments and plotted over the out-of-plane component of the electron current density. The whole profiles of energy and the displacement Δz are shown in the right two panels. The simulation itself is two dimensional in x and y but the displacement Δz can be calculated by integrating v_z over time. The background images are just snapshots taken during each segment, so care should be taken when comparing the trajectories with the images. The time of each image is chosen so that it best reflects the characteristics of each trajectory.

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Particle 1 is first energized at one of the few X-lines formed within the first coalescence stage (Figure 6(a)). Scatterings take place in the immediate downstream of the X-line, but the particle soon starts to travel along the rim of the island associated with the X-line and is further scattered when it reaches near the merging point of two islands approaching to each other (Figure 6(b)). While wandering around the merging point, the coalescence progresses to form a region of localized current in which the particle is confined to experience the second rapid energization (Figure 6(c)). Note that this localized current corresponds to the anti-reconnection in which the electric field direction is reversed. The energization persists even after the particle is ejected out of the central anti-reconnection region and the particle is pitch angle scattered afterward (Figure 6(d)). The first and the second rapid energization occur in a very localized region (Δy ∼ 0.1d_i) associated with the reconnection and anti-reconnection, respectively (Figure 6(e)). What is important here is that the particle moves toward the positive z-direction up to Δz ∼ 50d_i, but by the reversed electric field, it turns its direction toward the negative z-direction. As a result, it passes through the original position and reaches Δz ∼ −10d_i by Ω_cit = 104.0.

Figure 7 shows the details of the behavior of particle 1 during the first rapid energization. The trajectory is further divided into three time periods according to its characteristics. The time profiles of the particle energy as well as the electric fields felt by the particle are also displayed.

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Figure 7(a) illustrates the period during which the particle approaches the X-line and starts to be energized. During this phase, we can identify the polarized electric field mainly parallel to the reconnection plane (i.e., X–Y plane) and the particle is energized as it passes through this region. The polarized electric field felt by the particle reaches ∼V_AB₀ (Figure 7(e)). However, the corresponding energy increase is very small, Δε ∼ 0.1m_ec² (Figure 7(d)). In order to better visualize the energy increase, we multiplied the energy profile by 10, as shown by the dashed curve.

Figure 7(b) illustrates the "X-line phase" during which the particle stays at the very center of the diffusion region or the X-line and is rapidly energized. Within a relatively short period of time Ω_ciΔt ∼ 2, the energy increases nearly an order of magnitude. Note also that this energization takes place in the so-called inner diffusion region. Recent simulations revealed that the out-of-plane current is localized only at the center of the diffusion region ("inner diffusion region") so that the reconnection rate remains fast while electrons form a high-velocity jet in the "outer diffusion region" (Daughton et al. 2006; Fujimoto 2006; Shay et al. 2007; Karimabadi et al. 2007; Phan et al. 2007). Electrons are unmagnetized in these regions so that a snapshot of the z-component of E + V_e × B makes clear in which region the electron can be energized. The inner diffusion region (colored blue) is at the X-line and the outer diffusion region (colored red) extends toward the downstream. It is evident that the electron is rapidly energized within the "inner diffusion region." A close look at the particle trajectory reveals a meandering motion in this region (Figure 7, inset).

Figure 7(c) illustrates the "post-X-line phase" during which the particle moves toward the downstream and is pitch angle scattered. The rate of energy gain decreases, but the particle is energized up to ∼0.7m_ec². The energization takes place near the edge of the outer diffusion region or the outflow exhaust. This is where magnetic field magnitude starts to increase. Also, because electrons are already pre-energized at the X-line, they have gyroradii comparable to the curvature of the magnetic field lines of this region. Concurrently, the gradient B and curvature B drift motion plays an important role for the additional energization during this phase. In the later half of the post-X-line phase, particle 1 is reflected by a mirror force at (x, y) ∼ (3.5d_i, 74d_i). The particle again passes through the edge of the diffusion region at (x, y) ∼ (0, 76d_i) but the energy increase is very small. Finally, it escapes away from the diffusion region.

We now explore the details of the second rapid energization of particle 1. Figure 8 shows the last three of the four phases described in Figure 6. The enlarged views of the trajectory are shown in the upper three panels, and the time profile of the particle energy as well as the electric field felt by the particle is shown in the lower two panels.

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Figure 8(a) illustrates the "pre-anti-X-line phase" during which the particle approaches toward the merging point of two large islands. The particle is reflected twice by the mirror force but is not energized. The reversed electric field is already large at this phase reaching to ∼0.3V_AB₀ and the non-magnetized region starts to appear at the merging point.

Figure 8(b) illustrates when the anti-reconnection is taking place ('anti-X-line phase"). The particle is drawn into the anti-reconnection region and is energized significantly. The duration of energization, however, is somewhat longer than the time duration of the first rapid energization (Ω_ciΔt ∼ 6). Note that the anti-reconnection is embedded in closed magnetic field lines and the outflow from the anti-X-line collides with these magnetic fields. Moreover, because of the first rapid energization, the gyroradii of the electron is large comparable to the scale of the anti-reconnection. The electron motion is thus decoupled from the magnetic field lines. The above features lead to trapping of the electron within the anti-reconnection region and hence significant energy gain. Yet, each "kick" occurs in the inner diffusion region as was the case during the first rapid energization. This is shown more clearly in the time profiles of the particle energy and the x-positions (Figure 8(d)). The oscillating feature with the timescale of Ω_ciΔt ∼ 1 corresponds to the gyromotion of the particle. During the anti-X-line phase, this oscillatory feature is modified and a continuous energy increase occurs whenever the x-position is within ±1d_i from the X-line.

Figure 8(c) shows the "post-anti-X-line phase." Note the change of the horizontal axis range. The particle is ejected out of the anti-reconnection region but continues to gain energy by the gradient B and curvature B drift acceleration. Because of the contracting motion of the merged island, the electric field in this region continues to increase and reaches 0.4V_AB₀. As a result, the particle energy reaches near 2m_ec².

In summary, particle 1 is first energized at and around the X-line generated during the nonlinear evolution of the tearing mode instability. It is then energized at and around the anti-X-line generated by the merging of two islands. In both cases, the particle is accelerated directly by the reconnection electric field. As a result, the energy reaches more than ε>m_ec².

We now show another example of electrons that behaved quite differently from particle 1 (hereafter particle 2). Figure 9 shows the trajectory with the same format as Figure 6 but the trajectory is divided into five segments instead of four, and the background images are replaced by the out-of-plane component of the electric field instead of the electron current density.

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The first energization occurs at y ∼ 88.5d_i where an X-line develops as a consequence of the first coalescence growth phase (Figure 9(a)). There is a small, secondary island at y ∼ 86d_i but it only modulates the particle orbit and plays a minor role for the energization. The particle travels until it reaches another X-line, which is also produced by the first coalescence growth (Figure 9(b)). There, the particle receives two "kicks" from the reconnection electric field. The second kick occurs because the particle is reflected by the magnetic fields being piled up on the pre-existing current sheet and comes back to where it can again receives energy from the reconnection electric field. The two kicks can be identified in Figure 9(f) at (ε/m_ec², y/d_i) ∼ (0.5, 45) and ∼(1.8, 50).

What makes particle 2 different from particle 1 is the behavior during the rest of the trajectory. After the two kicks, the particle is trapped within an island and circulates six times (Figure 9(c)), as can be counted from the zigzag orbit of the Δz-profile (Figure 9(g)). Note the fact that this island is moving toward the negative y-direction because this island and the other island at y ∼ 20d_i are attracting each other. Thus, the direction of the gradient B and curvature B drift is positive z at both ends of the island while the direction of the motional electric field is negative and positive at the upper (y ∼ 70d_i) and lower (y ∼ 50d_i) ends, respectively. As a result, the particle gains energy at the upper end but loses energy at the lower end. After all, the net energy gain of the particle is kept small. This is represented by the foldings and overlappings of the curve in Figure 9(f) at ε ∼ 1m_ec².

The particle eventually reaches the merging point where it receives energy at the anti-X-line (Figure 9(d)), but the anti-X-line has already been well developed so that the particle is not trapped in this region. The particle soon exits from the merging region and starts to circulate inside the newly created, large island (Figure 9(e)). This large island is contracting in this phase so that the particle can gain energy every time it passes through both ends of the island (Drake et al. 2006). There exists a reversed electric field in between the island edges so that the particle can gain energy through the gradient B and curvature B drift. As for the displacement along the z-direction, the particle moves toward the negative z-direction only during the period of energization by the anti-reconnection. The resultant displacement is Δz ∼ 100d_i.

In summary, particle 2 gains a significant amount of energy from the motional electric field due to the contracting motion of the merged island. The final energy reaches as high as that of particle 1, that is, ε ∼ 2m_ec².

Another important feature of coalescence is the bouncing motion of merged islands (Tajima et al. 1987; Wan & Lapenta 2008). The merged island still possesses a certain amount of kinetic energy so that a contracting motion of an island is followed by an expanding motion.

Figures 10(a)–(c) show the evolution of the merged island. At Ω_cit = 104, the anti-reconnection has stopped but the reversed E_z electric field remains because of the expanding motion in the x-direction (Figure 10(a)). The shape of the merged island is somewhat elongated in the y-direction and the contracting motion in the y-direction still continues as evident from the negative E_z region. At Ω_cit = 112, the merged island has already expanded in the x-direction and the contracting motion in the y-direction has become weak (Figure 10(b)). Notable here are the ripple structures that surround the island core region, as evidently shown by the enhanced, localized E_z regions. These structures are due to a turbulent motion within the island. The turbulent motion is generated because the converging flows from the top and the bottom are mixing with each other. The ripple structures propagate mainly in the y-direction away from the merging center. By the time Ω_cit = 135, the ripple structure disappears and the island has become round. It is very slowly expanding.

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Also shown in Figure 10 is a sample trajectory of electrons that pass through a ripple structure (hereafter particle 3). After being energized up to ε ∼ 1.3m_ec², particle 3 undergoes pitch angle scattering just outside the anti-X-line region, i.e., (x, y) ∼ (6, 38) (Figure 10(a)). It then travels toward the negative y-direction (Figure 10(b)). During this travel, the particle encounters the ripple structure identified at (x, y) ∼ (6, 31). This electric field is generated by a local flow toward the positive x-direction and the local magnetic field directed toward the negative y-direction. The electron is energized by the enhanced E_z in association with the gradient B/curvature B drift (Figure 10(d)). A gradient E drift is unlikely to be causing the energization because such drift motion is perpendicular to E_z, the only dominant component at this time.

In a separate paper, we reported an "island surfing" mechanism of electron acceleration during magnetic reconnection (Oka et al. 2010). This mechanism utilizes the secondary magnetic islands that are produced in the diffusion region. Inside each island, electrons are trapped for a significant period of time so that they are energized continuously by the reconnection electric field prevalent in the diffusion region. Although a pre-acceleration is required to keep electrons trapped within islands, the trapped electrons can receive an unlimited amount of energy as long as the island stays in the diffusion region.

Note again that while X-type acceleration takes place at any X-line, O-type acceleration occurs far away from X-lines by taking advantage of a closed field line geometry. The "island surfing" mechanism takes place in a diffusion region very close to the X-line, but at the same time, makes use of the closed field line geometry of a secondary magnetic island. As such, this mechanism falls into both categories described above and we rather consider it as a hybrid type.

The "island surfing" is a natural consequence of the secondary tearing instability of magnetic reconnection, and hence not directly related to multi-island coalescence. Therefore, we do not discuss its detail any further in this paper.

We performed 2D PIC simulations of multi-island coalescence with no guide field. By following the trajectories of 198 energetic electrons, we identified various energization mechanisms as have been summarized in Table 1. Note again that these electrons are extracted at the peak time of energetic electron flux with the criteria of ε>1.4m_ec². The first column shows the categorization of each mechanism. The second column shows mechanism nomenclature used throughout this paper. The third column shows the origin of the electric field from which particles gain their energies. The fourth column shows what we refer to as the "contribution probability" P_c, defined as P_c = 100n_acc/n_tot, where n_acc is the number of electrons accelerated by the mechanism and n_tot is the total number of electrons we analyzed. In the present case, n_tot = 198. For each trajectory of the 198 electrons, we checked which mechanism the particle had been energized through and summed up the number of trajectories each mechanism worked on. Because each particle experiences more than one different mechanism, the total of P_c does not equal to 100%. The fifth column shows the largest energy gain by each mechanism Δε_l. Finally, the last column of Table 1 indicates which literature discussed each mechanism. Below, we summarize each mechanism based on our simulation results and describe how we counted n_acc for each mechanism.

The "surfing" was difficult to identify in our simulation because of the negligible amount of energy increase (Δε_l/m_ec²< 0.1). This may be due to the undriven nature of our simulation that generates a polarization electric field E_p ∼ 1V_AB₀. The original study of the "surfing" mechanism used a driven magnetic reconnection that generates the polarization electric field of E_p ∼ 6V_AB₀ so that the Lorentz force can be balanced by the force from E_p. For Table 1, we simply did not count the number of the "surfing" mechanisms.

The "X-line" mechanism is the classical way of energizing electrons, but based on the recently revealed two-scale structure, we verified that significant energization takes place at the very center of the diffusion region or the "inner diffusion region" (Δε_l/m_ec² ∼ 0.8). This may be a matter of course given the fact that the out-of-plane current is localized within the inner diffusion region. The number of electrons energized by this mechanism n_acc was 117. When counting this mechanism, we made sure that the electrons passed through the inner diffusion region.

The "anti-X-line" mechanism is the most common energization mechanism among the trajectories we analyzed (n_acc = 169). It is often intense and the largest energy increment was Δε_l/m_ec² ∼ 1.8. The energization process itself is basically the same as the "X-line" mechanism but there are three major differences. One is that the anti-X-line is created in a driven manner by two separate magnetic islands approaching toward each other. Since each island is expelled in association with the outflow from an X-line, the anti-reconnection rate reaches as large as 1V_AB₀. The second major difference is that an anti-X-line is bounded by the closed field lines of the merged island. The size of the anti-reconnection region is relatively small but the electrons can easily be trapped at and around the anti-reconnection region. This effect leads to multiple numbers of interactions with the inner diffusion region of the anti-reconnection. As a result, the "anti-X-line" mechanism was found to be a significant energization mechanism, although the anti-reconnection is a transient process that takes place in a small region. The third major difference between the X-line and the anti-X-line is that the electric field is reversed at the anti-X-line. Because of this, electrons move toward the opposite direction while being energized. In fact, this feature was what we used to count the number of anti-X-line mechanisms in Table 1.

The "grad-B/curv-B drift" mechanism is also a common process for electron energization (n_acc ∼ 174) but each kick is not significant (Δε/m_ec²< 0.3). Typically, this process occurs after a particle exits from either an X-line or an anti-X-line. We identified this mechanism whenever the trajectory showed energy increase during a drift motion near either the X-line or the anti-X-line. A drift motion can also take place at each end of the magnetic island but, in such cases, the trajectory can be regarded as a part of circulation inside the island and are not counted as the "grad-B/curv-B drift" mechanism. The number of "grad-B/curv-B drift" mechanisms n_acc was 174.

The "contracting island" mechanism was identified in 79 trajectories. Since this mechanism requires energy gain at both ends of the island, the process appears only in the later phase of island coalescence. In the earlier phase of coalescence, the merging is not complete and both approaching islands are not contracting. We identified this mechanism by the zigzag orbit of the y–ε plot. The largest energy gain was Δε_l/m_ec² ∼ 0.6.

The "ripple" mechanism was found to be as important as the "contracting island" mechanism, n_acc = 61. During the bouncing motion of a newly merged island, turbulent flows appear inside the island so that many electrons pass through localized electric fields. We counted the number of this mechanism by checking the association of an energy increase with the localized electric field within a merged island.

The "island surfing" mechanism was very rare. Only four electrons were found to be energized by this mechanism. This is probably because of the relatively small reconnection electric field in the later phase of the simulation run when secondary islands are created. Moreover, the size of each X-line was limited due to the presence of magnetic islands. If an X-line were able to reach a steady state, it would continue to spawn a number of secondary islands, which situation would lead to more importance of this mechanism.

In Table 1, we classified each mechanism as either X-type, O-type, or a hybrid. An energization mechanism belongs to the X-type if it occurs in the diffusion region of magnetic reconnection and/or in the magnetic field pile-up region just downstream of the diffusion region. The source of energy is the reconnection electric field. On the other hand, an energization mechanism belongs to the O-type if it takes place within the closed geometry of the magnetic field or magnetic islands. The source of energy is the kinetic motion of the island. A hybrid mechanism refers to an energization within a secondary island located in the diffusion region.

The striking feature of electron acceleration at the anti-reconnection site is the direction of motion in the z-direction being opposite to that of electrons at the primary reconnection site. If electrons were magnetized and drifting, they lose their energy by the reversed electric field (Pritchett 2008). However, at the very center of the anti-reconnection, electrons are not magnetized and not drifting so that they can move parallel to the reversed electric field and receive substantial amount of energy.

Figure 11 shows the displacement of the most energetic electrons along the out-of-plane direction, Δz, obtained at Ω_cit = 43, 72, 102, and 115. The solid line indicates a hypothetical energization by the electric field of 0.1V_AB₀, which roughly corresponds to the primary reconnection electric field measured in the earlier phase of the simulation. It is also the typical value of steady-state magnetic reconnection (Shay et al. 2007). In the earlier phase of the simulation run (Ω_cit = 73, green marks), many electrons follow this solid line because they are energized at the primary reconnection regions. They can reach as far as 100d_i. Later on, electrons are accelerated by the anti-reconnection so that they move toward the opposite direction (Ω_cit = 102, blue marks). By the time of peak electron flux (Ω_cit = 115, black marks), the asymmetry of the particle distribution remains, but most particles are confined within 140 d_i from the initial positions.

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The fact that many electron positions are largely deviated from the line of energization by the primary reconnection, particularly in the later phase of the simulation run, indicates that these electrons experienced anti-reconnection. Some electrons showed unusually large displacement of Δz>140d_i. All of these particles are energized by the "island surfing" mechanism and so did not experience anti-reconnection and direction turnings.

Let us now discuss applications of our simulations. We used periodic boundary conditions under the assumption of infinitely long current sheet in the y-direction. Therefore, our simulation can probably be best applied to the solar flares which show evidence of very long current sheets (e.g., Sui & Holman 2003; Bemporad et al. 2006).

For the discussion, the energy spectrum in Figure 5 is revisited as shown in Figure 12 (left). A caveat here is that our simulation uses unrealistically high, initial temperature (ε_th) due to limited computational resources. The temperature of the initial background electrons ε_th = 9.216 × 10⁻³m_ec² ∼ 55 MK ∼5 keV, which is already hot in the solar corona. In Figure 12 (left), we re-normalized the energy by ε_th (top axis) and then converted to the unit of keV under the assumption of a typical electron temperature in the solar corona, ε_th = 2 MK (bottom axis). Moreover, because the typical energy range of a space-borne, hard X-ray detector is from a few keV to a few hundreds of keV, we only showed the higher energy end of the simulated spectrum. For the vertical axis, we used the number of electrons F(ε) counted in the simulation box. F(ε) can be representing a population n(ε) keV⁻¹ cm⁻³ of electrons in a fully ionized plasma of the energy release region. Since this is a simulation, the numbers can be scaled arbitrarily.

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As a reference, we reproduced in Figure 12 (right) photon spectra from the well-known, solar flare event of 2002 July 23 obtained by the RHESSI satellite (e.g., Lin et al. 2003; Asai et al. 2009). In the pre-impulsive phase, the spectrum showed a power-law-like form that extends up to >40 keV from a coronal source. Lin et al. (2003) interpreted this spectrum by the combination of an exponential (thermal) and double power-law (nonthermal) spectra. In the impulsive phase, the energy range below 40 keV is dominated by a single exponential form, indicating clearly that electrons are significantly heated during this phase. Beyond this "super-hot" component is the power-law spectrum that extends up to >100 keV that originates from the chromospheric footpoints of the flare loop.

It is worth emphasizing here that our simulation is not intended to reproduce any particular solar flare event. We must keep in mind that our simulation has many assumptions and limitations as described below. First, the spatial size of the simulation domain is very small, ∼100 times the ion inertia length d_i, that is, ∼10⁴ cm or 0.14 mas. This is much smaller than the typical size of the energy release region in solar flares, that is ∼10⁹ cm or 20 arcsec. Therefore, it is impossible to take into account the dynamical evolution of the solar flares. Second, the ion-to-electron mass ratio m_i/m_e = 25 is very small compared to the actual value of 1836. Third, our simulation is a 2D model in x and y so that spatial variations along the out-of-plane direction z are not taken into account. Both reconnection and anti-reconnection create strong current at and around the X-lines, and our simulation lacks possible consequences of current-driven instabilities that would occur in the third dimension at the X-lines. Finally, we assumed zero magnitude of the magnetic field in the out-of-plane direction B_z, or the "guide field." This assumption is merely for a simplicity. In order to better understand the physics of the solar flares, we must perform additional simulation runs with different guide field magnitudes.

Nevertheless, the fact that our PIC simulation was able to produce electrons of up to >30 keV suggests that multi-island coalescence is potentially important for the understandings of electron acceleration/heating during the solar flares. Note that a multi-island coalescence may appear in association with developed turbulence that creates a volume-filling magnetic island (Tajima & Shibata 1997; Drake et al. 2006; Retinò et al. 2007; Bemporad 2008). Since turbulence may be generated in any possible electron energization sites, our simulation may also be applied to these sites (Figure 12, bottom). The details of the possible electron energization sites as well as theories can be found elsewhere (e.g., Aschwanden 2002 and references therein). It is to be emphasized again that we used periodic boundaries but energetic particles that reached ε> 1.4m_ec² (or 26 keV when renormalized) did not cross the boundary more than once (e.g., Figures 6 and 9), indicating that electrons do not need to move more than 100d_i (or ∼10⁴ cm when renormalized) in the y-direction to reach the energy. Also, they do not need to move more than 140d_i in the z-direction as we showed in Figure 11. Note also that the total simulation time was ∼130Ω⁻¹_ci. If we assume the coronal magnetic field magnitude to be 100 G, our simulation suggests that electrons of up to 30 keV can be created within a spatial extent of 10⁴ cm within a timescale of ∼ 10 μs. Such a localized and quick acceleration of electrons is an important feature of the multi-island coalescence.

Our PIC simulation also showed that the total energy gained by both thermal and nonthermal electrons reaches more than 30% of the magnetic field energy released (Figure 3(a)). This relatively large number is the advantage of having multiple numbers of X-lines during the multi-island coalescence. Our simulation of single X-line reconnection with exactly the same parameters employed in the presented simulation showed the fraction to be 20% (not shown). Note that X-ray observations show that nonthermal electrons alone carry upward of 10%–50% of the released magnetic field energy (Lin et al. 2003 and references therein). Therefore, our simulation may not fully explain the energy budget during the solar flares, but the multi-island coalescence is an attractive way to develop future models of the solar flare energy budget.

Figure 12 suggests that the current PIC simulations produce similar spectra as observed in coronal hard X-ray sources, but cannot account for the flat electron spectrum needed to produce the hard X-ray spectrum observed in footpoints. In fact, if we were able to perform PIC simulations with a larger simulation domain, say 1000d_i × 1000d_i, the third spectral component described in Figure 5 may extend to a higher energy range to form a clear power law. Moreover, there would be a variety of different sizes of magnetic islands so that the resultant turbulence leads to a Fermi-like, stochastic acceleration of electrons. Therefore, we anticipate our results to be a starting point for more sophisticated models of particle acceleration during the explosive energy release phenomena.