Electron acceleration in a secondary magnetic island formed during magnetic reconnection with a guide field

The evolution of magnetic reconnection in large‐scale systems often gives rise to extended current layers that are unstable to the formation of secondary magnetic islands. The role of these islands in the reconnection process and the conditions under which they form remains a subject of debate. In this work, we benchmark two different kinetic particle‐in‐cell codes to address the formation of secondary islands for several types of global boundary conditions. The influence on reconnection is examined for a range of conditions and collisionality limits. Although secondary islands are observed in all cases, their influence on reconnection may be different depending on the regime. In the collisional limit, the secondary islands play a key role in breaking away from the slow Sweet‐Parker scaling and pushing the evolution towards small scales where kinetic effects can dominate. In the collisionless limit, fast reconnection can proceed in small systems (30× ion inertial scale) without producing any secondary islands. However, in large‐scale systems the diffusion region forms extended current layers that are unstable to the formation of secondary islands, giving rise to a time‐dependent reconnection process. These instabilities provide one possible mechanism for controlling the average length of the diffusion region in large systems. New results from Fokker‐Planck kinetic simulations are used to examine the role of secondary islands in electron‐positron plasmas for both collisional and kinetic parameter regimes. Simple physics arguments suggest the transition should occur when the resistive layers approach the inertial scale. These expectations are confirmed by simulations, which demonstrate the average rate remains fast in large systems and is accompanied by the continuous formation of secondary islands.

Magnetic islands are considered to play a crucial role in collisionless magnetic reconnection. We use particle‐in‐cell simulations to investigate electric field Ez structure in the magnetic islands (including primary and secondary islands) with and without a guide field during magnetic reconnection. It is found that the electric field has multilayers in the primary island and a large bipolar structure in the secondary island in the absence of guide field. The electric field is provided by the Hall term (J × B)z (mainly), the divergence of electron pressure tensor, and the convective term (Vi × B)z in the outer and the inner region of primary island, while the electric field is much smaller (~0) in the middle and the core region of primary island due to the cancelation of the three terms. The single bipolar electric field is primarily provided by the Hall term in the secondary island. In the presence of a guide field, the electric field has multiple layers in the primary island (similar to zero guide field case) and the secondary island. However, there still exists one single large sharp bipolar structure of electric field in the central region of the secondary island. The differences of electric field in the primary and secondary islands are essentially due to the variations of the current Jy. These features can be used as the observational criteria to identify different types of magnetic islands in the magnetosphere using the data of future mission, such as the Magnetospheric Multiscale mission.

In this paper, with a two-dimensional particle-in-cell simulation model, we compare the structures of a primary magnetic island to those of a secondary magnetic island, and these islands are formed during collisionless magnetic reconnection in a force-free current sheet. The out-of-plane magnetic field By is enhanced in the center of both the primary and secondary islands; however, a quadrupole structure of By with a good symmetry may be formed at the ends of the primary island. The in-plane electric field also exists in both the primary and secondary islands. The electric field Ex has a positive value in the left and a negative value in the right of the islands, while the electric field Ez has a positive value in the upper part and a positive value in the lower part of the islands. However, the in-plane electric field exists in the outer part of the primary island, while it exists in the center of the secondary island.

The interactions between magnetic islands are considered to play an important role in electron acceleration during magnetic reconnection. In this paper, two-dimensional particle-in-cell simulations are performed to study electron acceleration during multiple X line reconnection with a guide field. Because the electrons remain almost magnetized, we can analyze the contributions of the parallel electric field, Fermi, and betatron mechanisms to electron acceleration during the evolution of magnetic reconnection through comparison with a guide-center theory. The results show that with the magnetic reconnection proceeding, two magnetic islands are formed in the simulation domain. Next, the electrons are accelerated by both the parallel electric field in the vicinity of the X lines and the Fermi mechanism due to the contraction of the two magnetic islands. Then, the two magnetic islands begin to merge into one, and, in such a process, the electrons can be accelerated by both the parallel electric field and betatron mechanisms. During the betatron acceleration, the electrons are locally accelerated in the regions where the magnetic field is piled up by the high-speed flow from the X line. At last, when the coalescence of the two islands into one big island finishes, the electrons can be further accelerated by the Fermi mechanism because of the contraction of the big island. With the increase of the guide field, the contributions of the Fermi and betatron mechanisms to electron acceleration become less and less important. When the guide field is sufficiently large, the contributions of the Fermi and betatron mechanisms are almost negligible.

[1] Secondary islands are considered to play a crucial role in collisionless magnetic reconnection. Based on 2-D particle-in-cell simulations, we investigate the characteristics of the out-of-plane electron currents in magnetic islands formed during collisionless magnetic reconnection with an initial guide field. In a primary island (formed simultaneously with the appearance of the X lines), due to the acceleration of the trapped electrons, the direction of the formed out-of-plane electron current is reverse to the original one. In the secondary island (formed in the vicinity of the X-line), the out-of-plane electron current is generated due to the accelerated electrons by the reconnection electric field in the vicinity of the X line. In such a way, the direction of the out-of-plane electron current in a secondary island is found to be opposite to that in a primary island. Such characteristics are found to be related to the evolution of the magnetic islands and then electron dynamics in the islands, which are proposed in this paper to be a possible criterion to identify a secondary island formed during collisionless magnetic reconnection, especially in the magnetotail.

Numerical simulations have predicted that an extended current sheet may be unstable to secondary magnetic islands in the vicinity of the X line, and these islands can dramatically influence the reconnection rate. In this Letter, we present the first evidence of such a secondary island near the center of an ion diffusion region, which is consistent with the action of the secondary island instability occurring in the vicinity of the X line. The island is squashed in the z direction with a strong core magnetic field. Energetic electrons with anisotropic or field-aligned bidirectional fluxes are found in the ion diffusion region, and the enhancement of energetic electron fluxes is more obvious inside the secondary island.

Ion-scale flux ropes and plasmoids are secondary magnetic islands produced during magnetic reconnection in various heliospheric plasma environments. Here we study the structure of secondary islands and the particle dynamics within them using particle-in-cell simulations. Ion-scale flux ropes (secondary islands with a strong core field) are formed in a strong guide field regime, whereas ion-scale plasmoids (secondary islands with a weak core field) are formed in a weak guide field regime. Currents in both types of secondary islands are carried primarily by electrons. Both types of secondary islands have a magnetic tension force pointing radially inward toward their center. In the flux rope type, this inward tension force is balanced by an outward magnetic pressure gradient of the strong core field; in the plasmoid type, it is countered by an outward thermal pressure gradient caused by electron and ion energizations. The transition between these two types occurs when B g /B 0 = 0.1–0.2 (B g is the guide field, and B 0 is the asymptotic magnetic field).

Coalescence of multiple magnetic islands is recognized as an effective process to energize particles during magnetic reconnection, while its energy conversion process still remains unclear. Here, a two-dimensional fully kinetic simulation of multiple island coalescence with a small reconnection guide field is studied. In the analysis of energy conversion within a magnetic island, the dot product of is a useful quantity to compare with j · E = w 2, since the average work done by the Lorentz force on the circulating particles is negligible in the island and . A bipolar pattern of w 1 is found at a secondary island when the electrons are in circular motion inside the island. Significant energy dynamo (w 3 < 0) resulting from j ∥ E ∥ is found at the secondary island, which has not been reported before, where the parallel electric field E ∥ is highly correlated with w 3. Moreover, significant energy dissipation (w 3 > 0) due to is seen in the merging region between two coalescing islands. Both types of energy conversions are accompanied by enhancements in j ∥ and the parallel electron temperature T e∥. Three ion-scale magnetic islands (FR1, FR2, and FR3) observed by the Magnetospheric Multiscale spacecraft are compared favorably with the simulated signatures of energy dynamo and dissipation of an evolving secondary island. In particular, FR1 displayed a similar energy dynamo signature as that simulated in an early stage of the secondary island. FR2 and FR3 showed a dominant energy conversion similar to that obtained in a later stage of the secondary island.

Abstract. Two-dimensional (2-D) particle-in-cell (PIC) simulations are performed to investigate the evolution of the electron current sheet (ECS) in guide field reconnection. The ECS is formed by electrons accelerated by the inductive electric field in the vicinity of the X line, which is then extended along the x direction due to the imbalance between the electric field force and Ampere force. The tearing instability is unstable when the ECS becomes sufficiently long and thin, and several seed islands are formed in the ECS. These tiny islands may coalesce and form a larger secondary island in the center of the diffusion region.

Secondary islands have recently been intensively studied because of their essential role in dissipating energy during reconnection. Secondary islands generally form by tearing instability in a stretched current sheet, with or without guide field. In this article, we study the electric field structure inside a secondary island in the diffusion region using large-scale two-and-half dimensional particle-in-cell (PIC) simulation. Intense in-plane electric fields, which point toward the center of the island, form inside the secondary island. The magnitudes of the in-plane electric fields Ex and Ez inside the island are much larger than those outside the island in the surrounding diffusion region. The maximum magnitudes of the fields are about three times the B0VA, where B0 is the asymptotic magnetic field strength and VA is the Alfvén speed based on B0 and the initial current sheet density. Our results could explain the intense electric field (~100 mV/m) inside the secondary island observed in the Earth’s magnetosphere. The electric field Ex inside the secondary island is primarily balanced by the Hall term (j × B)/ne, while Ez is balanced by a combination of (j × B)/ne, −(vi × B), and the divergence of electron pressure tensor, with (j × B)/ne term being dominant. This large Hall electric field is due to the large out-of-plane current density jy inside the island, which consists mainly of accelerated electrons forming a strong bulk flow in the –y direction. The electric field Ey shows a bipolar structure across the island, with negative Ey corresponding to negative Bz and positive Ey corresponding to positive Bz. It is balanced by (j × B)/ne and the convective electric field. There are significant parallel electric fields, forming a quadrupolar structure inside the island, with maximum amplitude of about 0.3B0VA.

A key feature of collisionless magnetic reconnection is the formation of Hall magnetic and electric field structure in the vicinity of the diffusion region. Here we present multi‐point Cluster observations of a reconnection event in the near‐Earth magnetotail where the diffusion region was nested by the Cluster spacecraft; we compare observations made simultaneously by different spacecraft on opposite sides of the magnetotail current sheet. This allows the spatial structure of both the electric and magnetic field to be probed. It is found that, close to the diffusion region, the magnetic field displays a symmetric quadrupole structure. The Hall electric field is symmetric, observed to be inwardly directed on both sides of the current sheet. It is large (∼40 mV m−1) on the earthward side of the diffusion region, but substantially weaker on the tailward side, suggesting a reduced reconnection rate reflected by a similar reduction in Ey. A small‐scale magnetic flux rope was observed in conjunction with these observations. This flux rope, observed very close to the reconnection site and entrained in the plasma flow, may correspond to what have been termed secondary islands in computer simulations. The core magnetic field inside the flux rope is enhanced by a factor of 3, even though the lobe guide field is negligible. Observations of the electric field inside the magnetic island show extremely strong (∼100 mV m−1) fields which may play a significant role in the particle dynamics during reconnection.

The nonlinear growth of the internal kink mode is studied numerically using reduced magnetohydrodynamic equations in cylinder geometry. For low Lundquist numbers, S < 107, the already well-known results have been reproduced: a m/n = 1/1 magnetic island (m: poloidal, n: toroidal mode number) grows while the original core shrinks until full reconnection is achieved. For higher S values, however, the dynamics is found to be qualitatively different from the well-known Kadomtsev's model (Kadomtsev 1975 Sov. J. Plasma Phys. 1 389). The growth of the 1/1 island causes the development of a very thin current sheet which becomes tearing unstable. The current sheet is thus broken up and secondary islands (plasmoids) form. These plasmoids strongly speed up the reconnection and eventually coalesce into one secondary island. The formation of a large secondary island stops the fast reconnection process, leading even to a partial reversal of this process. The final state of sawtooth reconnection is thus no longer an axis-symmetric equilibrium as in the case of complete reconnection for low S values, but a helical equilibrium with two coexisting magnetic islands.

In this work, we compare gyrokinetic (GK) with fully kinetic Particle-in-Cell (PIC) simulations of magnetic reconnection in the limit of strong guide field. In particular, we analyze the limits of applicability of the GK plasma model compared to a fully kinetic description of force free current sheets for finite guide fields (bg). Here, we report the first part of an extended comparison, focusing on the macroscopic effects of the electron flows. For a low beta plasma (βi = 0.01), it is shown that both plasma models develop magnetic reconnection with similar features in the secondary magnetic islands if a sufficiently high guide field (bg ≳ 30) is imposed in the kinetic PIC simulations. Outside of these regions, in the separatrices close to the X points, the convergence between both plasma descriptions is less restrictive (bg ≳ 5). Kinetic PIC simulations using guide fields bg ≲ 30 reveal secondary magnetic islands with a core magnetic field and less energetic flows inside of them in comparison to the GK or kinetic PIC runs with stronger guide fields. We find that these processes are mostly due to an initial shear flow absent in the GK initialization and negligible in the kinetic PIC high guide field regime, in addition to fast outflows on the order of the ion thermal speed that violate the GK ordering. Since secondary magnetic islands appear after the reconnection peak time, a kinetic PIC/GK comparison is more accurate in the linear phase of magnetic reconnection. For a high beta plasma (βi = 1.0) where reconnection rates and fluctuations levels are reduced, similar processes happen in the secondary magnetic islands in the fully kinetic description, but requiring much lower guide fields (bg ≲ 3).

In this paper, with a two-dimensional particle-in-cell simulation model, we study the formation of power-law spectra of energetic electrons in multiple X line magnetic reconnection with a strong guide field. The processes of both magnetic reconnection and electron acceleration can be separated into two stages. In the first stage, two X lines appear at the border and center of the simulation domain, and then, two magnetic islands are formed. In this stage, electrons are accelerated mainly by parallel electric fields, and a power-law spectrum of energetic electrons is generated with the appearance of the second X line. In the second stage, the two magnetic islands are merged into one big island. Besides parallel electric fields, the Fermi mechanism also plays an important role in the production of energetic electrons, and its contribution is comparable to that of parallel electric fields when the electron energy is sufficiently large. In this stage, the generated power-law spectrum of energetic electrons becomes hard. In general, the acceleration efficiencies by both the parallel electric fields and Fermi mechanism become higher with the increase in electron energy, and the tendency is more obvious for the Fermi mechanism. Therefore, both the parallel electric fields and Fermi mechanism are important in the formation of power-law spectra of energetic electrons during multiple X line reconnection. We also investigate the influences of the ion-electron temperature ratio, guide field, and initial flux perturbation on the formed power-law spectra of energetic electrons.

Magnetic islands or flux ropes produced by magnetic reconnection have been observed on the magnetopause, in the magnetotail, and in coronal current sheets. Particle-in-cell simulations of magnetic reconnection with a guide field produce elongated electron current layers that spontaneously produce secondary islands. Here, we explore the seed mechanism that gives birth to these islands. The most commonly suggested theory for island formation is the tearing instability. We demonstrate that in our simulations these structures typically start out, not as magnetic islands, but as electron flow vortices within the electron current sheet. When some of these vortices first form, they do not coincide with closed magnetic field lines, as would be the case if they were islands. Only after they have grown larger than the electron skin depth do they couple to the magnetic field and seed the growth of finite-sized islands. The streaming of electrons along the magnetic separatrix produces the flow shear necessary to drive an electron Kelvin-Helmholtz instability and produce the initial vortices. The conditions under which this instability is the dominant mechanism for seeding magnetic islands are explored.