Particle-in-cell simulations of electron energization in laser-driven magnetic reconnection

Electrons can be energized during laser-driven magnetic reconnection, and the energized electrons form three super-Alfvénic electron jets in the outflow region (Lu et al 2014 New J. Phys. 16 083021). In this paper, by performing two-dimensional particle-in-cell simulations, we find that the electrons can also be significantly energized before magnetic reconnection occurs. When two plasma bubbles with toroidal magnetic fields expand and squeeze each other, the electrons in the magnetic ribbons are energized through betatron acceleration due to the enhancement of the magnetic field, and an electron temperature anisotropy T e ⊥ > T e ∣ ∣ ?> develops. Meanwhile, some electrons are trapped and bounced repeatedly between the two expanding/approaching bubbles and get energized through a Fermi-like process. The energization before magnetic reconnection is more significant (or important) than that during magnetic reconnection.


Introduction
Magnetic reconnection is a fundamental physical process in plasmas during which the topologies of magnetic field lines are rearranged and magnetic energy is converted to plasma kinetic energy [1][2][3]. Magnetic reconnection is widely believed to be responsible for various kinds of explosive phenomena in space and laboratory plasmas, such as solar flares [4][5][6], magnetospheric substorms [7][8][9], and sawtooth crashes in tokamaks [10,11]. The generation of energetic electrons is considered to be one of the most important signatures in magnetic reconnection [12][13][14]. Generally, two-dimensional (2D) particle-in-cell (PIC) simulations show that electrons can be accelerated by the parallel electric field when they move towards the X-line [15][16][17], and get further accelerated in the vicinity of the X-line by the reconnection electric field [18,19]. Moreover, Hoshino et al [20] suggest that the electrons can also be accelerated in the magnetic pileup region during the curvature and gradient drift motions after the acceleration in the vicinity of the X-line.
In addition to the above non-adiabatic acceleration mechanisms, adiabatic betatron acceleration and Fermi acceleration are also proposed to explain the generation of energetic electrons in plasmas, especially in geospace and astrophysical plasmas [21][22][23][24][25][26][27]. Betatron acceleration works on the magnetized electrons in the enhancing magnetic field. The enhancement of the magnetic field leads to the generation of the inductive electric field, and the magnetized electrons are thus accelerated in the perpendicular direction by the inductive electric field. In contrast, Fermi acceleration is mainly in the parallel direction during which the magnetized electrons gain energy though head-on collision with the magnetic field. The classic Fermi acceleration can also be comprehended through the second adiabatic invariant J v s d L 0 ò = taken over a bounce orbit. When the orbit shrinks/contracts, the conservation of J leads to the increase of v .
Dedicated laboratory experiments have been conducted to study magnetic reconnection, for example, the Todai Spheromak-3 [28,29], the Swarthmore Spheromak Experiment facility [30,31], the Magnetic Reconnection Experiment [32,33], and the Versatile Toroidal Facility [34,35]. More recently, magnetic reconnection experiments in laser-produced high-energy-density plasmas have also been conducted with the OMEGA laser facility [36,37] and the Vulcan laser facility at the Rutherford Appleton laboratory [38][39][40], which provide a new experimental platform to study magnetic reconnection. In the experiments, two (or more) plasma bubbles with high density ( 10 cm 20 3 )  and high temperature ( 1 keV) are generated by focusing two nanosecond-duration laser beams on a planar-target foil. During the laser-plasma interaction, nonparallel density and temperature gradients lead to the spontaneous formation of azimuthal magnetic fields around the laser focal spots [41]. The azimuthal n T e e ´ magnetic field is found on the order of megagauss (MG), and forms toroidal ribbons wrapping around the plasma bubbles. When the laser beams are close enough, the bubbles eventually encounter each other with magnetic fields of opposing signs, and magnetic reconnection will occur. The laser-driven magnetic reconnection experiment has also been conducted to at the Shenguang-II (SG-II) laser facility [42]. Electrons energized up to MeVs are observed in the outflow region of the reconnection experiment, and three well-collimated high-speed electron jets are generated. By performing 2D PIC simulations, Lu et al [43] have investigated the formation of the three high-speed electron jets, which are super-Alfvénic in laser-produced plasma reconnection. During magnetic reconnection, electrons can be accelerated by the reconnection electric field in the vicinity of the X-line. These accelerated electrons move away from the X-line, forming the upper and lower super-Alfvénic electron jets. Meanwhile, electrons can also penetrate into the reconnection pileup region, gyrate in a semi-circle, and get accelerated by the inductive electric field in the pileup region. These accelerated electrons form the center super-Alfvénic electron jet.
In this paper, we further demonstrate that, before magnetic reconnection occurs, the electrons can be energized significantly through betatron and Fermi-like acceleration. The expansion and approach of the plasma bubbles play essential roles in the electron energization processes. The remainder of the paper is organized as follows. Section 2 describes the simulation model, section 3 presents the simulation results, and section 4 gives the summary and discussions.

Simulation model
The simulation model used in the present study is a 2D PIC model in which the electromagnetic fields are defined on grids and updated by solving Maxwell equations with a full explicit algorithm. The positions and velocities of the ions and electrons are advanced relativistically in the electromagnetic fields. A first-order weighting is employed for the particle shape factor. The parameters and geometry in the simulations are all in accordance with the SG-II experimental setup [42]. The initial configuration of the simulation system is two expanding semicircular plasma bubbles, similar to the previous PIC simulations by Fox et al [44,45] and Lu et al [43,46]. The simulation domain is a rectangular in the x z , The two semicircular bubbles are centered at L 0, z ( ) and L 0, z ( ) respectively. Define the radius vectors from the center of each bubble, x z L r , where n b is the background density, and the density contribution from each bubble n i Here L n is the initial scale of the bubbles and n 0 is the initial peak bubble density. In the simulation, we choose n n 0.2 . b 0

=
The initial expansion velocity of the bubbles is V V , where B 0 is the magnitude of the initial magnetic field, and L B is the half-width of the magnetic ribbons. To be consistent with the plasma flow, an initial electric field E V B = -´is imposed, while the initial current density is determined by Ampere's law. Table 1 lists the reported or otherwise estimated parameters for the SG-II experiment [42]. Table 2 lists the numerical parameters used in the simulations. The measured electron density near the X-line of reconnection is 5 10 cm , 19 3´which is on the order of one tenth of the peak electron density based on the simulations. The peak electron density is thus about 5 10 cm . 20 3 -Given the average ionic charge Z 10, the peak ion density is about 5 10 cm . 19 3 -Therefore, the ion inertial length based on the peak ion density is where v A is the Alfvén speed based on B 0 and n . 0 The initial velocity distributions for the ions and electrons are Maxwellian with bulk velocity in the radial direction and drift velocity in the out-of-plane  Table 2. Simulation parameters.

Parameter
Simulation setup In the simulations, we choose c v 75 A = which is larger than that in some other PIC simulations [27,47] (nevertheless, it is still much smaller than the realistic value). Our purpose is to pick a small thermal speed in comparison to the speed of light. If the initial thermal speed is comparable to the speed of light, the electrons are thus already relativistic at the initial time, which is far from the realistic experimental setup. In PIC simulations, to accommodate the available computer resources, it is common to use a nonrealistic mass ratio (100 or even smaller), not only in the simulations of the geospace plasmas [27,47], but also in the simulations of the plasmas in laboratory [45,48]. The mass ratio m m 100 i e = can distinguish the motions of the electrons and the ions well. The kinetic physics is not sensitive to the mass ratio qualitatively, so the energization mechanisms discussed in this paper do not change with different mass ratios.  (3)). At the initial time, the portion of the energetic electrons is very low, about 5%. As the two plasma bubbles expand at a supersonic speed, they strongly squeeze each other, which piles up the magnetic flux in the inflow region before reconnection occurs. At t 1.1, i W = just before the beginning of the The corresponding temperature anisotropy is thus A 2. e » From t 0 i W = to 1.1, the electrons are mainly heated in the perpendicular direction. Figure 3(a) shows the velocity distribution of the electrons in the blue box in figure 2. The red arrows denote the directions of the local magnetic field which is mainly along the x direction. The electron distribution is bi-Maxwellian with the perpendicular temperature higher than the parallel temperature. Figure 3(b) shows the velocity distribution of the accelerated electrons in the red box in figure 2. The velocity distribution is spread out mainly in the z direction, which indicates that the acceleration in the field reversal region is mainly in the v z direction.

Simulation results
To further investigate the electron energization mechanism in the magnetic ribbons, we 'tag' the electrons in the blue box (see figure 2)  Note that the electron energization is accompanied by the magnetic field enhancement, and most of the energized electrons are magnetized in the magnetic ribbons, which suggest a process of betatron acceleration.
A representative electron, electron 1, is presented in figure 5 to further illustrate the electron betatron acceleration process in the compressed magnetic ribbons. Figures 5(a)-(c) show the trajectory of the electron during different time intervals. The background contours show the magnitude of the magnetic field, and the The enhancement of the magnetic field leads to the generation of the inductive electric field in the magnetic ribbons, and the magnetized electrons are accelerated in the perpendicular direction by the inductive electric field. Therefore, electron 1 is energized through betatron acceleration, and this kind of energized electrons forms a bi-Maxwellian distribution with the electron temperature anisotropy (T T , e e > see figure 3(a)). In the magnetic field reversal region between the two expanding bubbles, the electrons can also be energized significantly, especially in the v z direction (see figure 3(b)). We also trace the trajectories and energies of all the electrons in the red box (see figure 2) at t 1.1.   Figure 8 further shows the kinetic energy of electron 2 as a function of its z position. The electron gains energy as it is reflected by magnetic ribbons at the boundaries of the bubbles, which suggests that the energization process of electron 2 is a Fermi-like process.
The magnetic field shows a wavy structure along the toroidal direction (parallel to the magnetic field) of the compressed magnetic ribbons. Figure 9  pi w The wavy magnetic field perturbation grows stronger at t 0.9 i W = (see figure 9(b)). Besides the out-of-plane magnetic field B , y the in-plane magnetic field (indicated by the magnetic field lines) is also wavy. At t 1.8, i W = due to the Hall effect of magnetic reconnection, there forms the quadrupolar structure of B . y During the same time, the wavy magnetic field perturbation gradually decreases. At the edges of the magnetic ribbons, the magnetic field is weak and the electron temperature is anisotropic, which is favorable to the Weibel instability [49]. The wavy magnetic structure along the parallel direction is considered to be the consequence of the Weibel instability. The wavelength obtained in the simulation is found to be consistent with the analytical theory, and the growth rate 5.1 sim i g » W is also close to the analytical . th g The above consistency between the simulation and linear theory shows that the formation of the wavy B y perturbation is due to the Weibel instability driven by the electron temperature anisotropy.
After about t 1.1, i W = magnetic reconnection begins, and the electrons are further energized. Figure 11 shows the contour of the electron bulk velocity   figure 13. It is noted that the total energy is well conserved in the simulations. Based on the energy evolutions, the whole process of the laser-driven reconnection can be summarized as follows: the initial supersonic expansion of the plasma bubbles leads to the enhancement of the magnetic field, and the electrons are energized through betatron and Fermi-like processes. Therefore, at the early stage, the ion kinetic energy is converted to the magnetic energy and electron kinetic energy. At the second stage after magnetic reconnection begins, the increase of the magnetic energy slows down due to the dissipation caused by reconnection. At the last stage, the expansion of the bubbles is gradually stopped, reconnection becomes the dominant process. The magnetic energy is converted to the kinetic energies of the electrons and ions through magnetic reconnection.

Conclusions and discussion
In this paper, based on the geometry and parameters of the SG-II reconnection experiment, we performed 2D PIC simulations to investigate the electron energization in laser-driven magnetic reconnection. The electrons were found to be energized to high energy before magnetic reconnection occurs, and the following magnetic reconnection can further energize these electrons. The energization before magnetic reconnection is more significant than that during magnetic reconnection. Before magnetic reconnection occurs, the magnetic field of the magnetic ribbons is enhanced due to the expansion of the plasma bubbles and the compression between the bubbles, which leads to the betatron acceleration of the electrons therein. The betatron acceleration is mainly in the perpendicular direction, therefore, an electron temperature anisotropy T T e e > develops. The Weibel instability is excited by the electron temperature anisotropy, which generates a wavy magnetic structure at the  edges of the magnetic ribbons. At the same time, some electrons can be bounced repeatedly between the two expanding bubbles and get energized through a Fermi-like process. After magnetic reconnection begins, the electrons are further energized. Part of the electrons can be accelerated by reconnection electric field in the vicinity of the X-line, and then move away from the X-line. Electrons can also be accelerated in the pileup region   where they gyrate in a semi-circle, and then leave. During the semi-circle gyration, these electrons are accelerated by the inductive electric field in the pileup region. The Weibel instability (or filamentation instability) has been found to play an important role in the hot, dense plasmas produced by high-intensity ( 10 W cm , 18 2 )  picosecond-or femtosecond-duration laser pulses [51][52][53]. The development of the counter-streaming Weibel instability leads to the generation of the turbulent magnetic fields in the plasmas. On the other hand, in laser-driven magnetic reconnection experiments, the laser beams are nanosecond-duration, and their intensity is 10 10 Wcm . we have demonstrated that the Weibel instability also develops due to the electron temperature anisotropy. Therefore, a turbulent/wavy magnetic structure is also generated at the edges of the magnetic ribbons. The turbulent/wavy magnetic structure may have a significant effect on the reconnection onset.  Magnetic reconnection between laser-produced plasma bubbles is found to be in a strongly driven reconnection regime. Fox et al [44,45] have demonstrated that the upstream magnetic flux pileup plays an essential role in the sharply increase of the reconnection rate in the laser-driven magnetic reconnection. In the present study, we have further shown that the magnetic flux pileup is also very important for the electron energization. Before magnetic reconnection begins, the upstream electrons are pre-energized through betatron acceleration. These pre-energized electrons are easier to get further energized during magnetic reconnection [16]. As the two expanding plasma bubbles approach to and squeeze each other, a thin current sheet forms between them [46]. The electrons in the current sheet can also be pre-energized by reflections between the two bubbles, which is a Fermi-like process. Therefore, the current sheet is thermalized, corresponding to a high-beta magnetic reconnection which has a low efficiency at converting magnetic energy into the electron kinetic energy [54]. This is why the electron energization by magnetic reconnection is less significant than the enegization by the magnetic flux pileup and plasma bubble expansion. Here E e is the energy of the electric field which is very small and thus negligible.