Kinetic Scale Magnetic Reconnection with a Turbulent Forcing: Particle-in-cell Simulations

Turbulent magnetic reconnection has been observed by spacecraft to occur commonly in terrestrial magnetosphere and the solar wind, providing a new scenario of kinetic scale magnetic reconnection. Here by imposing a turbulent forcing on ions in particle-in-cell simulations, we simulate kinetic scale turbulent magnetic reconnection. We find formation of fluctuated electric and magnetic fields and filamentary currents in the diffusion region. Reconnection rate does not change much compared to that in laminar Hall reconnection. At the X-line, the electric and magnetic fields both exhibit a double power-law spectrum with a spectral break near local lower-hybrid frequency. The energy conversion rate is high in turbulent reconnection, leading to significant electron acceleration at the X-line. The accelerated electrons form a power-law spectrum in the high energy range, with a power-law index of about 3.7, much harder than one can obtain in laminar reconnection.


Introduction
Magnetic reconnection is a fundamental plasma process during which magnetic field topologies change and magnetic energy is converted to plasma energy (Yamada et al. 2010). Magnetic reconnection occurs ubiquitously in various plasma environments, such as the solar corona (e.g., Masuda et al. 1994;Tsuneta 1996;Somov & Kosugi 1997;Su et al. 2013), the solar wind (e.g., Gosling et al. 2005;Phan et al. 2006Phan et al. , 2021Wang et al. 2022), and Earthʼs magnetosphere (e.g., Øieroset et al. 2001;Burch et al. 2016;Wang et al. 2017;Torbert et al. 2018;Lu et al. 2022a). It is widely believed to be responsible for the energy conversion and dissipation processes in these environments, with scales ranging from large, magnetohydrodynamic (MHD) scales to small, particle kinetic scales.
In large, MHD scales, the classical Sweet-Parker reconnection model (Parker 1957;Sweet 1958) gives a too slow reconnection rate so that it cannot well describe the reconnection process, for example, in the solar corona. Later, Lazarian & Vishniac (1999) found that the presence of a turbulent component of the reconnecting magnetic field can increase the reconnection rate dramatically. Such type of fast magnetic reconnection was thereafter validated by MHD simulations of magnetic reconnection with an external turbulent forcing (e.g., Kowal et al. 2009Kowal et al. , 2012Loureiro et al. 2009;Yang et al. 2020;Sun et al. 2022). Self-generated turbulence via formation of plasmoids was also found to realize fast magnetic reconnection according to MHD theories and simulations (e.g., Loureiro et al. 2007;Bhattacharjee et al. 2009;Pucci & Velli 2014). Using solar corona images, Cheng et al. (2018) observed turbulent magnetic reconnection in a current sheet during a solar flare, providing direct evidence for the occurrence of MHD turbulent reconnection.
Simulation and observational efforts have been made to understand kinetic scale turbulent magnetic reconnection, and its cause has been attributed to the following instabilities, waves, and coherent structures: Particle-in-cell (PIC) simulations showed that secondary tearing instability can form secondary magnetic islands (or secondary flux ropes) to develop turbulent magnetic reconnection (e.g., Daughton et al. 2011;Lu et al. 2019). Such a scenario, turbulent reconnection consisting of secondary magnetic islands, was observed thereafter by spacecraft in the magnetotail (e.g., Wang et al. 2016;Fu et al. 2017;Lu et al. 2020). PIC simulations found that lower-hybrid drift instability can form turbulences during magnetic reconnection (e.g., Divin et al. 2015a;Price et al. 2016Price et al. , 2017, which was also confirmed by spacecraft observations (e.g., Divin et al. 2015b;Cozzani et al. 2021). Moreover, PIC simulations found that other instabilities can also cause kinetic scale turbulences in reconnection, such as the Buneman instability (Drake et al. 2003), current filamentation instability (Che et al. 2011), interchange instability (e.g., Lapenta et al. 2015;Pucci et al. 2017), and electron Kevin-Helmholtz instability (e.g., Huang et al. 2017;Che & Zank 2020). On the other hand, direct spacecraft observations showed that the turbulences in kinetic scale magnetic reconnection can be caused by magnetic holes (i.e., depletion in magnitude of magnetic field) and electrostatic and electromagnetic waves (Ergun et al. 2016(Ergun et al. , 2017(Ergun et al. , 2020.
The above spacecraft observations and numerical simulations show that the constitution of the turbulences in kinetic scale reconnection is diverse; therefore, in this paper, we adopt a turbulent forcing as a collective manifestation of the various instabilities, waves, and coherent structures, and we impose the turbulent forcing into two-dimensional PIC simulations to study kinetic scale turbulent magnetic reconnection. The simulation model is described in the following Section 2, the simulation results are presented in Section 3, and Section 4 is the conclusions and discussion.

Simulation Model
We use a PIC simulation model, and it is 2D in the x-z plane. The initial configuration is the Harris current sheet with magnetic field z B z L B e tanh where B 0 is magnitude of the asymptotic magnetic field, n b is the background density, n 0 is the peak density in the Harris current sheet, and L is halfthickness of the current sheet. In our simulations, we use n b = 0.2n 0 and L = 0.5d i , where d i is the ion inertial length evaluated using n 0 . The initial ion and electron temperatures are uniform, with T mV 0.4 , where m i is the ion mass and V A is the Alfvén velocity evaluated using B 0 and n 0 .
] is discretized into grid cells, with L x = 204.8d i , L z = 51.2d i , and the grid size is Δx = Δz = 0.05d i , corresponding to a grid number of N x × N z = 4096 × 1024. Electric field and magnetic field are defined on the grids and updated by solving Maxwellʼs equations using an explicit algorithm. Ions and electrons are treated as full particles, and their positions and velocities are advanced by solving their equation of motion. The ion-toelectron mass ratio m i /m e = 100, and the speed of light c = 20V A . The unit density n 0 is represented by 610 particles per species per grid cell. Periodic and perfect conductor boundary conditions are used in the x-and z-directions, respectively.
A small perturbation in magnetic field is added at (x, z) = (0, 0) to expedite the onset of magnetic reconnection. Once reconnection begins, to evolve magnetic reconnection into a turbulent regime, we impose a turbulent forcing on the ions through the following procedure. We give a turbulent ion flow velocity dV with Then we interpolate it to the position of each ion and obtain the turbulent forcing on each ion.
)  l max , dV 0 determines the magnitude of the turbulent forcing, f mn and ψ mn are random phases, and L 0 is the maximum spatial scale of the turbulent forcing. We set L 0 = 51.2d i , l 1 min = , and l 8 max = , which gives the wavenumber k of the turbulent forcing, 0.1 < kd i < 1, i.e., the turbulent forcing is on ion scales, and this is why we impose it on ions and let it cascade to smaller-scale electrons self-consistently.  Figure 1 shows a representative case with a turbulent forcing dV 0 = 0.03V A added after magnetic reconnection begins at Ω i t = 30. Because of the turbulent forcing, the magnetic field lines and the current density become fluctuated. Figure 2 shows the zoomed-in view of the current density at the fluctuating X-line at Ω i t = 70. The current density presents a filamentary pattern, and the current is in the x, y, z all three directions. The quadrupolar Hall magnetic field B y and the bipolar Hall electric field E x and E z are typical in laminar reconnection (dV 0 = 0, see Figures 3(a), (c), and (e)); in the case with dV 0 = 0.03V A , the Hall magnetic field persists but is interfered by the magnetic fluctuations (Figures 3b), whereas the Hall electric field is fully surpassed by the electric fluctuations (Figures 3(d) and (f)). The electric field in the out-of-plane direction E y is also fluctuated with an amplitude of about ∼2V A B 0 (Figure 3(h)), much larger than the typical magnitude of the reconnection electric field, ∼0.2V A B 0 (Figure 3(g)). The ion and electron outflows also become fluctuated but still retain the bidirectional patterns that are typical in laminar reconnection (Figures 3(i)-(l)). The energy conversion j · E is nonzero in the diffusion region and peaks at the reconnection fronts in laminar reconnection (Figure 3(m)), and j E¢ · (where E E V B e ¢ = +´) is nonzero only in the vicinity of the X-line (i.e., electron diffusion region) in laminar reconnection (Figure 3(o)). In the case with dV 0 = 0.03V A , both j · E and j E¢ · are nonzero not only at the X-line but also throughout the entire domain (Figures 3(n) and (p)). Therefore, although nonzero j E¢ · has been commonly used as a criterion for electron diffusion regions in laminar reconnection (e.g., Lu et al. 2022b), it should be used with caution in turbulent reconnection.

Simulation Results
We show time histories of the reconnected magnetic flux and the reconnection rate in Figure 4. Although the out-of-plane electric field E y is fluctuated with a large amplitude (Figure 3(h)) in the case with dV 0 = 0.03V A , the reconnection  4 rate is still about ∼0.2V A B 0 , similar to that in laminar reconnection with dV 0 = 0. The reconnected magnetic flux is also close in these two cases. This is different from MHD scale reconnection in which the turbulent forcing can dramatically increase the reconnection rate (e.g., Kowal et al. 2009;Loureiro et al. 2009;Kowal et al. 2012;Yang et al. 2020;Sun et al. 2022). The reconnection rate is decreased in the late phase because the electron diffusion region elongates (e.g., Daughton et al. 2006). Note that the turbulent forcing further decreases the reconnection rate. This may be because the turbulent forcing provides additional ion energization so that the high ion pressure in unreconnected regions can suppress magnetic reconnection. Figure 5 shows virtual spacecraft observations at the X-line, (x, z) = (0, 0). Right after the turbulent forcing is turned on at Ω i t = 30, the fluctuations emerge at the X-line. The three components of the magnetic field are fluctuated with δB x ≈ 0.4B 0 and δB y ≈ δB z ≈ 0.2B 0 (Figure 5(a)). The electric field fluctuations are strong, with a large magnitude of ∼2V A B 0 ( Figure 5(c)), larger than that of the reconnection electric field (Figure 3(g)) and the Hall electric field (Figures 3(c) and (e)). The fluctuated magnetic field is self-consistent with the filamentary current that is fully fragmented with well-separated spikes in current density ( Figure 5(e)). The filamentary current is mostly carried by the electrons in the y-direction, with the maximum V V 7 ey A » | | , whereas the amplitude of V ex and V ez fluctuations is about 3V A (Figure 5(d)), and the amplitude of V i fluctuations ≈V A (Figure 5(b)). The virtual spacecraft observations, which show more fine structures than in the snapshots (Figures 1-3), suggest that the observed filamentary currents and electromagnetic fluctuations originate from not only their spatial structures but also their temporal evolution.
The filamentary current density is well correlated with the spikes in j E¢ · ( Figure 5(f)). The maximum j E¢ · is larger than en V B 1.2 0 A 2 0 , much larger than the typical value of about en V B 0.2 0 A 2 0 in laminar reconnection (Figure 3(o)). Note that the work done by the Lorentz force The j E¢ · spikes are also well correlated with peaks in electron temperature ( Figure 5(g)) and enhancements in electron energy distribution functions ( Figure 5(h)). The electrons keep being heated and accelerated, as indicated by the temperature increase from Figure 5(g)) and the energy increase in the tail of its distribution function with energy exceeds 1.0m e c 2 ( Figure 5(h)). Figure 6 shows the power spectra of the magnetic field and the electric field from the virtual spacecraft observations at the X-line. The magnetic field and electric field both follow a double power-law spectrum with a spectral break near ω ≈ 3Ω i . The local magnetic field is fluctuating with an average magnitude of B * ≈ 0.22B 0 , so the local ion gyrofrequency 0.22 . Therefore, the spectral break is near the local lower-hybrid frequency. Between the local ion gyrofrequency and the local lowerhybrid frequency, the slope of the magnetic field power spectrum is about −1.99, and that of the electric field is about −1.18. Above the local lower-hybrid frequency, the magnetic field and the electric field have steeper spectra with slopes of about −4.41 and −2.91, respectively. The power-law spectra and the spectral break are typical in plasma turbulence (e.g., Alexandrova et al. 2009). Therefore, kinetic scale magnetic reconnection with the turbulent forcing evolves into turbulent reconnection, with energy cascade from large to small scales and dissipation at local lower-hybrid frequency.
We have shown in Figure 5(h) that the electrons keep being accelerated at the X-line in turbulent reconnection with dV 0 = 0.03V A . In Figure 7, we show the electron energy distribution for cases with different magnitudes of the turbulent forcing. In laminar reconnection with dV 0 = 0, the electrons are accelerated, and the high energy electrons form a soft powerlaw spectrum with a slope of −5.95. In turbulent reconnection with dV 0 = 0.03V A , the electron acceleration is more efficient in the high energy part, which also forms a harder power-law spectrum with a slope of −3.69. When the turbulent forcing is increased to dV 0 = 0.05V A , the electrons at the X-line are heated to higher energies, but the spectral slope of the energetic part remains −3.68. To understand the electron acceleration mechanism, we plot the electron velocity distributions in Figure 8. In laminar reconnection (dV 0 = 0), the electrons are accelerated by the reconnection electric field when they follow the Speiser-type meandering motion at the X-line, as shown by the two positive and negative v ez peaks (Figure 8(a)) and the negative v ey peak (Figure 8(b)). In turbulent reconnection (dV 0 = 0.03V A and dV 0 = 0.05V A ), the high energy electrons are isotropic, suggesting a stochastic acceleration in the turbulent electric and magnetic fields (Zank et al. 2014;Che & Zank 2020).

Conclusions and Discussion
Our PIC simulations show that turbulent forcing can transform kinetic scale magnetic reconnection from laminar to turbulent. In the kinetic scale turbulent reconnection, the electric field and magnetic field become fluctuated, and filamentary current structures are formed. The fluctuated electric field and magnetic field at the X-line both exhibit a double power-law behavior with a spectral break near local lower-hybrid frequency. The electromagnetic turbulence barely changes the reconnection rate. The filamentary current has peaks in current density that are well correlated with spikes of energy conversion j E¢ · . In general, j E¢ · is much larger in turbulent reconnection than laminar reconnection, leading to significant electron energization. At the X-line in turbulent reconnection, the electrons are accelerated to form a power-law spectrum in the high energy tail, with a slope of about -3.7, much harder than that in laminar reconnection. The velocity distributions of the energetic electrons are isotropic in turbulent reconnection, suggesting a stochastic acceleration mechanism.
Although kinetic scale turbulent magnetic reconnection has been commonly observed in Earthʼs magnetotail, the observations were mostly in the ion diffusion region or the outflow region (e.g., Eastwood et al. 2009). Recently, Li et al. (2022b) reported Magnetospheric Multiscale (MMS) observations of the X-line region in turbulent reconnection, which presented filamentary currents, turbulent electric-and magnetic fields, and superthermal electrons. In Table 1, we compare the X-line properties obtained from our PIC simulations of laminar reconnection (dV 0 = 0) and turbulent reconnection (dV 0 = 0.03V A ) and the MMS observations of turbulent reconnection in Li et al. (2022b). It is clear that the filamentary currents, the strong electromagnetic fluctuations, and the double power-law spectral behavior with a break at local lower-hybrid frequency in the MMS observations are well reproduced in our PIC simulations by imposing a turbulent forcing on ions. Nevertheless, there are some quantitative differences between the observations and simulations, for example, the energetic electron power-law index in the MMS observations is about 8.0, whereas our simulations give a much harder power-law index of about 3.7.
Our simulations show that electron acceleration at the X-line is more efficient in turbulent reconnection than laminar reconnection. In laminar reconnection, the electrons follow a Speiser-type meandering motion at the X-line and are accelerated in the meantime by the reconnection electric field (e.g., Fu et al. 2006;Ng et al. 2011). However, the X-line configuration in laminar reconnection cannot well confine the electrons because the electrons can escape from the X-line in the x-direction. Therefore, the acceleration is not sufficient enough to produce hard power-law spectra for the electrons, and indeed, our simulations give a soft power-law index of ∼6 at the X-line in laminar reconnection. For turbulent reconnection, the electron energy spectrum is much harder, with a power-law index of ∼3.7, which may be because the electrons are trapped at the X-line for a longer time in turbulent reconnection (Matthaeus et al. 1984), and this type of electron acceleration in the stochastic electromagnetic fields can be well  explained by the second-order Fermi acceleration mechanism (Zank et al. 2014;Che & Zank 2020). However, it requires further examination to understand quantitatively why the power-law index is ∼3.7 in the turbulent reconnection with dV 0 = 0.03V A and why the index does not change with a stronger turbulent forcing dV 0 = 0.05V A .
Previous PIC simulation results have suggested various selfconsistent instabilities, waves, and coherent structures to form turbulences in kinetic scale turbulent reconnection (see Section 1). Here, from a different perspective, we launch kinetic scale turbulent reconnection by imposing a turbulent forcing as a collective manifestation to imitate the effect of   Note. Here LH w* is the average local lower-hybrid frequency. From the MMS observations, the maximum j E¢ · , is 1.31 nW m −1 , the average |B| is 9.33 nT, the maximum (δB x , δB y , δB z ) is (9.78, 10.73, 11.92) nT, the average |E| is 42.2 mV m −1 , the maximum (δE x , δE y , δE z ) is (40.6, 62.9, 122.2) mV m −1 , and the peak in electron temperature is 9.3 keV. To compare with the simulation results, the observation results are normalized to simulation units using the observed parameters, B 0 = 25 nT and n 0 = 0.1 cm −3 .