Magnetic reconnection provides a mechanism for mass transport and energy conversion in many plasma physical settings. It is generally considered to be responsible for solar flares^[1] and coronal mass ejections^[2] in the solar atmosphere, substorms in the earth’s magnetotail,^[3–6] injections of the solar wind mass and energy into the earth’s magnetopause,^[7,8] and disruptions in the laboratory experiments.^[9,10] In these plasma environments, the mean free path of charged particles is usually very large; thus the classical Coulomb collision is negligible. A collisionless reconnection model is believed to be suitable in these plasma systems.^[11,12]

The reconnection electric field plays a pivotal role in both energy conversion and production of energetic particles during magnetic reconnection by doing work on particles.^[13–17] In steady-state reconnection, the normalized reconnection electric field can be employed to represent the reconnection rate, and it is equivalent to the ratio of the inflow speed to the Alfvén speed. In collisionless magnetic reconnection, the diffusion region has two layers: the ion diffusion region and the electron diffusion region. In the electron diffusion region (EDR), both electrons and ions are unmagnetized, and electrons become magnetized in the ion diffusion region. The reconnection rate in collisionless magnetic reconnection is mediated by the Hall effect resulting from such a kind of decoupled motions between ions and electrons, and the reconnection rate is around 0.1. Correspondingly, the reconnection electric field is about 0.1v_AB₀, where v_A is the Alfvén speed, and B₀ is the upstream magnetic field.^[11,18] However, in reality, magnetic reconnection is non-stationary and the reconnection electric field is time evolutionary. The evolution of the reconnection electric field has three stages. A seed electric field is firstly produced in the current sheet, and then it grows until a maximum value is attained; finally its evolution saturates. The seed electric field can be generated by instabilities in the current sheet, such as the tearing-mode instability,^[19–21] lower hybrid drift instability,^[22,23] Buneman instability,^[24,25] and Kelvin–Helmholtz instability.^[26] In simulations of magnetic reconnection, the seed electric field is usually provided by an initial perturbation in order to bypass the linear growth of instabilities in the current sheet and forces the system into the reconnection stage, where the reconnection electric field grows spontaneously, from the beginning of the simulations.^[11,13,27] When the initial perturbation is sufficiently small, it will not change the following evolution of the reconnection electric field.^[11]

Litvinenko^[28] studied steady magnetic reconnection in the framework of incompressible Hall magnetohydrodynamics, they find that the Hall effect plays an important role in the reconnection rate enhancement and the current sheet thinning, and obtain an analytical relation between the reconnection electric field and the outflow speed. In collisionless magnetic reconnection, the reconnection electric field in the EDR is dominated by the divergence of the electron pressure tensor.^[21,27] Hesse^[29] further found that the reconnection electric field in the EDR is related to the gradient of the electron bulk velocity. However, so far, there were very few studies on the spontaneous growth of the reconnection electric field from the seed electric field in magnetic reconnection. Lu et al.^[21] developed a theoretical model to explain the self-reinforcing process of the reconnection electric field in the EDR during anti-parallel magnetic reconnection. In the model, they proposed that the reconnection electric field is proportional to the electron outflow speed, while the electrons obtain their outflow speed from the acceleration by the reconnection electric field in the EDR. Such a self-reinforcing process leads to exponential growth of the reconnection electric field, which was further verified by two-dimensional (2D) particle-in-cell (PIC) simulations. Compared with anti-parallel reconnection, the more general case is guide filed reconnection, where the shear angle between the reconnecting magnetic field cross the current sheet is less than 180°.^[30] Previous studies indicated that the reconnection rate in guide field reconnection is only a little smaller than that in anti-parallel reconnection. However, the tearing-mode instability in the current sheet with a guide field and the structure of the diffusion region in guide field reconnection are greatly different from the anti-parallel case.^[27,31] Therefore, in the present paper we will investigate the spontaneous growth of the reconnection electric field in the EDR during guide field reconnection.

In this section, 2D PIC simulations are performed to investigate the evolution of the reconnection electric field during magnetic reconnection with a guide field. The simulation code has been successfully used to study magnetic reconnection and plasma waves.^[13,32–36] The simulations are conducted in the x–z plane, and the initial condition is the Harris-type current sheet with the magnetic field B(z) = B₀tanh(z/δ) e_x – B_g e_y, where B_g = B₀ is a uniform guide field. The density profile is n(z) = n₀sech²(z/δ) + n_b, where n_b = 0.2n₀ is the background plasma density. The half width of the current sheet is δ = 0.5d_i, where d_i is the ion inertial length based on n₀. The initial velocity distributions of ions and electrons are assumed to be Maxwellian with the temperature ratio T_i/T_e = 4, and the ion-to-electron mass ratio is set to be m_i/m_e = 64. The speed of light is assumed to be c/v_A = 15, where v_A is the Alfvén speed based on B₀ and n₀. The simulation domain is [–L_x, L_x] × [–L_z, L_z] with spatial resolution Δ x = Δ z = 0.025d_i, where L_x = 30d_i and L_z = 7.5d_i. The time step is , where Ω_i is the ion gyrofrequency based on B₀. Nearly 10⁹ computational particles per species are employed in the simulations. The periodic boundary condition is used in the x direction, while in the z direction the conducting boundary is assumed. In this paper, we at first run one case without an initial perturbation, and then investigate the influence of an initial perturbation on the evolution of the reconnection electric field. The form of the initial perturbation of the magnetic flux is expressed as Δ Ψ = Ψ₀sech²(x/4δ)sech²(z/δ). Here, magnetic flux function Ψ satisfies ∂ Ψ/∂ z = B_x and ∂ Ψ/∂ x = –B_z.

In the first case, we do not add any initial perturbation, or Ψ₀ = 0. The process of magnetic reconnection is clearly demonstrated in Fig. 1(a), which shows the magnetic flux Ψ/(B₀d_i) along the line z = 0 at different times. Figure 1(b) plots the time evolution of | Ψ_M | with different M calculated by the FFT of Ψ, where M = L_xk_x / π, and k_x is the wave number. Obviously, the current sheet is unstable to the tearing-mode instability, and the most unstable mode is M = 7 mode, corresponding to k_x δ = 0.37. It is in agreement with previous linear theory,^[31] where k_xδ = 0.3–0.5 for the most unstable mode. At about Ω_it = 60, the magnetic flux Ψ around x = –26d_i increases rapidly, and reconnection begins to occur. Unlike other simulations of magnetic reconnection,^[11,13] where an initial perturbation is added to bypass the excitation of the tearing-mode in the current sheet, in our simulations we can observe the occurrence of magnetic reconnection following the tearing-mode instability.

Fig. 1.

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	Fig. 1. (a) Time evolution of the magnetic flux function Ψ/(B0di) along z = 0. (b) Time evolution of |ΨM| with M = 5–9. In this case, there is no initial perturbation.

In this paper, we focus on the spontaneous growth of the reconnection electric field after reconnection onset occurs. Figure 2 shows the time evolution of the electric field in the y direction E_y / v_AB₀ (left panel) and electron current density in the y direction J_ey / en₀v_A (right panel) with in-plane magnetic field lines for the case without an initial perturbation, panels (a)–(d) are displayed at Ω_it = 50, 60, 65, and 70, respectively. Reconnection occurs around Ω_i t = 60, and at that time both E_y and J_ey begin to be generated around the X line. As reconnection proceeds, the amplitude of both E_y and J_ey increases. During such a process, the electrons are accelerated by the reconnection electric field around the X line, and a thin electron current sheet embedded in the background current sheet is then formed. However, due to the deflection of the electron outflow by the guide field, the electron current sheet is biased along the lower left and upper right branches of the separatrices (the outflow separatrices) out of the EDR. This is consistent with previous studies.^[15,27] The peak of the reconnection electric field is firstly located around the X line, and it gradually moves to the pile-up region, where the magnetic flux is accumulated by the high-speed electron flow away from the X line.^[21]

Fig. 2.

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	Fig. 2. The out-of-plane electric field Ey / vAB0 and electron current density Jey / en0vA at Ωi t = 50 (a), 60 (b), 65 (c), and 70 (d) respectively. Black curves show the in-plane magnetic field lines, and there is no initial perturbation in this case.

Using the electron momentum equation, one can easily find that the electric field can be expressed as

After knowing the position and velocity of each particle, we can easily calculate the electron bulk velocity and electron pressure tensor. At last, we obtain the electron convection term, electron pressure tensor term, and electron inertial term as described in Eq. (2). Figure 3 shows the time evolution of the reconnection electric field E_y and its three decomposed terms along z = 0 for the case without an initial perturbation. Obviously, the reconnection electric field in the EDR is dominated by the electron pressure tensor term, although both the electron convection and electron inertial terms are important outside the EDR. Also, the reconnection electric field and its three decomposed terms increase with the proceeding of magnetic reconnection, and they saturate at about Ω_it = 70.

Fig. 3.

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	Fig. 3. Time evolution of the horizontal cut along z = 0 of out-of-plane electric field Ey / vAB0 (the black lines), electron convection term (the red lines), electron pressure tensor term (the blue lines), and electron inertial term (the orange lines) at Ωit = 61 (a), 64 (b), 67 (c), and 70 (d). Electron diffusion region is denoted by gray color. In this case, there is no initial perturbation.

In general, the reconnection electric field at the X line is used to quantify the topological change of magnetic field lines, and it is also related to the conversion of magnetic energy into plasma kinetic energy and the production of energetic particles. Therefore, in this paper, we focus on the evolution of the reconnection electric field at the X line, or its self-reinforcing process, during magnetic reconnection. Figure 4 plots the time evolution of the reconnection electric field E_y at the X line for the case without an initial perturbation. The reconnection electric field grows exponentially at about Ω_it = 60, and saturates at about Ω_it = 70. The growth rate is estimated to be about 0.226Ω_i.

Fig. 4.

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	Fig. 4. Time evolution of the out-of-plane electric field Ey / vAB0 in the EDR. The dashed line shows the fitted exponential growth rate. In this case, there is no initial perturbation.

We also run two cases with initial perturbations, whose amplitudes of the magnetic flux are Ψ₀ / (B₀d_i) = 0.05 and 0.1, respectively. The evolution of the reconnection electric field E_y at the X line in these simulations is plotted in Fig. 5. Compared to the case without an initial perturbation, magnetic reconnection occurs much earlier when an initial perturbation is introduced. However, in all these cases the reconnection electric field follows an exponential growth later (from Ω_it = 10 to 20 and 10 to 15 when the amplitude of the initial perturbation is Ψ₀ / (B₀d_i) = 0.05 and 0.1, respectively), and the growth rate is almost the same. In the case without an initial perturbation, the growth rate is about 0.226Ω_i, while it is about 0.225Ω_i and 0.236Ω_i when the amplitude of the initial perturbation is Ψ₀ / (B₀d_i) = 0.05 and 0.1, respectively. Therefore, the spontaneous growth of the reconnection electric field, which is the focus in this paper, is not affected by the initial perturbation with small amplitude.

Fig. 5.

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	Fig. 5. Time evolution of the reconnection electric field Ey / vAB0 when the initial perturbation amplitude is Ψ0 / (B0di) = 0.05 (the black line) and 0.1 (the blue line) respectively. The dashed lines show the fitted growth rate.

A theoretical model is proposed in this section to describe the spontaneous growth of the reconnection electric field in the EDR during guide field reconnection. From the simulation, we see that the reconnection electric field around the X line is mainly balanced by the electron pressure tensor term, and it can be expressed as^[29]

Figure 6 depicts the sketch of the EDR. Usually, the length of the EDR is much larger than its width. So ∂² / ∂ z²≫ ∂² / ∂ x². Combined with J_ey = –en_eV_ey, we get ∇²(m_en_eV_ey) ≈ – (m_e / e)∂²J_ey / ∂ z². If we denote the half length and width of the EDR by L_e and δ_e, respectively, the reconnection electric field around the X line can be approximated as

Fig. 6.

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	Fig. 6. A sketch of the electron diffusion region.[21]

If we assume that the EDR is stationary during the spontaneous growth of the reconnection electric field, the parameters J_ey, δ_e, L_e, n_e, and r_L are kept as constants. Then, from Eq. (4), we can know that the reconnection electric field is in direct proportion to the electron outflow speed

In our theoretical model, we assume that the half length and width of the EDR L_e and δ_e, as well as the electron current density in the y direction J_ey, electron density n_e, and electron Larmor radius in the guide field r_L in the EDR are unchanged. Based on the simulations described in Section 2, we plot their time evolution in Fig. 7. Here, δ_e is calculated by the half width of the electron current sheet along x = 0 in the EDR. L_e is the half separation between the two peaks of the electron bulk velocity V_ex along z = 0, the average of the absolute value of the two peaks is defined as V_{e, out} (see Eq. (4)), r_L is the electron Larmor radius in the guide field defined as m_ev_the⊥ / e| B_y | in the EDR, where . Because an initial perturbation with small amplitude does not change the spontaneous growth of the reconnection electric field, the amplitude of the initial perturbation in this session is Ψ₀ / (B₀d_i) = 0.05. During Ω_i t = 10 to Ω_it = 20, when the reconnection electric field grows exponentially, δ_e, L_e, and r_L are nearly constant; the variations of J_ey and n_e are also much smaller than E_y, which is increased by a factor of 10 (see Fig. 5). Therefore, the assumptions used in our theoretical model are reasonable.

Fig. 7.

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	Fig. 7. Time evolution of (a) the y directional electron current density Jey / en0vA in the EDR, (b) the half width of EDR δe / di, (c) the half length of EDR Le / di, (d) the electron density ne in the EDR, and (e) the electron Larmor radius in the guide field rL in the EDR. The initial perturbation amplitude is Ψ0 / (B0di) = 0.05.

To examine the robustness of the theoretical model, we vary the amplitude of the guide field to B_g / B₀ = 1, 2, and 3. The other parameters are kept fixed as those described in Section 2. As shown in Fig. 8, the evolution of the reconnection electric field is similar in the three cases, which exhibit the exponential growth phase presented before the reconnection rate reaches maximum. The time evolution of the electron outflow speed V_{e, out} / v_A is also plotted. During the growth phase of E_y, V_{e, out} is approximately proportional to E_y, and grows exponentially with a growth rate a little smaller than that of E_y. The small difference between the growth rates of V_{e, out} and E_y should come from our assumption that J_ey and n_e are unchanged during this process. From Eq. (4) and Fig. 7 we can learn that the increase of J_ey and the decrease of n_e lead to a smaller growth rate of V_{e, out} than E_y. We also note that the growth rate of V_{e, out} decreases after around Ω_i t = 12, 13, and 15 in Figs. 8(a)–8(c). Because the electron outflow is not strictly along z = 0, there will be an error when we measure the electron outflow speed, causing the discrepancy between the growth rate of V_{e, out} and E_y. In general, we can conclude that the proposed model is reasonable to explain the spontaneous growth of the reconnection electric field in the EDR during guide field reconnection.

Fig. 8.

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	Fig. 8. Time evolution of the reconnection electric field Ey / vAB0 (the blue line) and electron outflow speed Ve, out / vA (the red cross) when the guide field is Bg / B0 = 1, 2, and 3. The growth rates of the reconnection electric field and electron outflow speed are also shown in each panel. The initial perturbation amplitude is Ψ0 / (B0di) = 0.05.

In this paper, based on both 2D PIC simulations and a proposed theoretical model, we have investigated the spontaneous growth of the reconnection electric field in the EDR during magnetic reconnection with a guide field. The reconnection electric field in the EDR is dominated by the electron pressure tensor term, while it is found to be directly proportional to the electron outflow speed. The time derivative of electron outflow speed is proportional to the reconnection electric field in the EDR because the outflow is formed after the inflow electrons are accelerated by the reconnection electric field in the EDR and directed away along the outflow direction. This kind of reinforcing process at last leads to the exponential growth of the reconnection electric field in the EDR. In this paper, we do not consider the influence of secondary islands which may be formed in the EDR during the evolution of magnetic reconnection on the spontaneous growth of the reconnection electric field, and this is our future plan.