Electron Surfing Acceleration at Rippled Reconnection Fronts

The reconnection front (RF), a transient structure with a scale of ion inertial length (Zhou et al. 2010; Lu et al. 2013, 2020; Wu et al. 2018; Fu et al. 2020) in the terrestrial magnetotail, is characterized by an abrupt strong enhancement of magnetic field (Nakamura et al. 2002; Runov et al. 2009; Sergeev et al. 2009; Zhou et al. 2009; Liu et al. 2013; Fu et al. 2020). Such structures are thought to be followed by a strong magnetic field region called a dipolarizing flux bundle (DFB; Liu et al. 2013, 2014) or flux pileup region (Khotyaintsev et al. 2011) and usually preceded by a small magnetic dip structure (Pan et al. 2015; Yao et al. 2015; Schmid et al. 2019). It is usually embedded in the bursty bulk flows (BBFs) or high-speed plasma jets in the terrestrial magnetotail (Cao et al. 2013; Karlsson et al. 2015; Pritchett & Runov 2017), separating hot tenuous plasma from the ambient cold dense plasma. The transient reconnection has been suggested as a possible mechanism responsible for the formation of RFs in both simulations (Sitnov et al. 2009) and observations (Fu et al.2013).

RFs can propagate from the magnetotail to Earth over a long distance more than 10 R_E together with BBFs behind them (Runov et al. 2009; Cao et al. 2010). Studies have suggested that RFs are crucial regions for particle acceleration, pitch-angle evolution, wave–particle interactions, and electromagnetic energy conversion during their Earthward propagation. For instance, rapid increases in energy fluxes of electrons and ions from tens to hundreds of keV are a typical feature of RF events (Khotyaintsev et al. 2011; Liu et al. 2013, 2018c, 2021a, 2022b; Zhou et al. 2018; Liu & Fu 2019; Gabrielse et al. 2021), pitch-angle distribution of suprathermal electrons can evolve dramatically around RFs (Runov et al. 2013; Liu et al. 2020), strong particle and wave activity can occur in the vicinity of RFs (Ono et al. 2009; Zhou et al. 2009, 2014; Fu et al. 2014; Breuillard et al. 2016; Greco et al. 2017; Yang et al. 2017), and RFs are associated with energy conversion from electromagnetic fields to particles (Sitnov et al. 2009; Huang et al. 2015; Khotyaintsev et al. 2017; Liu et al. 2018a, 2022a). The energetic plasma in the vicinity of RFs plays a key role in connecting the magnetotail with the inner magnetosphere because they carry a large amount of energy and can be injected into the inner magnetosphere to affect the ring current and radiation belt (Gabrielse et al. 2012; Duan et al. 2014; Turner et al. 2014). Possible mechanisms responsible for the energization of particles around RFs have been widely investigated based on both spacecraft observations and numerical simulations during the past decade. The strong convection electric field induced by the strong magnetic field gradient of RFs provides significant adiabatic acceleration of the ambient particles (Birn et al. 2004, 2013, 2015; Gabrielse et al. 2012, 2014, 2016; Ganushkina et al. 2013; Liu et al. 2016; Turner et al. 2016). Nonadiabatic effects, caused by particle reflection ahead of the RFs (Zhou et al. 2018), resonance with RFs (Ukhorskiy et al. 2013, 2017), and scattering by wave emissions (Zhou et al. 2009; Greco et al. 2017), are also significant for particle energization. These above studies usually assumed that the RF surface has a planar boundary at a typical thickness comparable to the ion gyroradius and below (Nakamura et al. 2002; Sergeev et al. 2009; Zhou et al. 2009; Schmid et al. 2011; Liu et al. 2013; Vapirev et al. 2013). Divin et al. (2015b) revealed that the RF surface is unstable to instabilities ranging from electron scales to ion scales. Simulation studies found that RFs can be unstable to interchange instability and that finger-like structures on ion–electron hybrid scales can develop at the RF (Vapirev et al. 2013). Such finger-like structures are found to play a role in modulating the electron acceleration process (Wu et al. 2018). Bai et al. (2022) also reported significant ion trapping acceleration at the RF with ion-scale ripples. Unlike these surface structures with ion or ion–electron hybrid scales, Liu et al. (2018b) recently reported that the RF layer has electron-scale density gradients, currents, and electric fields, based on the MMS mission, which consists of four spacecraft separated by 30 km. Such electron-scale ripple structure can be generated by lower hybrid drift instability (Divin et al. 2015b; Pan et al. 2018). Liu et al. (2021c) presented a detailed investigation of energy flux densities at two RFs with/without the electron-scale surface ripples and indicated that surface ripples may play an important role in the particle dynamics. But how such electron-scale RF structure impacts the electron energization and transport still remains unknown. In this paper, with the aid of observation-based test-particle simulation, we aim to investigate in detail the effect of the front surface ripples on the local electron dynamics.

The MMS mission consists of four spacecraft with identical instruments (Burch et al. 2016). We use magnetic field data from the fluxgate magnetometer (Russell et al. 2016), electric field data from the fields electric double probes (Ergun et al. 2016), and particle data from the fast plasma investigation (Pollock et al. 2016) in this study.

We investigate two typical RF events, without and with rippled surface structure, observed by MMS. The RF with planar surface structures was observed on 2017 May 28 when the spacecraft were located at [−19.6, −12.6, 2.1] R_E in the Geocentric Solar Magnetospheric (GSM) coordinates, and the RF with rippled surface structures was observed on 2017 June 24 when the spacecraft were located at [−20.5, −0.6, 2.8] R_E. Note that, during both events, spacecraft separation is close to several d_e (local electron inertial length). Figure 1 shows the overview of the two RF events, the one with planar surface structures on the left (Figures 1(a)–(f)) and the one with rippled surface structures on the right (Figures 1(g)–(l)). Local coordinate systems are established by transforming the vector data from GSM to LMN coordinates based on minimum variance analysis (Sonnerup & Scheible 1998) of B. It yields L = [0.20, 0.11, 0.97], M = [0.63, −0.78, −0.04], and N =[0.75, 0.63, −0.22] for the RF case with planar surface structure and L = [0.15, −0.27, 0.95], M = [0.31, −0.90, −0.31], and N = [0.94, 0.34, −0.05] for the RF case with rippled surface structure.

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The RF with planar surface structures was observed during 06:06:23.5–06:06:25.0 UT on 2017 May 28. The four spacecraft provided quite similar measurements. During the whole interval, B_L increases from 6 to 16 nT (Figure 1(a)) and density drops from 0.5 to 0.3 cm⁻³ (Figure 1(b)). Electron temperature gradually increases across the front (Figure 1(c)). The electron energy spectrum at both low and high energy ranges (Figures 1(d) and (e)) and relative flux gain ((flux − flux0)/flux0, where flux is the current energy flux and flux0 is the initial energy flux; Figure 1(f)), i.e., the change of energy flux with respect to the pre-RF flux, barely change during the RF interval. On the other hand, the RF with rippled surface structures was observed during 23:58:26.7–23:58:28.8 UT on 2017 June 24. Across the RF boundary, this event exhibits very similar properties to those in the RF with planar surface structures, in terms of B_L increase (from −1 to 20 nT; Figure 1(g)), density drop (from 0.6 to 0.2 cm⁻³; Figure 1(h)), and electron temperature increase (Figure 1(i)). But around the RF boundary, the four spacecraft show evident differences in the magnetic field despite the small separation among the spacecraft, as marked by the red circle in Figure 1(g). The magnetic field measurements between the four spacecraft differ by 3 nT, which is about 14% relative change. This suggests that electron-scale (i.e., 4 d_e , or 30 km, the separation distance between the spacecraft) structure is present at the RF. The electron energy flux at high energies increases during the RF interval, as shown in Figure 1(k). The relative flux gain after 23:58:28 increases by a factor of 2–5 (Figure 1(l)). Note that both RF events occur in the magnetotail, where the global electromagnetic environment is observed to be similar. Therefore, the notable relative change of the electron energy flux in the second case is likely a consequence of the interaction between the electrons and the small-scale structures at the RF. Next, we investigate the role of electron-scale rippled structure on the electron energization around the RF. We use the characteristics and properties of the above events to set up our model, as described below.

To numerically solve the particle motion and investigate the effects of the electron-scale rippled RF surface on the particle dynamics, we use a two-dimensional plasma sheet model, superposed by an RF profile, to describe Earth's magnetotail in the equatorial plane. The initial magnetotail field is obtained by an asymptotic magnetotail vector potential A_0y given by Pritchett & Coroniti (1995), with F(x) in the form introduced by Zhou et al. (2018),

$$\begin{eqnarray}&&{A}_{0y}={{LB}}_{0}\mathrm{ln}\left\{\cosh [F(x)(z/L)]/F(x)\right\}\end{eqnarray} \tag{ 1 }$$

$$\begin{eqnarray}&&F(x)=\exp \left[\displaystyle \frac{{B}_{z\infty }(x-{x}_{0})}{{B}_{0}L}+\displaystyle \frac{({B}_{z0}-{Bz}\infty )({x}^{3}-{x}_{0}^{3}){x}_{0}}{3{B}_{0}{{Lx}}^{3}}\right],\end{eqnarray} \tag{ 2 }$$

where x₀ is the boundary of the plasma sheet near Earth's side, Bz ∞ is the equatorial B_z at x = −∞ , B₀ is the lobe field strength at x₀, L is the current sheet half-thickness at x = x₀, and B_z0 is the equatorial B_z at x₀. In our simulations, based on previous studies (Pritchett & Coroniti 1995; Zhou et al. 2011, 2018), we adopt the following parameters to represent a thick plasma sheet: Bz ∞ = 5 nT, B₀ = 50 nT, B_z0 = 15 nT, L = 2 R_E, x₀ = −10 R_E.

We further superpose an RF field on the plasma sheet by first prescribing the RF density and then solving the RF magnetic field profile by assuming local force balance. The RF density n₀ decreases from 0.6 to 0.25 cm⁻³ across the RF, as

$$\begin{eqnarray}&&{n}_{0}(X,Y)=-0.175\tanh \left(\displaystyle \frac{{X}^{* }}{D}\right)+0.425,\end{eqnarray} \tag{ 3 }$$

where X^* = X − X_f − V_f (t − t_i ) indicates the Earthward propagation of a coherent RF structure at the speed of V_f = 120 km s⁻¹ calculated from the time delay between the spacecraft. X_f = − 20 R_E is the location of the RF center, and D is the half-thickness of the RF. For a rippled RF boundary, we further include an addition of density perturbation n_p as follows:

$$\begin{eqnarray}&&{n}_{p}(X,Y)=A\cdot \exp \left(-\displaystyle \frac{{{X}^{* }}^{2}}{{a}^{2}}\right)\sin \left(\displaystyle \frac{2\pi }{\lambda }\cdot Y\right),\end{eqnarray} \tag{ 4 }$$

where A is the amplitude of perturbation of the density, λ is the ripple's wavelength, and a is the characteristic width of the Gaussian envelope. We choose the following parameters based on the observations: A = 0.03 cm⁻³, D = 0.05 R_E, λ = 120 km, a = 0.025 R_E. The ion temperature is represented by a planar hyperbolic-tangent profile from 5 to 7.5 keV, while the electron temperature changes from 1.2 to 2 keV, and the temperature disturbance is anticorrelated with the density disturbance. By assuming pressure balance, we then obtain the magnetic field profile Δ B to represent the RF field. In our simulations, the prescribed electric field consists of the dawn−dusk convective electric field (Runov et al. 2009; Zhou et al. 2018) and Hall electric field (Runov et al. 2011; Pan et al. 2015), as

$$\begin{eqnarray}&&{\boldsymbol{E}}=-{{\boldsymbol{V}}}_{{\boldsymbol{f}}}\times {\boldsymbol{B}}+{\boldsymbol{J}}\times {\boldsymbol{B}}/{en}.\end{eqnarray} \tag{ 5 }$$

With the above settings, we carry out two simulations: one with planar RF surface structures, and the other one with rippled surface structures. We trace electrons in the above-prescribed electromagnetic field by solving the relativistic Lorentz equation with a fourth-order Runge–Kutta scheme and determine the energy flux at the spacecraft location. Models that employ particle tracing can trace particle trajectories either forward in time until they reach the spacecraft location or backward in time from the spacecraft location to their initial positions. Tracing particles forward in time requires tracing thousands to millions of particles that all start at different locations in order for sufficient particles to arrive at the spacecraft for the calculation of local flux. On the other hand, backward tracing, or backtracing, allows one to obtain the local flux by following only a few particles starting from the spacecraft to their initial locations. Our study uses the backtracing method. Following Liouville's theorem (Birn et al. 2004; Gabrielse et al. 2016; Yin et al. 2021), the energy flux f(E) at the spacecraft location is calculated from

$$\begin{eqnarray}&&f(E)=\frac{{W}_{f}^{2}}{2\pi {m}^{2}}\mathrm{PSD}({W}_{i},{X}_{i},{Y}_{i})\frac{\mathrm{keV}}{{\mathrm{cm}}^{2}\,{\rm{s}}\,\mathrm{keV}\,\mathrm{sr}},\end{eqnarray} \tag{ 6 }$$

where m is the electron mass, PSD is the phase space density at the initial location (X_i , Y_i ) for a particle with an initial energy of W_i , and W_f is particle energy at the spacecraft location. This initial PSD is assumed to follow a Maxwellian-like distribution in regions far away from the RF. Such an assumption has been widely used in previous studies (Nakamura et al. 2002; Zhou et al. 2009; Pritchett & Coroniti 2010; Li et al. 2011; Divin et al. 2015a; Liu et al. 2019, 2021b). In summary, our tracing process is as follows: electrons with pitch angles of 90° and energy of W_f are followed backward in time, from the spacecraft position to its initial position (X_i , Y_i ) (a position well into the background field after passing through the RF or hitting the simulation boundary). At that initial position, we determine the initial energy (W_i ) based on the initial velocity calculated from the Lorentz equation. With these parameters, the PSD (W_i , X_i , Y_i ) is formulated and the energy flux at the starting position (i.e., at the spacecraft position) can be calculated from Equation (6).

As described in Section 3, we set up the initial electromagnetic field conditions for the two cases in the equatorial plane, as displayed in Figure 2. The RF with planar surface structures is shown on the left side, and the one with rippled surface structures is shown on the right side. Figures 2(a) and (e) show B_L profiles based on the model. In the RF with planar structures, B_L gradually increases along the N-direction and is smooth along the M-direction. In the RF with rippled structures, waves with a Gaussian envelope in the N-direction are added from the planar structures. Four virtual spacecraft are used to resemble the four spacecraft of the MMS mission. The magnetic field B_L and electric field E_x and E_y along the paths of four virtual satellites (corresponding to MMS separations) are shown in Figures 2(b)–(d) and (f)–(h). Around the RF with planar surface structures, the electromagnetic fields are the same along the four virtual satellites. But around the RF with rippled surface structures, the four virtual satellites show evident differences in the electromagnetic fields despite the small separation between the spacecraft. The fields along the four virtual satellites exhibit similar characteristics to the observational fields in both planar and rippled cases. After fitting the model to the observations, we conduct backward-tracing test-particle simulations for electrons at various energies starting from the virtual MMS2 to the background field (or the simulation boundary). The magnetic fields B_L along the paths of virtual satellites are shown in the first row (Figures 3(a), (b)). The energy flux of the electrons along the trajectories is calculated and shown in the second row of Figure 3. In the RF with planner structures, electron energy flux (Figure 3(c)) remains relatively steady, with only a slight increase after crossing the RF. In the RF with rippled structures, electron energy flux (Figure 3(d)) dramatically increases by 2–5 orders of magnitude. The modeled electron energy flux (Figures 3(c), (d)) is roughly in agreement with the observed energy flux (Figures 1(e), (k)).

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To explore the physical mechanism of electron acceleration by the rippled surface structures of RF, we investigate the temporal evolution of electrons' positions, energy, and electromagnetic fields along their trajectories. Figures 4(a) and (d) show backward-traced electron trajectories in the equatorial plane with the same final location (X = −20.06 R_E and Y = 0.21 R_E) and the same final energy (17 keV) in the two simulations. The background color represents the B_z field at t = −8 s. The color on the curves represents the kinetic energy of the electrons along their trajectories. The white stars show the location of the electrons at that time. To better understand the time evolution, Figures 4(b1)–(b8) and (e1)–(e8) show forward in time the positions of the electron X and Y, the local magnetic field B_Z , electric fields E_X and E_Y , the work done by electric fields W_Ex and W_Ey , and the corresponding kinetic energy E_K along their paths. Figures 4(c1)–(c8) show zoomed-in gyrations. In the simulation with flat surface structures (top), the electron undergoes both Earthward E × B drift and duskward ▽B drift. The duskward ▽B drift is well visualized from the trajectory in Figure 4(a) after the electrons move from the initial location (Y = −0.24 R_E) to the final location (Y = 0.21 R_E) in 10 s. As the RF has a sharp enhancement of B_Z at the boundary, the tailward gradient leads electrons to move duskward. During its interaction with the RF, the work done by E_X , as shown in Figure 4(b6), decreases, indicating that the electrons are decelerated by the Earthward E_X as they move Earthward. While the duskward electric field E_Y is expected to play a role in decelerating electrons, the work done by the E_Y is actually positive along the paths (Figure 4(b7)). This is because when the magnetic field of the RF changes at small scales, the electron no longer behaves adiabatically. In scales of gyrations, when electrons move dawnward as their Y decreases (Figure 4(c2), green shaded area (1)), they are accelerated by the duskward electric fields (Figure 4(c7)) as the work done by E_Y increases. Conversely, they are decelerated by the duskward electric field during their duskward gyration (Figure 4(c2), yellow shaded area (2)). The positive work done (0.0776 keV) by the duskward electric field is larger than the negative work done (−0.0759 keV) within one gyration, though it is not obvious during each full gyration. With time, electrons are finally accelerated (Figures 4(b6), (b7), and (b8)) by duskward electric field as long as they are interacting with the RF.

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In the simulation with rippled surface structures (bottom), the electrons undergo very similar behaviors, in terms of the direction of the overall motions. However, the electrons in this simulation are trapped at the RF and start surfing in the X-direction (Figure 4(d)). During the interaction with the rippled RF, the electrons experience oscillating E_x and E_y (Figures 4(e4) and (e5)). The electrons are decelerated by E_x during the Earthward surfing along the rippled surface of the RF as X increases, but the work done by E_x decreases (Figure 4(e6), blue shaded area (3)). In contrast, they are accelerated during the tailward surfing as X decreases (Figure 4(e6), purple shaded area (4)). The positive work done (11.420 keV) by E_x is larger than the negative work done (−11.096 keV) within one gyration. Over time, the electrons are accelerated by the Earthward electric field E_x along their Earthward motion (Figure 4(e6)). Meanwhile, the electrons move duskward and are accelerated (Figure 4(e), blue shaded area (5)) and decelerated (Figure 4(e), purple shaded area (6)) by the dawnward and duskward electric field, respectively. Over time, the electrons are accelerated by the duskward electric field along their duskward motion (Figure 4(e7)). In general, the work done by E_x in the simulation with flat surface structures is negative (Figure 4(b6)), while it is positive (Figure 4(e6)) in the simulation with rippled surface structures. As a result, the electrons undergo surfing motion and gain more energy (initial energy: 5 keV; final energy: 17 keV) than the typical drift motion in the first case (initial energy: 13 keV; final energy: 17 keV), which explains the evident flux enhancement in the second case in the observations. The smaller initial energy actually indicates a larger energy flux according to Equation (6), which is consistent with the conclusion in Figure 3. To better visualize and understand the surfing motion, we also show in Figure 4(f) the electrons' trajectories in the rest frames of the Earthward-moving RF. In the rest frame of the Earthward-moving RF, the electrons are surfing along the rippled surface, which can be easily understood as ▽B drift in the oscillating background magnetic field.

RFs have been well recognized to be transient structures with a scale of ion inertial length and play an important role in particle acceleration (Zhou et al. 2009; Gabrielse et al. 2012; Liu et al. 2013; Turner et al. 2014; Wu et al. 2018; Fu et al. 2020; Lu et al. 2020). Recent investigations into RFs have suggested that they are not planar, as commonly assumed, but rather rippled with electron scales (Liu et al. 2018b, 2020; Pan et al. 2018). The knowledge of how such electron-scale RFs impact the electron energization in the magnetotail remains elusive. In this study, we use high-resolution data from MMS spacecraft and test-particle tracing techniques to study electron dynamics in association with the rippled RFs and compare the results to those with planar RF surface structures. In both cases, the dominant motion of these electrons is the ▽B drift during their interactions with the RF. Meanwhile, the sharp magnetic field gradients in the tailward direction result in a duskward drift. We find that the energy flux of the tens of keV electrons increases significantly as they traverse the rippled RFs. In contrast, the energy flux barely changes in the flat RFs. The main acceleration mechanism is electron surfing acceleration around the ripples, in which electrons are confined at the RFs by the rippled magnetic field gradient and surfing in the X-direction. During the surfing motion, electrons are accelerated by the electric field.

Both events occur at about −20 R_E along the X_GSM-direction, and the spacecraft separation is close to several d_e . The difference in the flux variation of the electron energy flux is probably mainly attributed to the different surface structures at RFs, considering that the global effect due to RF Earthward transport is likely to be similar for the two events given their similar locations. The above simulations were initiated with the same electrons but under different kinds of RFs. The largely different acceleration efficiency of these electrons indicated that surface ripples can play an important role in regulating the kinetic motion of the electrons. We also tested electrons with other initial energies. For electrons with lower energy (a few keV or a few hundreds of eV), the E × B drift is dominant around the RF with flat surface structures and pushes electrons to the DFB quickly. In contrast, around the RF with rippled surface structures, electrons are trapped for a longer time than that in the flat-RF case and surf longer to gain more energy. For electrons with higher energy (above 50 keV), the magnetic gradient drift is dominant in their motion. It should be noted that the surfing acceleration is sensitive to the electron velocity along the X-direction. If the initial velocity along the X-direction is large enough, then electrons would move across the RF boundary easily, rather than being reflected by E_x and surfing along the ripples. Our results imply that such electron-scale structure could play an important role in electron acceleration.

Discussion with Dr. Chengming Liu is appreciated. This work was supported by the NSFC grant 41821003.