Direct Evidence of Electron Acceleration at the Dipolarization Front

The dramatic changes in the magnetic field at the dipolarization front (DF) provide a suitable environment for electron acceleration, which usually can cause the flux enhancement of energetic electrons behind the front. However, it is unknown whether energetic electrons observed at the DF are energized locally, and which mechanism accelerates the electrons at the DF is unclear. Our study performs a direct quantitative analysis to reveal the acceleration process of energetic electrons at the DF using the high-time-resolution data from NASA's Magnetospheric Multiscale mission. The fluxes of energetic electrons at 90° are enhanced at the front. Under adiabatic conditions, our quantitative analysis indicates that these electrons at the front could be locally accelerated to over 100 keV by betatron acceleration. Eventually, the electron temperature anisotropy formed via the betatron mechanism could provide the free energy to excite whistler waves at the DF. Our quantitative study provides, for the first time, strong direct evidence for the local electron acceleration at the DF.


Introduction
The dipolarization front (DF) is a sharp transient boundary featured by a significant increase of the Z component of magnetic field B z in a short time period (Nakamura et al. 2002;Runov et al. 2009). The DF can separate the hot and sparse, bursty plasma from the cold and dense background plasma (Zhou et al. 2009;Runov et al. 2011;Huang et al. 2012;Fu et al. 2020aFu et al. , 2020bJiang et al. 2020;Liu et al. 2021;Wei et al. 2021;Xu et al. 2021Xu et al. , 2022. The scale of the DF is usually on the order of the ion inertial length or the ion gyroradius (Runov et al. 2009;Zhou et al. 2009;Fu et al. 2012;Huang et al. 2012Huang et al. , 2015aHuang et al. , 2015b. The DF is closely related to the dynamic processes of the magnetotail and it plays a vital role in particle acceleration (Deng et al. 2010;Fu et al. 2011Fu et al. , 2013Fu et al. , 2019Huang et al. 2012Huang et al. , 2019Zhou et al. 2013;Nakamura et al. 2021), energy conversion (Huang et al. 2012(Huang et al. , 2015bAngelopoulos et al. 2013;Zhong et al. 2019), as well as magnetic flux and mass and energy transport (Angelopoulos et al. 1994(Angelopoulos et al. , 2002Nakamura et al. 2009;Liu et al. 2014) in the magnetotail. Previous studies suggested that the DF is a signature of unsteady magnetic reconnection Fu et al. 2013).
The pitch angle distribution (PAD) of particles at and behind the DF is typically of various types, and it could be divided mainly into five types to date: isotropic, cigar, pancake, rolling-pin, and butterfly (Wu et al. 2013;Liu et al. 2017;Xu et al. 2018;Fu et al. 2019;Huang et al. 2019;Wei et al. 2022). The electron anisotropy around the front usually provides favorable conditions for the generation of electron anisotropy instabilities (Deng et al. 2010;Huang et al. 2012Huang et al. , 2019Zhang et al. 2018Zhang et al. , 2019, especially the whistler instability driven by perpendicular electron temperature anisotropy (T e⊥ > T e|| ). The whistler instability could excite whistler waves (Deng et al. 2010;Huang et al. 2012Huang et al. , 2019Huang et al. , 2020bFu et al. 2014;Jiang et al. 2019;Zhang et al. 2019;Grigorenko et al. 2020;Chen et al. 2021). Recently, the formation of various types of electron PADs behind DF and the acceleration mechanism of these electrons have attracted much interest. Betatron acceleration and Fermi acceleration mechanisms are the core mechanisms for forming different types of electron PADs. Previous studies have adopted the Liouville mapping model to quantify the energy gain caused by the adiabatic electron acceleration behind the DF (Fu et al. 2011(Fu et al. , 2013Liu et al. 2017;Xu et al. 2018). They assumed the electrons in the quiet plasma sheet to be source electrons and then fit the best Fermi and betatron acceleration factors by their mapping models. Nevertheless, the determination of acceleration factors may depend on the different selections of the source region, and there are multiple sources of particles around the DF (Ashour- Abdalla et al. 2011;Eastwood et al. 2015). Thus, there could be a discrepancy between the fitting results from the Liouville mapping model and the satellite observations. Based on a particle tracking analytical model, Gabrielse et al. (2017) also studied electron betatron acceleration around the DF, but they did not quantify the betatron acceleration of electrons at the DF based on the observational results. The electron acceleration around the DF has been investigated in previous studies (Fu et al. 2011(Fu et al. , 2013Gabrielse et al. 2017); however, whether and how energetic electrons with energy above 100 keV are generated directly under the local plasma environment at the DF remains unclear.
Here, we perform a quantitative study of electron acceleration at the DF in light of the local plasma parameters measured by the Magnetospheric Multiscale (MMS) mission. This quantitative analysis demonstrates that the energetic electrons at the DF could be locally accelerated to more than 100 keV by the betatron mechanism.

Event Overview
The data in burst mode used in this work are provided by the onboard instruments of the MMS mission. The Fluxgate Magnetometer (FGM) measures the magnetic field data (128 Hz; Russell et al. 2016), and the Search-Coil Magnetometer (8192 Hz; SCM) quantifies the fluctuated magnetic field data with high time resolution (Le Contel et al. 2016). The Electric Double Probe (EDP) records the electric field data (8192 Hz; Ergun et al. 2016;Lindqvist et al. 2016). The Fast Plasma Instrument (FPI) provides the data of plasma moments as well as 3D particle distribution functions (resolution is 30 ms for electrons and 150 ms for ions; Pollock et al. 2016). The Flyʼs Eye Energetic Particle Spectrometer (FEEPS) records the data of plasma with energy above 30 keV (resolution ∼1/3 s; Blake et al. 2016;Mauk et al. 2016). The data measured by the four MMS missions are similar; thus we only use the data from MMS1. Figure 1 presents the overview of the DF event. The MMS1 was located at [−16.8, 7.3, 0. Figure 1(e)), and large plasma β (>0.5 in Figure 1(h)) indicate that MMS was immersed in the plasma sheet (Cao et al. 2006;Wei et al. 2019;Huang et al. 2020aHuang et al. , 2022. The B l sharply increases from 10.5 to 24.8 nT from 06:04:14.4 to 06:04:22.6 UT (Figure 1(a)), as marked by the magenta shadow. Simultaneously, the plasma density decreases from 0.34 to 0.22 cm −3 . This magnetic field increase is accompanied by an earthward ion burst flow, with the N component rising up to 214 km s −1 (Figure 1(c)). Meanwhile, the electron fluxes above 89.45 keV show a significant increase, from the quiet ambient region to the region where the B l enhances obviously (around 06:04:20.4 UT). All these features imply that MMS observed a typical DF. Utilizing the method of timing analysis (Russell et al. 1983), the propagation velocity of the DF is obtained as 294 × [0.8242 −0.4830 0.2957] km s −1 in LMN coordinates. By multiplying the propagation velocity acquired with timing analysis by the duration of the DF (∼8.2 s), the spatial scale of the DF is estimated as 2411 km, i.e., 5.9 λ i and 2.5 ρ i (λ i ∼410 km is the ion inertial length and ρ i ∼ 968 km is the ion gyroradius), which are calculated from the undisturbed ambient parameters B t = 12.3 nT, N i = 0.31 cm −3 , and T i = 6834 eV. Thus, the observed DF belongs to the ion-scale structure.
The B l remained flat within 1.7 s (from 06:04:16.1 to 06:04:17.8 UT) at the center of DF, implying that there could be a substructure at the DF. After being investigated in detail, the possibility that this event is related to flux rope (Poh et al. 2019;Sun et al. 2019) or electron vortex (Jiang et al. 2020) proposed by previous studies can be ruled out. Considering that the electron density decreases at two regions at the DF, and the decreases of density are accompanied by the increase of magnetic field with the emergence of electron jets (Figure 1(f)), implying that there could be DF structure at both regions. The N component of the ion flow velocity V in (Figure 1(c)) at the latter DF is larger than that at the former DF, which can cause the latter DF to compress the former one, resulting in the coalescence of the two dipolarizing flux bundles (DFBs). Thus, it could be inferred that this substructure is likely formed by the coalescence of two DFBs. Figure 2 shows the electron pitch angle distributions (PAD) and the betatron acceleration rate around the DF in the GSM coordinates. It can be found that the high-energy electrons (above 2 keV) recorded by FPI and FEEPS instruments (Figures 2(f)-(k)) are mostly concentrated in the direction perpendicular to the magnetic field during 06:04:14.6-06:04:18.3 UT. Simultaneously, the fluxes of energetic electrons with energy up to 150 keV experience an obvious increase (Figure 2(c)). These imply that the electrons may be accelerated at the DF.

Quantitative Analysis of the Acceleration Rate at DF
The acceleration of electrons in the direction perpendicular to the magnetic field may be caused by the betatron acceleration mechanism (Fu et al. 2011(Fu et al. , 2013. The motion of electrons will follow the guiding center approximation in the adiabatic plasma environment. Under the adiabatic condition, the acceleration efficiency of an electron in the perpendicular direction caused by the spatiotemporal variation of the magnetic field can be quantified by the following equation (Dahlin et al. 2014;Ma et al. 2020;Zhong et al. 2020;Jiang et al. 2021): whered w dt is the perpendicular kinetic energy gain of an electron via betatron acceleration in unit volume and unit time, W ⊥0 is the electron energy before acceleration, E is the electric field (Figure 2(l)), B is the magnetic field (Figure 2(b)), and ∇B is the magnetic field gradient, which is calculated by using the magnetic field vectors measured by four MMS satellites and the position vectors of four satellites (Paschmann & Daly 1998). It is assumed that magnetic field B varies linearly with the position vector r within a spatial tetrahedron composed of four satellites. The separation among the four satellites dr is ∼18 km. The spatial scale of DF is 2411 km (as estimated above), which is much larger than dr. Therefore, the assumption that B varies linearly with position r is reliable when calculating the gradient of B within a tetrahedron composed of four MMS satellites at the DF. Summing over all electrons within a unit volume, the total energy gain attributed to the betatron acceleration could be quantified as where W b is the energy gain of electrons caused by betatron acceleration mechanism in unit volume and unit time, and P e⊥ is the perpendicular electron pressure calculated by P e⊥ = N e kT e⊥ (N e is the electron density, k is the Boltzmannʼs constant, and T e⊥ is the perpendicular electron temperature). The ¶ ¶ B t is calculated by using the magnetic field difference within 0.0078 s divided by 0.0078 s. The temporal variation of the magnetic field cannot be strictly separated from the spatial variation based on the observation data. But in Equation (2), the total contribution of both temporal and spatial changes in the magnetic field to the betatron acceleration rate is considered simultaneously.
Only when the adiabatic condition is satisfied can Equations (1) and (2) be true. Generally, the parameter κ is used to examine whether the particles are adiabatic. It is defined as where R c is the curvature radius of the magnetic field, and R L is the Larmor radius of the particle calculated based on the thermal velocity. Figure 2(m) presents the adiabatic parameter κ for the electrons in 1, 30, and 150 keV energy channels. It can be seen that all κ are larger than 5, implying that the electrons with energy below 150 keV are adiabatic during the DF event (Büchner & Zelenyi 1989;Vasko et al. 2014). Therefore, Equation (2) can be used to estimate the betatron acceleration rate. The betatron acceleration rate W b based on Equation (2) is displayed in Figure 2(n). The W b enhances significantly with a peak of 400 eV s −1 cm −3 when the PADs of high-energy electrons are close to 90°at the DF, which indicates that the betatron acceleration mechanism can lead to the enhancement of electron fluxes in the perpendicular direction at the DF.
For a more quantitative and detailed analysis of the betatron acceleration process at the DF, Figure 3 presents the betatron acceleration process of electrons at the DF from 06:04:15.4 to 06:04:15.9 UT. Assuming that these electrons have no parallel velocity and the energy gain is only attributed to the betatron acceleration process, we conduct an iterative calculation analysis to quantify the efficiency of multiple acceleration processes under the local plasma environment. This model assumes that the electrons could be continuously accelerated inside the flow channel of the DF, and the change in the background magnetic field, as well as the difference of the magnetic field gradients experienced by electrons with different energy, could be ignored. A time step Δt (∼0.0375 s) is set as an iterative time interval that represents the timescale of the magnetic field variation, and the Δt is much larger than the electron gyro-period (∼0.0024 s) to conserve the first adiabatic invariant. For the first step of electron acceleration, the betatron acceleration rate of an electron is calculated according to Equation (1) based on the mean value of the measured parameters. Then, the energy of this electron after the first acceleration time step W ⊥1 is obtained by adding the energy gain (ΔW ⊥1 = Δt · dW ⊥1 /dt) and to the energy before the acceleration (W ⊥0 = E initial ) in the first step: W ⊥1 = W ⊥0 + ΔW ⊥1 . Next, in the second acceleration time step, all other parameters remain unchanged, except that the electron energy before acceleration needs to be updated using the total electron energy obtained after the first step: W ⊥0 = W ⊥1 . The energy gain in the second acceleration step ΔW ⊥2 could be calculated by utilizing the updated particle parameter based on Equation (1), and then the energy after the second acceleration step W ⊥2 could be calculated: W ⊥2 = W ⊥1 + ΔW ⊥2 . After each acceleration time step, the perpendicular energy of the electron will increase. This iterative calculation process will continue until the electron is accelerated to the target energy level. This iterative method will take 39 s to accelerate an electron with the initial energy of 0.5 keV (0.5 keV is close to the lower limit of the typical energy range (0.2-12 keV) of electrons in the Earthʼs magnetotail, as suggested by Bame et al. 1967) to 147.1 keV, as illustrated by the black line in Figure 4(a). It should be noticed that the energy gain of electrons is dependent on the initial energy in accordance with Equation (1). The partial derivative of energy to time increases along with the increase of energy (red line in Figure 3(a)), suggesting that the electrons with higher initial energy will get more energy gain.
The betatron acceleration process of electrons with different initial energy under the local plasma environment at the DF is shown in Figure 3(b). The initial electron energies below 30 keV marked on the left Y-axis are set according to the energy channels of the FPI instrument. For electrons with energy levels over 30 keV, to ensure the acceleration process is continuous, five initial energies are set within the energy measurement range of FEEPS. Having experienced 53 Δt (2 s) of betatron acceleration, the electron with an initial energy of 5.471 keV can be accelerated up to 7.289 keV. Similarly, the second measurement energy level of FPI is 7.161 keV, which is set as the initial energy of the electrons for the second energy channel. And the energy of electrons in this energy channel is accelerated to 9.541 eV after 2 s. Calculated in the same way, the energy of an electron ranging from 5.471 to 30 keV in the present energy channel can be accelerated to the next energy channel of FPI, as presented in Figure 4(b). Besides, the electrons with energy lower than 5.471 keV can also be accelerated to the adjacent high-energy channels in the same time period (not shown here). Furthermore, the electrons with energy higher than 30 keV in each energy level will be further accelerated to an energy level as high as 150 keV after 2 s through the betatron acceleration mechanism. This quantitative analysis indicates that electrons at the 90°pitch angle in Figures 2(f)-(k) could be accelerated locally from the lowenergy level at the DF within a short time period (∼2 s).   Figure 4 shows the wave activities around the DF. The enhancements of magnetic field power spectral density (Figure 4(b)) and electric field power spectral density (Figure 4(c)) between 0.1 f ce (electron cyclotron frequency) and f ce , the right-hand circular polarization (Figure 4(d)), and the fieldaligned propagation (propagation angles <20°in Figure 4 Figure 4(g) displays the spectrogram of the flux anisotropy parameter (F ⊥ /F || ) of electrons, where F ⊥ and F || are the average phase space densities in the perpendicular direction (PAD at 80°-100°) and parallel/antiparallel direction (PAD at 0°-30°and 150°-180°), respectively. There is an obvious enhancement of F ⊥ /F || in the energy range of 6-30 keV when the whistler waves are observed, implying that the electron temperature anisotropy is attributed to electrons with energy ranging from 6 to 30 keV. To find vital evidence that these observed whistler waves are locally excited, we conduct a kinetic analysis by adopting the PDRK (kinetic dispersion relation solver for magnetized plasma) code (Xie & Xiao 2016). The electron phase space densities (PSDs) detected at two time periods are fitted according to a drifting bi-Maxwellian distribution (Zhao et al. 2019), and the corresponding fitting results are shown in Figures 4(h) and (j). Tables 1 and 2 display the fitting parameters, which are adopted as the input of the PDRK. The growth rate of whistler-mode instability shows positive peaks at the frequency of 0.43ω ce and 0.45ω ce , respectively (Figures 4(i) and (k)), which is consistent with the observational result. The main contributor of the whistler-mode instability is investigated by removing the fitting components one by one. It is found that component 2 with largest temperature anisotropy has the largest contribution, and only when this component is reserved, a positive growth rate will appear. Moreover, for the region where no whistler waves are observed, although the electron temperature anisotropy exists, no positive growth rate appears (not shown). Therefore, these kinetic analyses prove that the observed whistler waves at the DF could be locally excited by the electron temperature anisotropy instability.

Discussions and Conclusions
The betatron acceleration process of electrons at the DF may be caused by the compression of the local magnetic field by the ion flow. The N component of the ion flow velocity V in (Figure 1(c)) exhibits an increasing trend when the energy flux enhancement of energetic electrons in the perpendicular direction is observed at the former part of the DF (06:04:14.6-06:04:18.3 UT). The increase of V in indicates that the flux tube at the former part of the DF is probably compressed, thus resulting in the significant betatron acceleration at the DF, as suggested by Fu et al. (2011). Moreover, our quantitative analysis reveals it will take 39 s to accelerate the electrons from 0.5 to 147 keV by the betatron mechanism. This acceleration, which can last up to 39 s, is possible because the same DF could maintain in the magnetotail for up to 3 minutes, as reported by Runov et al. (2009). It can be noted that the increase in V in (lasts to 06:04:19.5 UT) lasts longer than the enhancement in electron flux at 90°(lasts until 06:04:18.3 UT). This temporal inconsistency may be due to other acceleration processes, such as Fermi acceleration or parallel electric field acceleration, which accelerates most electrons to the fieldaligned direction after 06:04:18.3 UT.
There are two possible reasons for the electrons to be trapped in the DF region for sustained acceleration: (1) The local magnetic field gradient around the sharp DF could trap some electrons to spiral around the DF region and maintain these electrons within the flow channel (Gabrielse et al. 2016(Gabrielse et al. , 2017, as described in Figure 8 of Section 3.3 in Gabrielse et al. (2017). (2) The Y-component of the drift velocity of the electron at 20 keV is estimated about 1000 km s −1 at the DF region. If we assume that 1000 km s −1 is the average value of drift velocities of energetic electrons, as a consequence, the length of the drift paths of the electrons within 39 s in dawndusk direction can be estimated to be 39,000 km (∼6 R E ), which is comparable to the dawn-dusk scale of the DF (Huang et al. 2015a(Huang et al. , 2015b, indicating that the electrons could stay in the DF region continuously within 39 s.
In summary, we conduct a quantitative analysis of electron acceleration to explain the formation of pancake type PADs of energetic electrons at the DF for the first time. Corresponding to the pancake type PADs, the betatron acceleration rate increases and the peak value can reach up to 400 eV s −1 cm −3 . The quantitative calculations suggest that the electrons at the DF could be locally accelerated to over 100 keV through the betatron acceleration mechanism, leading to the enhancement of energetic electron fluxes with 90°pitch angle. In addition, it is demonstrated by the kinetic analysis that the electron temperature anisotropy due to betatron acceleration could excite the whistler waves locally at the DF. Our study reveals direct evidence for the local acceleration process of electrons at the DFs.