Simultaneous Observations of Localized and Global Drift Resonance

Abstract

In this study, we present Van Allen Probe observations showing that seed (hundreds of keV) and core ( 1 MeV) electrons can resonate with ultra-low-frequency (ULF) wave modes with distinctive m values simultaneously. An unusual electron energy spectrogram with double-banded resonant structure was recorded by energetic particle, composition, and thermal plasma (ECT)-magnetic electron ion spectrometer (MagEIS) and, meanwhile, boomerang stripes in pitch angle spectrogram appeared at the lower energy band. A localized drift resonance with m = 10 wave component was responsible for the resonant band peaked at ∼200 keV while a global drift resonance with m = 3 component gave rise to the upper band resonance peaked at ∼1 MeV. Time-Of-Flight on boomerang stripes suggested that the localized drift resonance with ∼200 keV electrons was confined within the plasmaspheric plume. Electron flux modulations were reproduced by numerical simulations in good consistency with the observations, supporting the scenario that localized and global drift resonance could coexist in the outer belt electron dynamics simultaneously.

Key Points

• We, for the first time, report the banded drift resonance between outer belt electrons and ULF waves
• ULF waves with small and moderate azimuthal wave numbers can resonate with core and seed electrons simultaneously
• Azimuthally localized and global drift resonances are able to coexist

1 Introduction

The acceleration mechanism of radiation belt electrons has been one of the most hotly debated topics in magnetospheric physics area for decades. Wave-particle interactions between electrons and various types of plasma waves are believed to have their own contribution to the acceleration of outer belt electrons (e.g., Horne et al., 2005; O'brien et al., 2003; Reeves et al., 2013; Summers et al., 1998; Ukhorskiy et al., 2006; Zong et al., 2009). With their period comparable with the drift period of outer belt electrons, ultra-low-frequency (ULF) waves in Pc4-5 band have been found to accelerate electrons efficiently via a nondiffusive process termed as drift resonance (Southwood & Kivelson, 1981, 1982; Zong et al., 2007). As the drift resonance drives radial transport of electrons with their first and second adiabatic invariant unchanged, it has also been considered as the first step when investigating the outer belt dynamics (Mann et al., 2013, 2016; Su et al., 2015).

Electrons in the outer radiation belt are classified into a “source” population (a few to tens of keV), a “seed” population (hundreds of keV), and a “core” population ( 1 MeV) according to the current understanding (e.g., Boyd et al., 2014, 2016; Jaynes et al., 2015). Drift resonances between ULF waves and both seed and core populations have been reported. During the Van Allen Probes mission, ULF waves with the azimuthal wave number m ranging from 1 to 10s have been diagnosed to resonantly accelerate the particles with energy ranging from 60 keV to ∼3 MeV (Claudepierre et al., 2013; Foster et al., 2015; Hao et al., 2014, 2017, 2019; Zhou et al., 2016). The resonant energy and the waves' m value corresponding to the resonance differ significantly from case to case. Therefore, it seems enigmatic to form a general idea about which population is likely to be accelerated via drift resonance and what acts as its controlling factor.

Besides their ability to resonate with electrons in a broad energy range, ULF waves also leave the imprints of their temporal evolution and spatial distribution to the electron flux modulations. Zhou et al. (2016) discussed electron signatures modulated by a growing and damping ULF wave packet. In comparison with the classic steady 180° phase shift (Southwood & Kivelson, 1981; Zong et al., 2007), growing phase shift across the resonant energy channel that finally exceeds 180° has been attributed to the damping of the ULF wave. A peculiar type of modulation stripes in electron pitch angle spectrogram, termed as “boomerang stripes,” was also recognized in the outer radiation belt by Hao et al. (2017). Drift resonance localized in azimuthal extent has been found to be responsible for these boomerang stripes. Additional observations from BD-IES (Li et al., 2017) and Arase (Teramoto et al., 2019) have also indicated the existence of azimuthally localized drift resonance. In the companion paper of this work (Zhao et al., 2020), smoking-gun evidence of localized drift resonance has been provided by a case study showing simultaneous observations of straight and boomerang stripes taken from azimuthally separated Van Allen twin probes.

Although both localized and global drift resonances have been diagnosed in individual cases over years, they have not been found to coexist simultaneously before. On the other hand, despite of multiple pieces of evidence indicating that ULF waves with more than one m value can be excited at the same time (Barani et al., 2019; Degeling et al., 2018; Sarris & Li, 2017), simultaneous drift resonances between such multi-m waves and electrons have not yet been reported. In this paper, with detailed analysis on Van Allen Probes observations on 21 April 2015 and numerical modeling, we prove that localized and global drift resonance could happen to outer belt electrons simultaneously and the coexisting different m values of the wave modes give rise to a banded modulation structure in electron energy spectrogram.

2 Instrumentation

In this study, we focus on the energetic electron measurements from magnetic electron ion spectrometer (MagEIS) (Blake et al., 2013) of energetic particle, composition, and thermal plasma (ECT) suite (Spence et al., 2013) onboard Van Allen Probes. Magnetic field data from the electric and magnetic field instruments suite and integrated science (EMFISIS) (Kletzing et al., 2012) are also used for ULF wave observations. Prediction of the position and shape of the plasmapause is given by plasmapause test particle (PTP) simulations (Goldstein et al., 2014). Electron measurements from synchronous orbit particle analyzer (SOPA) (Belian et al., 1992) onboard LANL-97A satellite have also been investigated to check the drift resonance at geosynchronous orbit.

3 Banded Drift Resonances

A corotating interaction region (CIR) structure in solar wind passed Earth's magnetosphere between 20 and 22 April 2015 (see Figure 1a and1b). The velocity of solar wind remained ∼400 km/s on 20 April and jumped to >550 km/s at the beginning of 21 April. A compression region of high plasma density was recorded before the high-speed stream (HSS). The trailing edge of the compression region corresponded to a discontinuity of solar wind flow direction. As shown in Figure 1a, atan(V_y/V_x) in GSM coordinate changed by ∼5° at ∼2340 UT on 20 April, while the proton density dropped to precompression level at the same time. All these features indicate the passage of a CIR structure (Gosling, 1996; Hundhausen, 1973).

Figure 1

Wave-particle interactions observed by Van Allen Probe A and the solar wind conditions from OMNI data set during the interval. (a) Solar wind speed (black line) and its direction atan(V_y/V_x) (red line). (b) Proton density profile during the time interval. (c) Radial (orange line) and azimuthal (violet line) component of the magnetic field perturbations measured by EMFISIS on Van Allen Probe A, decomposed in main-field-aligned (MFA) coordinate. (d) Spectrogram of differential electron flux with α_local = 90° measured by MagEIS-A. (e) Spectrogram of residual flux , where J is the differential flux presented above and J₀ is the 15-minute running average of J calculated in each energy channel.

During the HSS passage, a set of unusual energetic electron flux oscillations was observed in the outer radiation belt. Figures 1c–1e shows the spectrogram of α_local = 90° electrons and magnetic field perturbations measured by Van Allen Probe A at dusk sector during the time interval marked with dashed lines in Figures 1a and 1b. Magnetic field perturbations with the period of ∼300s were clear in both azimuthal (toroidal) and radial (poloidal) components. Electric field perturbations also showed periodic oscillations with the same period (not shown). Energy spectrogram of electrons also showed modulations in the same period, lasting from 1145 UT to 1245 UT throughout the energy range 100 keV ∼ 2 MeV. Unfortunately, Van Allen Probe B was orbiting at the L shell ∼2 R_E smaller than spacecraft A during the time interval (outbound through slot region), not being able to provide joint wave and particle observations. Panel (e) presents the background-trend-removed spectrogram of electrons to address the ULF modulations. Residual flux was calculated with a 15-min running boxcar averaged flux J₀ (Claudepierre et al., 2013). The stripes of Pc5 oscillations broke down into two distinctive bands unexpectedly. One band below 500 keV show large phase shift across energy channels, henceforth referred to as R1. The other band which located above 500 keV show smaller phase shift (∼180° in the first three clear cycles) are referred to as R2 namely. To the best of our knowledge, such double-banded structure has not yet been reported in ULF wave-electron coupling processes.

Banded structures in electron spectrogram modulated by ULF waves hint that more than one resonant process happened simultaneously. Squared wavelet power (e.g., Grinsted et al., 2004) of the residual flux in each energy channel were integrated between 1130 UT and 1245 UT among Pc4–5 range (45s – 600s). As presented in the top panel of Figure 2, there were two distinctive peaks of flux oscillations among MagEIS energy channels, corresponding to the double-banded resonance. The resonant band R1 peaked at 183.4 keV and the resonant band R2 peaked at 1,077.7 keV. The peaks R1 and R2 were remarkably discrete, as the minimum flux oscillations between them (namely, 464.4 keV channel) were of ∼10% the power of the peaks R1 and R2.

Figure 2

Top: Normalized squared wavelet power of residual flux oscillations in each energy channel for α_local = 90° electrons, integrated among Pc4-5 range. Bottom: Estimation of the azimuthal wave number m for each resonant peak. Solid and dashed lines trace the energy dependence of drift frequency divided by m at L = 5.0 and L = 6.0 respectively. Blue lines denote T_drift/m for m = 10 and red lines for m = 3. Areas filled with tilted lines mark the frequencies of half maximum squared wavelet power of 183.4 keV blue and 1,077.7 keV channel red.

As for the flux oscillations of 100 keV to several MeV electrons modulated by Pc5 ULF waves, drift resonance is the physical process that is most likely responsible for the resonant peaks (e.g., Hao et al., 2014, 2017, 2019; Zong et al., 2007, 2008). The resonant condition of drift resonance follows

(1)

where the ω is the angular frequency of the wave, m is the azimuthal wave number and ω_d is the angular drift velocity of electrons (Southwood & Kivelson, 1981). ω_d largely depends on the particles' energy (see Equation A4). Therefore, for a given wave frequency ω, the azimuthal wave number m determines the resonant energy. The wave frequency ω of fundamental wave mode could be regarded as an eigen mode solution to the MHD equations, which is not likely to vary too much under certain plasma and field condition. Therefore, two distinctively different mvalues that coexisted simultaneously are likely to explain such double-banded drift resonant structure observed by MagEIS in this event.

The m values corresponding to the drift resonances R1 and R2 could be estimated with the resonant condition. Equation 1could be converted to T_drift/m = T_wave. T_wave was estimated according to the periodicity of electron flux oscillations (cf. Hao et al., 2017), which did not differ much from the local wave frequency in this case. The energy dependence of T_drift were calculated with Equation A4, with equatorial pitch angle converted from local measurements. As shown in the bottom panel of Figure 2, T_drift/m with m = 10 fitted the peak of resonance R1 at 183.4 keV well and m = 3 fitted the peak of resonance R2. Therefore, we estimate that there were two azimuthal wave modes, namely m = 10 and m = 3, which drove drift resonances that peaked at ∼200 keV and ∼1 MeV, respectively.

4 Boomerang Stripes at the Lower Resonant Band

In this section, we focus on the observational details of the drift resonant band R1 (<500 keV in energy). The resonant band R1 extended from ∼100 keV to ∼ 500 keV and peaked at the 183.4 keV channel. As the residual flux spectrogram in Figure 1 has indicated, resonant band R1 featured large phase shifts across the resonant energy channel, which exceeded the classic 180° phase shift predicted by Southwood and Kivelson (1981). Significant energy-dependent flux oscillating amplitude and phase shift >180° are also clear in the line plot of residual fluxes (see Figure 3(top)). Pitch angle evolution of R1 electrons has been detrended with the same method as 90° pitch angle channel; 90° led the phase of each set of flux oscillations throughout the R1 channels. Such type of pitch-angle-dispersive flux oscillations in Pc4-5 range has been termed as “boomerang stripes” by Hao et al. (2017).

Figure 3

From top to bottom: Residual flux of α_local = 90° electrons for energy channels near R1, peaks in 183.4 keV channel marked with vertical lines. Pitch angle spectrogram of the residual flux in channels 342.1, 226.1, and 143.5 keV. Spin-averaged flux of each channel are plotted in black lines.

Such boomerang-shaped stripes in pitch angle spectrogram have been recognized as a diagnostic characteristic for the azimuthally localized drift resonance, as interpreted and simulated by Hao et al. (2017). The phase shift >180° among energy channels at the very beginning of the modulation signatures was also attributed to localized drift resonance by a couple of following-up studies (Li et al., 2017; Teramoto et al., 2019). The proposed mechanism is that electrons have to drift to the satellite outside the azimuthally confined region of drift resonance to get measured, while the drift velocities of them are dependent of energy and pitch angle, which adds to extra phase shift in pitch angle and energy spectrograms.

Time-Of-Flight method on pitch angle spectrograms (PA-TOF) has been developed to estimate the origin of localized drift resonance in a very recent work (Zhao et al., 2020). Main idea of the PA-TOF is to trace the boomerang stripes backward to their initially dispersionless state. As shown in Figure 4 (left, inset), pitch-angle-dependent arriving time of the peaks and valleys t(α) were picked out in the top panel, and their drift period T_drift(α) were calculated and plotted in the format of T_drift versus t in the middle panel. Note that the local pitch angles have been converted to equatorial pitch angles when calculating T_drift(α) with Equation A4. By extrapolating the data points (t(α),T_drift(α)) of each stripe to T_drift = 0 linearly, the time t₀ when the stripe was generated was estimated (e.g., Sergeev et al., 1992). Angular distance between the satellite and the origin of boomerang stripes could also be derived from the drifting time t(α)−t₀. Please check our companion paper (Zhao et al., 2020) for technical details and validations from both simulations and multisatellite observations of PA-TOF method.

Figure 4

Left: Origin of boomerang stripes in R1 channels derived from PA-TOF method. Orbit of Van Allen Probe A from 11:30 to 12:45 is highlighted in red. Dark area depicts the plasmapause predicted by Plasmapause Test Particle (PTP) Simulations. (Figure 4 left, inset) An example of pitch-angle-TOF on boomerang stripes. Right: Schematic illustration of the banded drift resonance. Orange and blue arrows show the angular displacement of electrons with the energy ∼1 MeV and ∼200 keV in terrestrial dipole field over a fixed time interval. m = 3 global wave and m = 10 localized wave resonate with ∼1 MeV and ∼200 keV electrons simultaneously, leading to two distinctive resonant bands in energy spectrogram. A satellite azimuthally away from the excitation region of m = 10 ULF waves will see boomerang-shaped stripes in channels near R1 due to pitch angle dispersion.

Figure 4 (left) gives the origins of boomerang stripes in R1 estimated by PA-TOF method. Boomerang stripes in six energy channels ranging from 108.3 to 342.1 keV were traced. According to our estimation, electron flux modulations in R1 resonant band originated from the local time range 1500LT-1800LT. A plasmasphere topology derived from Plasmapause Test Particle (PTP) simulation (Goldstein et al., 2014) has been plotted on the equatorial plane. We suggest that the PTP result seems reasonable during this time interval, as the upper hybrid line measured by EMFISIS-HFR indicated a plasmapause crossing of probe A at ∼1130UT (not shown), which was consistent with the PTP prediction. PTP simulation also indicated a plume structure stretching out at the postnoon sector at the epoch. The origins of boomerang stripes estimated by PA-TOF mostly located inside the plume structure. If it was not a coincidence, such an unexpected good spatial overlap may imply that the drift resonance R1 was confined inside the region with high plasma density.

5 Interpretations

In previous sections, we have drawn preliminary conclusions as follows:

1. Double-banded drift resonance between outer belt electrons and Pc5 ULF waves during the high speed solar wind passage. Resonant band R1 peaked at ∼200 keV and R2 peaked at ∼1 MeV, which corresponded to m = 10 and m = 3, respectively.
2. Boomerang stripes implied that the resonant band R1 was a localized drift resonance in azimuthal extent. Estimation by PA-TOF method indicated that boomerang stripes originated from 1500LT-1800LT region, at which a plasma plume structure was expected.
3. Phase shift across resonant band R2 of ∼180° indicated a possible global drift resonance with m = 3 ULF waves, which is also evidenced by the straight stripes during the first three cycles in pitch angle spectrogram (see observations from 901 keV channel in Figure 5).

Figure 5

Comparison between observations and simulations. Left panels show the energy and pitch angle spectrograms of residual flux measured by MagEIS-A; 183.4 keV channel and 901.8 keV channel correspond to resonant band R1 and R2, respectively. Right panels show the normalized energy change ΔW/W of electrons in the same format as right panels. Drift phase mixing effect has been taken into account in right panels by averaging over the energy bandwidth of MagEIS channels.

Based on the observational facts listed above, we present Figure 4 (right) to summarize our scenario on the double-banded drift resonance. An m = 10, eastward propagating wave packet confined at postnoon sector resonates with ∼200 keV electrons. Assuming a satellite outside the wave packet, electron populations modulated by the localized wave drift to get detected, which will lead to >180° phase shift in energy spectrogram and boomerang stripes in pitch angle spectrogram. An m = 3 wave with a global extent modulates the outer belt electrons simultaneously, giving rise to another resonant peak at ∼1 MeV.

The driving force of the ULF waves in this event is possibly attributed to the high speed solar wind. During this event, the solar wind speed was above 550km/s (see Figure 1a). HSS imposes strong flow shear to the magnetopause, which is in favor of Kelvin-Helmholtz (K-H) instability to grow. K-H instability is able to produce large-scale vortices at the magnetopause and henceforth drives ULF activities through field line resonances (Claudepierre et al., 2008; Mathie & Mann, 2001; Pu & Kivelson, 1983; Rae et al., 2005). The eastward propagation of the waves and their moderate azimuthal wave number m = 3 and 10 are also consistent with the MHD simulations (e.g., Claudepierre et al., 2008).

In terms of the resonant band <500 keV, we suggest that the azimuthal inhomogeneity of plasma density may account for the localized drift resonance. Numerical simulations by Degeling et al. (2018) suggested that the plasma plume can significantly alter the field line resonance point as it introduces asymmetry to the Alfvén speed distribution. They also suggested that a waveguide (or local cavity) embedded inside the plasmaspheric plume structure could lead to inward penetration of ULF wave power. As to this event, plume structure was also predicted by PTP simulations and was found to overlap with the origin of boomerang stripes. Therefore, a plausible explanation is that the m = 10 wave mode powered by magnetopause K-H instability was able to penetrate into the L shell of Van Allen Probe A through the plume structure and resonated with ∼200 keV electrons. We suggest that such scenario of inward penetration might be reasonable, as drift resonance peaking at ∼200 keV was also observed at geosynchronous orbit in noon sector by LANL-97A satellite during the same interval (see Figure S1 in the supporting information).

6 Reproducing Banded Drift Resonance

To validate our scenario of simultaneous localized and global drift resonance, numerical modeling has been conducted and compared with the observations in this section. As a first order estimation of the particles' energization, the total energy gain ΔW from the wave electric field is calculated along the undisturbed trajectories of electrons, which has been also used in plenty of studies (e.g., Hao et al., 2017, 2019; Li et al., 2017; Southwood & Kivelson, 1981; Zhou et al., 2016). For simplicity, dipole field is used to calculate the undisturbed drifting trajectory of electrons. Sinusoidally oscillating azimuthal electric field is given by analytical expressions (see Equation A1). Please check the Appendix for details about idealizing the wave model and calculating the energy gain of particles.

Figure 5 presents the comparison between observations and our simulations. Double-banded structure in energy spectrogram for α_local = 90° electrons were well reproduced. The modulations peaked at ∼200 keV and ∼1 MeV separately in the simulated spectrogram. The different types of phase shifts among R1 and R2 band were also consistent with the observations. One may argue that as the drift resonance R1 has been assumed to be localized in local time, there should be a drift echo of modulations arriving at the virtual probe after ∼1235UT, as Hao et al. (2017) and Li et al. (2017) have shown in their simulations but it was not observed in this event by MagEIS. We suggest that such a missing drift echo of ULF modulations can be attributed to the drift phase mixing effect within the MagEIS energy bandwidth (Schulz & Lanzerotti, 1974). Once the energy bandwidth of the equipment was taken into account (same method with Hao et al., 2019; Zhou et al., 2016), the drift echo also vanished in the simulated energy spectrogram (see the right top panel of Figure 5).

Pitch angle spectrograms for the resonant band R1 and R2 were also simulated. As shown in the second row of Figure 5, the modulation stripes in 901.8 keV channel were mostly straight at the the first 20 min (from 1155UT to 1215UT) and evolved to boomerang-shaped later, which has been reproduced by our simulations. Since 901.8 keV are mostly affected by the global wave with an m value of 3, there is no surprise that initial pitch angle dispersion due to time-of-flight effect did not appear. One may notice that the first bunch of modulation stripes in the simulated 901.8 keV electron pitch angle spectrogram are not as straight as the observations. We suggest that such slight difference might be cause by the deviation between electrons' actual drift velocity in magnetosphere and the dipole field estimation. In terrestrial dipole field, 901.8 keV electrons are close to the drift resonance condition for T = 300, m = 3 wave at L = 5.5. As the electrons with the energy from 817.8 keV to 949.2 keV have been simulated and mixed numerically to simulate the 901.8 keV stripes, there exist a portion of electrons oscillating with phase shift across the drift resonance, which might account for the slight deviation from straight stripes in our simulation. Such effect is hard to quantify accurately and could be rather sensitive to the parameters, especially to the electrons' drift velocity. Boomerang stripes in R1 resonant band were also reproduced by the simulations. In the bottom row, pitch angle dispersion appeared at the very beginning of flux modulations in 183.4 keV channel. The shape of simulated boomerang stripes were also quite similar with the observations. The PA-TOF method has also been conducted to the simulated boomerang stripes. Their origin turned out to be consistent with the azimuthal distribution of the m = 10 wave in our wave model and also fitted with the observations well (see Figure S2 for details).

In brief, our simulation reproduced the electron behaviors in highly consistency with MagEIS observations in both energy and pitch angle spectrograms. Therefore, we argue that the mixture of a localized, moderate m wave mode, and a global, low m wave mode, is enough to explain the peculiar electron modulations in this event. The existence of ULF waves with other m values during the time interval we studied has not been absolutely excluded but is not likely to be responsible for the discrete resonant bands observed in MagEIS energy range.

7 Discussion and Conclusions

Based on comprehensive data analysis and simulations on the 21 April 2015 event, we now come to the conclusion that ULF waves with different azimuthal wave number m are able to exchange energy with electrons in an extended energy range through banded drift resonance. Electrons' simultaneous drift resonance with localized, moderate m and global, low m ULF waves (in this case, m = 10 and 3, respectively) has been recognized for the first time. We would like to suggest that azimuthal wave number m is a critical parameter in particle energization and can be multivalued. Such multi-m, coherent wave field can transport and energize electrons nondiffusively, differentiating from the current radial diffusion treatment (e.g., Fei et al., 2006; Ozeke et al., 2012). Consequently, it is possible that seed electrons (tens to hundreds keV (e.g., Boyd et al., 2016; Jaynes et al., 2015)) and “killer” electrons ( 1 MeV, the core population of the radiation belt (e.g., Horne, 2007; Wrenn, 2009; Zong et al., 2011)) get energized simultaneously at the presence of ULF waves.

The banded drift resonance event we focused on in this paper is not a unique case. For instance, another double-banded drift resonant spectrogram was also recorded by Van Allen Probe A on 18 April 2015 (see Figure S3), which presented quite similar modulations. Therefore, we would like to imply that the simultaneous drift resonances driven by ULF waves with a moderate and a low m value might be a common feature in outer radiation belt. Further investigations on Van Allen Probes data set are demanded for a more precise and statistical picture.

As Mann et al. (2016) has argued, the correct ULF wave physics is critical to explain the radiation belt dynamics with elegance. With this case study, we have demonstrated that spatially cross-scale ULF waves are able to drive the electron energization processes over broad energy range. Further studies are needed to examine the possibility of electrons to get cross-m energized, that is to say, the chance for an electron to get resonantly energized by a higher m wave and then fit in the resonant condition for a lower m wave. Electrons experiencing cross-m acceleration may have the possibility to increase their energy by order of magnitude (e.g., from one hundred keV to several MeV) with a time scale of <1 hr, similar to the RTA mechanism for chorus waves (Furuya et al., 2008). Nonlinear behavior of near-resonant particles has to be taken into account for this issue (e.g., nonlinear trapping island(s) Li et al., 2018 and their overlapping Degeling et al., 2007; 2019). Such a cross-m energizing scenario has been implied by previous numerical simulations (Degeling et al., 2007, 2008; Ukhorskiy & Sitnov, 2008), but direct observational evidence is still missing. Ultra-high resolution energy channels of MagEIS based on Pulse Height Analysis (PHA) (Hartinger et al., 2018) may shed light on this topic in the future. Accurate measurements of the ULF wave m spectrum (both the power and the phase information) (e.g., Barani et al., 2019; Sarris & Li, 2017) and ULF wave model with realistic plasma and field information (e.g., Degeling et al., 2018; Mager & Klimushkin, 2008; Leonovich & Kozlov, 2019) are also essential to depicting the cross-energy coupling between ULF wave activities and radiation belt electrons.

Acknowledgment

Y. X. H and X. X. Z contributed equally to this work. This work was supported by National Natural Science Foundation of China (41421003, 41627805, 42011530080). R. Rankin acknowledges financial support from the Canadian Space Agency and NSERC. Processing and analysis of the MagEIS and REPT data was supported by Energetic Particle, Composition, and Thermal Plasma (RBSP-ECT) investigation funded under NASA's Prime contract no. NAS5-01072.

Appendix A: Electrons' Energy Gain from ULF Wave Field Along Their Undisturbed Trajectories

With the parameters estimated from the observations, we assume an equatorial electric field defined by

(A1)

where ϕ is the longitude, t₀ gives the epoch of maximum wave amplitude, τ gives the waves' damping time and ω is their central angular frequency. The azimuthal wave number m₁ and m₂ are set to be 10 and 3, respectively for resonant band R1 and R2, according to the observations. and τ = 600s are adapted to reproduce the observations, which is reasonable according to the waveform analysis on both magnetic field and 183.4 keV flux pulsations.

The local time confining of drift resonance R1 is realized by introducing a spatial envelope g₁, which is given by a von-Mise-like function (Mardia & Jupp, 2009)

(A2)

Here, ϕ₀ defines the center of the wave excitation region in local time (set to be 1630LT according to PA-TOF results), and ξ modulates the concentration of the envelop. Similar formulas have been utilized to model the impact of spatial distribution (Li et al., 2017) and time evolution (Zhou et al., 2016) on the electrons' drift resonance with ULF waves. Spatial envelope in the form of von Mise function does not lead to significant difference in electron flux patterns in comparison with a boxcar envelope (e.g., Hao et al., 2017; Zhao et al., 2020). With such a set of equations, wave form composed by a global low m wave and a localized moderate m wave has been defined. A visualization of the wave form has been depicted in Figure 4(right), in which the fraction E₁/E₂ has been set to be 3.

The energy electrons gain from wave field (ΔW) can be calculated by integrating

(A3)

where R_E = 6371km is the Earth radius, e is the charge of electron, a(α_eq) is a pitch angle dependent factor denoting the equivalent electric field amplitude integrated between the mirror points (cf. Hao et al., 2017), and the undisturbed angular drift velocity of an electron is given by

(A4)

Here, B_E is the equatorial surface magnetic field strength of Earth, c is the speed of light, β and γ are relativistic factors of the electron and m₀ stands for its rest mass (Roederer & Zhang, 2014).

Open Research

Data Availability Statement

All RBSP-ECT data are publicly available at https://www.RBSP-ect.lanl.gov/. LANL data are also available at https://rbsp-ect.lanl.gov/data_pub/LANL-GEO/data/. Predicted plasmapause by PTP simulations is available at https://enarc.space.swri.edu/PTP/. Solar wind parameters are available at https://omniweb.gsfc.nasa.gov/.