Realistic Electron Diffusion Rates and Lifetimes Due to Scattering by Electron Holes
Abstract
Plasma sheet electron precipitation into the diffuse aurora is critical for magnetosphere-ionosphere coupling. Recent studies have shown that electron phase space holes can pitch-angle scatter electrons and may produce plasma sheet electron precipitation. These studies have assumed identical electron hole parameters to estimate electron scattering rates (Vasko et al., 2018, https://doi.org/10.1063/1.5039687). In this study, we have re-evaluated the efficiency of this scattering by incorporating realistic electron hole properties from direct spacecraft observations into computing electron diffusion rates and lifetimes. The most important electron hole properties in this evaluation are their distributions in velocity and spatial scale and electric field root-mean-square intensity (
). Using direct measurements of electron holes during a plasma injection event observed by the Van Allen Probe at 
, we find that when 
4 mV/m electron lifetimes can drop below 1 h and are mostly within strong diffusion limits at energies below 
10 keV. During an injection observed by the THEMIS spacecraft at 
, electron holes with even typical intensities (
1 mV/m) can deplete low-energy (
a few keV) plasma sheet electrons within tens of minutes following injections and convection from the tail. Our results confirm that electron holes are a significant contributor to plasma sheet electron precipitation during injections.
Key Points
-
Realistic electron diffusion rates are computed by incorporating the observed electron hole distributions
-
Electron hole distributions in velocity and parallel scale and E-field intensities control the scattering efficiency
-
Realistic lifetime estimates suggest efficient plasma sheet electron losses due to electron holes in plasma injections
Plain Language Summary
Recent studies have revealed that a class of nonlinear time domain structures, consisting mostly of electron phase space holes, which appear as broadband electrostatic emissions in the frequency domain, may pitch-angle scatter plasma sheet electrons, producing diffuse auroral precipitation. Although how significantly electron holes contribute to the global diffuse aurora remains unclear, our study attempts to address the most fundamental question—what are realistic electron loss rates due to electron hole scattering in the terrestrial magnetosphere? Incorporating realistic electron hole distributions directly observed by various spacecraft missions into a newly developed methodology of calculating scattering rates, we have obtained realistic electron lifetimes for two injection events. In these events, broadband electrostatic emissions produced by electron holes are associated with sudden electron energy flux enhancements in the energy range of a few hundred eV up to more than 100 keV. The calculated electron lifetimes substantiate that electron holes play a significant role in generating low-energy (less than a few keV) plasma sheet electron precipitation from the magnetotail following electron injections and convection. Although in the inner magnetosphere this electron hole induced precipitation is not as much as estimated from previous studies, appreciable precipitation can be produced by sufficiently intense electron holes which are not uncommon.
1 Introduction
Plasma sheet electrons below a few keV provide the dominant contribution to diffuse auroral precipitation (Hardy et al., 1985; Newell et al., 2009; Ni et al., 2016; Swift, 1981). Accounting for nearly 75% of the total particle energy flux from the magnetosphere to the ionosphere (Dombeck et al., 2018; Newell et al., 2009), this precipitation critically affects the ionospheric conductance and hence provides significant feedback to the convection electric field in the magnetosphere (Hardy et al., 1987; Wiltberger et al., 2017; Yu et al., 2016). Precipitation of low-energy plasma sheet electrons to the ionosphere has been suggested to be mainly driven by electron scattering into the loss cone from whistler-mode chorus and electron cyclotron harmonic (ECH) waves in the magnetosphere (Koskinen, 1997; Ni et al., 2011; Thorne et al., 2010; X.-J. Zhang et al., 2015). Spacecraft measurements have shown that ECH waves are the most plausible drivers of plasma sheet electron precipitation at radial distances 
from Earth, whereas upper- and lower-band chorus waves (whistler-mode waves) drive substantial diffusive precipitation at 
(see, e.g., a review by Ni et al., 2016). Nevertheless, ECH and chorus waves might not be the only wave activities contributing to diffuse auroral precipitation. Recent theoretical analyses and spacecraft measurements have indicated that time domain structures (TDSs), which appear as broadband electrostatic fluctuations in the frequency domain, may also contribute to diffuse auroral precipitation (Mozer et al., 2015; Shen et al., 2020; Vasko, Agapitov, Mozer, Artemyev, Krasnoselskikh, & Bonnell, 2017; Vasko et al., 2018). These TDSs comprise mostly electron holes (Malaspina et al., 2014, 2018; Vasko, Agapitov, Mozer, Artemyev, & Jovanovic, 2015; Vasko, Agapitov, Mozer, Artemyev, Drake, & Kuzichev, 2017), and, more rarely, electron-acoustic double layers (Dillard et al., 2018; Mozer et al., 2015; Vasko, Agapitov, Mozer, & Artemyev, 2015; Vasko, Agapitov, Mozer, Bonnell, et al., 2017). This paper will focus on electron holes as a potential driver of the diffuse auroral precipitation.
Early electric field spectra measurements in the plasma sheet have shown the presence of broadband electrostatic fluctuations in the frequency range below a few hundred Hz (Gurnett, 1976; Koskinen, 1997; Lakhina et al., 2000; Scarf et al., 1974). Waveform measurements have demonstrated that broadband electrostatic fluctuations in the plasma sheet are produced predominantly by electron phase space holes (Cattell et al., 2003, 2005; Ergun et al., 2015; Lotekar et al., 2020; Matsumoto et al., 1994), which are electrostatic solitary waves formed in a nonlinear stage of various electron streaming instabilities (Che et al., 2010; Drake et al., 2003; Goldman et al., 1999; Omura et al., 1996; Pommois et al., 2017; Umeda et al., 2006). Electron holes have bipolar parallel electric fields with parallel spatial scales of a few to more than 10 Debye lengths and typically propagate with velocities of a fraction of the local electron thermal velocity (Cattell et al., 2003, 2005; Lotekar et al., 2020; Mozer, Agapitov, Giles, & Vasko, 2018; Tong et al., 2018), although some electron holes may have velocities smaller than or comparable to the local ion thermal velocity (Lotekar et al., 2020; Norgren et al., 2015). There have also been reports of electron holes propagating with relativistic (close to the speed of light) velocities and exhibiting magnetic field signatures (Andersson et al., 2009; J. B. Tao, Ergun, et al., 2011).
THEMIS (Angelopoulos, 2008) and MMS (Burch et al., 2016) measurements have shown that electron holes in the plasma sheet are always associated with fast plasma flows (Deng et al., 2010; Ergun et al., 2015; Lotekar et al., 2020; Viberg et al., 2013). Recent measurements from Van Allen Probes (VAP) (Mauk et al., 2013) have shown that electron holes are highly prevalent around injection fronts in the inner magnetosphere (Malaspina, Wygant, et al., 2015; Mozer et al., 2014; 2015). The VAP orbits allowed measurements of electron holes around the equatorial plane within magnetic latitudes 
. These nonlinear electrostatic structures were also measured at higher latitudes 
by the FAST and Polar spacecraft (Ergun, Carlson, McFadden, Mozer, Delory, et al., 1998; Franz et al., 2000, 2005; Muschietti et al., 1999).
The most likely sources of electron holes are electron beam instabilities of the bump-on-tail type; the corresponding electron beams are either of ionospheric origin (Abel et al., 2002; Artemyev et al., 2020; Ergun, Carlson, McFadden, Mozer, Delory, et al., 1998; Klumpar, 1993; Lönnqvist et al., 1993; Moore & Arnoldy, 1982) or produced locally by whistler-mode waves (An et al., 2019; Drake et al., 2015) and kinetic Alfvén waves (An et al., 2020; Artemyev et al., 2015; Génot et al., 2004; Kletzing, 1994; Watt & Rankin, 2009). The latter scenario is more likely, because electron holes around injection fronts and fast plasma flows are always associated with broadband electromagnetic fluctuations below a few tens of Hz, which have been interpreted in terms of kinetic Alfvén wave turbulence (Chaston et al., 2012, 2015; Moya et al., 2015).
Observations of a few injection fronts from multiple spacecraft (Van Allen Probes and THEMIS-D) radially separated by a few Earth radii have shown that electron holes may be generated around injection fronts continuously in time (Malaspina, Wygant, et al., 2015). In addition, previous numerical simulations have demonstrated that electron holes produced around the equatorial plane are not likely to reach the high latitudes, at which Polar measurements were carried out (Franz et al., 2000, 2005), because of limited lifetime (Goldman et al., 1999), braking by a non-uniform magnetic field, or both (Kuzichev et al., 2017; Vasko, Kuzichev, et al., 2017). Thus, it is most likely that electron holes are produced locally and continuously at various latitudes as the associated injection front propagates toward Earth.
In the formation of electron holes, the nonlinear development of electron bump-on-tail instabilities in three-dimensional (3D) magnetized plasma is critical (Goldman et al., 1999; Lu et al., 2008; Umeda et al., 2006; Wu et al., 2010). Nearly 1D electron holes appear early in these nonlinear stage of bump-on-tail instabilities, with their perpendicular spatial scales much larger than parallel spatial scales. But then 1D electron holes break up into 3D ones with parallel and perpendicular scales related roughly as 
, where 
and 
are electron plasma and cyclotron frequencies (Lu et al., 2008; Umeda et al., 2006; Wu et al., 2010). This scaling relation implies that the amplitudes of perpendicular and parallel electric fields are related as 
. Such a scaling relation was originally reported by Franz et al. (2000) based on Polar spacecraft measurements at high latitudes in the inner magnetosphere. Measurements from the Van Allen Probes at lower latitudes have shown that the perpendicular electric field is statistically a few times smaller than the parallel electric field, consistent with the scaling relation (Malaspina et al., 2018; Vasko, Agapitov, Mozer, Artemyev, Drake, & Kuzichev, 2017). The validity of the scaling relation has also been recently confirmed by multi-satellite measurements of electron holes from the Magnetospheric Multiscale (MMS) mission (Holmes et al., 2018; Steinvall et al., 2019; Tong et al., 2018). The scaling relation states that the perpendicular spatial scale of electron holes should be of a few thermal electron gyroradii, although the validity and physics behind that scaling relation are debated (Hutchinson, 2020).
Theoretical studies have shown that perpendicular electric fields of electron holes can efficiently pitch-angle scatter plasma sheet electrons, thereby producing electron loss and precipitation to the diffuse aurora (Vasko, Agapitov, Mozer, Artemyev, Krasnoselskikh, & Bonnell, 2017; Vasko et al., 2018). Diffuse auroral precipitation has hitherto only been attributed to chorus and ECH waves (Ni et al., 2016). These two wave modes are suggested to be the main drivers because (Ni et al., 2016; Thorne et al., 2010): (a) global magnetic local time (MLT) distributions of these waves are correlated with the intensity of the diffuse aurora; (b) both these waves and diffuse auroral intensities are elevated during higher geomagnetic activities as represented by the 
and 
indices; (c) ECH and chorus waves observed in the magnetosphere can resonantly scatter electrons with energies below 
10 keV with typical pitch-angle scattering rates of 
–
(Ni, Thorne, & Ma, 2012); (d) ECH and chorus waves scatter low-pitch-angle electrons preferentially over perpendicular electrons, which results in formation of pancake-shaped and anisotropic pitch-angle distributions of electrons at less than 10 keV in the magnetosphere following injections of plasma sheet (
100 eV to 
30 keV) electrons (Meredith et al., 2000; X. Tao, Thorne, et al., 2011); (e) several case studies have shown correlated measurements of these waves in the magnetosphere and conjugate diffuse auroral activities in the ionosphere (Liang et al., 2011; Mozer, Agapitov, Blake, & Vasko, 2018; Ni, Liang, et al., 2012; Ni et al., 2014; Nishimura, Bortnik, et al., 2010).
That electron holes contribute to diffuse auroral precipitation is supported by several arguments that are identical to the arguments (a–e) for chorus and ECH waves. The statistical analysis by Malaspina, Wygant, et al. (2015) has shown that electron holes and broadband electrostatic fluctuations are associated with particle injection fronts. Like the diffuse auroral precipitation, the MLT distribution of electron holes in the inner magnetosphere extends from premidnight sectors (
18 MLT) to morning sectors (
6 MLT) (Newell et al., 2009; Petrinec et al., 1999; Thorne et al., 2010). Because of their close association with particle injection fronts, electron holes are also intensified during higher geomagnetic activity. These features are similar to those of ECH and chorus waves known to be correlated with injections (see, e.g., Meredith et al., 2000, 2009; X. Zhang & Angelopoulos, 2014; X. Zhang et al., 2018) except that the MLT distributions of chorus waves occur well into the dayside sectors (
6 MLT) while electron holes have a slight preferential occurrence rate in the premidnight sectors (Ergun et al., 2015; Malaspina, Wygant, et al., 2015). Thus, arguments (a and b) also apply to electron holes.
One has to note that, however, correlation is not causation. For example, previous simulations and observations have shown that electron holes can form as a result of nonlinear electron trapping by whistler-mode chorus waves (An et al., 2019; Drake et al., 2015), which may suggest that electron holes are only a by-product of nonlinear interactions between chorus and electrons. Although this process is a plausible explanation for certain events, it is unlikely to explain the majority of electron holes associated with injections. The reasons are: (a) electron holes are predominantly observed in the pre-midnight sectors (Ergun et al., 2015; Malaspina, Wygant, et al., 2015), whereas the occurrence of chorus waves favors the post-midnight sectors and maximizes on the dayside (Li, Thorne, Angelopoulos, Bortnik, et al., 2009); (b) electron holes are frequently observed right at the injection fronts where low-energy field-aligned electrons prevail (Malaspina, Wygant, et al., 2015; Mozer et al., 2015). In contrast, chorus waves are usually observed behind the injection fronts or in the magnetic flux pileup regions where perpendicularly accelerated energetic electrons dominate and provide the anisotropy needed to drive chorus waves (Deng et al., 2010; Li, Thorne, Angelopoulos, Bonnell, et al., 2009; Viberg et al., 2014; X. Zhang et al., 2018).
Besides the observational correlation between electron holes and diffuse auroral precipitation, Vasko, Agapitov, Mozer, Artemyev, Krasnoselskikh, and Bonnell (2017) (hereafter as Vasko2017) have theoretically demonstrated that electron holes can pitch-angle scatter electrons and generate electron precipitation. Considering an ensemble of identical electron holes distributed in latitude around an injection front, Vasko2017 computed electron bounce-averaged diffusion coefficients and demonstrated that broadband electrostatic fluctuations comprising identical electron holes in the inner magnetosphere can scatter electrons with energies below a few keV with typical pitch-angle scattering rates of 
–
. The pitch-angle scattering rate is stronger at lower pitch angles than around 
, resulting in formation of pancake distributions, a process similar to that for chorus waves (X. Tao, Thorne, et al., 2011). Shen et al. (2020) have recently presented conjugate observations of electron holes in the plasma sheet by the THEMIS spacecraft and precipitation of a few hundred eV electrons by the e-POP spacecraft (Yau & James, 2015) at low altitudes. Although a larger number of conjunction events is highly desirable to further verify the contribution of electron holes to the diffuse aurora, arguments (c–e) for chorus waves may also apply to electron holes.
The above discussions suggest that broadband electrostatic fluctuations and the associated electron holes may contribute to diffuse auroral precipitation. A critical task is to quantify the contribution of electron holes and to compare with those of ECH and chorus waves at various 
-shells and MLT sectors. To do that, we must first compute electron diffusion coefficients using realistic statistical properties of electron holes. Vasko2017 estimated these diffusion coefficients using a highly simplifying assumption that broadband electrostatic fluctuations consist of an ensemble of identical electron holes. The parameters of electron holes within the broadband fluctuations are not identical, however, and extend over some distributions (Andersson et al., 2009; Lotekar et al., 2020; Malaspina et al., 2018; Vasko, Agapitov, Mozer, Artemyev, Drake, & Kuzichev, 2017). Because of 3D localization each electron is to develop a formalism of computing the diffusion coefficients for realistic electron holes as observed from both the inner magnetosphere and the magnetotail. The study is motivated in part by (a) the analysis of Malaspina et al. (2018), who have recently underscored the importance of using continuous burst electric field measurements, rather than short bursts triggered by high-amplitude events, to obtain unbiased distributions of electron hole parameters; and (b) the analysis of Vasko et al. (2018), who have recently demonstrated that electron diffusion coefficients driven by electron holes can be computed using the standard quasi-linear theory (see, e.g., Sagdeev & Galeev, 1969 and Kennel & Engelmann, 1966), substantially simplifying the computation using realistic electron hole distributions.
This paper is organized as follows. In Sections 2 we present the formalism of computing diffusion coefficients and electron lifetimes from scattering by realistic distributions of electron holes. In Section 3 we present realistic case-specific lifetimes based on direct observations of electron holes around two injections from the Van Allen Probes in the inner magnetosphere (
6) and from the THEMIS spacecraft in the outer magnetosphere (
). The results are summarized in Section 4.
2 Data-Driven Electron Diffusion Rates From Electron Holes
of perpendicular electron velocities by an ensemble of stationary and statistically uniform electron holes is given as follows (Lyons, 1974; Vasko et al., 2018)
(1)
(2)
and 
are electron velocities parallel and perpendicular to the local magnetic field, 
is the electron cyclotron frequency, 
is the Bessel function of the 
th order, 
, 
is the dispersion relation of electrostatic structures contributing to the broadband electric fields 
, and 
is the spectrum of fluctuations occupying volume 
. The spatially averaged root-mean-square intensity 
of electron holes is determined as follows
(3)
. With statistically uniform and stationary fluctuations, however, the spatially averaged intensity coincides with temporally averaged intensity, which can be computed using spectral measurements onboard a spacecraft. We first clarify how we determine the broadband spectrum produced by identical electron holes. Each electron hole has an electrostatic potential 
localized in 3D space. For example, the electrostatic potential of electron holes is often described by the Gaussian model 
, where the 
-axis is along the local magnetic field and 
is the electron hole velocity in the spacecraft rest frame (Chen et al., 2005; Tong et al., 2018; Vasko, Agapitov, Mozer, Artemyev, Drake, & Kuzichev, 2017). Because of 3D localization each electron hole is equivalent to superposition of monochromatic electrostatic waves with a dispersion law 
propagating at various wave normal angles to the local magnetic field, 
. We choose 
to represent the number of electron holes in the volume 
and assume that the distance between individual electron holes is at least a few times larger than electron hole spatial scales so that electron holes do not overlap. Using Equation 3 we determine the number of electrons holes per unit volume and the broadband spectrum:
(4)
(5)
. Similar to electrostatic spectra of ECH waves (Lyons, 1974), the momentum and pitch-angle diffusion coefficients are straightforwardly related to 
due to conservation of electron energy in the wave rest frame, 
. The general formula for local pitch-angle diffusion rates driven by identical electron holes is given as follows (Vasko et al., 2018)
(6)
and 
are the electron velocity and pitch angle, 
, 
is the 
-th order modified Bessel function of the first kind, and
(7)The various terms in the sum in Equation 6 correspond to cyclotron resonances of 
th order, 
, with oblique electrostatic waves contributing to the spectrum of electron holes. Importantly, the diffusion coefficient (Equation 6) does not depend on electric field amplitudes of individual electron holes, but only on the root-mean-square intensity 
of electron holes, which takes into account the occurrence of electron holes.
, 
, 
and 
or, equivalently, 
which is the parallel electric field amplitude. In what follows we assume that perpendicular and parallel scales of electron holes are related as (Franz et al., 2000):
(8)
in 3D parameter space with 
or 
. In accordance with the assumption of stationary and statistically uniform broadband fluctuations, 
is independent of time and space, so that at any moment the root-mean-square intensity 
is assumed to be produced by electron holes with parameters distributed according to 
. The electrostatic potential of electron holes with parameters around 
is denoted 
. In the volume 
the number of electron holes with parameters in the range 
is 
, where 
is the total number of electron holes in that volume. Using Equation 3 we determine the number of electron holes per unit volume and the broadband spectrum
(9)
(10)
can be identified as the same as Equation 4, and 
(Vasko et al., 2018). We further consider that the electric field amplitude is statistically independent of 
and 
: 
, where 
. In that case the amplitude of the electrostatic potential is statistically dependent on the parallel scale, because 
. Strong experimental evidence as well as theoretical predictions support a positive correlation between 
and 
(Cattell et al., 2003; Ergun, Carlson, McFadden, Mozer, Muschietti, et al., 1998; Hutchinson, 2017; Lotekar et al., 2020; Muschietti et al., 1999). No similar correlation has been observed between 
and 
. Using the above assumption and the broadband spectrum given by Equation 10, we find that the diffusion coefficients for realistic electron holes can be computed as follows
(11)
is the diffusion coefficient given by Equation 6, as a result of scattering by electron holes with identical parameters around 
, while 
, where 
is normalized such that 
= 1. By intuition, the weighting factor 
accounts for interaction time between electrons and the ensemble distribution of electron holes. Larger 
corresponds to longer interaction time and therefore a larger contribution to the net local diffusion rates. Importantly, the diffusion coefficient does not depend on the distribution of individual amplitudes of electron holes, 
. Thus, to estimate the diffusion coefficients we need only to know the root-mean-square intensity of broadband fluctuations 
and the statistical distribution 
of electron hole velocities and spatial scales. The evaluation of 
should be based on sufficiently large number of electron holes so as to accurately reflect the occurrence rate for averaging within a certain time or spatial region. This condition is commonly satisfied for spacecraft spectrum measurements, of which time resolutions are usually larger than 1 s, nearly a thousand times the millisecond-scale electron hole in the spacecraft frame.
(12)
, and 
is the length of the field line as measured from the equator to the magnetic latitude of 
. The local pitch angle is related to the equatorial pitch angle 
as follows: 
, where 
is the equatorial magnetic field (
). Note that a dipole field for bounce-averaging local diffusion coefficients is quite valid for the inner magnetosphere, but we may considerably underestimate 
for the plasma sheet where the magnetic field becomes increasingly non-dipolar (see discussions of the non-dipole field effects on bounce-averaged diffusion from, e.g., K. G. Orlova & Shprits, 2010).In calculating bounce-averaged diffusion coefficients, we take into account the realistic distribution of electron hole parameters 
emerging from statistical spacecraft observations in both the inner and the outer magnetosphere (Andersson et al., 2009; Lotekar et al., 2020; Malaspina et al., 2018; J. B. Tao, Ergun, et al., 2011; Vasko, Agapitov, Mozer, Artemyev, Drake, & Kuzichev, 2017; Vasko et al., 2018). According to measurements of electron holes collected around an injection front from the Van Allen Probes in the inner magnetosphere (Malaspina et al., 2018), 
falls roughly between 1,000 and 11,000 km/s with a peak at 3,000 km/s, and 
resides approximately within 0.1–3.0 km with a peak around 0.4 km. Similar statistical measurements of electron holes have been reported in the magnetotail (
20 
) (Lotekar et al., 2020), in which case, velocities 
of more than 1,000 fast electron holes, potentially driven by bump-on-tail instabilities (see, e.g., Omura et al., 1996; Pommois et al., 2017; Umeda et al., 2006), were obtained in the range of 0.1–0.7
, where 
is the electron thermal velocity. The associated parallel scales 
are generally larger than their counterparts in the inner magnetosphere, varying from sub km up to 10 km, but similar to them in the normalized units, that is, parallel scales vary from about one to 10 Debye lengths 
. In addition, using coordinated THEMIS observations in the plasma sheet at 
, Andersson et al. (2009) and J. B. Tao, Ergun, et al. (2011) reported 67 magnetized fast electron holes with relativistic velocities (
105 km/s) within a dipolarization front. These relativistic electron holes have elongated parallel scales up to 150 km, almost 30 times the local Debye length. This special collection of relativistic electron holes may be particularly relevant to dipolarization events, so we will also use distributions of these elongated fast holes. Note that the aforementioned three sets of observed statistical electron hole distributions generally have an uncertainty of better than 20% in the hole velocity estimation. These observed distributions together with the synthesized two-dimensional distributions representing 
in calculations are demonstrated in Supporting Information S1.
Once 
has been specified, we obtain the distribution of 
from the scaling relation in Equation 8. Generally, we also apply the field line electron density model in the inner magnetosphere from Denton et al. (2004) to obtain the ratio 
; however, in specific cases where observations are available from the Van Allen Probes and THEMIS spacecraft in the following section, we use measured densities rather than those inferred from the model. Note that the statistical spread in 
will only result in a variation factor of less than one for the local diffusion rate (Equation 7).
Figure 1 displays the comparison of bounce-averaged diffusion coefficients calculated from realistic and identical electron hole distributions in the dipole field geometry at 
. For the identical electron hole case (Figure 1b), bounce-averaged electron pitch-angle diffusion rates are quite significant at energies below 
10 keV, of the order of 
to 
. This efficiency of scattering is consistent with previous calculations (Shen et al., 2020; Vasko, Agapitov, Mozer, Artemyev, Krasnoselskikh, & Bonnell, 2017; Vasko et al., 2018). Comparing the differences in scattering rates between realistic and identical electron holes (Figures 1c), realistic electron diffusion rates are stronger by more than 100% at energies above a few keV and pitch angles less than 
, as well as at energies below approximately 100 eV and pitch angles greater than 
. This intensification of diffusion rates can be partly explained by the expansion of the cyclotron resonance region due to the spread in electron hole velocities and parallel spatial scales. Conserving constituent wave power of electron holes, the broadened resonance region will produce enhanced diffusion rates at both higher and lower resonant energies. As a result, the scattering region effectively expands to lower and higher energy regions, which can be identified by comparing Figures 1a and 1b. In addition, the weighting factor of 
shown in Equation 11 indicates that wider 
will have relatively larger 
for realistic electron hole broadband fluctuations. Because the cyclotron resonant energy is of the order of 
, higher resonant energies will have larger contributions to the local diffusion rates than those in the identical case.
Comparison of bounce-averaged diffusion coefficients 
obtained from realistic (Panel a) and identical (Panel b) electron holes at 
. Realistic electron holes have a distribution of velocities 
and parallel scales 
specified according to observations from Malaspina et al. (2018) (which is demonstrated in Supporting Information S1), whereas idealized (or identical) electron holes have constant velocities 
= 3,000 km/s and parallel scales 
= 0.4 km. The root-mean-square intensity of electron hole 
in the magnetosphere is assumed to be 1.5 mV/m. Electron holes are uniformly distributed within a latitudinal extent of 
. Panel c shows the percentage difference in 
, calculated using (a and b)/a
100%.
In contrast, at energies between 100 and 5 keV, realistic electron holes produce considerably weaker diffusion rates, where 
mostly declines to slightly above 
. This is expected to some extent because the spread of electron hole parameters also reduces diffusion rates at the resonant energies and pitch angles dictated by identical electron hole parameters. Such a reduction can be clearly identified in Figure 1c. More importantly, this substantial variation in diffusion rate strongly suggests that the distribution of the microscopic electron hole parameters of 
and 
has a significant influence on electron scattering rates and therefore should be determined from individual events of broadband electrostatic fluctuations to accurately evaluate its impact on electron losses.
3 Case-Specific Lifetimes Associated With Injections
(13)This lifetime expression remains valid even when diffusion curves have wide, deep minima at intermediate pitch angles, in which case the widely used simple estimate of lifetime by 
(
is the loss cone diffusion coefficient) is inappropriate (Albert & Shprits, 2009; Shprits et al., 2006). We will also compare calculated electron lifetimes with the lifetime associated with the strong diffusion limit, which is defined as 
(Kennel, 1969), where 
is the equatorial loss cone and 
is the electron bounce period.
Because broadband electrostatic fluctuations and electron holes are closely associated with injections in the magnetosphere (Malaspina, Wygant, et al., 2015), we calculate electron lifetimes driven by electron holes based on direct observations of two injection events from the Van Allen Probes (VAP) in the inner magnetosphere (
= 6) and from THEMIS in the magnetotail (
), respectively.
3.1 Instrumentation
We use VAP measurements from the following instruments: (a) the Helium, Oxygen, Proton, and Electron (HOPE) Mass Spectrometer providing pitch-angle-resolved electron measurements in the energy range of a few eV to 50 keV (Funsten et al., 2013); (b) the Electric and Magnetic Field Instrument Suite and Integrated Science (EMFISIS) instrument including observations of electric and magnetic field spectra data in the frequency range of 10 Hz–12 kHz and intermittent short-burst (0.5 s or 
s) AC-coupled waveform data in 35,000 samples per second (sps), in addition to magnetic field observations at 64 sps from a fluxgate magnetometer (FGM) (Kletzing et al., 2013); and (c) the Electric Fields and Waves (EFW) instruments which provide additional burst-mode waveform recordings at 16,384 sps (Wygant et al., 2013). Note that electron density data can usually be inferred by tracking wave emissions near the upper-hybrid line in the spectrograms measured by EMFISIS (Kurth et al., 2015). We estimate electron densities mainly from spacecraft potentials measured by EFW (Wygant et al., 2013) from the event reported in this paper. When spacecraft potentials are not suitable for directly inferring electron densities, however, we interpolate densities from neighboring regions where such estimates are legitimate.
We use THEMIS routine data products from these instruments: the Electrostatic Analyzers (ESA), which measure pitch-angle-resolved electrons and ions below 25 keV (McFadden et al., 2008); the Fluxgate Magnetometer (FGM), which measures DC vector magnetic fields at 0.25 sps in fast mode (Auster et al., 2008); the Search Coil Magnetometer (SCM) (Roux et al., 2008), the Electric Field Instrument (EFI) (Bonnell et al., 2008), and the Digital Fields Board (Cully et al., 2008), together provide electric and magnetic field power spectra with frequencies up to 4 kHz and short-burst (a few to tens of seconds) waveform data at 8,192 sps (or 16,384 sps for AC-coupled electric fields). For THEMIS, electron densities are inferred from spacecraft potentials (Bonnell et al., 2008).
3.2 Case 1 in the Inner Magnetosphere
Malaspina et al. (2018) have reported VAP observations of a large number of various nonlinear wave structures, including electron holes, in broadband electrostatic fluctuations associated with a strong injection event in the inner magnetosphere at 
on December 21, 2016. Using continuous recordings of waveform electric fields from the EFW instrument at 16,384 samples per second (sps) for almost half an hour, they provided an unbiased data set of electron hole statistics. Here we examine further the characteristics of broadband electrostatic fluctuations (or electron holes) in this event and use the relevant hole parameters to obtain realistic electron lifetimes controlled by electron holes associated with injections in the inner magnetosphere.
Figure 2 presents a summary of this injection event observed by Van Allen Probe B at 
6 in the premidnight sector. The magnetic field dipolarization and electron injection (Figures 2a and 2b) are associated with various linear and nonlinear wave fluctuations (Figures 2c and 2d), including electromagnetic and electrostatic broadband fluctuations, lower- and upper-band chorus waves, and electrostatic electron cyclotron harmonic (ECH) waves. We will focus on broadband electric fields, consisting of dispersive or kinetic Alfvén waves (KAWs) in frequencies roughly below 100 Hz (Chaston et al., 2015) and nonlinear solitary structures comprising mostly electron holes. To ensure that we use an appropriate root-mean-square intensity 
in the following calculation of electron lifetimes, we need to separate wave power of electron holes from those of KAWs and sometimes whistler-mode waves. To properly isolate wave power, we investigate in detail several collections of waveform electric fields involving broadband electrostatic fluctuations at near 20:50:17 UT. As will be shown later through wavelet analyses, electron hole wave power can be well captured within the frequency range of 
0.1–3 kHz in this event while KAWs wave power concentrates at below 100 Hz. In general, the frequency band of electron holes depend on the duration of individual holes in the time domain measurements. Wave power of longer holes will peak at lower frequencies whereas shorter holes maximize at higher frequencies. Figure 2e presents the resulting root-mean-square intensities 
(one sample per 6 s, with a duty cycle of 0.5 s) of electron hole electric fields in the frequency ranges of 0.1–3 kHz (black) and 0.05–3 kHz (red). We have eliminated occasional wave power contributions of whistler waves to the broadband signal by excluding waves which have high ellipticity (
), high degree of polarization (
), and high degree of planarity (
) in the frequency band of 
to 3 kHz. Detailed wave properties analyzed for this event are provided in Supporting Information S1. The root-mean-square intensities 
during intense intervals are observed mostly in the range of 2–8 mV/m. Note that intense electron hole electric fields may be slightly underestimated in Figure 2e because electron holes may also have finite power residing within the whistler wave frequencies which we have excluded.
Figure 3 demonstrates the burst waveform measurements that we use to identify the proper bandwidth of electron holes during this event. Figures 3a and 3b show a 0.1-s observation of waveform magnetic and electric fields in field-aligned coordinates measured at 16,384 sps after filtered with a passband of 5–3,000 Hz. Note that we use only two components of electric fields (
, 
) in the spin plane to obtain the electric fields in field-aligned coordinates and have therefore excluded one component measured along the spin axis (
) due to potential uncertainties associated with it (Wygant et al., 2013). The spin axis is roughly 
away from the background magnetic field direction, so by ignoring 
perpendicular electric fields will be underestimated. We can identify a number of magnetized electron holes appearing as solitary bipolar parallel electric fields (
) of up to 20 mV/m in tandem with roughly unipolar parallel magnetic field (
) fluctuations of approximately 0.01 nT on time scales of a millisecond. These features have also been reported in Malaspina et al. (2018) (Figure 5, panel D). The electron hole signatures can be distinguished from the low-frequency electromagnetic (potentially Alfvénic) oscillations (Figure 3c) using a bandpass filter of 0.1–3 kHz, the result of which is shown in Figure 3d.
Injection and broadband electrostatic fluctuations observed by the Van Allen Probe B at 
. This event has been previously reported by Malaspina et al. (2018). (a) Background magnetic field showing 
dipolarizations or magnetic compression. (b) Omni-directional electron energy fluxes with energies from 15 to 50 keV showing an electron injection. (c and d) Wave magnetic and electric field power spectra in the 10 Hz up to 12 kHz frequency range. The features of broadband fluctuations, lower- and upper-band chorus, and electrostatic electron cyclotron harmonic (ECH) waves are all present in this event. The red and black traces indicate half (1.58 kHz) and one tenth the electron gyrofrequency 
along the spacecraft trajectory. (e) Integrated broadband electric field root-mean-square intensity (
) integrated over the frequency ranges of 0.1–3 kHz (black) and 0.05–3 kHz (red), which capture most broad fluctuations due to electron holes in the time period of interest. The whistler-mode wave electric field intensity has been excluded from the broadband fluctuations by examining wave polarization, the analysis of which is demonstrated in Supporting Information S1. The wave burst interval we show in this event took place near 20:50:17 UT as indicated in the gray-shaded region. (f) Ratios of 
.
Zoomed-in view of the burst waveform measurements indicated in Figure 2, demonstrating features of solitary bipolar electric fields and unipolar magnetic fields associated with electron phase space holes within the broadband fluctuations observed near 20:50:17 UT. Only two-component spin-plane electric fields are used to obtained field variations in the field-aligned coordinates. (a) Magnetic field perturbations at 16,384 sps in the B-field-aligned coordinates, which have been bandpass-filtered to be within 5–3,000 Hz. (b) Electric field fluctuations at 16,384 sps bandpass-filtered to be in the bandwidth of 5–3,000 Hz. (c) Electric field fluctuations bandpass-filtered to be within 5–100 Hz, showing mostly Alfvénic fluctuations. (d) Electric field fluctuations bandpass-filtered to be within 0.1–3 kHz, highlighting solitary bipolar parallel electric fields (
) in electron holes. (e) Ratios of the parallel and perpendicular components of electron hole electric fields. The ratios are calculated only at locations where 
4 mV/m. (f) Wavelet spectrogram of 
for an interval of 1 s near 20:50:17 UT. The white dashed demarcation line of 100 Hz distinguishes localized solitary structures of electron holes from Alfvénic wave power in lower frequencies. (g) Wavelet spectrogram of 
. (h) Zoomed-in view of electron hole electric fields as indicated by the gray-shaded interval in Panel (d).
Figure 2f displays the ratio 
, which is near 4.2 during the burst of interest, suggesting that 
is smaller than 4.2 based on the expected scaling relation in Equation 8 (Franz et al., 2000). This is roughly consistent with the measured ratios 
in Figure 3e which are mostly less than 3. The underestimation can be partially attributed to the fact that the ratio 
is measured locally by the spacecraft at some distance away from the center of the electron hole, where the ratio reaches a maximum (Hutchinson, 2021). Another plausible explanation is that electron densities inferred at this time by interpolation from neighboring regions may be overestimated within a factor of 2.
Figures 3f and 3g present wavelet spectrograms of 
and 
measured during a 1-s interval (20:50:16.7 UT to 20:50:17.7 UT). A notable cluster of intense, localized, broadband emissions can be clearly identified in the frequency range of 100 Hz up to approximately 4 kHz. These features can be easily distinguished from electromagnetic fluctuations at lower frequencies. Such a bandwidth provides an important reference for extracting proper wave power of electron holes from broadband fluctuations. Figure 3h demonstrates one of the electron hole spikes after we isolate wave power in this frequency band. It is worth noting that electron hole wave power will spread over a larger range of frequencies in Fourier spectra than in wavelet spectra, which show better localization features of electron holes. For this reason and to account for potential lower-frequency power of electron holes in Fourier spectra that we use (Figure 2d), we also display in Figure 2e the root-mean-square 
in the frequencies from 0.05 to 3 kHz, again excluding the contribution from whistler waves. Comparing Figures 2c and 2d, we observe that evident electrostatic wave power is presented at low-frequencies down to approximately 50 Hz.
The aforementioned root-mean-square electric field intensity 
and the ratio 
are applied to our case-specific calculations of electron lifetimes (
) driven by electron hole scattering during this injection event. Figure 4 displays the corresponding electron lifetimes using three different electric field intensities, 
= 2, 4, and 8 mV/m, according to the observed values in Figure 2e. The statistics of electron hole distributions collected by Malaspina et al. (2018) in this event are directly used in the calculation. With typical weak electron hole intensities, 
= 2 mV/m, electron lifetimes are larger than 4.8 h at energies above 100 eV. Electron losses on such time scales are generally larger than the minimum lifetime (
one hour) associated with strong diffusion limits (the magenta dash-dotted line). When 
is elevated to 4 or 8 mV/m (the blue and red curves), however, electrons with energies less than approximately 10 keV are within the strong diffusion limit. The electron lifetimes drop to about 15 min to one hour, so appreciable diffuse auroral electron losses are expected to take place during these intense intervals. Given that these relatively powerful (
4 mV/m) electron holes are probably not uncommon in injection events, such as in the event shown in Figure 2, electron holes may effectively scatter plasma sheet electrons below a few keV into the loss cone and generate electron precipitation from the inner magnetosphere. Questions remain, however, as to whether electron hole distributions observed in this event are representative of injection events in the inner magnetosphere and whether electron hole root-mean-square intensities are generally strong enough to induce significant diffuse auroral precipitation from the inner magnetosphere globally.
Electron lifetimes (
) caused by electron hole scattering at 
using the observed electron hole root-mean-square intensities of 
(black line), 4 (blue line), and 8 mV/m (red line), along with background 
nT and 
at the equator as measured from the Van Allen Probe. Electron holes are within the latitudes of 
with hole velocities 
(1,000, 11,000 km/s) and 
(0.1, 3.0 km) according to the statistical observations of electron holes in (Malaspina et al., 2018). The detailed distribution is provided in Supporting Information S1. Lifetimes associated with the strong diffusion limit are shown by the magenta dash-dotted line.
3.3 Case 2 in the Plasma Sheet
We next consider realistic electron lifetimes driven by electron holes during injections in the outer magnetosphere. Figure 5 presents an overview of an injection event accompanied by strong broadband fluctuations in the premidnight magnetotail at 
. Starting from 20:52:30 UT, the background magnetic field develops a feature of dipolarization, increased 
(Figure 5a). This field topology change is associated with increased electron energy fluxes with energies larger than 1 keV (Figure 5b), primarily due to adiabatic acceleration (Birn et al., 2014; Runov et al., 2013). Broadband fluctuations and whistler-mode waves can be also identified in Figures 5c and 5d. At the leading edge of the dipolarization front, we observe compressional magnetic field oscillations with a period of approximately 20 s, mainly in the 
component and less in 
and 
components; these oscillations correlate with electron flux modulations and also with the most intense broadband electrostatic fluctuations (Figures 5d and 5e). These compressional fields are probably slow magnetosonic waves as reported in several previous studies (Cao et al., 2013; Zhou et al., 2014).
Our focus here is mainly on the broadband fluctuations, comprising Doppler-shifted kinetic Alfvén waves (KAWs) (Chaston et al., 2014) and solitary structures of electron holes. To distinguish electron holes from KAWs, we have examined in detail wavelet spectra of electron holes and KAW (as will be presented in the following) and compared the measured 
spectra to the KAW dispersion relation (not shown here)(Stasiewicz et al., 2000) using burst waveform data. We find that Doppler-shifted KAWs due to ion flows across the spacecraft can only explain the measured spectra up to 30 Hz and that a bandpass filter of 
0.1–4 kHz can sufficiently isolate spectral intensities of electron holes from those of kinetic Alfvén waves within the broadband spectra. In addition, because we do not have detailed wave polarization information in survey mode of THEMIS data, we exclude the whistler wave power contribution from the respective broadband spectra by identifying and eliminating frequencies which have relatively strong (
/Hz) magnetic fluctuations in Figure 5c. Figure 5e shows that the resulting root-mean-square intensity 
(the black line, one sample per second) of the broadband fluctuations ranges from near 1 to more than 4 mV/m during this interval. Similar to the case in the inner magnetosphere, we also show 
evaluated in the frequencies of 0.05–4 kHz (excluding whistler waves) to indicate the potential upper limit of electron hole wave power during this event. In Figure 5f, the ratio 
is close to 4.7 at this time, indicative of oblate electron holes with a ratio 
close to 0.2.
Figure 6 displays burst-mode waveform electric and magnetic fields from 20:52:33.5 UT to 20:52:33.4 UT to highlight the fine structures of electron holes. Figures 6c and 6d show electric field variations of kinetic Alfvén waves in the frequency range of 5–100 Hz and electron holes in the frequency range of 0.1–4 kHz. The latter frequencies encompass most wave power from electric field spikes of electron holes while suppressing most Aflvénic fluctuations as shown in Figures 6c and 6d. Figure 6e examines the ratio 
, the variations of which are less than 5, approximately consistent with the anticipated proportionality with 
(Franz et al., 2000). Wavelet-transformed spectra of 
and 
in Figures 6f and 6g demonstrate the distinct features of solitary structures and broadband emissions of electron holes at above 100 Hz, which are distinguishable from the intense fluctuations of KAWs at below 100 Hz. An enlarged view of one of the electron holes is shown in Figure 6h. The identified average root-mean-square intensity of electron holes will be an input in the following calculations of electron lifetimes. Note that electron scattering by KAWs is thought to work at energies much larger than the energies we are considering here relevant to diffuse auroral precipitation (Chaston et al., 2018).
Injection and broadband electrostatic fluctuations observed by THEMIS at 
. (a) Background magnetic field showing 
dipolarizations and compressional 
fluctuations (probably slow magnetosonic waves). (b) Field-aligned electron energy fluxes with energies from 100 eV up to 16 keV. (c and d) Wave magnetic and electric field power spectra in the frequency range of 10–4,000 Hz. The red and black traces indicate half and one tenth the electron gyrofrequency 
along the spacecraft trajectory. (e) Integrated wave electric field root-mean-square intensity (
) at 1 sample per second (sps) over the frequency ranges of 0.1–4 kHz (black) and 0.05–4 kHz (red), which encompass most broad wave power caused by electron holes in this event. By identifying relatively strong (
/Hz) magnetic fluctuations in (c), the frequency band of whistler waves has been mostly excluded from wave power integration in the broadband fluctuations. (f) Ratios of 
assumed to be dictating the ratio 
of electron holes. The wave burst interval near 20:52:33 UT is shown in the gray-shaded region.
Zoomed-in view of the burst waveform measurements indicated in Figure 5, showing features of kinetic Alfvén waves (KAWs) and electron holes within the broadband fluctuations observed by THEMIS at 
. (a) Magnetic field perturbations at 8,192 sps in B-field-aligned coordinates and bandpass-filtered to be within 5–4,096 Hz. (b) Electric field fluctuations at 16,384 sps in the bandwidth of 5–4,096 Hz. (c) Electric field fluctuations bandpass-filtered to be within 5–100 Hz, showing mostly Alfvénic fluctuations. (d) Electric field fluctuations bandpass-filtered to be within 0.1–4 kHz, highlighting solitary bipolar 
fluctuations of electron holes. (e) Ratios of the parallel and perpendicular components of electron hole electric fields. The ratios are calculated only at the locations where 
4 mV/m and 
1 mV/m. (f) Wavelet spectrogram of 
. The white dashed demarcation line of 100 Hz divides electron holes and KAWs in frequency. (g) Wavelet spectrogram of 
. By comparing the measured 
spectrum with that predicted by the KAW dispersion relation (not shown here), we find that the broadband low-frequency spectra can be explained by Doppler-shifted KAWs due to ion flows only up to 30 Hz. (h) Zoomed-in view of electron hole electric fields within one of the spikes indicated by the gray-shaded interval in Panel (d).
The observed magnetic field, electron hole root-mean-square electric field intensity, and plasma parameters are then used to constrain our calculations of electron lifetimes 
as a result of scattering by electron holes. For this injection event at 
, we use electron hole velocity and parallel scale distributions obtained from the MMS mission at 
and from the THEMIS mission at 
(Andersson et al., 2009; Lotekar et al., 2020; J. B. Tao, Ergun, et al., 2011). Along with the normalized 2D distributions (
) that we use in lifetime calculations, these electron hole statistics are displayed in Supporting Information S1. Because these observations of electron holes are collected not exactly at the same region of our current injection event, we rescale 
and 
according to their general statistical scaling with 
and 
as have been shown in Lotekar et al. (2020). We did not readjust relativistic electron hole velocities measured by Andersson et al. (2009).
Figure 7 illustrates the calculated electron lifetimes as a function of energy using 
= 1 (black) and 4 mV/m (red), respectively. Comparing non-relativistic (solid lines) and relativistic (dashed lines) electron hole scattering, we observe commensurate electron lifetimes from a few hours to just several minutes at energies less than 10 keV. Small differences in lifetimes also exist; relativistic electron holes seem to exert a stronger scattering effect on thermal (
1 keV) electrons and suprathermal (
tens of keV) electrons, but the scattering rates of relativistic electron holes fall below those of non-relativistic electron holes for diffuse auroral (several-keV) electrons. But comparing electron lifetimes calculated at 
and 
as shown in Figures 4 and 7, lifetimes are notably smaller in the magnetotail, by nearly one order of magnitude, than those derived in the inner magnetosphere, given the same electron hole root-mean-square intensity. When the average 
goes up to 4 mV/m, as has been observed here, electrons with energies roughly less than 5 keV can get lost within less than 15 min. Such an efficiency of scattering is remarkable and sufficient to remove freshly injected or convecting low-energy electrons from the magnetotail during substorms. As a result, electron scattering by electron holes from the magnetotail can sometimes provide a significant, if not dominant, supply of diffusive electron precipitation into the ionosphere. In general, efficient scattering (
less than 1 h) by electrostatic fluctuations can be expected only at energies below a few keV.
Electron lifetimes (
) due to electron hole scattering at 
with 
(black lines) and 4 mV/m (red lines) with the background parameters of 
nT and 
from THEMIS observations near the equator. Electron holes are within the latitudes of 
with distributions set to 
(2,000, 14,000 km/s) and 
(0.6, 6.4 km) based on MMS observations from Lotekar et al. (2020) as shown by the solid lines, and set to 
km, 1.4
km/s
and 
(3.5, 18 km) based on THEMIS observations from Andersson et al. (2009) and J. B. Tao, Ergun, et al. (2011), as shown by the dashed lines. Note that velocities from Lotekar et al. (2020) are adapted to the THEMIS position in our event based on physical scaling with 
2,100 km/s. Similarly, electron hole parallel scales 
are rescaled using 
0.6 km measured at the THEMIS position. The distributions applied to the lifetime calculation are demonstrated in detail in Supporting Information S1.
4 Summary and Discussion
Recent theoretical work by Vasko, Agapitov, Mozer, Artemyev, Krasnoselskikh, & Bonnell, 2017 and Vasko et al. (2018) as well as observations from Shen et al. (2020) have demonstrated that electron phase space holes have a significant potential to pitch-angle scatter low-energy (less than a few keV) electrons into the loss cone, thereby contributing to diffuse auroral precipitation. Previously diffuse auroral precipitation was thought to be driven only by electron cyclotron harmonic (ECH) waves and chorus waves in the equatorial magnetosphere (Ni et al., 2016; Thorne et al., 2010). As discussed in the Introduction, electron holes share similar features with chorus waves and ECH waves in driving diffuse auroral precipitation (Ergun et al., 2015; Malaspina, Wygant, et al., 2015; Vasko, Agapitov, Mozer, Artemyev, Krasnoselskikh, & Bonnell, 2017). The similarities include the statistical distribution of broadband electrostatic fluctuations in magnetic local time (MLT) in the inner magnetosphere (Malaspina, Wygant, et al., 2015), their association with and intensification by plasma sheet electron injections (Malaspina, Wygant, et al., 2015), the pattern of scattering into the loss cone for low pitch-angle electrons at energies less than a few keV (Vasko, Agapitov, Mozer, Artemyev, Krasnoselskikh, & Bonnell, 2017), and the occasionally reported correlation with diffuse auroral precipitation (Shen et al., 2020). Although global statistical observations of broadband electrostatic fluctuations (or electron holes) and conjugate diffuse auroral precipitation are necessary to accurately quantify the contribution of electron holes to aurora precipitation, this paper amounts to a steppingstone for achieving such a goal by accurately evaluating the electron diffusion rates and lifetimes driven by realistic electron holes.
We have presented the methodology of estimating electron scattering rates driven by realistic electron hole distributions. One important result is that to compute the diffusion coefficients we need only to know the root-mean-square intensity of electron holes, which can be determined using spectral measurements, along with the statistical distribution of velocities and spatial scales of electron holes. Individual amplitudes of electron holes are not needed for this evaluation. We have incorporated the observed statistical distributions of the electron hole velocities and spatial scales, collected from both the inner magnetosphere and the magnetotail (Andersson et al., 2009; Lotekar et al., 2020; Malaspina, Wygant, et al., 2015; J. B. Tao, Ergun, et al., 2011), into a newly developed methodology of computing electron diffusion rates caused by electron hole scattering.
As a result of using realistic distributions of electron holes rather than assuming identical holes, we have obtained drastically different diffusion rates in the inner magnetosphere (
6) from those obtained previously. Although the bounce-averaged diffusion rates are mostly augmented (
100%) at energies above 10 keV and below 100 eV, the efficiency of scattering diminishes by almost an order of magnitude, from roughly 
to lower than 
, for electrons with energies of 
100 eV to 
10 keV, an energy range crucially important for diffuse auroral precipitation. Such a substantial modification implies that the microscopic electron hole parameters of 
and 
critically affect electron scattering rates and therefore need to be determined from individual broadband electrostatic fluctuation events to accurately evaluate their impact on electron losses and diffuse auroral precipitation.
Our preliminary investigations of electron lifetimes driven by electron holes show that realistic electron hole distributions with typical root-mean-square (RMS) intensities, 
= 2 mV/m, does not induce electron losses as efficiently as previously expected in the nightside inner magnetosphere (
6). However, as the radial distance 
increases (i.e., the magnetic field strength decreases) or as 
increases, electron lifetimes drop to be less than one hour and are mostly within the strong diffusion limit at energies below 
10 keV. Applying our realistic electron lifetime calculations directly to two injection events observed by the Van Allen Probes at 
and the THEMIS spacecraft at 
, we find that intense electron holes (
4 mV/m) are not uncommon and may play a role in diffuse auroral precipitation from the inner magnetosphere. In the magnetotail, however, based on estimates of electron lifetime, electron holes with typical intensities (
1 mV/m) can efficiently deplete low-energy (
a few keV) plasma sheet electrons within tens of minutes following injections and convection from the magnetotail.
The slight preference of strong scattering to the outer magnetospheric regions by electron holes is in parallel with ECH waves, which have been suggested to play a dominant role in generating diffuse auroral precipitation beyond 
(Liang et al., 2011; Ni et al., 2011, 2016; X.-J. Zhang et al., 2015). Our results suggest that electron holes may be as competitive as ECH waves in producing electron losses from the magnetotail, especially considering that electron holes effectively scatter electrons at an energy (a few eV up to approximately 10 keV) and pitch angle range (up to 
) that is, much broader than ECH waves (scattering electrons at energies between 
100 eV to 
5 keV and pitch angles of up to 
). Therefore, electron holes tend to be more effective in scattering a larger fraction of plasma sheet electrons from the outer magnetosphere. In addition, electron holes are probably active in controlling electron losses from the magnetotail transients, such as plasma injections and bursty bulk flows (Angelopoulos et al., 1992, 2013; Gabrielse et al., 2014), where intense broadband electrostatic fluctuations are frequently observed (Ergun et al., 2015; Malaspina, Claudepierre, et al., 2015). A relevant scenario in which this electron precipitation may be taking place is degeneration of equatorward-moving auroral streamers into structured diffuse auroras at the poleward boundary of the diffuse auroral regions (Henderson, 2012; Henderson et al., 1998; Nishimura, Lyons, et al., 2010; Sergeev et al., 1999; Wendel et al., 2019; Zesta et al., 2000). To systematically evaluate the role of broadband electrostatic fluctuations and electron holes in driving electron precipitation from the magnetosphere and in controlling dynamic magnetosphere-ionosphere coupling, a full investigation of electron hole distributions in velocity, parallel scale, and root-mean-square intensity is needed.
Acknowledgments
We acknowledge NASA contract NAS5-02099 for use of data from the THEMIS mission. Y. Shen and A. Artemyev acknowledge the support by NSF grant No. 1902699. The work of I. Vasko, A. Artemyev, X. Chu and D. Malaspina was supported by NASA Heliophysics Supporting Research grant No. 80NSSC20K1325. I. Vasko also thanks NASA for support through grant No. 80NSSC19K1063 and National Science Foundation for support through grant No. 2026680. We are greatly indebted to H. Aryan for providing the numerical chorus-driven electron lifetime results from Aryan et al. (2020). We appreciate the Van Allen Probes team for the use of fields and electron data. We thank K. H. Glassmeier, U. Auster, and W. Baumjohann for the use of FGM data provided under the lead of the Technical University of Braunschweig and with financial support through the German Ministry for Economy and Technology and the German Aerospace Center (DLR) under contract 50 OC 0302.
Open Research
Data Availability Statement
The Van Allen Probe HOPE data used in this study were accessed via https://rbsp-ect.lanl.gov/data_gpub/. The EFW and EMFISIS data are accessible from http://www.space.umn.edu/rbspefw-data/and https://emfisis.physics.uiowa.edu/data/index. Data for the THEMIS satellites were accessed from http://themis.ssl.berkeley.edu/. Data analysis was done using SPEDAS V4.1, see Angelopoulos et al. (2019).
