Superposed Epoch Analyses of Electron-Driven and Proton-Driven Magnetic Dips
Abstract
In this study, we statistically investigate the features of magnetic dips by constructing superposed epoch analysis on Van Allen Probe data. Based on the values of electron and proton plasma betas, we categorize dips into two types: electron-dominant and proton-dominant. The global distributions of dips are obtained. Superposed epoch analysis on two types reveals a correlation between the magnetic fluctuations and plasma betas. Moreover, the occurrences of butterfly pitch angle distributions of relativistic electrons driven by the magnetic dips are confirmed on a statistical basis. Our results reveal the statistical characteristics of magnetic dips and establish the relationship among the magnetic fluctuations and background plasma parameters, indicating the potentially important role of magnetic dips in the inner magnetosphere dynamics.
Key Points
-
Two types of magnetic dips are statistically investigated based on the Van Allen Probes observation
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Superposed epoch analysis is adopted to analyze the relation between the electromagnetic fields of magnetic dips and plasma betas
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The relationship among the depth of magnetic dips and several background parameters is produced
Plain Language Summary
Magnetic dips are the minima of magnetic field intensities with several-minutes duration in the inner magnetosphere. Previous simulation and observation studies analyzed the proton-driven or electron-driven magnetic dips case by case. In this study, comprehensive statistics on the two types of magnetic dips are produced for the first time. The global spatial distributions of magnetic dips and their dependences on a geomagnetic index are obtained. Superposed epoch analysis on two types reveals a correlation between the magnetic fluctuations of magnetic dips and plasma betas. The formation of butterfly pitch angle distributions of relativistic electrons driven by the magnetic dips is also confirmed by the superposed epoch analysis. This study comprehensively investigates the generation of magnetic dips and their effect on relativistic electrons and indicates an important role of magnetic dips in the ring-current-radiation-belt-coupling.
1 Introduction
A local minimum of magnetic field strength in the inner magnetosphere (named as localized magnetic dip), may play an important role in ring-current-radiation-belt coupling (Ebihara et al., 2008; Xiong et al., 2017). Magnetic dips are suggested to be driven by the ring currents' pressure during ion injections. He et al., 2017 used multiple-Geostationary-satellites constellation to find that the magnetic dip can drift azimuthally westward at a speed comparable to the drift speed of energetic ions that make up most of ring current pressures. Ukhorskiy et al. (2006) used an empirical magnetic field model (Tsyganenko & Sitnov, 2005) to illustrate a statistically significant magnetic depression over magnetic local time (MLT) ranging from 23 to 18 and 
ranging from 3 to 7 during storm times. Xia et al. (2017, 2019) built up a self-consistent equilibrium model to analyze the formation of magnetic dips in magnetospheric plasma conditions and analyzed the dependence of magnetic dip parameters on the plasma beta profiles. Recently, Zhu et al. (2020) reported a magnetic dip event driven by suprathermal electrons (as opposed to protons) and found that its spatial scale is smaller than the scale of magnetic dip driven by the protons. However, a full investigation of the magnetic dips associated with electrons is still required.
Magnetic dips are suggested to be capable of producing the butterfly pitch-angle distribution of relativistic electrons (Ebihara et al., 2008; Xiong et al., 2017). These are characterized by a minimum phase space density (PSD) near 
pitch angle, in contrast to a normal distribution with a PSD peak at 
. Several mechanisms, for example, drift-shell splitting, wave-particle interaction with magnetosonic waves, and field line curvature scattering, have been proposed to account for the formation of butterfly distribution in different regions (Albert et al., 2016; Artemyev et al., 2015; Chen et al., 2015; Horne et al., 2007; Maldonado et al., 2016; West et al., 1973). In the magnetic dips, the relativistic electrons move toward the Earth in gradient drift, and the force of the gradient drift is proportional to the first adiabatic invariant of the electron. Therefore, at the same energy, the electrons at higher pitch angles are more subject to the influence of magnetic dips than those at lower pitch angles. Such pitch-angle-dependence produces the butterfly distribution in the presence of a negative radial gradient of electron fluxes (Xiong et al., 2017, 2019). In addition, a test-particle simulation (Yin et al., 2021) suggests that magnetic dip can also change the behaviors of ions to maintain the dispersionless structure of ion injections on the front side and produce the dispersive reduction on the trailing side. In this study, we will statistically collect magnetic dips from the Van Allen Probes observations in 2013–2018 (Mauk et al., 2013). We categorize the magnetic dips into two types: electron-dominant and proton-dominant magnetic dips and use superposed epoch analysis to systematically study the generation of two types of magnetic dips and their effects on relativistic electrons.
2 Instrumentation
(1)
is the mass of particles, 
is perpendicular velocity, 
is pitch angle, and 
is velocity distribution function of particles, provided by the Helium, Oxygen, Proton, and Electron (HOPE) Mass Spectrometer measurement (Funsten et al., 2013). The lower and upper limits of the integrals correspond to the energies of 0.1 and 51.8 keV, respectively. Then, we obtain the corresponding perpendicular beta values of electrons 
and protons 
, respectively, where 
and 
are the electron and proton perpendicular plasma pressures; 
is magnetic pressure; 
is ambient magnetic field strength and 
is the vacuum permeability. In this study, we categorize all dip events into two types: (I) proton-dominant dips with 
; (II) electron-dominant dips with 
, where 
and 
are the maximum values of the electron and proton betas within the 10-min interval at the center of magnetic dips, respectively. We find 597 type I events and 35 type II events, with the remaining 9 of 641 events that have no HOPE data available. Figure 1 shows examples of type I and II magnetic dip events observed by the Van Allen Probe B on October 10, 2013 and May 8, 2018, respectively. The event on October 10, 2013 was observed in the afternoon sector and the event on May 8, 2018 was observed in the pre-dawn sector. Figures 1a and 1e represent the ambient magnetic strengths and detrended magnetic field strengths. The minima of magnetic strengths are clearly seen, which are indicated by the vertical dotted lines. The magnetic strength in the first event has a sharper front than the second event. The electron and proton betas are shown in Figures 1b and 1f. In the first event, the proton betas are much larger than the electron betas but in the second event, the electron betas are larger than the proton betas near the center of the magnetic dip. The enhancement of plasma betas and pitch angle distributions of proton at energy of 24.1 keV (shown in Figure 1c) and electrons at energy of 8.26 keV (shown in Figure 1g) are clearly seen, respectively. The butterfly distributions of relativistic electrons at the energy of 1.8 MeV in response to the magnetic dips are shown in Figures 1d and 1h, which have been reported by the previous studies (He et al., 2017; Xiong et al., 2017; Zhu et al., 2020).
Two examples of magnetic dip event observed by the Van Allen Probe B on (a–d) October 10, 2013 and (e–h) May 8, 2018: (a and e) ambient magnetic intensity (black line) and detrended magnetic intensity (red line); (b) electron beta (black line) and proton beta (red line); (c) pitch angle distributions of protons at energy of 24.1 keV; (g) pitch angle distributions of electrons at energy of 8.26 keV; (d and h) pitch angle distributions of electrons at energy of 1.8 MeV. The vertical dotted lines indicate the centers of magnetic dip.
3 Observations and Analysis
Figure 2 shows the spatial distributions of magnetic dips on the equatorial plane and meridian plane of Solar Magnetic coordinate system. The blue and red circles represent the type I and II magnetic dips, respectively. It is found that most events fall in 
. The outermost boundary is determined by the orbits of the Van Allen Probe. The innermost boundary at 
supports that the formation of magnetic dips is driven by the injection of suprathermal charge particles from the magnetotail. Moreover, with the increase of 
, the ambient magnetic field strengths decrease and thus the plasma beta values tend to increase, which favors the formation of magnetic dips. In MLT, considering that the dwell times of the Van Allen Prob B are MLT-symmetrical (not shown here), the type I dips mainly occur in the duskside and nightside, whereas the type II magnetic dips are mainly observed in the nightside and pre-dawnside, consistent with the drift paths of injected protons and electrons (Birn et al., 1997). On the meridian plane shown in Figure 2b, most magnetic dips are located near the equatorial region with latitudes between 
and 
. This near-equatorial preference likely has to do with the minimum in magnetic field strength along a field line at the magnetic equator, thus skewing toward higher beta and favoring the generation of magnetic dips. Figure 2c shows the histogram of magnetic dip counts for four different 
ranges of 
100 nT, (100 nT, 300 nT), (300 nT, 500 nT), and 
500 nT, respectively, where 
is the maximum value of AE (auroral electrojet index) in the previous 3 hr. It is seen that the numbers of magnetic dips increase with the enhancement of 
.
Magnetic dip events projected on the (a) X-Y plane of Solar Magnetic coordinate system; (b) 
-Z plane; histogram of magnetic dip event numbers for different (c) 
values. The blue and red circles in panels (a and b) represent type I and II magnetic dips, respectively.
Figure 3 shows the results of the electric and magnetic field fluctuations and plasma betas for the type I and type II events, respectively, by means of superposed epoch analysis. The zero epoch time is taken as the time when the minima of the detrended magnetic field strengths are observed. The gray solid lines represent the individual events. The red lines are the mean values, the blue solid lines are the median values, and the blue dashed lines are lower and upper quartiles, respectively. Figures 3a and 3i represent the detrended magnetic field strengths as functions of 
. For both types of magnetic dips, the mean and median profiles have pronounced minima at 
. Though the detrended magnetic field of a given individual event has various fluctuations, the mean and median profiles exhibit smooth variation. At the centers of magnetic dips (
), the mean values of type I and II events are −20.2 and −15.0 nT, respectively, and the median values of type I and II events are −17.6 and −13.9 nT, respectively. Figures 3b–3d and 3j–3l show the fluctuations of magnetic field components in the magnetic field-aligned coordinate system (MFA). The field-aligned (or compressional) direction in the MFA coordinate system is determined by the 500-s running average of the magnetic field, the azimuthal direction is obtained by the cross product of the field-aligned vector and satellite position vector, and the radial direction completes the triad (Takahashi et al., 2015; Zhu et al., 2019). It is shown that for both type I and type II events, the fluctuations of magnetic fields are almost along the field-aligned direction. The fluctuations along the radial and azimuthal directions can be ignored compared to the fluctuation along the field-aligned direction and their mean and median values are close to zero even at the centers of magnetic dips. This statistical result is expected for a system exhibiting magnetohydrodynamics equilibrium between plasma pressure and magnetic pressure (Xia et al., 2017). Figures 3e, 3f, 3m and 3n present the radial (outward) and azimuthal (eastward) components of electric fields in the MFA coordinate system. The electric field data are measured by the Electric Field and Waves Suite instrument (Wygant et al., 2013) using only the spin plane components. The assumption of 
is adopted to obtain the electric field along the spin axis, allowing the full 3D field to be rotated into the MFA coordinate system. If there are any flagged electric field data during the time interval of any magnetic dip events, such events are precluded. In order to keep the accuracy of the electric field data in the MFA coordinate system, the 
assumption is used only when the spin axis component is more than 15° away from the magnetic field direction. Magnetic dip events that do not satisfy the above criteria, within the time interval of interest (from −30 to 30 min) are precluded as well. Thus, only 91 type I dip events and 2 type II dip events are finally valid for the electric field analysis and we mainly discuss the features of the electric fields of type I events. In the radial direction, there is a positive peak at the centers of magnetic dips (pointing away from the center of the Earth). These radial electric fields can be considered as the magnetospheric part of subauroral polarization streams (SAPS) electric fields near the equator region (Anderson et al., 1993; Nishimura et al., 2008; Wang et al., 2019). The azimuthal components of the electric fields show a positive (eastward) fluctuation, followed by a weak negative (westward) fluctuation. The max value (0.87 mV/m) of the mean azimuthal electric field is slightly smaller than (0.94 mV/m) that of the mean radial component. Note that the electric field features associated with magnetic dips are independent on the satellite orbit, for example, inbound or outbound. Figures 3g and 3o show the perpendicular plasma betas of electrons for type I and II dip events, respectively. It is shown that the mean and median profiles of electron plasma betas have the opposite trend compared to the magnetic strengths of magnetic dips shown in Figures 3a and 3i. Figures 3h and 3p represent the perpendicular plasma betas of protons for type I and II dip events. Similar to the plasma betas of electrons, the mean and median profiles of protons also show the opposite trend to the magnetic strengths. Note that the 
-axes of the electron and proton betas have different ranges. For the type I dip events, the mean and median values of proton betas are larger than those of electron betas at one order of magnitude.
Superposed epoch analysis of magnetic dip events for type I (a–h) and II (i–p) events: (a and i) detrended magnetic field strengths; (b–d, j–l) detrended radial, azimuthal, and compressional components of magnetic field in the magnetic field-aligned (MFA) coordinate system; (e, f, m, n) radial (outward) and azimuthal (eastward) components of electric field in the MFA coordinate system; (g and o) electron perpendicular betas; (h and p) proton perpendicular betas. The red and blue solid lines represent the mean and median profiles. The blue dashed lines indicate the lower and upper quartiles.
Figure 4 shows the superposed epoch results of the normalized unidirectional fluxes of relativistic electrons at local pitch angle of 
at three energies of 899 keV, 1.8 MeV, and 2.1 MeV for the type I and II dip events, respectively. The electron fluxes are normalized by the fluxes at the centers of magnetic dips. The fluxes at the energy of 899 keV are provided by the Magnetic Electron Ion Spectrometer instrument (Blake et al., 2013) and the fluxes at the energies of 1.8 and 2.1 MeV are provided by the Relativistic Electron-Proton Telescope instrument (Baker et al., 2012). It is shown that prior to the centers of magnetic dips (
), the electron fluxes gradually decrease and after the centers of magnetic dips, the electron fluxes gradually increase. For both the mean and median profiles of electron fluxes at different energies, there are significant minima at the centers of magnetic dips, which suggests the positive correlation between the occurrence of magnetic dips and the dropout of locally equatorial relativistic electron fluxes. Such positive correlation can be seen for both type I and II dip events, which is consistent with previous case studies and numerical simulations (Xiong et al., 2017; Zhu et al., 2020). There are also two noticeable differences between the electron fluxes for the two types of dip events. First, the mean, median, lower and upper quantiles for type I events are smoother than those for type II events since the number of type II events is fewer. Second, the net changes of the electron fluxes decrease after across the type I dips, whereas the net changes of the electron fluxes increase across the type II dip events.
Superposed epoch analysis of magnetic dip events for type I (a–e) and II (f–j) events: (a and f) detrended magnetic field strengths; (b–d, g–i) relativistic electron fluxes normalized by the fluxes at the center of magnetic dips at local pitch angle of 
and energies of 899 keV, 1.8 MeV, and 2.1 MeV; (e and j) the median profiles of normalized electron fluxes at three energies. The red and blue solid lines in panels (a–d, f–i) represent the mean and median profiles. The blue dashed lines in panels (a–d, f–i) indicate the lower and upper quartiles.
The relationship among the depth of magnetic dips and several parameters (including spatial location, geomagnetic indices, and plasma parameters) at the centers of magnetic dips are shown in Figure 5. The red lines indicate the mean values and the red error bars indicate the standard deviations. The black and blue circles represent the type I and type II magnetic dip events, respectively. Note that considering the variability of the spatial locations of magnetic dips, the depth of magnetic dips here is characterized by 
, where 
and 
are the detrended and ambient magnetic field strength at the centers of magnetic dips. Figures 5a–5c show the dependence of magnetic dip depth on magnetic latitude 
, 
-shell, and MLT. It is shown that the depths of magnetic dips maximize at the magnetic equator region and decrease with the increase of the absolute values of magnetic latitude. The depths show a weak dependence on the 
-shell, and also show a relatively weak dependence on MLT except in the MLT range from 5 to 10, consistent with the drift paths of charged particles (Birn et al., 1997). The dependence of magnetic dip depths on the geomagnetic index 
is plotted in Figure 5d. It is seen that the magnetic dip depths gradually increase with increasing 
. Figures 5e and 5f show the depths of type I magnetic dips versus proton betas at the centers of magnetic dips and the depths of type II dips versus electron betas at the centers of magnetic dips, respectively. Note that the x axes of Figures 5e and 5f are in logarithmic scales. It is shown that the proton betas vary from 0.001 to 2 and their mean values are in general positively correlated with the mean values of depths of type II magnetic dips. The electron betas shown in Figure 5f range from 0.01 to 1 and their mean values are positively correlated with the mean values of depths of type I magnetic dips. Compared Figures 5e and 5f, the depths of type I magnetic dips are larger than those of type II dips. We also investigate the pitch angle anisotropy 
of suprathermal plasma shown in Figure 5g, where 
and 
are the parallel and perpendicular plasma pressure contributed by both ring current protons and electrons. The values of anisotropy are mainly located from −0.2 to 0.6 and show no correlation with the depths of magnetic dips. Previous study by (Xia et al., 2017) suggests the pitch angle anisotropy may change the threshold of magnetic dip formation but have a slight influence on the depths of magnetic dips. The independence of magnetic dip depths on pitch angle anisotropy obtained from the observational supports the model conclusions. Figure 5h represents the scatter plot of magnetic dip depths versus the relativistic electron flux ratios at local pitch angle of 
and the energy of 1.8 MeV. Here the flux ratios are the fluxes at epoch time 
minute to the fluxes at epoch time 
minute, which represents the effects of magnetic dips on the relativistic electrons. It is shown that the flux ratios of most scatters are larger than 1, indicating that at the magnetic dip centers the relativistic electron fluxes are significantly reduced. This result is also consistent with superposed epoch analysis shown in Figure 4. Moreover, the mean profile suggests a moderate and positive correlation between the flux ratios and the depths of magnetic dips.
The relationship among the magnetic dip depths and several parameters: magnetic dip depths versus (a): magnetic latitude; (b) magnetic latitude; (c) magnetic local time; (d) 
; (e) proton betas for type I events; (f) electron betas for type II events; (g) pitch angle anisotropy; (h) the normalized electron fluxes at energy of 1.8 MeV. The black and blue circles represent type I and II magnetic dips, respectively. The red solid lines represent the mean profiles.
4 Conclusions and Discussion
-
Based on the max values of electron betas
and proton betas 
around the centers of magnetic dips, we separate the dip events into two categories: type I events with 
and type II events with 
. The number of type I events is much larger than the number of type II events. Both types of events are observed between 
, in which type I events exist from noonside to nightside and type II events exist just in the nightside, consistent with the drift orbits of electrons and protons. -
The fluctuations of magnetic dips are mainly along the field-aligned direction, consistent with theoretical estimations. Superposed epoch analysis finds a significant and negative correlation between the detrended magnetic field strengths and proton betas for type I events, and a moderately negative correlation between the detrended magnetic field strengths and electron betas for type II events. The electric fields related to the type II magnetic dips have an outward radial component near the centers of dips, which may be associated with the SAPS. Note that magnetic dips are moving with the ring current protons or electrons which account for the dip formation, the inductive electric field is expected. However, a 3D plasma-driven self-consistent magnetic field model or an ideal theoretical expression of magnetic dips are required to further obtain the corresponding inductive electric field. This work is left for future.
-
Superposed epoch analysis finds that the relativistic electron fluxes at local pitch angle of
show a significantly positive correlation with the detrended magnetic field strengths of magnetic dips, which supports the previous observations that magnetic dips produce the butterfly distributions of relativistic electrons. Particularly, for type I dip events, the electron fluxes after the centers of magnetic dips are lower than those fluxes before the center of magnetic dips. -
The depths of magnetic dips show a significant dependence on magnetic latitude and
. At the magnetic equator, the depths maximize and decrease with the increase of latitude. With the increase of 
, the depths of magnetic dips gradually increases.
Previous studies (He et al., 2017; Xiong et al., 2017) focused on the magnetic dip events driven by the enhancement of ring current proton betas. Zhu et al. (2020) reported a case study in which an electron-driven smaller-scale magnetic dip is embedded within a proton-driven larger-scale magnetic dip, which suggests the potential impotence of superthermal electrons in the generation of magnetic dips. In our study, we for the first time collect the magnetic dip events driven by the protons (as type I events) and electrons (as type II events), respectively. It is found that the event counts of type II events are much lower than the counts of type I events. It explains that electron-driven magnetic dips are rarely reported. Moreover, most of the type I events are observed in the duskside and nightside while the type II events are observed in the nightside, which is consistent with the drift orbits of protons and electrons (Birn et al., 1997), respectively. In addition, both the types of magnetic dips are able to produce the dropouts of relativistic electrons at the local pitch angle of 
, indicating the occurrence of butterfly distributions.
Previous studies (He et al., 2017; Xiong et al., 2017; Zhu et al., 2020) built up the correlation between the magnetic field fluctuations of magnetic dips and plasma betas (or pressures) case by case. In this study, we use a superposed epoch analysis to systematically investigate their relationship. The moderate and negative correlation between the detrended magnetic field strengths and plasma pressures provides statistical support for the generation of magnetic dips. Particularly, it is shown that the fluctuations of magnetic fields are mainly field-aligned, which is consistent with magnetohydrodynamics theory.
Xiong et al. (2019) systematically investigated the occurrence of the butterfly distributions of relativistic electrons during the magnetic dips. In our study, we use a different method (superposed epoch analysis) to study the variations of relativistic electron fluxes caused by the magnetic dips. Statistically, the fluxes of relativistic electrons at local pitch angle of 
and at different energies gradually decrease when the satellites are approaching the centers of magnetic dips and the electron fluxes increase when the satellites are away from the magnetic dip centers. These results are consistent with the conclusion of Xiong et al. (2019). Moreover, the median profiles of the electron fluxes suggest that accompanied with type I magnetic dips, there may be some additional non-adiabatic mechanisms to produce the decreases of the electron fluxes, which requires further investigation in future work.
We also investigate the relationship among the depths of magnetic dips and several parameters. It is found that the depths maximize at the equatorial plane and gradually decrease with the increasing of magnetic latitudes. This is because along a given field line, plasma beta maximizes at the magnetic equator, which favors the magnetic dips with the largest depths. With the 
range of 4–6, the depths slightly decrease with increasing 
, which may be associated with the 
range of ring current populations. As for the geomagnetic indices, the depths of magnetic dips increase with the increasing of the index 
, as expected, since the pressures of the ring current populations which drive the generation of magnetic dips are positively correlated with geomagnetic activities.
Acknowledgments
We acknowledge the support by the NASA grants NNX17AI52G, 80NSSC20K0689, and 80NSSC19K0283, and the NSF grant AGS-1702805.
Open Research
Data Availability Statement
The fluxgate data of Van Allen Probes are obtained from http://emfisis.physics.uiowa.edu/Flight/. The electric field data of Van Allen Probes are obtained from http://www.space.umn.edu/rbspefw-data/. The particle data of Van Allen Probes are obtained from https://cdaweb.gsfc.nasa.gov/pub/data/rbsp/. The geomagnetic indices are obtained at the CDAWeb (http://cdaweb.gsfc.nasa.gov/cdaweb/).
