Simulations of Electron Flux Oscillations as Observed by MagEIS in Response to Broadband ULF Waves

Abstract

Coherent electron flux oscillations of hundreds of keV are often observed by the Van Allen Probes in the magnetosphere during quiet times in association with ultralow frequency (ULF) waves. They are observed in the form of periodic flux fluctuations, with a drift frequency that is energy dependent, but are not associated with drift echoes following storm- or substorm-related energetic particle injections. Instead, they are associated with the resonant interaction of electrons with ULF waves and are an indication of ongoing electron radial diffusion. To investigate details of such flux oscillations, particle-tracing simulations are conducted under the effect of realistic, broadband ULF electric and consistent magnetic fluctuations. Virtual detectors are simulated along spacecraft orbits and the results are compared to measurements. Through a parametric study, it is found that the width of electron energy channels is a critical parameter affecting the observed amplitude of flux oscillations, with narrower energy channel widths enabling the observation of higher-amplitude flux oscillations; this potentially explains why such features were not observed regularly before the Van Allen Probes era, as previous spacecraft generally had lower energy resolution, which only enabled the observation of large-amplitude drift echoes following a storm or substorm. Results are confirmed using the Magnetic Electron Ion Spectrometer (MagEIS) ultrahigh energy resolution data. Energy width effects are quantified through a parametric simulation study that matches flux oscillation observations during a period that is characterized by extremely quiet conditions, where the Van Allen Probes observed flux oscillations over multiple days.

Key Points

• Electron flux oscillations at 100 s keV energy are often observed in the radiation belts, associated with broadband ULF waves
• Observations of electron flux oscillations are dependent on detector energy channel width
• Electron flux oscillations are indicators of ongoing radial transport processes

1 Introduction

It is well known that, following an injection and energization of electrons and ions in the magnetosphere during a storm or substorm, periodic enhancements of the initial injection will appear as the injected particles drift around the earth following a constant magnetic field strength contour: these periodic flux enhancements are termed drift echoes, are energy dependent, they are generally observed to have a smaller amplitude than the initial injection, and will gradually phase mix and decrease in amplitude while broadening in time, over a time of a few drift periods (Lanzerotti et al., 1967; Li et al., 1993; Liu et al., 2009; Reeves et al., 1991; Sarris et al., 2002). Such drift echoes have also been used to characterize discrepancies between different measurements of particle flux and improve the intercalibration of different sensors (O'Brien et al., 2015). The amplitude of these drift echoes is believed to contain information of the initially injected population as well as of the radial transport of these electrons: The association between the appearance of electron flux oscillations and radial transport processes has been proposed early on by Schulz and Lanzerotti (1974), where they derived an expression that relates the peak-to-peak amplitude of electron flux oscillations with the radial diffusion coefficient D_LL for a dipole magnetic field. They identified drift echo events in data from the ATS-1 geosynchronous satellite and used their formula to obtain an order-of-magnitude approximation of the diffusion coefficient, which they found to be D_LL ~ 5 × 10⁻⁹ L¹⁰ day⁻¹ at geosynchronous altitude, where L represents the distance from the center of the dipole magnetic field on the equatorial plane, expressed in number of Earth radii.

Through both modeling and observation, Sarris et al. (2017) have shown that small-amplitude electron flux oscillations can also be produced without injections of energetic particles, through the drift resonance of energetic electrons with broadband magnetic and electric field waves. This resonant interaction is more prominent when waves have enhanced power at frequencies that are close to the electrons' drift frequency around the Earth, which lies in the ultralow frequency (ULF) range of waves, and in particular in the Pc4 and Pc5 ranges of frequencies (see, e.g., the classification of ULF waves by Jacobs et al., 1964). Such fluctuations lead to radial transport and to an increase or decrease in the flux, depending on the radial gradient of the local Phase Space Density (PSD) (e.g., Schulz & Lanzerotti, 1974), while producing oscillations in the electron flux, whose frequency is roughly the electrons' drift frequency.

The electron flux oscillations associated with radial diffusion are distinctly different from the drift echoes following an injection: The latter are characterized by a large initial flux enhancement and a gradual decrease of the subsequent drift periodic peaks, due to phase mixing (see, e.g., Birn et al., 1997; Patel et al., 2019; Reeves et al., 1996; Sarris & Li, 2005) the former are characterized by flux oscillations of significantly smaller magnitude, which may increase, decrease, or remain steady, depending on the field oscillations and the PSD gradient. These oscillations are a unique identifier of radial transport, since other acceleration mechanisms, such as local heating by wave-particle interactions, cause oscillations at the electron's cyclotron frequency, which is much faster than a typical electron detector's response. Contrary to radial transport, local acceleration is expected to be characterized by a relatively smooth increase of electron flux with no oscillation near the electrons' drift frequency.

As it will be demonstrated below through measurements and simulation, a key factor in being able to detect flux oscillations is energy resolution of the particle detectors. The underlying principle is that, if the energy range of each channel of a detector is large, then electrons in each channel with different energies and drift periods will partially phase mix, obscuring the apparent amplitude of flux oscillations. The high-energy resolution of the Van Allen Probes' Magnetic Electron Ion Spectrometer (MagEIS) instrument (Blake et al., 2013) mitigates phase mixing and allows flux oscillations to be observed more clearly than has been possible in the past. Flux oscillations at the electrons' drift periods are often seen in multiple energy channels, particularly at the spacecraft apogee. Their amplitude varies between channels, indicating an energy dependence of the radial diffusion process, as discussed below.

The appearance of electron flux oscillations under the effect of broadband electric and magnetic field oscillations with frequencies in the range of the drift frequency of the electrons, which is in the ULF range, has recently been demonstrated through particle tracing simulations by Sarris et al. (2017). In that study, it was demonstrated that the amplitude of the flux oscillation closely follows the local gradient of PSD. Thus, large PSD gradients will result in large-amplitude oscillations, whereas a flat, zero-gradient PSD will result in no observable flux oscillations. In this study we extend the initial demonstration of Sarris et al. (2017) by performing a parametric study, aiming to determine the dependence of the flux oscillations on the energy width of the electron detectors. The particle tracing simulations are performed for electron populations with realistic PSD gradients and energy spectra, under the effects of realistic electric and magnetic field fluctuations. The electron fluxes reproduced through these simulations have amplitudes and frequencies that closely mimic electron flux oscillation measurements.

2 Observations of Flux Oscillations

Measurements of relativistic electrons from MagEIS (Blake et al., 2013) onboard both Van Allen Probes A and B are used to detect and parameterize flux oscillations that can be associated with ULF wave-driven radial diffusion. MagEIS measures electrons from 20 keV to 4 MeV, and, as it will be shown, it is ideally suited to investigate these flux oscillations, due to its fine energy resolution (typically ΔE/E < 30%). Such observations of flux oscillations are ubiquitous in the magnetosphere and are more easily identified during quiet geomagnetic conditions. In the present study, a prolonged period of extremely quiet solar and geomagnetic conditions is identified, between a high-speed stream that occurred on March 1, 2013 and a solar storm that occurred on March 17, 2013. The dynamics of the outer radiation belt during this period in March 2013 have been described in detail in Baker et al. (2014), Li, Hudson, et al. (2014), Li, Thorne, et al. (2014), Reeves et al. (2016), and Ripoll et al. (2016). In particular, Baker et al. (2014) specifically note that the March 1 to 17, 2013 interval has the characteristics of radial diffusion uninterrupted by injections until the strong storm of March 17.

Electron fluxes for four MagEIS energy channels with energies of 183, 342, 593, and 742 keV for the month of March 2013 are shown in Figure 1, plotted as a function of L-shell. The high-speed stream and the solar storm events, which mark respectively the beginning and the end of this prolonged quiet period, are indicated in this figure with vertical red lines. This period between the two events is characterized by a lack of intense magnetospheric perturbations caused by external sources, and also a lack of new injections, as seen through electron fluxes in Figure 1. We note that, during the period between the two events, electron fluxes gradually decrease, particularly in the lower L region. This is likely due to precipitation loss and it might also possibly mean that radial diffusion is not very efficient in energizing electrons during such quiet period, potentially due to the relatively low amplitude of electric and magnetic field fluctuations.

Figure 1

Electron fluxes during the month of March 2013 as recorded in (a–d) four sample energy channels by MagEIS onboard Van Allen Probe A. The two vertical red lines mark a high-speed stream (HSS) that occurred on March 1 and a solar storm that occurred on March 17.

During this period, electron flux oscillations are observed over subsequent passes by the Van Allen Probes A and B: In Figure 2 we plot time series of electron flux measurements when the Van Allen Probes were located around apogee, at L-shells greater than ~4. Flux measurements are plotted in nine MagEIS energy channels, from ~110 to ~893 keV for Probe A and ~103 to ~875 keV for Probe B, as marked. We can see that flux oscillations are observed in multiple energy channels over subsequent orbits on March 10 and 11. It is noted that the MagEIS instruments on the two probes A and B have slightly different energies.

Figure 2

MagEIS fluxes from successive apogee passes of the Van Allen Probes (a) A and (b) B, from March 10 to 12, 2013, for times when the spacecraft are located at L ~ 4 and above.

In Figure 3 we zoom-in on one apogee pass of Van Allen Probe B, on March 11, 2013; clear electron flux oscillations are observed in most energy channels, with frequencies that match the drift frequencies of electrons at each channel.

Figure 3

MagEIS electron fluxes from one apogee pass of Van Allen Probe B, on March 11, 2013.

The orbits of the two Van Allen Probes are shown in Figure 4. The locations of Van Allen Probes A and B at 01:00 on March 11, 2013 are shown in red and blue circles respectively. The direction of motion and the location of the two probes every hour from 01:00 until 10:00 are shown with arrows along the orbit. From this figure it can be seen that these flux oscillations are observed in the night side, close to the midnight region. This is further discussed below. It is noted that this orbital configuration does not change significantly over the period studied, aside from the latitudinal coverage.

Figure 4

Orbits of the two Van Allen Probes, A and B, on March 11, 2013, in GSE coordinates. Arrows indicate the direction of motion of the spacecraft, every 1 hr.

3 Overview of Particle Tracing Simulation

3.1 Test Particle Simulation

Particle-tracing models can provide valuable insight into the dynamics and evolution of charged particles in the Earth's magnetosphere. For example, such models have been used to model the enhancement of electrons in the outer radiation belt during a strong interplanetary shock (Li, Baker, et al., 2003; Li et al., 1993), to model electron and proton injections and drift echoes during substorms (Li et al., 1998; Li, Sarris, et al., 2003; Liu et al., 2009; Sarris et al., 2002) and to model radial diffusion of relativistic electrons under the effect of broadband ULF waves (Sarris et al., 2006), which were simulated as a superposition of a large number of randomly initiated pulse fields. In this study, a guiding-center code is used to trace the orbits of relativistic electrons, initially distributed on a range of L-shells, under the effect of the Earth's background magnetic field as well as fluctuating fields in the ULF range of frequencies. Equatorially mirroring electrons are considered, whose drift velocity is given by the relativistic guiding-center equation (Northrop, 1963) as equation 1:

(1)

δB_θ is the total magnetic field vector, which is given as the sum of the Earth's background magnetic field, Β_Earth (considered to be a simple dipole in the simulation) and superimposed magnetic field fluctuations, B is the magnitude of the magnetic field, δB_θ; c is the speed of light; q is the electron charge; γ is the relativistic correction factor γ = (1 − v²/c²)^−1/2; μ is the electron's relativistic first adiabatic invariant and ∇_⊥ is the gradient perpendicular to the local magnetic field direction.

3.2 Initial Distribution of Electrons in the Simulation

Particles in the simulation are initialized randomly in L, MLT, and μ. In order to minimize simulation run time, we do not trace particles that cannot be recorded by the electron detectors under consideration; this is done by investigating the range of μ values that correspond to the energy channels where flux oscillations are observed. In this study, electron detectors at energies 103, 164, 168, 235, 339, 459, and 584 keV are simulated and compared to measurements.

Figure 5 gives an overview of the energies and initial L of the electrons traced in the simulation. The thick black and red dashed lines in Figure 5a/5b shows the energy/drift frequency as a function of L of electrons of μ = 40 and 800 MeV/G respectively, which are the lower and upper limit of μ values of the traced electrons in the simulation. Circles on the μ = 40 MeV/G and μ = 800 MeV/G dashed lines and the corresponding labels within the plot give indicative values of the energy/drift frequency of electrons at L = 5, 6, and 7, for reference. In Figure 5a the MagEIS energy channels that are plotted in Figure 3 for Van Allen Probe B are marked with horizontal lines, colored in the same rainbow color scale from magenta (103 keV) to red (875 keV), and the median energy of each of the energy channels is written in the right side of the figure. In Figures 5b and 5c the corresponding drift periods and drift frequencies, respectively, are given as a function of L for the same electron energy ranges shown in Figure 5a.

Figure 5

(a) Solid horizontal lines correspond to MagEIS energy channels simulated through particle tracing, whereas dash-dotted lines correspond to energy channels not considered in the simulation. Energy vs. L for μ = 40 MeV/G, the lower limit of μ values of the traced electrons, is plotted with a thick black dashed line. Energy versus L for μ = 800 MeV/G, the upper limit of μ values of the traced electrons, is plotted with a thick red dashed line. (b) The corresponding ranges of drift periods, T, and (c) drift frequencies, ω of electrons in the simulation are shown for the limits of μ values considered in the simulation. Values of energy, drift period and drift frequency are given in the three panels, respectively, at L = 5, 6, and 7, at the locations marked in hollow circles.

Thus, from Figure 5c we can see that, for the range of energies recorded by the energy channels of MagEIS, drift resonance can occur with waves of frequencies from 0.13 to 1.3 mHz, if these waves have a mode number m = 1. We note that, as demonstrated in Sarris et al. (2017), the appearance of flux oscillations at the drift frequency (and not at multiples of the drift frequency) indicate that probably the broadband fluctuations during this event are dominated by an m = 1. In the simulation, electric and magnetic field fluctuation frequencies are assigned values randomly in the range from 0.13 to 1.3 mHz, as described in the following section, to resonantly interact with the entire range of energies under consideration.

3.3 Model Electric and Magnetic Fields

The model fluctuations in the electric field in the ULF range of frequencies follow the model fields used in Sarris et al. (2017), which are produced by a sum of sinusoidal oscillation according to equation 2:

(2)

where i denotes the index of each individual sinusoidal wave, E_0i is the amplitude of the i^th sinusoidal wave, is the azimuthal direction, r is the radial distance, number p determines the radial dependence (e.g., if p > 0 then E_φ increases with r, as observed by, e.g., Liu et al. (2016)), d is used for normalization, m is the azimuthal mode number, φ is the longitude on the equatorial plane, ω_i and ψ_i are the wave frequency and phase of the i^th sine wave, respectively. Subsequently, Faraday's law is used to obtain the consistent model magnetic field fluctuations corresponding to the electric field. The resulting fluctuating magnetic field takes the form of equation 3:

(3)

A similar formulation for model fluctuating magnetic fields has been used in Sarris et al. (2017). Together the oscillating and background fields satisfy _φ · ( _Earth +  _θ) = 0 & ∇ · ( _Earth +  _θ) = 0. It is noted that at the equatorial plane, the radial component of the magnetic field becomes _r = 0.

Electric and magnetic field oscillations during the quiet times of this event are possibly related to small-amplitude solar wind pressure fluctuations, as described in, for example, Kepko and Viall (2019). Electric and magnetic field measurements that were used in the particle tracing simulations were compared in amplitude and power spectral density to those provided by the Electric and Magnetic Field Instrument Suite and Integrated Science (Kletzing et al., 2013) on the Van Allen Probes. Electric and magnetic field fluctuations that were used as inputs to the particle-tracing model were adjusted by varying values E_0i in equation 3. Data from the above instruments are available online (http://cdaweb.gsfc.nasa.gov/istp_public/).

The model fluctuating magnetic field δΒ_θ according to equation 3 and the corresponding power spectra around a simulated apogee location of a virtual Van Allen Probe B for 4 hr of simulation time are plotted in the upper left panels of Figure 6; time series and power spectra of magnetic field measurements from Van Allen Probe B for 4 hr around the apogee pass on March 11, 2013, along the orbit shown in Figure 3, are plotted in the upper right panel of Figure 6. From the comparison we can see that the amplitude of the model fluctuations and the spectral characteristics at the frequencies of interest (as described in the previous, section 3.2) closely match the measured magnetic field fluctuations. It is noted that some higher frequency fluctuations are observed in the time series of the magnetic field that are not captured in the model field fluctuations, due to the selected frequency ranges; these are at a higher frequency than the drift period of the electrons under investigation, as discussed in section 3.2, and do not impact the particle populations under consideration.

Figure 6

Time series and corresponding spectra of model (left) and data (right) magnetic field fluctuations, δΒ_θ (top) and electric field fluctuations, δΕ_φ (bottom). Model and data are detrended using a 100-point moving average, to reveal the fluctuations in the ULF range of frequencies.

The model electric field fluctuations δΕ_φ that are consistent with the model magnetic field fluctuations and the corresponding spectra are shown in the lower left panels of Figure 6; time series and power spectra of electric field measurements from Van Allen Probe B for 4 hr around the apogee pass on March 11, 2013 are plotted in the lower right panel of Figure 6. From the comparison we can see that the average amplitude of the model fluctuations is close to the average measured amplitude of the electric field fluctuations, except for a period around 03:00 UT, when there are no electric field fluctuations observed. It is noted that even during this time period around 03:00 UT flux oscillations are observed, indicating that the interaction with ULF waves has taken place elsewhere along the electron drift paths; this is further elaborated below. In addition, there appear to be higher frequency electric field fluctuations that are not captured in the model. Similarly to the magnetic field fluctuations, these are of higher frequency than the drift frequency of electrons under consideration; thus they are not in resonance with the electron population that is studied herein, and are not expected to affect the observed flux oscillations. It is also noted that wave power in the model decreases with frequency, which is consistent with past observations (see, e.g., Perry et al., 2005, 2006, and references therein). Finally, it is noted that there is no radial dependence on the amplitude of the model fluctuations and thus in the wave power in the model and that the azimuthal distribution is assumed to be uniform; both of these assumptions are made due to a lack of measurements to constrain the model radially and azimuthally.

3.4 Electron Weighting Factors

Each electron in the simulation is representative of an ensemble of electrons in the electron distribution, and is thus multiplied by a weighting factor according to its contribution in the initial spatial and energy distribution. This methodology has been followed in, for example, the particle simulations of Li et al. (1993), Sarris and Li (2005), Sarris et al. (2002). In order to convert individual electron orbits that are traced in the simulation into electron flux, a virtual detector is placed at the instantaneous location of the Van Allen Probes satellites, and electrons are recorded in the simulation when they enter within a region defined as L ± ΔL and MLT ± ΔMLT, where L is the instantaneous L-shell of the spacecraft and MLT is the local time of the spacecraft. In the simulation, ΔL is set to 0.125 and ΔMLT is set to 0.25. Subsequently, each electron that enters the area of the virtual detector is multiplied by a weighting factor, based on the relative significance of that electron in the initial energy distribution function and the radial distance distribution function.

The energy weighting function is assigned according to the following exponential function:

(4)

where E_i is the initial energy of each electron in the simulation and Q_f, g_n are constants; similar exponential functions have been used as energy weighting functions by, for example, Elkington (2000) and Sarris et al. (2017). In order to assign values to the constants Q_f and g_n, we calculated the energy spectra from MagEIS measurements in terms of flux vs. energy; these values are plotted in colored squares in Figure 7a. As shown in this figure, the optimal fit is achieved if two separate power law functions are used. The values of the constants Q_f and g_n that best fit the observations in the present simulation are Q_f = 4 × 10⁸ and g_n = 3.35 for energies below 221 keV and Q_f = 8.6 × 10⁵ and g_n = 1.25 for energies above 221 keV. Such distinct energy distribution functions have been used in the past to characterize energetic electrons in the radiation belts: for example, Cayton et al. (1989), based on geosynchronous satellites, derived energy distribution functions that had distinctively different parameters for lower-energy (30–300 keV) and higher-energy (300–2,000 keV) electrons.

Figure 7

(a) Energy weighting, Q_g, based on energy spectra from MagEIS data on March 11, 2013 when Van Allen Probe A was located at apogee, and (b) radial distribution weighting, Q_r.

Subsequently, a radial distribution weighting function is assigned according to the following spatial distribution function:

(5)

where r_i is the initial location of each electron in the simulation; such radial distribution has been used for radiation belt simulations in, for example, Sarris et al. (2002) and Sarris and Li (2005). The constants in this equation take the values: a₀ = 3, a_0d = 6, nl = 2, ml = 6. The radial distance weighting is scaled to a maximum of 1 at L = 4.5. The constants in equation 5 were adjusted so that the low-frequency flux trends at the various channels best match the observations at Van Allen Probes' perigee. It is noted that the above spatial distribution function as described in equation 5 is expected to be energy dependent; such energy dependence is not being considered in this study.

Figure 7 shows schematically the two distribution functions in energy and radial distance, according to equations 4 and 5 respectively. In Figure 7a the energy weighting function is plotted according to the energy spectrum of electrons; colored squares show the flux vs. energy measurements for Van Allen Probe B MagEIS on March 11, 2013, when the spacecraft was located at apogee. The color coding of the squares is according to energy, and it is similar to the colors selected for different energy channels in Figures 2b, 3, and 4a. The equations that best fit through these points are also written on the figure, together with the values for R², which indicates the proportion of the variance in the flux that is predictable from the energy based on the fitted equation; the resulting values of R² indicate an excellent fit of the selected power law functions. In Figure 7b the radial distribution weighting function is plotted as a function of L.

3.5 Simulating Electron Fluxes Along the Van Allen Probes' Orbits

In order to reconstruct electron flux time series and to compare them with MagEIS electron flux measurements, electrons are traced and recorded when they cross the azimuthal location of virtual detectors at the simulated instantaneous location of the Van Allen Probes A or B. The spatial length of the detectors determines the spatial resolution of the simulation; an L-shell bin size of 0.125 R_E is set in the present simulation. Each recorded electron is multiplied by the weighting factors as described by equations 4 and 5, according to, respectively, its initial energy and initial L. The weighted electrons are subsequently binned according to final energy and arrival time at the detector. The time binned, summed, and weighted electrons, recorded whenever they crossed the virtual spacecraft location measurements, form the time series of the simulated electron flux; in this simulation they are formed so as to simulate 1-min time resolution time series of the fluxes at various energy channels of the MagEIS instrument. Figure 8 presents the results of the simulated weighted flux along a virtual Van Allen Probe orbit on March 11, 2013. To produce the simulated electron flux as a function of time along the simulated orbit of Van Allen Probe A, as shown in Figure 8, the energy channels were formed for similar energies and energy channel widths as in the measurements shown in Figure 3; flux results are also color coded in the same way as in Figure 3. We can see that the simulation results are able to capture to a large extent both the relative flux levels and their separation and also the amplitudes of the flux oscillations along the orbit of Van Allen Probe B.

Figure 8

Simulated weighted flux along a virtual Van Allen Probe orbit on March 11, 2013 for six MagEIS energy channels, color coded as marked. Solid thin lines correspond to simulation results with a 1-min resolution, whereas thick dashed lines correspond to median values based on a 3-min time window.

4 On the Relationship Between Electron Flux Oscillations and Electron Detector Energy Width

In the following, we investigate the relationship between the simulated flux oscillation amplitudes and the energy resolution of the electron detectors that observe these flux oscillations. To explore the underlying principle, if we consider an infinitely thin electron detector energy width, which essentially focuses on a single particle's energy, then the observed oscillating flux would correspond to a delta-like function of that particle crossing through the detector once every drift period; on the other hand, if we consider an infinitely wide detector that captures particles of all energies, then there would be no flux oscillations, as particles of all energies (and thus all drift periods) would phase mix, resulting in the smearing out of any flux oscillations at the particles' drift frequency. Thus, the finite energy width is expected to impact the observed flux oscillation amplitude.

In the MagEIS design, silicon detectors consist of pixels of unequal size, and the surface area of each pixel of the detector corresponds to the energy range of the corresponding energy channel. This surface area increases as the pixel number increases, which leads to wider energy channels as the pixel number (and hence the corresponding energy channel number) increases. In MagEIS this is done to maintain uniform count rates across all energy channels, by compensating for the combined effects of a falling spectrum and the decrease of the geometry factor with increasing energy (Blake et al., 2013).

To explore the direct relationship between electron detector energy width and flux oscillation amplitude, we performed particle tracing simulations under the effect of broadband ULF waves for different energy channel widths. This is done in the postprocessing phase (i.e., after running the particle traces under the effect of the model ULF waves), through accumulating electrons in successively wider ranges of energies around the detector's center value. We then measure the corresponding amplitude of the observed flux oscillations at each energy width and plot the simulated relative flux oscillation amplitude (i.e., average flux oscillation amplitude divided by the average low-frequency flux trend), as a function of the energy channel half-width. In the parametric study that was conducted, energy channel widths from 5 to 60 keV were simulated, with a step of 5 keV. The results are shown in Figure 9, where the relative flux oscillation amplitude at a simulated spacecraft location corresponding to the Van Allen Probes' apogee is plotted as a function of the simulated energy channel width, for six different energy channels with median energies of 110, 145, 184, 232, 458, and 592 keV, color coded as marked. In Figure 9, the resulting relative flux oscillation amplitude for each simulated width is marked with a small square, colored according to energy channel. It can be seen that the relative flux oscillation amplitude decreases exponentially for increasing energy widths for each of the energy channels. The corresponding exponential fits of flux oscillation amplitudes versus width for each of the simulated energy channels are plotted as solid, colored lines, and the functions of these fits are written next to each channel's corresponding median energy indication, in the right-hand side of the plot. When comparing the six curves for the different energy channels, it is seen that the relative flux oscillation amplitude is energy dependent, with higher energy channels having higher relative amplitudes for the same energy widths.

Figure 9

Simulated relative flux oscillation amplitudes (flux oscillation amplitude over average flux) are plotted as a function of simulated energy channel width (in keV), for six MagEIS electron channels, as marked. The exponential fits through the simulated energy widths are also plotted. The expected values of relative flux oscillations for the MagEIS widths of each energy channel are plotted in a black hollow circle. The actual values of relative flux oscillations for the MagEIS widths of each energy channel are plotted in a colored circle, as marked.

Subsequently, we simulate the expected flux oscillation amplitude for the actual energy channel energy widths for each of the six energy channels of Figure 9. The actual widths of the MagEIS channels are as follows, in order of ascending energy of the channel: 27, 32, 37, 42, 78, and 88 keV. The resulting simulated relative flux oscillations for the above widths are plotted on top of the exponential curves with black hollow circles.

It is noted that the first four energy channels correspond to energy channels of the MagEIS Low Chamber unit; the corresponding simulated flux oscillation amplitudes, marked with hollow black circles, are connected with a solid black line in Figure 9. The last two hollow black circles, which are also connected with a solid black line, correspond to energy channels of the MagEIS Medium Chamber unit (Blake et al., 2013); this is further discussed below.

Measurements of the average actual flux oscillations during the March 2013 events studied herein for each energy channel are plotted with colored circles, where the color corresponds to each channel's energy, as marked. The average over all observed flux oscillations is calculated centered around the apogee of both Van Allen Probes A and B. We can see that the colored circles (observations) are generally higher than the hollow circles (simulations). This could be due to electric and magnetic field effects beyond the locations where fields and fluxes are observed: electron flux oscillations, even though observed in situ, are a result of ULF broadband waves acting along their entire drift path; the flux oscillations thus carry information of transport processes elsewhere, similar to the information of radial transport that is contained in drift echoes following a storm or substorm injection. Thus, it is possible that there were fluctuations of higher amplitudes along the electrons' drift paths, higher than the fields used in the simulations, which mimic the field amplitudes and frequency content of the in situ measured fields. It is noted that apogee of both Van Allen Probes during this event is located close to the midnight region, as shown in Figure 4. Equatorially mirroring electrons in the nightside will drift to higher radial distance in the dayside via drift shell splitting (Schulz & Lanzerotti, 1974), so as to experience constant magnetic field along their orbit. At the same time, ULF wave power often appears to be higher at higher L: This has been shown, for example, by the study of Mathie and Mann (2001) based on ground magnetometers and also by the study of Sarris and Li (2016) based on multispacecraft analysis, including Van Allen probes, THEMIS and GOES spacecraft. This means that electrons measured by MagEIS may have been subject to higher ULF wave power on the dayside than what is measured in situ near midnight, which could explain the above discrepancy between observations and simulations.

The observed exponential decrease of the flux oscillation amplitudes with channel width for each of the energy channels means that beyond a certain energy channel width the flux oscillations will not be distinguishable from the energy channel's noise levels, or from naturally occurring random variations in the flux. This means that past energetic particle instruments, which generally had wider energy channel widths, would not be able to detect but the largest of flux oscillations, such as are observed as drift echoes after a storm or substorm. We also observe that higher energy channels generally have higher relative flux oscillation amplitudes than lower energy channels. This means that, for similar values of energy channel width, a higher energy channel should have higher flux oscillations. This, at first, appears to be in contrast to observations of flux oscillations, which most often show flux oscillations only in the lowest MagEIS energy channels, such as in the examples shown in Figures 2 and 3, where the channels with energies 459 keV and above show smaller oscillation amplitudes compared to lower energy channels. However, this can be explained due to the increasingly larger energy widths for higher energy channels: MagEIS energy channel widths vs. energy channel median energies are shown in Figure 10. In this figure we note in particular a sharp discontinuity between the widths of energy channels at or below ~230 keV, which correspond to the Low Chamber pixels of the MagEIS instrument, and those of energy channels above ~240 keV, which correspond to the Medium Chamber pixels (Blake et al., 2013).

Figure 10

MagEIS electron energy channel widths for MagEIS electron channels of increasing energy. On the lower left side the Low Chamber channels are connected with a solid black line. On the upper side the Medium Chamber channels are connected with a black line.

5 High-Resolution MagEIS Data

In the following we explore similar flux oscillation amplitudes as observed through a novel MagEIS data product: ultrahigh energy resolution electron measurements. Claudepierre et al. (2017) and Hartinger et al. (2018) have recently demonstrated a technique to obtain ultrahigh energy resolution electron measurements, which are derived from the histogram data used for background removal of the main rate channels. It is a nonstandard data product that is possible due to the designed functionality of the MagEIS instruments. These ultra-high resolution measurements are obtained using a data product that improves the energy resolution by roughly an order of magnitude, i.e. from about ΔE/E = 20–30% to 2–3%. The technique is based on the fact that, whereas the standard MagEIS energy channels are obtained from the full pulse-height spectrum from each MagEIS detector or pixel over a narrow pulse-height range centered on the pixel response (see Claudepierre et al., 2017), a subsampling of the full pulse-height spectrum is also retained, via onboard look-up tables, to produce the MagEIS histogram data product; these data can be used to obtain ultrahigh energy resolution fluxes when count rates are large enough. In the study of Hartinger et al. (2018), these ultra-high-resolution measurements have revealed a range of complex dynamics that cannot be resolved by standard measurements. Furthermore, as Hartinger et al. (2018) demonstrated using the histogram channels, the flux modulation amplitudes as observed using the standard-resolution MagEIS energy channels appear lower than the ultrahigh resolution channels, and can differ by as much as a factor of 2. In the following we explore the different flux oscillation amplitudes as they are observed via the ultrahigh resolution and the standard-resolution MagEIS data.

An example of the flux oscillations in the ultra-high-resolution measurements is given in Figure 11, for the same pass of Van Allen Probe B on March 11, 2013 as that shown in Figure 3. The energy of each channel that is plotted in Figure 11 is marked on the right side of the figure. It is noted that only one every four energy channels are plotted here for clarity, out of a total of 139 energy channels.

Figure 11

MagEIS electron flux measurements over one apogee pass of Van Allen Probe B, on March 11, 2013, using the ultrahigh energy resolution data according to Hartinger et al. (2018).

In Figure 12 we plot the absolute observed flux oscillation amplitudes as a function of energy channel width for a large number of ultrahigh energy resolution data; the flux oscillation amplitudes show a power law dependence on the corresponding energy width, which is assumed here to be 3% of the energy channel value. The energy channels of the corresponding standard MagEIS energy resolution are color coded as in previous figures, for reference.

Figure 12

Ultrahigh resolution electron flux oscillation amplitudes on March 11, 2013 as a function of energy channel width, for the same apogee pass of Van Allen Probe B as that shown in Figure 11. The energy channels of the MagEIS standard energy resolution are also shown, color coded as marked, for reference.

The observed relative flux oscillation amplitudes of the same energy channels as plotted above in Figure 9 are plotted as a function of energy channel width in Figure 13, using both the standard and the ultrahigh energy resolution data: Circles mark the standard MagEIS energy resolution whereas squares mark the ultrahigh energy resolution. Circles and squares are colored according to energy channel, as marked in the upper part of the figure. We can see that the flux oscillation amplitudes observed by the ultrahigh resolution channels (colored squares) are higher than the corresponding flux oscillations observed by the standard energy resolution channels, confirming that flux oscillations are more easily identifiable with narrow energy channels; however, we can also observe that the observed flux oscillation amplitudes are considerably smaller than what is predicted by extrapolating the simulated energy channel width dependence, as plotted in Figure 9. This could be due to the fact that the ultra-high energy channels used herein have a coarser time resolution than the standard time resolution (Hartinger et al., 2018), which leads to smoothing out and dampening of the flux oscillation amplitudes. It is noted that such tradeoff in time and energy resolution is not unique to MagEIS measurements but is common also with other measurement techniques, including for example Bremsstrahlung X-rays generated by precipitating electrons (see, e.g., Table 1 of Woodger et al., 2015).

Figure 13

Comparison of measured relative flux oscillation amplitudes using the standard (circles) and the ultrahigh (squares) energy resolution data of MagEIS. Energies are color coded, as marked.

6 Summary and Conclusions

We have demonstrated through particle tracing simulations that flux oscillations at the drift frequency of electrons, observed at different energy channels, are a direct consequence of the effects of ULF electric and magnetic field oscillations on electron populations, and we studied the dependence of these flux oscillations on the widths of the energy channels that observe these oscillations. Energy channel widths were demonstrated to be a critical parameter, with narrower widths enabling the observation of higher-amplitude flux oscillations; this could explain why such features were not observed before the Van Allen Probes era (except for large-amplitude drift echoes following a storm or substorm). The observations of such flux oscillations are enabled by the excellent energy resolution of the MagEIS electron flux measurements onboard the Van Allen Probes.

It is noted that there are other effects that affect the appearance of flux oscillations, such as the slope of the local PSD, and the amplitude of electric and magnetic field fluctuations; furthermore, the effects of some of these parameters could be coupled. A further parametric study is required to fully decouple, for example, the dependence of flux oscillation amplitudes on simultaneously PSD gradients and energy channel energy widths. This should lead to an empirical relationship that predicts the appearance of such flux oscillations, taking into account the dependence on all the above factors.

Since the parameters that affect flux oscillations also directly determine the rates of radial diffusion and transport, and since these flux oscillations are a direct manifestation of ongoing radial diffusion, this parameterization could lead to the formation of an empirical relationship, through which important information of the radial transport processes could be investigated by observations of flux oscillations. It is noted in particular that the radial transport rates are currently calculated through electric and magnetic field measurements, which rely on having satellites with appropriate instrumentation exactly at the place and time where radial diffusion occurs. However, electron fluxes continue to oscillate everywhere along the electron drift paths; thus electrons carry the oscillation information (and hence the radial diffusion information) beyond the region where measured field fluctuations occur. A comparison of model results and in-situ measurements of fluxes oscillations in the nightside has shown higher amplitudes of flux oscillations in the observations; these are attributed to electrons drifting to higher radial distances in the dayside, where ULF wave power is expected to be higher than what is measured in-situ in the nightside. At the same time, flux oscillations were observed even at times when there are no electric field oscillations observed in situ. Thus, electron flux oscillations could offer an additional means of probing the magnetosphere beyond the specific locations and orbits where there are actual measurements of ULF waves and related radial transport processes.

Acknowledgments

We thank the MagEIS team for making available electron flux data. MagEIS data are available online (https://cdaweb.sci.gsfc.nasa.gov/). We thank the EMFISIS team for making available magnetic and electric field data. EMFISIS data are available online (https://cdaweb.sci.gsfc.nasa.gov/). This work was supported in part by NASA Grants NNX15AF56G and NNX17AD85G and by NASA/RBSP-ECT and -EFW funding through JHU/APL Contract 967399 under prime NASA Contract NAS5-01072 (Van Allen Probes), and NASA Contract NAS5-02099 (THEMIS).