On the Similarity and Repeatability of Fast Radiation Belt Loss: Role of the Last Closed Drift Shell

Abstract

Properly characterizing fast relativistic electron losses in the terrestrial Van Allen belts remains a significant challenge for accurately simulating their dynamics. In particular, magnetopause shadowing losses can deplete the radiation belt within hours or even minutes, but can have long-lasting impacts on the subsequent belt dynamics. By statistically analyzing seven years of data from the entire Van Allen Probes mission in the context of the last closed drift shell, here we show how these losses are much more organized and predictable than previously thought. Once magnetic storm electron dynamics are properly analyzed in terms of the location of the last closed drift shell, not only is the loss shown to be repeatable but its energy-dependent spatio-temporal evolution is also revealed to follow a very similar pattern from storm to storm. Employing an energy-dependent ULF wave radial diffusion model, we further show for the first time how the similar and repeatable fractional loss of the pre-storm electron population in each storm can be reproduced and explained. Empirical characterization of this loss may open a pathway toward improved radiation belt specification and forecast models. This is especially important since underestimates of loss can also create unrealistic sources in models, creating phantom electron radiation and leading to the prediction of an overly harsh radiation environment.

Key Points

• Magnetopause compression, characterized by inward motion of the last closed drift shell, results in rapid loss of radiation belt electrons

• The associated loss develops in a similar and repeatable manner depleting the radiation belt by the same fraction in every event studied

• The L^∗ and energy dependence of the loss is consistent with magnetopause shadowing losses enhanced by outward ULF wave radial diffusion

1 Introduction

Since the discovery of the Earth's Van Allen radiation belts over 50 years ago (Van Allen & Frank, 1959), determining the physical processes responsible for their dynamics have been a major focus of space physics research (e.g., Friedel et al., 2002). The dynamics of the trapped relativistic electron population with energies from hundreds of kilo electronvolts (∼100 keV) to millions of electronvolts (∼1 MeV) during geomagnetic storms is of particular interest because of their potential impact on satellite electronics through deep-dielectric charging (Baker et al., 1994). Electrons can be lost at the beginning of the storm before the subsequent acceleration of a source population to MeV energies, which can partially or completely replace those which are lost, in some cases creating radiation orders of magnitude stronger than that which existed pre-storm (e.g., Reeves et al., 2003). In particular, interaction with high-frequency plasma waves, such as chorus (e.g., Lee et al., 2013; Shprits et al., 2007), electromagnetic ion cyclotron (EMIC) (e.g., Ukhorskiy et al., 2010; Usanova et al., 2014), or plasmaspheric hiss (e.g., Thorne et al., 1973) waves, can cause loss into the atmosphere. Meanwhile, ultra-low frequency (ULF) plasma waves (Mann et al., 2016) can cause particles to be transported outwards to the last closed drift shell (LCDS) and lost at the magnetopause into the solar wind in a process known as magnetopause shadowing (e.g., Olifer et al., 2018; Turner et al., 2012). The relative importance of these processes continues to be hotly debated (Mann et al., 2018; Shprits et al., 2018). This paper, in particular, focuses on investigating whether the LCDS dynamics can explain the fast magnetopause shadowing losses.

Recent studies have shown that a compression of the outer magnetospheric boundary can cause intense electron radiation belt losses occurring on very short (1 day) timescales. For example, Olifer et al. (2018) showed that the LCDS location of the Van Allen belt electrons and its dynamics can explain rapid and intense dropouts of relativistic electrons in some of the most intense geomagnetic storms of the last decade. It is also interesting to investigate on a statistical basis whether a substantial compression of the magnetopause, characterized by inward motion of the LCDS, can cause a radiation belt dropout in every storm and whether the response is repeatable and/or predictable. In this research, we perform a statistical superposed epoch analysis of 69 events which contain rapid and deep magnetopause compressions and reveal how outward radial transport to the compressed magnetopause is the dominant cause of rapid particle depletions in the heart of the Van Allen radiation belt for such storms. Our results also reveal for the first time a remarkable repeatability and indeed similar radiation belt response to such solar wind forcing.

2 Results

Recent event-specific studies of radiation belt dropouts (e.g., Olifer et al., 2018; Ozeke et al., 2020; Tu et al., 2019; Turner et al., 2012) commonly associate the electron loss with the rapid compression of the LCDS into the heart of the radiation belt. In particular, such studies reveal, using in situ observational data and radiation belt modeling, that the dropouts in some storm events are caused by magnetopause shadowing loss enhanced by outward radial diffusion and not by precipitation into the atmosphere. Here, we extend these studies by statistically studying the effects of the LCDS compression. We select events with a single isolated intense compression of the LCDS below the L* value of 5.8 over the course of six days. Notably, the threshold value of 5.8 was selected to obtain the largest number of isolated events with strong magnetopause compression whilst removing any impacts from compound events. For the purposes of this study, we analyze the LCDS dynamics during the entirety of the Van Allen Probes mission (September 2012 until July 2019) and select 69 isolated events with a single LCDS compression. We calculate the location of the LCDS using the neural network LANL* code (Yu et al., 2012) which allows for fast detection of the times with singular LCDS compression. Once the low LCDS events have been identified, accurate L* locations of the LCDS are then determined using the full calculation by the LANLGeoMag library (Henderson et al., 2017) for every storm. By superposing the selected events with respect to the minimum LCDS in each event, we analyze the median radiation belt response to the LCDS compression and its variability. The list of selected events is shown in Table S1 in Supporting Information S1.

2.1 Overview of the Storms

Figure 1 shows the superposed epoch analysis of selected solar wind conditions and geomagnetic indices for these 69 isolated events. Panels a through f show the mean value across different storms in red, median value in black, and upper and lower quartiles in gray as a function of the superposed epoch time in units of days. The panel g shows the median spin-averaged electron flux in the 2.6-MeV channel as a function of the superposed epoch and L* parameter (Roederer, 1970). In this research, the solar wind data were taken from the Qin-Denton dataset provided by the Energetic Particle, Composition, and Thermal Plasma (RBSP-ECT) team (Spence et al., 2013) and the radiation belt flux measurements data were taken from both Van Allen Probes A and B for both of the REPT (Baker et al., 2012) and MagEIS (Blake et al., 2013) instruments. Notably, 66 of the selected 69 events can be considered geomagnetic storms of weak to severe intensity (Dst nT) according to the typical storm classification by Loewe and Prölss (1997). The three remaining events, the May 2015 (minimum Dst = −28 nT), the July 2016 (minimum Dst = −23 nT) and the June 2019 (minimum Dst = −17 nT), still contained a substantial compression of the LCDS (minimum values of 5.5, 5.3, and 5.4 in L*, respectively) and due to the low value of the LCDS are retained in the statistical analysis.

Figure 1

Superposed epoch analysis of selected solar wind parameters, geomagnetic indices, and radiation belt response for the selected geomagnetic storm events. (a) solar wind speed; (b) dynamic pressure; (c) southward component of the interplanetary magnetic field (IMF); (d) geomagnetic storm-time disturbance index (Dst); (e) planetary Kp activity index; (f) L* of the LCDS; (g) median 2.6-MeV spin-averaged electron flux in units of cm⁻²s⁻¹sr⁻¹ MeV⁻¹. For panels (a)–(f), the mean value across all events is shown with the red line, with the black line and two gray lines representing the median value and the lower and upper quartiles, respectively.

Figure 1g shows that the selected low LCDS events display clear signs of fast radiation belt losses, with maximum depletion occurring within half a day and reaching relatively low L* regions of L* ≈ 4 well inside the envelope of the LCDS (Figure 1f). These fluxes are reduced by almost an order of magnitude relative to the pre-storm levels (cf., Figure 3). To further analyze the radiation belt dynamics and determine the dominant mechanism that causes such rapid and intense loss we perform similar superposed epoch analysis for multiple energy channels from both REPT and MagEIS instruments. However, because different energies and L*, as well as different storms, have different pre-storm electron flux levels it is beneficial to perform superposed epoch analysis of the relative flux change, with respect to the pre-storm levels.

Figure 2 shows the results from a superposed epoch analysis of the fractional flux loss, with different energies shown in different panels. Note that the color scale is the same for the different energies and represents the fraction of the flux changes with respect to the pre-storm level. The yellow color represents no changes with respect to the pre-storm flux, and it is the dominant color on the left side of each panel until half a day before the minimum of the LCDS (zero epoch). This result shows that the pre-storm flux in the selected events is a stable population suitable for subsequent loss analysis. The loss across all panels in Figure 2 is represented by the color change from yellow through green to blue. This loss happens earlier for higher L* regions and later for lower L*, but occurs with approximately the same timescale across all energy channels. Meanwhile, the normalized minimum flux level reached during the loss period is lower for higher energies. This indicates that all of the energies react at the same time, starting on the higher L* regions and with the loss propagating inward, but with a loss intensity which increases with energy. On the other hand, the recovery for different energies differs strongly in both intensity and timescale with the lower energies recovering earlier and reaching stronger intensities as compared to the pre-storm flux.

Figure 2

Superposed epoch analysis of the normalized spin-averaged electron flux. Different panels represent different energies that are shown on the bottom left corner of each panel. The color scheme in each panel is the same and represents the median relative change of the flux with respect to the pre-storm levels as a function of L* and superposed epoch days. Electron flux data was taken from both MagEIS and REPT instruments on board both the Van Allen Probes A and B. The mean L* value of the last closed drift shell across all events is shown with the red line, the black line and two gray lines show the median value and the lower and upper quartiles, respectively.

2.2 Similarity and Repeatability of the Radiation Belt Loss

To further investigate the loss dynamics and the self-similarity between the storm events, we analyze the evolution of the fractional reduction of the pre-storm flux at a fixed L* as a function of superposed epoch time. To assess any potentially competing adiabatic effects on the observed flux at fixed energy we also analyze the fractional changes in superposed epoch phase space density (PSD) at fixed first and second adiabatic invariants at the same fixed L*. Figure 3 shows examples of these fixed L* superposed epoch time series at L* = 4.25 with flux plotted in the panels in the left column and PSD plotted in the right columns. Different energies (left) and first adiabatic invariants (right) are plotted in different rows. In each row, the first adiabatic invariant value on the right corresponds to the energy value on the left at L* = 4.25 at the superposed epoch time zero. The median normalized values are shown with black dots and the upper and lower quartiles are represented by the error bars. Such a representation of the flux and PSD dynamics provides an unexpected but extremely important result about the Van Allen belt loss dynamics during intense magnetic storms. The panels in the left column of Figure 3 show that the fractional loss in electron flux is extremely self-similar from one storm to the next. This feature is clearly represented by the small separation between the upper and lower quartiles, which indicates the low level of variability between the different storms during the dropout. Notably, the size of the error bars, representative of the separation between the quartiles, in normalized flux sharply increases from being a factor of three at most up to end of the loss phase, to approximately two orders of magnitude during the recovery. In addition, during the pre-storm phase (up to 3 days before the minimum of the LCDS), the error bars remain small across all energies which indicates that the pre-storm phase of the selected storms is mostly quiet with no significant additional loss processes. Overall, the left column of Figure 3 shows that the temporal profile of the fractional loss of pre-existing radiation belt electrons follows almost exactly the same time profile during the period of loss, independent of the magnitude of the pre-storm flux. Rather unexpectedly the flux evolves such that the relative amount of the particle loss in each event is the same, and depends only on the electron energy and L*.

Figure 3

Storm-time evolution of the fractional change of electron flux (left) and phase space density (PSD) (right) at fixed L*. The left column shows the fractional change of the spin-averaged electron flux with respect to the pre-storm levels at fixed L* = 4.25, with different fixed energies in different rows. The right column shows the fractional change of the electron PSD at fixed first adiabatic invariants at fixed L* = 4.25 and for fixed second adiabatic invariant of K = 0.1 R_EG^1/2. Different first adiabatic invariants are shown in different rows. Note that the first adiabatic invariants shown on the right correspond to the energy channel on the left at the end of the loss phase for the selected L* and K. Energy and invariant values are shown in the top left corner of every panel. The median values are shown with black dots and the upper and lower quartiles are represented with error bars. Additionally, each panel shows a two-sided Gaussian fit to the data plotted with a red line, and with the maximum loss intensity defined in the bottom left corner of each panel.

Similar behavior is seen in the right panel of Figure 3 for the observed evolution of the normalized median PSD and the respective 25% and 75% quartiles. The same observation in PSD demonstrates that the observed changes in flux represent genuine losses and are not explained purely by adiabatic effects, but rather by the LCDS reaching low L*. It is also interesting to note that the variability of both flux and PSD during the acceleration phase remains symmetrical around the median for all studied energies and first adiabatic invariants. In addition to that, the final median value of the flux of more energetic (≥1.8 MeV) electrons which are reached at the end of the acceleration phase is close to the pre-storm level. A natural interpretation of this result is that the level of flux recovery is strongly dependent on the solar wind conditions and the availability of the new source populations with the same probability for flux recovery to reach above or below the pre-storm flux. This result is in good agreement with the analysis of Reeves et al. (2003) where they reported that the properties of the 2-MeV post-storm flux at geosynchronous orbit was almost equally likely to have increased or decreased compared to the pre-storm level.

To further analyze the dynamics of the radiation belt during the dropout we perform a quantitative analysis of the timing and intensity of the loss by fitting a two-sided Gaussian to the L* = const flux and PSD profiles. This approach provides a quantitative method of analyzing loss dynamics. A detailed description is presented in the Supplementary Material. The two-sided Gaussian fits are shown for both flux and PSD in Figure 3 with red lines. The estimated intensity of the loss from the fit as characterized by the ratio of the pre-storm flux (PSD) to minimum flux (PSD), J_PS/J_min (f_PS/f_min), is shown in the bottom left corner in each panel. To confirm the self-similarity of the fractional loss of pre-existing radiation belt flux, we analyze 12 different energy channels from the REPT and MagEIS instruments between energies of 0.5 and 4.2 MeV and at 14 different fixed L* values between 3.75 and 5.57. Figure 4 shows a summary of the loss intensity, or by how much the pre-storm flux is larger than the minimum flux reached during the loss, for these fixed energies as a function of L*. Different energies are shown in different colors and the error bars represent uncertainties in the loss intensity, obtained from the numerical least squares fitting of the two-sided Gaussian profiles.

Figure 4

The intensity of the fractional flux loss for different energies and L* locations in the Van Allen radiation belt. Different energies are shown in different colors with data from the MagEIS instrument marked with an “x” and data from the REPT instrument with a circle. The error bars were obtained from the least-squares numerical fitting of the fractional flux loss with a two-sided Gaussian.

Figure 4 shows a remarkably clear structuring of the loss intensity in both energy and L*. First, it shows that the loss is more intense at higher L*. For different energy channels, the loss at L* = 5.5 is ∼30 times larger than at L* = 4.0 reaching values as large as ∼100 for the highest energies. The magnitude of this loss during low LCDS events is approximately an order of magnitude more intense than has been reported in some previous superposed epoch studies (Morley et al., 2010; Murphy et al., 2018; Olifer et al., 2021; Turner et al., 2019). Second, the fractional loss is very clearly much more significant at higher energies. For example, the loss of the 3.4-MeV population is an order of magnitude more intense than that at 0.6 MeV, across all values of L*. In combination with the L* dependence, it creates a remarkably well-organized picture of a self-similar and repeatable loss profile in energy-L* space. Moreover, the relatively small and non-overlapping error bars confirm that this structure repeats from one storm to the next. It shows that even though the initial pre-storm fluxes can differ significantly in magnitude from storm to storm, the relative change during the loss phase is very well-behaved and described by the repeatable pattern shown in Figure 4. Our discovery of substantial repeatability during radiation belt losses that occur during the LCDS compression has the potential to play a decisive role in the development of improvements to the accuracy of physics-based radiation belt specification and forecast models.

Figure S1 in Supporting Information S1 shows a similar plot to Figure 4 but calculated for PSD. Very similar structuring is also present in the PSD data with the higher losses being observed at higher first adiabatic invariants, and higher L*. Notably, the intensities of the dropouts in PSD are somewhat smaller than those measured in flux at fixed corresponding energies. This proves that the flux decreases are not caused only by adiabatic processes alone, but represent genuine repeatability of loss profiles associated with magnetopause shadowing, as shown below.

2.3 Probing the Origin of the Similarity and Repeatability of the Loss

The superposed epoch analysis performed in this study allows for an extensive investigation of the role of the compressed LCDS in driving losses in the heart of the Van Allen radiation belt, even at L* 5. In particular, it is interesting to investigate how the energy and L* dependence of the loss intensity (Figure 4) might be explained by outward radial diffusion to a compressed LCDS. The role of this process in radiation belt depletion has been examined in computer simulations in a number of recent case-studies (Hudson et al., 2014; Hudson et al., 2015; Mann et al., 2016; Ozeke et al., 2019; Ozeke et al., 2020; Tu et al., 2019; Wang et al., 2020). These studies have already concluded that magnetopause shadowing, perhaps enhanced by ULF wave outward radial diffusion, can explain the losses at L* 5. However, the discussion of whether the losses at lower L-shells (L* 5) can be explained in terms of ULF wave enhanced magnetopause shadowing still continues (Mann et al., 2018; Shprits et al., 2018). Here, we use our observations to investigate the signatures of the loss as defined by its timing, intensity, and location, and assess whether such an outward radial transport model can explain the repeatable characteristics of the observed dropouts.

Figure 5 presents the L* dependence of the timing of the flux loss as characterized by the time of the beginning of the loss, and the epoch of the minimum flux relative to the time of minimum LCDS at epoch time zero for different energies in the same format as Figure 4. It shows that the loss starts earlier at higher L* and propagates inward, with lower L* being affected later. Meanwhile, different energies are affected at the same time at fixed L*. Such behavior is consistent with the radial diffusion paradigm where the particles at the higher L* are affected and see loss effects earlier as they are transported outward to the more proximal magnetopause. The data from the highest L* = 5.57 observed by the Van Allen Probes in the statistics reported here shows that the loss starts half a day before the time that the minimum of the LCDS is reached. Interestingly, this is the same time as the median LCDS typically starts to move inward (cf., Figure 1). In addition, Figure 5 also shows that the losses stop simultaneously for all energies and L* at the time of superposed epoch 0.047 ± 0.003 days, very close to the time at which the LCDS reaches its minimum L* location. These results show that the timing of the energy-independent loss is controlled by the dynamics of the magnetopause as defined by the LCDS location. This is very strong evidence that outward radial diffusion and magnetopause shadowing are likely the dominant cause of the observed losses.

Figure 5

Fractional flux loss timescales. The top panel shows the time before the minimum last closed drift shell (LCDS) when the dropout starts for different energies and L*. The value of this time when the loss begins was obtained from the two-sided Gaussian fitting as described in Supporting Information S1. The bottom panel shows the time inferred as the end of the loss phase based on the fit, compared to the epoch time zero at the time of the minimum L* location of the LCDS. This time also marks the beginning of the recovery. For both panels, different energies are shown in a different color with data from the MagEIS instrument marked with an “x” and data from the REPT instrument with a circle. Also, the timings at fixed L*, shown by the gray shadowing point on the x-axis, are grouped into the shaded regions and are spread out L* for the convenience of the reader. The actual L* value obtained from the analysis corresponds to the center of each bin and is represented in the bottom panel by where the shadowing reaches the x-axis.

To describe the global behavior and trends in the PSD evolution we analyze a statistical PSD which is calculated from the statistically well-behaved median flux dynamics. It is important, as the data of the superposed epoch PSD has limited L* coverage (Figure S2 in Supporting Information S1) and a larger deviation from the median (Figure 2), suggesting that additional variability in PSD profiles is introduced by the magnetic field model used to convert flux to PSD. The detailed description of this transformation is provided in the Supplementary Material. Figure 6 shows the resulting PSD profiles as a function of L* for different fixed first adiabatic invariants, μ. Each fixed μ is shown in different rows, with L* PSD profiles at different times shown in a different color. This statistical PSD is determined from the median flux using a superposed epoch Tsyganenko and Sitnov (2005) magnetic field model at the location of the spacecraft, assuming that the spin averaged flux from Figure 2 is representative of the flux at a local pitch angle of 90°.

Figure 6

Evolution of the phase space density (PSD) profiles. The PSD profiles calculated from the median superposed epoch flux (left column) and their relative change with respect to the pre-storm levels at each L* (right column) for fixed μ at different epochs in advance of the epoch time zero of minimum LCDS. Different rows show data for different values of the first adiabatic invariant, μ, with different epochs shown in a different color according to the legend on the top of the figure. The right column additionally shows a linear fit to the logarithm of the relative PSD losses as a function of L* used for the PSD profiles calculations (see text for details).

The PSD dynamics in Figure 6 again point to outward transport to the compressed LCDS as the dominant cause of the radiation belt depletion. The PSD profiles in the left column show that the population of particles is gradually depleted with time across a wider and wider range of L*, which causes the profiles to monotonically decrease in magnitude outwards toward the LCDS and with a gradient in L* which gets steeper with time. Meanwhile, the right column shows that the normalized relative loss in PSD at each L* is well-behaved and starts earlier and is larger at higher L*. The same result was already noted for the median flux and median PSD in Figure 4 and Figure S1 in Supporting Information S1, respectively. Note also that the PSD profiles become more structured in L* as the epoch approaches the time of the minimum LCDS. For example, the largest PSD fluctuations away from a monotonic profile of PSD decreasing with L* are present for PSD traces at superposed epoch time zero (brown line in Figure 6). According to Figure 1f, 75% of all events contain LCDS below L* = 5.8 at epoch time zero with the lowest LCDS reaching L* = 4.3. Hence, such fluctuations away from a monotonic PSD profile as a function of L* are representative of cases that may have the LCDS inside the L* domain.

To explain the energy dependence of the dropout intensity in terms of a radial diffusion model we turn to an analysis of the temporal evolution of the PSD profiles as a function of L*. We use the PSD data calculated for every analyzed storm to determine how fast the electron population at different μ and L* for fixed K = 0.1 R_EG^1/2 are transported outwards and compare it to that expected based on a model for outward ULF wave radial diffusion to the LCDS. In particular, we assess the normalized outward transport rate, in units of L*/day, that combines both PSD gradients and outward radial diffusion coefficients, D_LL, in one equation [defined by Schulz and Lanzerotti (1974)]:

(1)

where L is interpreted as the L*, D_LL is the radial diffusion coefficient in units of L*²/day, f is the PSD, K is the second adiabatic invariant, and the dot represents the time derivative. Radial diffusion coefficients were obtained from the Kp-parametrization in Ozeke et al. (2014). The gradients of the PSD in log-space are calculated for every individual storm from every inbound or outbound pass of the Van Allen Probes. Then, the superposed epoch transport rates are determined in the similar way as the superposed epoch flux (see Section 2 above and Supporting Information S1). Figure 7 shows the outward radial transport rate as a function of L* and superposed epoch days, with different panels showing different fixed first adiabatic invariants, μ. Note that in computing the rates in Figure 7 we used the L* dependence of PSD from the Van Allen Probe data, but used an assumed equivalence between dipole L and L* in Equation 1 and in the (Ozeke et al., 2014) D_LL.

Figure 7

Outward radial transport rate, . Median superposed epoch outward radial transport rates in units of L*/day with different first adiabatic invariants, μ, shown in different panels as a function of L* and superposed epoch. All data present is calculated for the second adiabatic invariant K = 0.1 R_EG^1/2 using Tsyganenko and Sitnov (2005) magnetic field model. The mean L* value of the last closed drift shell across all events is shown with the red line, the black line and two gray lines show the median value and the lower and upper quartiles, respectively.

Figure 7 shows that there is a clear structuring of the radial transport rates in μ-L* space, consistent with the structuring observed in the loss intensity of the flux as a function of energy and L* (Figure 4). Figure 7 confirms that populations with different energies (that are equivalent to different μ) are being transported outward with different rates even at the same L* because of the dynamical development of energy-dependent PSD gradients. Figure 7 also demonstrates that the maxima of the PSD profiles for a constant μ moves inward with time, consistent with outward radial diffusion driving the loss. Such behavior is of course associated with the appearance of the outward gradients with at lower L* at later times during the loss phase as clearly demonstrated in Figure 7. Significantly, our analysis shows how the energy dependence of the loss can be explained by only utilizing the outward radial diffusion paradigm. The energy dependence of the loss is often used to argue in favor of plasma wave-particle scattering into the loss cone. However, our results show that the energy dependence of the loss shown here can instead be naturally explained by outward radial diffusion to a compressed LCDS.

3 Discussion and Conclusions

In this paper, we assess the dynamics of Van Allen radiation belt losses observed during magnetic storms in which the LCDS penetrated to relatively low L* (L* ≤ 5.8). In contrast to some previous superposed epoch studies, where the median flux loss ranged from a factor of 2–10 with respect to the pre-storm levels, our results show that low LCDS-related losses can generate much stronger losses and deplete the Van Allen belt electron fluxes by up to two orders of magnitude, depending on energy and L*. For example, recent results presented by Turner et al. (2019) show rapid losses happening over a half-a-day period but with much smaller statistically averaged magnitudes, their median electron flux at most only decreasing by an order of magnitude. Similarly, results by Murphy et al. (2018) suggest that total radiation belt content for the relativistic electron population (0.5 MeV) decreases at most by a factor of 4. This suggests that the classical approach of assessing losses in storms selected using the storm-time disturbance index Dst (or its proxy SYM-H) statistically underestimate the strength of extreme losses.

We show for the first time strong evidence for the repeatability and indeed similarity and repeatability in the characteristics of rapid and strong radiation belt dropouts during intense magnetopause shadowing events. The fractional flux change with respect to the pre-storm levels during the loss is effectively statistically almost identical across the assessed events, differing only with energy and location and independent of pre-storm flux. The self-similarity is confirmed by small deviations in individual storms from the superposed epoch median fractional flux loss and quantified by studying the variability in fractional loss using the quartiles of the observed distribution. Notably, the pre-storm flux levels in the selected storms span approximately two or three orders of magnitude, depending on energy. However, the observed fractional loss of flux is effectively almost the same (within a factor of approximately 3 between the quartiles) from storm to storm when the LCDS reaches low L* (here using storms when the minimum LCDS is at L* ≤ 5.8). To our knowledge, this is the only study to date that presents such self-similar behavior of the radiation belt electron loss. The self-organisation and repeatability of the loss we show here can be contrasted with previous descriptions of a relative unpredictability of belt response and which is usually attributed to a delicate balance between acceleration and loss processes (e.g., Reeves et al., 2003). Instead, our results show that the magnitude of the losses is in fact rather predictable and repeatable, once examined in terms of the LCDS location as shown previously in four case studies by Olifer et al. (2018).

Moreover, we can also explain the dynamical evolution of losses in the heart of the belt (L* 5) in terms of outward radial transport and particle loss to the LCDS. Analysis of the PSD profiles reveals that the loss is associated with monotonic gradients in PSD which get steeper with time, with losses at high L* happening first across the L* range between 3 and 6, with more intense losses happening for higher L* and μ. This result is equivalent to that obtained for the electron flux, where a very clear structuring of the loss intensity as a function of energy and L* also exists (cf., Figure 4). The L* dependence of the loss is caused by the strong power-law dependence of the diffusion coefficients on L-shell (Brautigam & Albert, 2000; Olifer et al., 2019; Ozeke et al., 2014), with an energy dependence developing as a result of the radial transport rates in the radial diffusion model depending on the radial PSD gradients at fixed first adiabatic invariants, and thus different energies at fixed L*.

An important aspect of the observed dynamics is the preference for faster losses to develop at higher μ. How can this be explained by outward radial diffusion? The answer is that the impacts of the magnetopause shadowing are experienced more quickly per unit time at higher μ, as a result of the energy dependence of the electron drift. Just inside the LCDS, higher μ particles have more interactions with the magnetopause per unit time, such that the drift-averaged rate of loss per unit time is larger for higher μ. As a result, the rate of inward propagation of the loss (Equation 1) develops a μ dependence (and an energy dependence at fixed L*) due to the faster evolving ∂f/∂L at higher μ at the outer boundary at the LCDS, even if the radial diffusion coefficient, D_LL, inside the LCDS is independent of energy at fixed L*. In other words, the relevant time step for establishing a drift-averaged radial gradient at the LCDS at the outer boundary of the belt, , is the drift time of the electrons and that is shorter for higher μ.

The resulting self-organization of radiation belt loss under the action of radial diffusion from low L* to the LCDS acts across a wide range of energies larger than ∼0.3 MeV. However, this model may also be able to explain why there is no apparent loss for the lower energies below ∼0.3 MeV as well, except in the L* regions which actually lie outside the LCDS. The drift time of such particles, on relatively high L* close to the LCDS, is 1 hr; for example, a 150-keV electron at L = 6 has a drift period of one hour. As shown in Figure 1, although the LCDS penetrates to low L* below 5.8 in every storm included in our statistics, it does not stay at low L* for very long. For example, around 75% of the storms have the LCDS below L* = 6 for 6 hours. For electron energies with drift times on the order of this LCDS penetration timescale, drift-arranged losses and hence the gradients of are quite likely to be smaller. The PSD profiles at these energies do not have enough time to develop strong outward gradients in such cases at the LCDS. Therefore, the outward radial diffusion losses for that population would be expected to be rather slow. Moreover, the electron populations below 300 keV are known to be rapidly replenished at the very beginning of the storm from new source population available from the plasmasheet (e.g., Jaynes et al., 2015; Reeves et al., 2016). Such rapid recovery appears to be able to overcome any losses developing at the LCDS. Thus, the only losses for those particles would exist in the domain of direct magnetopause shadowing outside the LCDS, and only at L* locations where the particle drift time allows the entire drift shell to be emptied during the period of LCDS compression at the magnetopause.

Interestingly, the observed self-similarity of the belt dynamics does not extend into the recovery and acceleration phase. At that time, flux increases as it is partially or completely replenished, varying by orders of magnitude from storm to storm. This suggests a strong variability of this phase that is being controlled by the availability of the source population, as suggested for example by Jaynes et al. (2015), as well as by the strength of the acceleration process itself. Such behavior in the recovery phase is of course also similar to the results presented by Reeves et al. (2003) where they suggest, based on geosynchronous observations, that the recovery of the multi-MeV electron population is highly sensitive to the storm conditions and can vary from storm to storm by orders of magnitude. Additionally, the results we present in our Figure 3 nicely agree with the results of Reeves et al. (2003) where they showed that the MeV electron population usually recovers to levels with an equal probability of being above or below pre-storm levels, depending on the storm.

Overall, our study demonstrates that once characterized in terms of the dynamics of the LCDS, the fast loss of Van Allen belt electrons can not only be explained but also to a large degree predicted as a function of energy and L*. The processes that are responsible for the loss of electrons can be explained in terms of direct magnetopause shadowing at the LCDS, as well as at lower L* due to radial diffusion to the compressed magnetopause. This creates a self-organizing system that affects Van Allen belt electron losses in exactly the same manner from storm to storm. The discovery reported here that the fractional flux loss is essentially identical once LCDS dynamics are examined represents a significant advance towards more accurate physics-based radiation belt specification and forecast models. It promises a relatively simple method of defining the electron population which remains at the end of the loss phase in any storm since it appears that the extent of the loss of the pre-storm population can be defined by a factor that depends on energy and L* and which are connected clearly and repeatably to LCDS dynamics. All radiation belt models consist of the superposition of processes that result in the transport, acceleration, and loss of electrons within the simulation domain. When losses are incorrectly specified in these models, errors will develop either due to the transport and acceleration of phantom particles which should have been lost or due to an under-representation of flux levels of some populations due to unrealistic extreme losses. Consequently, a more accurate specification of the energy and L* dependent loss processes using approaches based on the importance of the LCDS dynamics reported here offers an important advance towards significantly improved radiation belt specification and forecast capabilities.

Acknowledgments

This work was supported by a Discovery Grant from Canadian NSERC to I.R.M., and by the Canadian Space Agency through the Geospace Observatory (GO) Canada program and by PWGSC contract 9F045-150826. The authors thank Harlan Spence and the ECT team for the Van Allen Probe data. Processing and analysis of the MagEIS and REPT data was supported by Energetic Particle, Composition, and Thermal Plasma (RBSP-ECT) investigation funded under NASA's Prime contract NAS5-01072.

Open Research

Data Availability Statement

All RBSP-ECT data and the Qin-Denton parameters are publicly available at the Website https://rbsp-ect.newmexicoconsortium.org/rbsp_ect.php.