Causal Inference in the Outer Radiation Belt: Evidence for Local Acceleration

Abstract

Currently, there is no clear understanding of the comprehensive set of variables that controls fluxes of relativistic electrons within the outer radiation belt. Herein, the methodology based on causal inference is applied for identification of factors that control fluxes of relativistic electrons in the outer belt. The patterns of interactions between the solar wind, geomagnetic activity and belt electrons have been investigated. We found a significant information transfer from solar wind, geomagnetic activity and fluxes of very low energy electrons (54 keV), into fluxes of relativistic (470 keV) and ultra-relativistic (2.23 MeV) electrons. We present evidence of a direct causal relationship from relativistic into ultra-relativistic electrons, which points to a local acceleration mechanism for electrons energization. It is demonstrated that the observed information transfer from low energy electrons at 54 keV into energetic electrons at 470 keV is due to the presence of common external drivers such as substorm activity.

Key Points

• Evidence of direct causality from relativistic into ultra-relativistic electrons, compatible with local acceleration in the outer belt

• Detection of information transfer unveils the mechanisms of energy transfer in radiation belts, important for space weather forecasting

• Information flow formulation of causality has a great potential for space physics discoveries

Plain Language Summary

Despite the fact that the discovery of the radiation belts occurred more than 60 years ago, a comprehensive understanding of the physical processes that are involved in the dynamics of the fluxes of relativistic electrons in the outer radiation belt is still lacking. Development of a thorough physical model accounting for the radiation belt dynamics has the potential to assist mitigation of spacecraft hazards caused by energetic particles. Herein, an information-theoretical approach, based on the methodology of causal inference, is applied for identification of factors that control fluxes of relativistic electrons in the outer belt. The patterns of interactions between the solar wind, geomagnetic activity and belt electrons have been investigated. We have found that inward radial transport from an external source is a less favorable mechanism than local acceleration for the energization of outer radiation belt electrons from relativistic to ultra-relativistic energies.

1 Introduction

The acceleration of electrons in the magnetosphere is one of the fundamental but still unsolved problems of magnetospheric physics. This interesting scientific problem also has important applications. Both lower energy ring current electrons and relativistic radiation belt electrons pose hazards for spacecrafts in the magnetosphere. Electrons in the energy range from a few keV to a few hundred keV can cause surface charging and electrostatic discharges (Garrett, 1981; Garrett & Whittlesey, 2001). Relativistic electrons with energy above 1 MeV are able to penetrate spacecraft shielding and cause internal electric charging and discharges (Frederickson et al., 1991; Wrenn, 1995; Wrenn et al., 2002). In spite of the fact that the discovery of the radiation belts occurred more than 60 years ago, a comprehensive understanding of the physical processes that are involved in the dynamics of the fluxes of relativistic electrons in the outer radiation belt is still lacking.

Among different mechanisms that have been proposed for acceleration of electrons to the relativistic and ultra-relativistic energies (Reeves & Daglis, 2016; Reeves et al., 2003; Turner et al., 2013), local acceleration caused by chorus waves and inward radial diffusion facilitated by ultra-low frequency (ULF) waves are widely considered as the two main ones (Balasis et al., 2016; Reeves et al., 2013; Selesnick & Blake, 1998; Thorne et al., 2013). In fact during intense substorms a seed population with energies of tens to a few hundred keV can be formed and then they can be accelerated to relativistic energies (500 keV up to 2 MeV) by chorus waves (Bortnik et al., 2016; Horne et al., 2005; Thorne et al., 2013). On the other hand, they can further be accelerated to ultra-relativistic energies (above 2 MeV) through gaining energy with inward diffusion driven by ULF waves (Millan & Baker, 2012). The relative importance of these two processes may depend upon the electron energy and location within the radiation belts (Wing et al., 2022). For example, local acceleration by chorus waves in the center of the outer radiation belt can produce populations of electrons with energies of 1–2 MeV. Later, the outward radial diffusion of these populations can bring some of these electrons to the periphery of radiation belts, that is, geostationary orbit (GEO).

The relative impact of these two mechanisms is still under great debate and it is hard to distinguish the driving effects of each mechanism, especially during geomagnetic disturbances. For example, it has been shown that the observed electron acceleration during St. Patrick's storm event of 2015 can be underestimated if we only take into account the radial diffusion mechanism (Li et al., 2016). Ma et al. (2016) have also shown that the observed evolution of high energy electron flux of 0.59 and 3.4 MeV needs both local acceleration and radial diffusion during the 1 March 2013 storm. Accordingly, focused upon a key period in August–September 2014, it was proposed that ULF waves can enhance the electron fluxes up to several MeV (Jaynes et al., 2015). Recently, one study indicated that the radial diffusion of pre-accelerated seed electrons causes the observed electrons enhancement in the energy range of 1.8–3.4 MeV during the 2–3 October 2013 event (Liu et al., 2018).

Identification of the factors that control the evolution of these fluxes of energetic electrons is important because this knowledge provides information to discriminate between various scenarios and models for radiation belt dynamics. Initially, a simple correlation was employed by Paulikas and Blake (1979) to demonstrate the presence of a correlation between the solar wind velocity and fluxes of relativistic electrons with energies >0.7, >1.55, and >3.9 MeV at GEO. Reeves et al. (2011) demonstrated that such a correlation is “one of the best known results in the study of the Earth's radiation belts.” However, in the case of nonlinear relationships, the results from the use of the correlation function can be misleading. Based on 20 years of GEO electron flux data, it was shown that the relationship between the solar wind velocity and fluxes of electrons in the energy range 1.8–3.5 MeV is much more complex than can be described assuming a simple linear dependence (Reeves et al., 2011). The Error Reduction Ratio (ERR) has been used to identify the relative contribution of various factors to the evolution of fluxes of relativistic electrons at GEO (Balikhin et al., 2011; Boynton et al., 2013). The ERR enables the identification of nonlinear relationships between the input parameters, a role similar to that of the correlation function in the linear case. The results obtained by ERR analysis have been used to assess relative contributions of radial diffusion and local wave particle interactions to the dynamics of relativistic electrons at GEO (Balikhin et al., 2011; Boynton et al., 2013). While these results were obtained only for GEO, the current availability of the Van Allen Probes (also known as Radiation Belt Storm Probes: RBSP) mission data allows investigations of the factors that affect the dynamics of relativistic electrons within the entire region of radiation belts.

Recently, information theory has proven itself as a powerful approach to study causal relationships among various coupled complex systems. Due to the information transfer between interacting systems, one tries to find a proper measure capable of inferring the net direction of information transfer and causality, among which conditional mutual information (CMI) (Paluš et al., 2001) and transfer entropy (TE) (Schreiber, 2000), as a lag-specific variant of CMI, have played a key role (Kathpalia et al., 2022; Manshour et al., 2021; Runge et al., 2018; Stumpo et al., 2020; Wing et al., 2016). For example, Wing et al. (2016) used TE to show that information transfer from solar wind velocity and density to geosynchronous electrons with energy range of 1.8–3.5 MeV have a 2-day and 1-day delay, respectively. Following their previous work, Wing et al. (2022) employed CMI to untangle and rank the effects of individual solar wind and magnetosphere variables on the radiation belt electrons with energy range of 480 keV to 4.8 MeV. By calculating CMI, as a generalization of partial correlation, they estimated the time scale of the dependence of the radiation belt electrons on solar wind parameters.

Causality measures like CMI or TE can also be fooled due to the presence of indirect influences or common drivers. Imagine two variables, X and Y, that appear connected when they both share a common cause, C. In this scenario, bi-variate CMI might show a causal link from X to Y, even though there is no direct effect of X on Y. This misleading connection arises because both X and Y are influenced by C. To avoid such confusion, we need to consider the influence of C. Conditioning techniques, like using CMI conditioned also on the third variable C, can help us exclude these spurious causal links and reveal the true relationship between X and Y. For instance, it has been shown by Manshour et al. (2021) that bi-variate CMI yields a spurious causal path from the Auroral Electrojet (AE) to the geomagnetic disturbances at mid-latitudes, and using CMI properly conditioned on the vertical component of the interplanetary magnetic field, B_z, as the common driver, no causal relationship between magnetospheric substorms and geomagnetic storms can be detected.

In the present study, we use CMI as a measure of causality which is, indeed, the mutual information between the cause and the future of the effect variable, conditioned on the history of the effect variable, to investigate the possible coupling direction and pattern of interactions among different electron fluxes of different energies and various solar wind variables and geomagnetic activity indices. We find a direct causal path from low energy electrons into high energy ones. We also find the driving impacts of solar wind and geomagnetic activity on such electron populations. By searching for the presence of possible common drivers of electron fluxes with different energies, we show that relativistic electrons (470 keV) can be accelerated into ultra-relativistic electrons (2.23 MeV) purely by internal mechanisms, pointing to local acceleration.

The present study is organized as follows: Section 2 presents the methods, focusing on causal inference using the information-theoretic framework. In Section 3, we introduce the data set, consisting of solar wind data, fluxes and magnetic activity indices, while in Section 4, we present the results of the analysis. Finally, Section 5 summarizes our findings.

2 Methods: Causal Inference Using Information Theory

Let a discrete random variable X takes values in a set Ξ, and is defined by a probability distribution p(x), then its information content is obtained by the Shannon entropy H(X) (Shannon, 1948), defined as

$$$H(X)=-\sum\limits _{x\in {\Xi }}p(x)\log \,p(x).$$$ (1)

Here we use the natural logarithm and then the entropy is expressed in nats. Suppose Y represents another discrete random variable with the set of values ϒ, and probability distribution of p(y), and also p(x, y) represents their joint probability distribution, then the joint entropy of X and Y can be obtained as

$$$H(X,Y)=-\sum\limits _{x\in {\Xi }}\sum\limits _{y\in {\Upsilon }}p(x,y)\log \,p(x,y).$$$ (2)

The conditional entropy of Y given X is simply defined as

$$$H(Y\vert X)=-\sum\limits _{x\in {\Xi }}\sum\limits _{y\in {\Upsilon }}p(x,y)\log \,p(y\vert x)$$$ (3)

where p(y|x) represents the conditional probability of Y given X. Accordingly, one can find H(X, Y) = H(Y|X) + H(X). Mutual information I(X; Y), which is the common information of two variables X and Y,

$$$I(X\mathrm{;}Y)=H(X)-H(X\vert Y)=H(X)+H(Y)-H(X\mathrm{;}Y),$$$ (4)

can be expressed in terms of probability distributions as

$$$I(X;Y)=\sum\limits _{x\in {\Xi }}\sum\limits _{y\in {\Upsilon }}p(x,y)\log \frac{p(x,y)}{p(x)p(y)}.$$$ (5)

If variables X and Y are independent then p(x, y) = p(x)p(y) and so I(X; Y) = 0, whereas the presence of any dependence among X and Y results in I(X; Y) > 0. Due to the symmetric nature of I(X; Y), one cannot use mutual information as an appropriate measure of causality. However, the conditional version of mutual information may be a proper candidate to infer causality. The CMI of the variables X and Y given the variable Z is defined as

$$$I(X;Y\vert Z)=H(X\vert Z)+H(Y\vert Z)-H(X,Y\vert Z),$$$ (6)

which represents general dependence between two variables of X and Y if we know another third variable Z. One can reformulate the CMI to infer causal relation in an unidirectionally coupled dynamical system (Paluš et al., 2001). In fact, by assuming a time delay between two processes X and Y, the net information about the τ-future of the process Y contained in the process X can be obtained as

$$$I\left(X(t);Y(t+\tau )\vert Y(t)\right),$$$ (7)

which is indeed a nonlinear generalization of the Granger causality (Granger, 1969). This asserts that if, for a particular time delay τ, $$I\left(X(t);Y(t+\tau )\vert Y(t)\right) > 0$$ then there exists a causal link from X to Y, denoted as X → Y. This measure is a powerful tool for analyzing causality between time series and has been applied in a wide variety of systems (Hlaváčková-Schindler et al., 2007). Sometimes, the observed causal relationship of X → Y may be a spurious link due to the presence of a common driver C that drives both X and Y, making a false causal-like relationship between X and Y. One can also use CMI to address such a more complex problem, by considering the effects of one or more common drivers as

$$$I\left(X(t);Y(t+\tau )\vert Y(t),{C}_{1}(t),{C}_{2}(t),\text{\ldots }\right)$$$ (8)

where C_i represents the ith possible common driver. Note that Equation 8 can include delayed variables of C_i(t − τ′), in order to consider all possible delayed effects of C_i. However, the observed time delays in our upcoming results remain close to one day and thus we only include C_i(t) to avoid increasing the computational complexity in CMI functional.

Note also here that the absolute values of CMI may not be informative due to some biases such as finite size effect and give nonzero values even for uncoupled systems. In order to correctly identify the causal links we should compare the absolute value obtained from the original system with that of an uncoupled surrogate system, provided that this surrogate preserves some distinct important statistical features of the studied system (Theiler et al., 1992). Then, if the computed CMI for the original system is significantly different from the values obtained for the surrogate system, one can conclude that a true causal link exists; otherwise the null hypothesis, that the system variables are uncoupled is accepted. A wide range of surrogate tests have been introduced, among which the circularly time-shifted surrogates has proved itself to be well suited for causal inferences (Manshour et al., 2021; Quiroga et al., 2002). In this work, we generate a set of 100 different realizations of the surrogate series and consider its 99th percentile as our significance threshold. Thus, if the CMI values obtained from original processes are larger than this threshold, we consider it as a significant causal link.

3 Data

For the studies of the dynamics of the trapped electron population we used data from the RBSP mission. The RBSP are twin spacecrafts (designated RBSP-A and RBSP-B) designed and deployed to study the radiation belts and the inner magnetosphere. The probes were launched in August 2012 in near equatorial highly elliptic orbit with 10.2° inclination (perigee at 618 km and apogee at about 30,000 km). The mission officially ended in 2019, with the deactivation of RBSP-B in July 2019 and of RBSP-A three months later. One of the main instruments for energetic particle radiation studies is the Magnetic Electron Ion Spectrometer (MagEIS). Magnetic Electron Ion Spectrometer consists of four magnetic spectrometers, which together capture electrons with energies from 20 keV up to 4.8 MeV (Blake et al., 2013; Claudepierre et al., 2021). Specifically for this study we used data derived from MagEIS on board the RBSP-B satellite, from the beginning of 2014 up to the middle of 2019, where the mission officially ended. Data from 2013 were excluded, since they come from the early phases of the mission. All measurements are omnidirectional, differential electron fluxes, in units of s⁻¹ cm⁻² sr⁻¹ MeV⁻¹. Magnetic Electron Ion Spectrometer provides 25 energy channels, but for simplicity the data set was limited to three characteristic energies, namely at 54 keV, 470 keV, and 2.23 MeV, spanning from the very low energy electrons at a few tens of keV to relativistic electrons at ∼500 keV and up to super-relativistic electrons at >2 MeV. The energy widths of the selected channels are 13, 71, and 480 keV respectively. Only fluxes for L* values from 4.5 to 5 and equatorial pitch angle from 80° to 90° were considered, so these are electrons that remain close to the magnetic equator, near the peak of the outer radiation belt. Finally, to avoid short-term fluctuations, all fluxes have been reduced to daily-averaged values. So for each day, there is one flux value for each selected energy channel all within the constraints outlined above. The electron flux data are accompanied by daily-averaged solar wind parameters from NASA's OmniWeb (Papitashvili & King, 2020), namely the z component of the magnetic field B_Z (in Geocentric Solar Magnetospheric—GSM coordinate system), Solar Wind Proton Density (N_sw), Solar Wind Velocity (V_sw) and Pressure (P_sw). Finally, the data set is coupled with daily averaged values of the typically used geomagnetic activity indices, that is, Disturbance storm-time (Dst), ap, AE, and polar cap (pc). We note here that, due to the daily averaging, both the local time dependence of the interactions involved as well as other phenomena of smaller time scales, as are the injection of particles due to substorms and other faster transient variations (Baker et al., 1994; Blake et al., 1992), will not be covered by the present study and are left for future work.

4 Results

In order to check the possible causal relationships between electron fluxes, we calculate the CMI of Equation 7 for all pairs of (F₁, F₂), (F₁, F₃), and (F₂, F₃), where F₁, F₂, and F₃ denote 54 keV, 470 keV, and 2.23 MeV fluxes, respectively. Figure 1a shows information transfer of the form F₁ → F₂ and Figure 1b represents possible causal relationship for the reverse direction of F₂ → F₁, and the same applies to other two pairs. The dashed lines demonstrate the significance threshold which is the 99th percentile of the corresponding CMI values calculated for 100 surrogates discussed in the previous section. The values well above this threshold can be considered as a significant causal relationship. Figure 1 demonstrates that a significant information transfer from low energy to high energy electrons exists. On the other hand, time delay for the maximum information transfer is about 1 day for F₁ → F₂, 2 days for F₂ → F₃, and 3 days for F₁ → F₃. We note here that the observed causal relationships among these three electron fluxes are not due to a general transitivity rule and we demonstrate in the following that only the causal relationship F₂ → F₃ remains significant and the rest are spurious due to the influence of common drivers.

Figure 1

The conditional mutual information (CMI) of Equation 7 between different pairs of electron fluxes, for the information flow directions of (a) F₁ → F₂, (b) F₂ → F₁, (c) F₁ → F₃, (d) F₃ → F₁, (e) F₂ → F₃, and (f) F₃ → F₂, where F₁, F₂, and F₃ denote the 54 keV, 470 keV, and 2.23 MeV fluxes, respectively. The red dashed lines represent the significance threshold obtained as 99th percentile of CMI values for a set of 100 circularly time-shifted surrogates.

As we mentioned above, the acceleration mechanism of low energy electrons into high energy ones can be due to solar wind as well as geomagnetic activity. To check this, we calculate CMI of Equation 7 between all these external variables and different electron fluxes. For example, in Figure 2, we represent the effect of one of the most significant drivers on all electron fluxes, that is, AE. This indicates that substorm activity significantly drives electron fluxes with different energies, and the information transfer time delay is around 1 day or less for F₁ and F₂, and is around 2 days for ultra-relativistic electron fluxes F₃. Figures S1 to S3 (see Supporting Information S1) represent the effect of all other possible external drivers of B_z, V_sw, P_sw, N_sw, Dst, a_p, and pc, on electron fluxes with different energies. We observe that electron flux of very low energy, F₁, can be weakly driven by P_sw, a_p, and significantly driven by AE. The time delay for all of them is around 1 day or less (due to the minimum time step limit of 1 day in our data sets). As can be seen in Figure S2 in Supporting Information S1, there exists a significant information transfer from all external drivers into electron fluxes with relativistic energy, F₂. The time delay for all drivers is ≤1 day except for P_sw which is ≈2 days. The same behavior is observed in Figure S3 in Supporting Information S1 for electron flux of F₃ with ultra-relativistic energy, except for time delays which is mostly longer than before. For example, information transfer takes about 2–3 days for V_sw, Dst, a_p, AE, and pc. Note here that the observed 2-day time delay for transfer of information from solar wind velocity V_sw to MeV electrons is in agreement with previous findings (Wing et al., 2016).

Figure 2

The conditional mutual information (CMI) of Equation 7 between one of the most influential drivers, that is, Auroral Electrojet and electron fluxes with different energies of (a) a very low energy, F₁, (c) relativistic energy, F₂, and (e) ultra-relativistic energy, F₃. (b), (d), and (f) show the corresponding reverse causal directions. The red dashed lines represent the significance threshold obtained as 99th percentile of CMI values for a set of 100 circularly time-shifted surrogates. The notations are the same as in Figure 1.

As we observed so far, there is a significant information transfer from low energy electron fluxes into high energy ones. On the other hand, such electron populations can be driven by external drivers as indicated in Figures S1–S3 in Supporting Information S1 as well as in Figure 2. This triggers an important question: Are the observed causal paths from low to high energy electrons due to some internal mechanisms or driven by common external drivers such as solar wind or geomagnetic activity? To answer this question, we can include the effects of possible common drivers into the causality measure of CMI, as in Equation 8. Accordingly, we investigate the effects of all external forces, one by one, on causal relationships among pairs of electron fluxes observed in Figure 1. For example, Figure 3 represents the effect of AE as one of the most significant (possible) common drivers on different electron fluxes with different energies. Note here that since the obtained values of CMI are strongly dependent on its functional dimension as well as the probability distributions used in that functional, we compare the CMI of Equation 8 (Left column in Figure 3) with its shuffled version of the same dimension (Right column in Figure 3), in which the common driver of C_i = AE (see Equation 8) has been shuffled by randomly permuting C_i timeseries to eliminate its possible driving effect, which, after averaging over 100 different realizations, yields the uncorrected causal paths of F₁ → F₂ (Figure 3b), F₁ → F₃ (Figure 3d) and F₂ → F₃ (Figure 3f). This means that if, in the presence of a common driver C_i, the causal link X → Y vanishes (compared with the shuffled version of C_i), then one can conclude that this observed link of X → Y is a spurious one, emerged due to the presence of that common driver C_i. Figure 3a–3f demonstrate that after including AE as a possible common driver, the previously observed causal paths of F₁ → F₂ and F₁ → F₃ will be destroyed, however, the causality from F₂ to F₃ still remains significant. Figures S4–S6 in Supporting Information S1 also represent the effect of all other possible common drivers on causal paths already observed in Figure 1. As we can see in Figure S4 in Supporting Information S1, the causal path of F₁ → F₂ vanishes when including the effects of some external forces as the possible common drivers (ap, AE, and pc), indicating that this observed causal relationship may be due to the presence of common drivers. Similarly, Figure S5 in Supporting Information S1 shows the possible common drivers (a_p and AE) of the causal path F₁ → F₃. However, in Figure S6 in Supporting Information S1, the observed causal relationship from relativistic electrons into ultra-relativistic ones, that is, F₂ → F₃, is not destroyed in the presence of all external variables.

Figure 3

The conditional mutual information (CMI) of Equation 8 between different electron fluxes of (a) F₁ → F₂, (c) F₁ → F₃ and (e) F₂ → F₃, after including the effect of one of the most influential possible common driver C₁ = Auroral Electrojet (AE). These CMI values (left column) are compared with the corresponding values in which the effect of the common driver AE has been eliminated by shuffling (right column). Panels (g, h) are similar to (e) and (f), but the effect of the other two most influential (possible) common drivers of C₂ = V_sw and C₃ = Dst are also included. The red dashed lines represent the significance threshold obtained as 99th percentile of CMI values for a set of 100 circularly time-shifted surrogates. The notations are the same as in Figure 1.

We also include the role of the three most influential external variables of C₁ = V_sw (as a solar wind parameter), C₂ = Dst (as a magnetic storm index), and C₃ = AE (as a magnetospheric substorm index) into the CMI of Equation 8, noting that the choice of these variables are also consistent with those found in Wing et al. (2022). Figures 3g and 3h demonstrate that the observed causal relationship of F₂ → F₃ still remains significant for a range of time delays and thus can be considered as a robust causal relationship. This asserts that the electron fluxes of relativistic energies can be accelerated into ultra-relativistic energies purely by some internal mechanisms (see also Figure S8 in Supporting Information S1).

5 Conclusion and Discussion

The evolution of fluxes of energetic electrons in the magnetosphere involves a multitude of physical processes operating on a broad range of spatial scales, from hundreds of meters to tens of thousands of kilometers (e.g., magnetopause shadowing). Understanding the acceleration mechanism of outer radiation belt electrons to relativistic and ultra-relativistic energies, remains a central research theme for magnetospheric physics (Lejosne et al., 2022). In particular, determining the relative importance of local acceleration due to whistler-mode chorus waves, versus large-scale acceleration associated with inward radial diffusion due (mainly) to ULF waves is a topic of intense study, because of its great impact on the parameterization of space weather models.

Recently developed approaches in the field of nonlinear time series analysis offer great potential for exploring the complex (input-output) dynamical system of the Earth's magnetosphere (see the recent review by Balasis et al. (2023) and references therein). In this context, causal discovery based on information theory is a new key application to magnetospheric research. In this article, we investigated the information transfer within electron fluxes as well as between external drivers and electron fluxes in the outer radiation belt. Using CMI as a causality measure, we showed that there is information transfer from low energy electrons into high energy ones, and also from solar wind and geomagnetic activity, as external drivers, into such fluxes. Further, we indicated that the observed transfer of information from very low energy electrons (≈54 keV) into relativistic electrons (≈470 keV) is due to the presence of some common external drivers (such as AE) and does not represent a direct causal path. This is in agreement with a series of studies suggesting that the most significant enhancements of relativistic electron flux occur either during time intervals of prolonged substorm activity, as quantified by the AE index (Meredith et al., 2003) or during the period of High-Intensity Long-Duration Continuous AE Activity events (Hajra et al., 2015; Tsurutani et al., 2006). However, we found that the statistically significant causality in the direction from relativistic electrons to ultra-relativistic ones (≈2.23 MeV) is a direct causal relationship, after including the effects of all solar wind and geomagnetic activity indices, as possible external drivers. Our finding provides a hint for local acceleration of relativistic electrons into ultra-relativistic ones, since radial diffusion is associated with externally driven ULF waves. This result is consistent with recent studies, in which the local acceleration have been found to be the dominant acceleration mechanism at L* = 4 − 5.5 (Boyd et al., 2018; Wing et al., 2022).

In summary, our findings suggest, at least based on the analyzed data of fluxes, solar wind parameters and magnetic indices, that inward radial transport from an external source is a less favorable mechanism than local acceleration for the acceleration of outer radiation belt electrons from relativistic to ultra-relativistic energies.

Acknowledgments

The authors would like to thank M. A. Balikhin for valuable discussions. The authors acknowledge the Kyoto World Data Center (WDC) for Geomagnetism and the observatories that produce and make SYM-H and AE indices available at http://wdc.kugi.kyoto-u.ac.jp. We also thank B. Blake, J. Fennell, S. Claudepierre, D. Turner for the use of the MagEIS data. This research was supported by the International Space Science Institute (ISSI) in Bern, through the ISSI International Team project #455 “Complex Systems Perspectives Pertaining to the Research of the Near-Earth Electromagnetic Environment.” PM and MP acknowledge support by the Czech Science Foundation, Project No. GA19-16066S and by the Czech Academy of Sciences, Praemium Academiae awarded to M. Paluš.

Open Research

Data Availability Statement

For the electron fluxes we used data derived from the MagEIS instrument on board the RBSP-B satellite (Blake et al., 2013), which are available at the RBSP-ECT website via https://rbsp-ect.newmexicoconsortium.org/science/DataDirectories.php. The electron flux data are accompanied by daily-averaged solar wind parameters from NASA's space physics data facility OMNIWeb (Papitashvili & King, 2020) via https://doi.org/10.48322/5fmx-hv56, as well as with daily averaged values of the typically used geomagnetic activity indices which were also retrieved from the same source.