Ensemble modeling of long-term radiation belt dynamics: accounting for variability in radial diffusion

This study presents ensemble modeling of outer radiation belt dynamics observed by Van Allen Probes throughout 2017 using DREAM3D simulations. We utilize a recently developed empirical radial diffusion coefficient ($D_{LL}$) model with statistical distributions, employing two outer boundary (OB) conditions: Van Allen Probes data at $L^{*}=5.5$ and GOES data at $L^{*}=6$, for two magnetic moments $\mu =512keV/G$ and $\mu =1237keV/G$. The results demonstrate that OB location critically influences model performance. A model using Van Allen Probes OB at $L^{*}=5.5$ better reproduces observations, particularly for high $\mu$ electrons. In this scenario, radial diffusion alone can largely explain the dynamics because the OB is located near the phase space density (PSD) peak region. Stronger $D_{LL}$ outside the plasmasphere effectively captures inward/outward radial diffusion to/from the OB, while an intermediate $D_{LL}$ around the 30th percentile best matches electron PSD inside the plasmasphere. With GOES OB at $L^{*}=6$, not only does strong $D_{LL}$ remain crucial for achieving good performance, but chorus heating also becomes critical. Model performance inside the plasmasphere is the best for the intermediate $D_{LL}$ but consistently worse than that outside the plasmasphere, suggesting a need for improved $D_{LL}$ and loss inputs in the plasmasphere.

1 Introduction

The dynamics of relativistic electrons in the outer radiation belt are strongly affected by plasma waves including ultra-low frequency (ULF) waves, such as the Pc5 ULF waves and the electromagnetic ion cyclotron (EMIC) waves, and very-low frequency (VLF) waves such as hiss and chorus waves. Wave-particle interactions play important roles in outer radiation belt dynamics by balancing between source and loss processes (e.g., Li and Hudson, 2019 and references therein). These physical processes are characterized by their diffusive nature which can be described by a Fokker-Planck equation. As an example, the global 3-D diffusion model, Dynamic Radiation Environment Assimilation Model (DREAM3D), is employed to analyze the dynamics of radiation belts by solving the 3-D Fokker-Planck equation, specifically focusing on the diffusive behavior of energetic electrons (Reeves et al., 2012; Tu et al., 2013; 2014; Lee et al., 2024).

A significant challenge in developing deterministic diffusion models lies in representing ULF and VLF wave power, which typically follows a log-normal distribution (Li et al., 2011; Tu et al., 2013; Aryan et al., 2014; Spasojevic et al., 2015; Ali et al., 2016; Liu et al., 2016; Meredith et al., 2021). The mean and median of such distributions can differ by orders of magnitude, and the choice between them critically impacts model performance. For example, Lee et al. (2024) demonstrated that a radial diffusion model based on the geometric mean (i.e., median) of ULF wave power from Ali et al. (2016) was too weak to reproduce observed electron dynamics, whereas models based on the arithmetic mean from Ozeke et al. (2014) performed comparably better. It indicates that a fundamental sensitivity in radiation belt modeling that the outcome is highly dependent on the statistical representation of wave power. Previous studies have quantified uncertainties arising from parameters like electron density or chorus wave latitude (Camporeale et al., 2016; Hua et al., 2022; Hua et al., 2023), but a systematic investigation into the sensitivity of global electron dynamics to the statistical distribution of radial diffusion rates has been lacking.

Another approach is by setting up physical models in a stochastic way rather than adopting deterministic models. Thompson et al. (2020) applied the stochastic parameterization of radial diffusion in the context of “variability” for its spatial and temporal scales. They suggested that spatial and temporal variability, informed by measurements, contributes to the variation in electron phase space density (PSD) observed in numerical experiments. For example, if radial diffusion coefficients vary rapidly (i.e., with short temporal variability), the radial diffusion rates increase in average compared to the deterministic radial diffusion model due to the fact that the deterministic model is determined by the median wave power but the average wave power is much higher in log-normal distributions. Although the study introduced a novel approach, its application to realistic radiation belt dynamics is still absent and the proof-of-concept study was limited to a short-term (a few days) investigation.

Furthermore, Camporeale et al. (2016) investigated the uncertainties regarding the radiation belt simulations. Based on the uncertainty of three parameters: Kp-index, the maximum latitude extent of chorus waves, and the electron density, they conclude that the electron density is the most sensitive factor for the flux variation of outer radiation belt electrons. However, this study has limitations in that they only focused on pitch angle-energy diffusion without cross diffusion effects at a single L-shell, L = 4.5. Hua et al. (2022), Hua et al. (2023) also conducted an ensemble study on the pitch angle-energy diffusion including cross diffusion at a single L-shell. Specifically, Hua et al. (2022) conducted an ensemble modeling of the electron flux decay by hiss waves, and Hua et al. (2023) did similar research on the electron acceleration by chorus waves, quantifying the uncertainty from input parameters such as wave amplitude, frequency, background magnetic field, and electron density. They concluded that wave amplitudes contribute the most to the uncertainty in the flux decay by hiss waves, while the background magnetic field contributes the least. For the electron acceleration by chorus waves, the background magnetic field also contributes the least to the uncertainty of the modeled flux, while the other three parameters result in comparable uncertainty levels.

As for radial diffusion, several radial diffusion coefficients ($D_{LL}$) have been calculated using different approaches and incorporated into simulations over a few decades (Brautigam and Albert, 2000; Ozeke et al., 2014; Liu et al., 2016; Ali et al., 2016). Wang et al. (2019) and Drozdov et al. (2021) investigated the performance of various empirical models of radial diffusion coefficients. Lee et al. (2024) compared four empirical models of radial diffusion coefficients, focusing on their effects on the model performance. The study concluded that the radial diffusion coefficients from Ali et al. (2016) are too weak to reproduce the observed electron dynamics in the outer radiation belt because the Ali model used the geometric mean values (close to median) of ULF wave power to estimate the $D_{LL}$. On the other hand, the other three empirical models resulted in comparable levels of electron variations because they all used the mean values of ULF wave powers in $D_{LL}$ estimation. These results suggest that the selection of mean or median value from a wave power distribution to represent an empirical model of $D_{LL}$ is critical for the model performance.

This study aims to fill that gap by performing a comprehensive ensemble modeling study. We leverage the new empirical model for radial diffusion coefficients from Murphy et al. (2023). Unlike previous models, this model not only provides a deterministic median value but also quantifies the statistical distribution of residues (bias and standard deviation). Moreover, this linear regression model follows the median value of ULF power distribution similar to Ali et al. (2016) and parameterizes $D_{LL}$ in the physical $L^{*}$ coordinate (which follows the electron drift shells) rather than the dipole L or McIlwain L used in other empirical $D_{LL}$ models. Note that full theoretical treatment of radial diffusion in background magnetic fields where L* is not equal to dipole L is given in Cunningham (2016). In addition, the Murphy model uses Sym-H as a geomagnetic factor, which is different from the other radial diffusion coefficient models adopting the Kp-index. The Murphy model also includes three more parameters of solar wind conditions: Z-component of interplanetary magnetic field ($B_z$), solar wind speed ($V_{sw}$), and dynamic pressure ($P_{dyn}$). The radial diffusion coefficient is calculated by a median multi-regression method based on these five parameters. Moreover, they provided the bias and the standard deviations at each $L^{*}$ bin of $D_{LL}$ so that we can reproduce the gaussian-fitted residue distributions of the radial diffusion coefficients along $L^{*}$. This unique feature allows us to transform the deterministic model into a stochastic one, enabling the generation of an ensemble of $D_{LL}$ values for a range of percentiles from the underlying ULF wave power distribution. By using this ensemble, we can systematically quantify the sensitivity of the radiation belt system to different levels of radial diffusion intensity.

Furthermore, we extend our sensitivity analysis to a less obvious but equally critical factor: the outer boundary condition. The location and data source for the outer boundary condition can significantly influence the electron flux that is transported into the radiation belts. In this study, therefore, we investigate how the electron radiation belt response to varying radial diffusion rates is modulated by different, commonly used outer boundary conditions. By comparing simulations with an outer boundary condition derived from Van Allen Probes against those with an outer boundary condition derived from GOES, we explore the interplay between the external source and internal transport mechanisms.

This paper is organized as follows. In Section 2, we briefly introduce the DREAM3D model, and various empirical models of radial diffusion coefficients compared to the model from Murphy et al. (2023). In Section 3, we present a set of ensemble simulations during the entire year of 2017 using the outer boundary (OB) condition at $L^{*}=5.5$ derived from Van Allen Probes data. In Section 4, we changed the OB condition location from $L^{*}=5.5$ (based on Van Allen Probes data) to $L^{*}=6$ (derived from GOES data) and investigated the differences in the simulation results based on the ensemble runs.

2 Simulation model and inputs

The 3-D Fokker-Planck equation is the governing equation of the DREAM3D model (Schulz and Lanzerotti, 1974):

$\begin{array}{ll}\hfill \frac{\partial f}{\partial t}=& \frac{1}{G}\frac{\partial }{\partial \alpha }\left({GD}_{\alpha \alpha }\frac{\partial f}{\partial \alpha }\right)+\frac{1}{p^2}\frac{\partial }{\partial p}\left({p^{2}D}_{pp}\frac{\partial f}{\partial p}\right)+\frac{1}{G}\frac{\partial }{\partial \alpha }\left({GD}_{\alpha p}\frac{\partial f}{\partial p}\right)\hfill \\ \hfill & +\frac{1}{p^2}\frac{\partial }{\partial p}\left({p^{2}D}_{p\alpha }\frac{\partial f}{\partial p}\right)+L^2\frac{\partial }{\partial L}\left(\frac{D_{LL}}{L^2}\frac{\partial f}{\partial L}\right)-\frac{f}{\tau }\hfill \end{array}(1)$

where $f$ is the electron phase space density (PSD), $\alpha$ and $p$ are the equatorial pitch angle and the momentum of electron. $D_{\alpha \alpha }$, $D_{pp}$, and $D_{\alpha p}\left(=D_{p\alpha }\right)$ are the bounce and drift-averaged pitch angle, momentum, and mixed pitch angle-momentum diffusion coefficients, respectively. The electron lifetime, $\tau$, includes the effects of the electron loss by Coulomb collision with the atmosphere (Tu et al., 2013) and the loss outside the last closed drift shell (LCDS) of electrons (Tu et al., 2019). For further information and more details about this model, please see the previous papers (Tu et al., 2013; Tu et al., 2014; Lee et al., 2024). In this study, we adjust the magnitude of radial diffusion coefficients in Equation 1 for the ensemble simulations, while keeping the 2-D pitch angle-momentum diffusion coefficients unchanged for all ensemble runs, to focus on the model uncertainty from radial diffusion. We use an electron lifetime model by hiss waves from Orlova et al. (2014) and the chorus wave model from Tu et al. (2014). The 2-D pitch angle-momentum diffusion coefficients of chorus waves are calculated using a Pitch Angle and Energy Diffusion of Ions and Electrons (PADIE) code (Glauert and Horne, 2005) based on CRRES statistical wave data, although DREAM3D can also calculate diffusion coefficients that resolve problems with the formulation in PADIE, as described in Cunningham (2023).

For the simulation runs presented in the following two sections, we implement two different outer boundary conditions. First, we set up a data-driven boundary condition at $L^{*}=5.5$ using Van Allen Probes PSD data. Second, we changed the data-driven outer boundary condition to $L^{*}=6$ and derived it using the GOES PSD data. Baker et al. (2019) performed a statistical comparison of electron flux data from Van Allen Probes and GOES and showed that the flux values measured by GOES-15 sensors matched up very well with Van Allen Probes sensors when the spacecraft were physically close to one another, especially at the energy of E > 0.8 MeV. This justifies the use of GOES outer boundary condition to drive the simulations for Van Allen Probes electron measurements. The implementation of the two outer boundary conditions is motivated by our statistical data analysis shown in Figure 1, where we calculated the gradients of logarithmic electron PSD over $L^{*}$ using Van Allen Probes data over the entire year of 2017, only including data outside the plasmasphere. Figure 1a presents the occurrence (i.e., number of samples) of different gradient values along $L^{*}$ at $\mu =510MeV/G$, while Figure 1b shows the same for $\mu =1237MeV/G$ at a constant $K=0.117G^{1/2}{R}_E$. For the lower $\mu$ case, PSD gradients for $5<L^{*}<6$ show a median around zero, but the distribution is skewed towards positive gradients as shown in Figure 1a, indicating a very small positive slope with respect to $L^{*}$. The gradients then shift positive at lower $L^{*}$. In contrast, for the higher $\mu$ case, the gradient values are mostly negative for $5.5<L^{*}<6$, around zero for $5<L^{*}<5.5$, and positive values at $L^{*}<5$. This suggests that the PSD vs. $L^{*}$ distribution exhibits a local peak around $5<L^{*}<5.5$, with PSD values decreasing on either side of this peak. These statistical PSD vs. $L^{*}$ distributions at different $\mu$ values indicate that the choice of $L^{*}=5.5$ versus $L^{*}=6$ for data-driven outer boundary conditions could significantly influence our simulation results, an effect that will be explored in the next section.

Figure 1

Figure 1. The occurrence of gradients of observed electron PSD (in logarithmic space) over $L^{*}$ outside of plasmasphere at (a) $\mu =510MeV/G$ and $K=0.117G^{1/2}{R}_E$ and (b) $\mu =1237MeV/G$ and $K=0.117G^{1/2}{R}_E$ using Van Allen Probes PSD data over the entire year of 2017. The time resolution is 3-h so that the total number of samples are 2,920.

To specify the variability of radial diffusion with these different outer boundary conditions, we conduct an ensemble modeling for the entire year of 2017 with 11 members (1^st, 10th, 20th, 30th, 40th, 50th, 60th, 70th, 80th, 90th, 99th percentiles) of the radial diffusion coefficient $D_{LL}$ model from Murphy et al. (2023). Figures 2a–d shows the time history of input parameters for $D_{LL}$ from Murphy et al. (2023). In Figure 3, we show the 11 percentiles of the radial diffusion coefficient from Murphy et al. (2023) in dashed purple lines. The thick purple and pink curves indicate the median and mean values, respectively. The mean curve in pink is located close to the 60th percentile at low $L^{*}$ and 70th percentile at high $L^{*}$ for both quiet time in Figure 3a and storm time in Figure 3b. Note that the mean is always higher than the median (the 50th percentile) in a log-normal distribution. To directly compare Murphy’s Sym-H and solar wind dependent $D_{LL}$ model (thick purple and pink curves) with the other Kp-dependent $D_{LL}$ models (see Kp index in Figure 2e), i.e., from Brautigam and Albert (2000), Ozeke et al. (2014), Liu et al. (2016), and Ali et al. (2016) (red, black, green, and blue curves in Figure 3 respectively), two sample periods are selected in Figure 3. Figure 3a is for a quiet time on 15 Feb 2017 with 3-h averaged geomagnetic input parameters: $Kp=0,V_{sw}=308km/s,P_{dyn}=1.53,B_z=0.53nT,$ and $SymH=-2.3nT$, and Figure 3b for a storm time on 8 Sep 2017 with parameters: $Kp=8,V_{sw}=722km/s,P_{dyn}=3.4nT,B_z=-12.8nT$, and $SymH=-132$. During the quiet time, the comparison in Figure 3a shows that the magnitude of radial diffusion coefficients of all the listed models agrees within an order of magnitude. During the storm time as shown in Figure 3b, however, Ali’s model and Liu’s model greatly underestimate the coefficients compared to the others due to their weaker dependence on Kp (Drozdov et al., 2021; Lee et al., 2024), while the others are still in agreement within an order of magnitude. Interestingly, during both quiet and storm times, the median $D_{LL}$ values from the Murphy model are higher than those from the Ali model, despite Ali et al. (2016) also using median ULF wave power values. This could be due to different data sources used in these two models and different parameterization methods. Overall, the comparison in Figure 3 highlights that the choice between mean and median in empirical $D_{LL}$ models, as well as the selection of geomagnetic indices for parameterization, can significantly impact the estimated $D_{LL}$ values, especially during storm time.

Figure 2

Figure 2. Time history of daily averaged solar wind parameters: (a) IMF Bz, (b) solar wind speed (Vsw), (c) dynamic pressure, and geomagnetic indices: (d) Dst index, and (e) Kp index are shown for the entire year of 2017.

Figure 3

Figure 3. Radial diffusion coefficients of each empirical model along $L^{*}$. The thick purple and pink curves indicate the median (the 50th percentile) and mean of radial diffusion coefficients from Murphy et al. (2023), denoted as $D_{LL,med}^{Murphy}$ and $D_{LL,mean}^{Murphy}$, respectively. The ten dashed purple curves indicate the 1^st, 10th, 20th, 30th, 40th, 60th, 70th, 80th, 90th, 99th percentile of radial diffusion coefficients from Murphy et al. (2023), denoted as $D_{LL,perc}^{Murphy}$. For comparison, the radial diffusion coefficients from Brautigam and Albert (2000), Ali et al. (2016), and Ozeke et al. (2014) are shown in red, blue, and black curves, respectively. The radial diffusion coefficients from Liu et al. (2016) are also shown in the green solid curve and green dashed curve for lower $\mu =523keV/G$ and higher $\mu =1237keV/G$, denoted as $D_{LL}^{Liu,low}$ and $D_{LL}^{Liu,high}$, respectively. (a) The left panel show an example coefficients distribution along $L^{*}$ during a quiet time on 02.15.2017 with 3-h averaged geomagnetic input parameters: $Kp=0,V_{sw}=308km/s,P_{dyn}=1.53,B_z=0.53nT,$ and $SymH=-2.3nT.$ (b) The right panel during a storm time on 09.08.2017 with parameters: $Kp=8,V_{sw}=722km/s,P_{dyn}=3.4nT,B_z=-12.8nT,$ and $SymH=-132$.

3 Ensemble modeling with Van Allen Probes outer boundary condition at ${\mathit{L}}^{*}=\mathbf{5.5}$

We first examine the simulation results at a lower value of the first adiabatic invariant, $\mu =523MeV/G$ (with fixed $K=0.117G^{1/2}{R}_E$). Figure 4a1 shows the observed PSD from combined GOES and Van Allen Probes data over $L^{*}=2.5-5.5$ throughout 2017. Note that because GOES satellites typically covers $L^{*}>5.5$ while Van Allen Probes covers $3<L^{*}<5.5$, Figure 4a1 mostly show Van Allen Probe observation results (see Figures 7a1,b1 for the larger coverage). Figures 4a3–a7 display the corresponding DREAM3D simulation results using increasing percentiles (10th–90th) of radial diffusion coefficients ($D_{LL}$), while Figure 4a2 shows the case with 10th percentile $D_{LL}$ while turning off the chorus waves in the model. From panel (a2) to (a7), we see that the electrons get transported inward faster with higher percentiles of $D_{LL}$ and the electron PSD enhancement becomes stronger, indicating more efficient inward radial diffusion from the outer boundary. Meanwhile, stronger $D_{LL}$ also facilitates more rapid PSD dropouts through outward radial diffusion, as seen during the early May period. By comparing the model results in Figures 4a2,a3 both with 10th percentile of $D_{LL}$ but one with chorus waves and one without, we find that the chorus wave contribution results in slightly higher PSD compared to the no-chorus case at the same $D_{LL}$ percentile, but the difference is marginal. This suggests that, at this low $\mu$, radial diffusion dominates the overall PSD enhancement with Van Allen Probes outer boundary (OB) condition at $L^{*}=5.5$.

Figure 4

Figure 4. PSD distribution (in unit of ${\left(c/MeV/cm\right)}^3$) and error analyses of different magnitudes of radial diffusion coefficients for (a1–a8) $\mu =510MeV/G$ and $K=0.117G^{1/2}{R}_E$ and (b1–b8) $\mu =1237MeV/G$ and $K=0.117G^{1/2}{R}_E$ using Van Allen Probe outer boundary condition at $L^{*}=5.5$. Panels (a1, b1) show the observed PSD data from GOES and Van Allen Probes satellites. The black solid lines show the location of plasmapause. (a2, b2) show the simulated PSD from DREAM3D with the 10th percentile of radial diffusion coefficient excluding chorus waves. On the other hand, simulated PSD from DREAM3D simulations with chorus waves includes results with (a3, b3) 10th, (a4, b4) 30th, (a5, b5) 50th, (a6, b6) 70th, and (a7, b7) 90th percentiles of radial diffusion coefficients. Panel (a8, b8) show the calculated MSA as a function of $L^{*}-L_{pp}$ with the purple lines for 10th, blue lines for 30th, green lines for 50th (median), orange lines for 70th, and red lines for 90th percentiles. The other percentiles are shown in gray lines for better visibility for the others.

In addition, by comparing the model results with various $D_{LL}$ percentiles with the PSD data on the top, we find the model performance is different inside and outside the plasmasphere. To quantitatively evaluate model performance, we use the Median Symmetric Accuracy (MSA) defined as (Morley et al., 2018):

$MSA\left(L^{*}\right)\left[\%\right]=100\left(e^{M\left(\left|\mathit{ln}\left(m_i/d_i\right)\right|\right)}-1\right)(2)$

where $M$ is a median function over time with index $i$, and $m_i$ and $d_i$ are arrays of model results and observation data at a given $L^{*}$ over different times, respectively. To assess dependence on location relative to the plasmapause ($\left.L_{pp}\right)$ calculated by the empirical relation from Carpenter and Anderson (1992), MSA is plotted as a function of $L^{*}-L_{pp}$ in Figure 4a8. The MSA results show that outside the plasmasphere (i.e., $L^{*}-L_{pp}>0$), stronger $D_{LL}$ yields better model performance, with the 99th percentile performing best. However, improvements become less sensitive beyond the 50th percentile (green curve). Inside the plasmasphere, the model performance is worse overall compared to outside the plasmasphere. Here, the best performance is achieved near the 30th percentile, indicating that weaker radial diffusion better reproduces the observed PSD inside the plasmasphere (i.e., $L^{*}-L_{pp}<0$). The MSA results inside the plasmasphere also show more sensitive behavior to $D_{LL}$ compared to outside.

This trend is further illustrated in Figure 5, which shows the time series of modeled PSD at fixed $L^{*}$ values with respect to plasmapause in the left panels. The simulation results with different percentiles of $D_{LL}$ are shown in different colors with the data points in black. In addition, the distributions of logarithmic values of the model to observation data ratio, which is the exponent of MSA - see Equation 2, are shown in the right panels of Figure 5. Four locations with respect to the plasmapause ($L^{*}=L_{pp}+1.0,L_{pp}+0.5,L_{pp},$ and $L_{pp}-0.5$) are selected for illustration in Figure 5. Outside the plasmasphere ($L^{*}>L_{pp},$ top two rows, we find that lower $D_{LL}$ leads to underestimation of PSD, which is suggested by the higher tails at negative $\mathit{ln}\left(m_i/d_i\right)$ parts of the distributions in Figures 5b,d. This is also illustrated in Figures 5a,c. For example, we see that a lower $D_{LL}$ model results in weaker PSD enhancements compared to the data in early March and mid-August. Interestingly, we also find that a higher $D_{LL}$ sometimes also better reproduces the outward radial diffusion loss to the outer boundary, such as the case in May in Figures 5a,c, which shows lower $D_{LL}$ leads to overestimation of the observed PSD. Note that the current model generally well produces the observed PSD enhancement and loss over the year of 2017, while some abrupt PSD drops are not captured by the model, mostly at higher $L^{*}$ regions, e.g., some of the sharp PSD dips in the data in Figure 5a are not captured by the model.

Figure 5

Figure 5. PSD variations (in unit of ${\left(c/MeV/cm\right)}^3$) and error distributions at $\mu =510MeV/G$ and $K=0.117G^{1/2}{R}_E$. (a,c,e,g) PSD of observation data in black line with dots and each percentile in dotted line with colors shown on the right legend panel at different locations, (a) $L^{*}=L_{pp}+1.0$, (c) $L^{*}=L_{pp}+0.5$, (e) $L^{*}=L_{pp}$, and (g) $L^{*}=L_{pp}-0.5$, respectively. (b,d,f,h) The distributions of logarithmic ratio between modeled PSD and observed PSD corresponding (a,c,e,g), respectively.

Near and inside plasmasphere, the MSA results in Figure 4a8 show that the overall model performance is worse compared to those outside the plasmasphere. This is further illustrated in Figures 5e–h where more model results are largely different from the data and the error distributions in Figures 5f,h show more spread over the x-axis compared to those outside the plasmasphere. The results show that lower $D_{LL}$ or weaker radial diffusion leads to underestimation of the observed PSD with higher tails at negative $\mathit{ln}\left(m_i/d_i\right)$. This is also illustrated in Figure 5g, where the observed PSD peaks in February and May are underestimated by the model when the radial diffusion is too slow. On the other hand, higher $D_{LL}$ tends to overestimate PSD showing the widespread distribution of errors and higher tails at positive $\mathit{ln}\left(m_i/d_i\right)$ parts in Figures 5f,h. Clear examples of this overestimation with high $D_{LL}$ is shown in Figures 5e,g in April and the end of November. The high $D_{LL}$ model results also overestimate PSD during the observed PSD drop periods as shown in Figures 5e,g too. Consequently, consistent with the MSA results shown in Figure 4a8, the 30–40th percentile $D_{LL}$ mode results perform best overall, with the corresponding error distribution more narrowly centered around zero in Figures 5f,h for near and inside the plasmasphere.

For the model results at higher $\mu$ value ($\mu =1237MeV/G$), shown in Figure 4b1–b7; Figure 6, the features remain largely consistent with those at lower $\mu .$ Radial diffusion is shown to dominate the PSD enhancement over chorus waves, with only marginal improvement seen when chorus is included by comparing Figure 4b2,b3. Outside the plasmasphere, model performance improves with increasing $D_{LL}$, with the 99th percentile again showing the best agreement as shown in the MSA results in Figure 4b8. Compared to the low $\mu$ case, the high $\mu$ case outside the plasmasphere exhibits slightly more sensitivity to $D_{LL}$. Inside and near the plasmasphere, model performance is best at around the 20th percentile as shown in the MSA results, indicating similar $D_{LL}$-dependent features between under- and overestimation. These patterns are similarly illustrated by the error distribution plots in Figure 6.

Figure 6

Figure 6. Similar to Figure 5, but for the higher $\mu =1237MeV/G$.

4 Ensemble modeling with GOES outer boundary condition at ${\mathit{L}}^{*}=\mathbf{6}$

To investigate the effects of different outer boundary conditions on the simulated radiation belt dynamics, in this section we use electron PSD data from GOES-13 and GOES-15 satellites to specify the data-driven model outer boundary condition at $L^{*}=6$. Figure 7 presents the simulation results in the same format as Figure 4, but with $L^{*}$ ranging from 3 to 6 and applying the GOES OB condition at $L^{*}=6$. Again, the top two plots in Figure 7 show observed electron PSD variations using the combination of GOES and Van Allen Probes observations. In Figure 7a2, a simulation is run with the 10th percentile of $D_{LL}$ without chorus heating, while the bottom plots include chorus waves in the simulations but with increasing $D_{LL}$ levels.

Figure 7

Figure 7. Similar to Figure 3 but using GOES OB condition at $L^{*}=6$.

For the lower $\mu$ case on the left, similar to the Van Allen Probes OB cases in Figure 4, stronger $D_{LL}$ transports electrons further inward across a wider $L^{*}$ range. However, unlike Figure 4 with the Van Allen Probes OB, Figure 7 with the GOES OB shows that weaker $D_{LL}$ cases (e.g., Figure 7a3) result in overestimated PSD at larger $L^{*}$. This is well illustrated in Figures 8a–d outside $L_{pp}$, where lower $D_{LL}$ consistently overestimates the observed PSD rather than underestimation with the Van Allen Probes OB, and the corresponding error histograms shift to the positive $\mathit{ln}\left(m_i/d_i\right)$ side with low $D_{LL}$ values as shown in Figures 8b,d. Clear illustrations of this overestimation are also shown in February and May in Figures 8a,c, which suggests that stronger $D_{LL}$ is necessary to reduce the overestimated PSD to better match observations. By comparing between the model results in Figure 7a2 without chorus and those in Figure 7a3 with chorus, we find that this overestimation at low $D_{LL}$ percentile is mainly due to the effect of chorus heating. An interesting question to ask here is: why does chorus heating not lead to overestimation in the Van Allen Probes OB case in Figure 4? The key difference is the location of the outer boundary. Because the Van Allen Probes OB at $L^{*}=5.5$ is located further in than the GOES OB at $L^{*}=6$, the outward radial diffusion loss to the OB is more effective in the Van Allen Probes OB case. Note that radial diffusion will only result in a loss if the PSD at the outer boundary is less than the PSD in the interior, since the PSD is fixed at the outer boundary. If the PSD at the outer boundary is higher than in the interior, then radial diffusion will be a source. Similar to the Van Allen Probes OB case at lower $\mu$ value, Figure 7a8 shows that simulations with higher percentile $D_{LL}$ or stronger radial diffusion perform better outside the plasmasphere, with the 99th percentile of $D_{LL}$ yields the best performance (MSA becoming insensitive to $D_{LL}$ beyond the 70th percentile). In addition, the GOES OB simulation results are more sensitive to the radial diffusion levels outside $L_{pp}$ compared to the Van Allen Probes OB results in Figure 4a8. Therefore, the model results with GOES OB suggest that it is the balance of chorus heating and outward radial diffusion that controls the electron dynamics at large $L^{*}$ outside the plasmasphere. This is different from our findings from the Van Allen Probes OB simulations, where radial diffusion from the outer boundary plays a dominant role. These results are very interesting, highlighting the effects of outer boundary conditions in the interplay between chorus heating and radial diffusion at large $L^{*}$ regions of radiation belt. For simulation results inside and near $L_{pp}$ (Figures 8e–h), the simulation behavior is similar with the Van Allen Probes OB case overall. Lower $D_{LL}$ simulations underestimate the observed PSD while higher $D_{LL}$ simulations overestimate the PSD, with best performance occurring around the 30th percentile of $D_{LL}$ (see Figure 7a8).

Figure 8

Figure 8. Similar to Figure 5 but GOES OB condition at $L^{*}=6$ is applied.

For the high $\mu$ case (right panels of Figure 7), the model results with different percentiles of $D_{LL}$ generally underestimate the observed PSDs at large $L^{*}$, which could be due to insufficient acceleration from chorus heating. This is different from the low $\mu$ case (left panels of Figure 7) and Van Allen Probes OB cases (Figure 4). Comparison between the simulation results with and without chorus (Figure 7b2 vs. Figure 7b3) shows that chorus heating enhances the electron PSD but remains insufficient to match observations. This underperformance contrasts with the Van Allen Probes OB case, where the OB itself is located at the local PSD peak around $L^{*}=5.5$ as shown in Figure 1b for the high $\mu$ case. Therefore, the Van Allen Probes OB condition already includes the heating effect which provides sufficient electron source for inward radial diffusion. On the other hand, because the GOES OB at $L^{*}=6$ is away from the local peak of PSD by active chorus heating, the data-driven OB does not include enough local heating. This demonstrates that local heating plays an important role in radiation belt dynamics at high $\mu$ and the current statistical chorus model is not sufficient in producing the local PSD enhancement. Therefore, these results demonstrate that different data-driven OB conditions can affect the simulated radiation belt dynamics even with the same model.

Another possible reason for this insufficient heating is that the total electron density used in our model may not be realistic enough when estimating the local heating due to waves. Recent studies have shown that chorus diffusion coefficients are highly sensitive to the background plasma density (Camporeale et al., 2016; Allison et al., 2021). The electron density model used in this study is obtained from Sheeley et al. (2001). However, as demonstrated by Allison et al. (2021), this model represents statistical average conditions and fails to reproduce the extremely low-density ($\sim 10c{m}^{-3}$) environments observed during major acceleration events. It is precisely in these low-density regions that chorus heating becomes significantly more efficient. Therefore, the chorus heating calculated in this study, being based on the Sheeley et al. (2001) model, could underestimate the level of local acceleration, contributing to the discrepancy with observations at high $\mu$.

As a result, the MSA plot in Figure 7b8 and the detailed time series of PSD at different $L^{*}$ values with respect to $L_{pp}$ in Figure 9 show very different features compared to the Van Allen Probes OB case. We find that the modeled PSD at high $\mu$ with GOES OB generally leads to worse performance compared to the Van Allen Probes case (e.g., comparing Figure 7b8 vs. Figure 4b8 at large $L^{*}$ regions and the error distributions in Figures 9b vs. Figure 6b). The underestimation with GOES OB is obvious in Figure 9 (in both the time series plots on the left and the error distributions on the right), especially near plasmapause and outside the plasmasphere. The MSA values for the lower percentiles (e.g., the 10th percentile of $D_{LL}$ in purple in Figure 7b8) are high at $L^{*}>L_{pp}+0.5$ close to the OB due to the overestimation. This is shown in, for example, May and late October in Figure 9a, where low percentiles $D_{LL}$ cannot drive sufficient loss through outward radial diffusion after the PSD enhancement. And the errors are largely insensitive for higher than the 30th percentile $D_{LL}$ as shown in Figure 7b8. This is because there is not enough electron source from the OB, stronger $D_{LL}$ cannot bring in more electron PSD. As a result, the model still under-reproduces the observed PSD with high $D_{LL}$ as shown in Figures 9a,b.

Figure 9

Figure 9. Similar to Figure 8, but for the higher $\mu =1237MeV/G$.

For regions just outside plasmasphere from $L_{pp}$ to $L_{pp}+0.5$, the model results become more sensitive to $D_{LL}$ as shown in Figure 7b8, with the best performance at the 30th percentile of $D_{LL}$ plotted in blue. This is further supported by Figure 9c–f, we find that when radial diffusion is too fast, it leads to an underestimation of observed PSD due to limited chorus heating and rapid outward and inward diffusion from the local peak (e.g., May in Figure 9e). Conversely, when radial diffusion is too slow, there is insufficient outward radial diffusion for the losses (e.g., November in Figure 9e). By further comparing Figure 9f with Figure 6f, we find that this behavior contrasts with the Van Allen Probes OB case at the same region, where low $D_{LL}$ leads to underestimation and high $D_{LL}$ to overestimation. This difference is attributed to differing dominant mechanisms: radial diffusion is primary in Van Allen Probes OB cases, while in GOES OB cases, both chorus and RD processes play a significant role.

Finally, for regions inside the plasmasphere ($L^{*}=L_{pp}-0.5$ in Figures 9g–h), the model performance is similar to the Van Allen Probes OB case at high $\mu$, showing sensitivity to $D_{LL}$. The best performance is again found at 30–40th percentiles of $D_{LL}$, with higher $D_{LL}$ values leading to overestimation and lower values resulting in underestimation. Interestingly, we find that overall for the high $\mu$ case with GOES OB, the 30th percentile of $D_{LL}$ consistently provides the best performance across all $L^{*}$ regions.

5 Conclusions and discussion

In this study, we performed a set of ensemble simulations on the dynamics of radiation belt electrons using the DREAM3D diffusion model. The radial diffusion coefficients are obtained from Murphy et al. (2023), which provides both a deterministic relation with a multi-linear regression and the residuals as standard deviations. The investigation of different outer boundary conditions and varying levels of radial diffusion coefficients helps us understand radiation belt dynamics both inside and outside the plasmasphere.

First, we find that the location of data-driven OB can significantly influence model performance. Models utilizing Van Allen Probes OB at $L^{*}=5.5$, which encompasses the chorus heating region, demonstrate better performance in terms of lower MSA compared to models employing GOES OB at $L^{*}=6$. Although it is not surprising that using a boundary closer to the heart of the radiation belt can produce a better result, our results show that the interplay between radial diffusion and chorus heating can be strongly affected by different, realistic outer boundary conditions as demonstrated in our results. When Van Allen Probes OB is applied at $L^{*}=5.5$, the majority of the observed dynamics can be reproduced through radial diffusion. Stronger $D_{LL}$ yield better results outside the plasmasphere, effectively capturing the strong inward and outward radial transport to and from the outer boundary. Conversely, intermediate $D_{LL}$ around the 30th percentile provides optimal performance inside the plasmasphere, accurately accounting for the appropriate PSD levels in this region. These findings are consistent for both low and high $\mu$ electron population.

In contrast, when GOES OB conditions are implemented at $L^{*}=6$, chorus heating effects become increasingly important since they are not encompassed within the OB coverage. The chorus wave acceleration in the model appears to overestimate PSD enhancement for low $\mu$ electrons while underestimating enhancement for high $\mu$ electrons. Consequently, outside the plasmasphere, low $\mu$ cases require strong $D_{LL}$ to generate sufficient outward transport and losses to the outer boundary. Similar requirements apply to high $\mu$ cases, though at larger $L^{*}$ ($L^{*}>L_{pp}+0.5$) closer to the outer boundary. Additionally, for high $\mu$ electrons at just outside $L_{pp}$, intermediate diffusion levels around the 30th percentile are necessary to maintain PSD enhancement from local heating while providing adequate outward transport to the outer boundary for loss processes. Inside plasmasphere, the results are consistent with the Van Allen Probes OB case, where intermediate RD levels (∼30th percentile) best reproduce the observations.

The coupling between the external source and internal transport mechanisms is investigated by comparing the Van Allen Probes and GOES OB configurations. The different results reveal the critical physical role of the plasmapause region. When the primary source of acceleration (chorus heating) is included in the boundary condition (Van Allen Probes at $L^{*}=5.5$), the system dynamics are dominated by the subsequent radial transport. However, when the boundary is moved outward (GOES at $L^{*}=6.0$), the model reveals a complex interplay between local heating by chorus waves and radial diffusion, demonstrating that these two processes are highly competitive and their balance is essential for accurately describing the state of the outer belt.

Furthermore, the consistent model discrepancy inside the plasmasphere, regardless of the boundary condition or electron energy, points to a fundamental gap in our physical models for this region. The need for intermediate diffusion levels suggests that the observed electron PSD is maintained by a delicate balance of weak transport and loss. Our results imply that the empirical hiss wave model from Orlova et al. (2014) used, consistent with our previous work (Lee et al., 2024), do not provide sufficient loss. This emphasizes the potential importance of other loss mechanisms not yet included, such as electron scattering by EMIC waves, which are known to be significant near the plasmapause.

Finally, the simulations with the GOES boundary suggest that the empirical chorus heating model used may not be sufficient to produce the observed local acceleration up to high energies, a conclusion also supported by Lee et al. (2024). The local heating rate is sensitive not only to wave properties but also to background plasma parameters (e.g., Tu et al., 2014; Hua et al., 2022; Hua et al., 2023). Therefore, the model’s underperformance in this configuration is not simply a parameter adjustment issue but rather an indicator that more physically realistic models for both chorus waves and background plasma density are required to fully investigate the competitive balance between local acceleration and radial diffusion that governs the outer radiation belt.

Data availability statement

Van Allen Probes HOPE, MagEIS, REPT data and the combined ECT level 3 data are available from the Science Operations and Data Center (https://rbsp-ect.newmexicoconsortium.org/rbsp_ect.php). Geomagnetic activity indices were obtained from the NASA OMNI Web (https://cdaweb.gsfc.nasa.gov/). The PSD data analyzed in this study and the DREAM3D simulation results generated in this study are deposited on the data repository (Lee et al., 2025), which is publicly available.

Author contributions

S-YL: Methodology, Writing – review and editing, Software, Writing – original draft, Visualization, Conceptualization, Investigation, Formal Analysis, Data curation. WT: Resources, Funding acquisition, Supervision, Writing – review and editing, Conceptualization, Methodology, Project administration. GC: Software, Validation, Writing – review and editing. MC: Writing – review and editing, Resources, Project administration.

Funding

The author(s) declared that financial support was received for this work and/or its publication. This work was supported by the NASA grants 80NSSC21K1312, 80NSSC21K2008, and 80NSSC24K1112, DOE grant DE-SC0020294, and NSF grant AGS 2247856.

Conflict of interest

The author(s) declared that this work was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Generative AI statement

The author(s) declared that generative AI was not used in the creation of this manuscript.

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