On the Energy-Dependent Deep ( L < 3.5) Penetration of Radiation Belt Electrons

Deep penetration of outer radiation belt electrons to low L (


Introduction
Energetic electrons in the inner magnetosphere are normally trapped in two regions: the inner and outer radiation belts.The inner belt is relatively stable, while the outer belt is highly dynamic, with electron fluxes varying by orders of magnitude within days during geomagnetic storms.The slot region, usually devoid of energetic electrons, separates the two belts.During geomagnetically active times, outer belt electrons can extend to lower L (distance to the center of Earth in units of Earth radii in the equatorial plane) and even fill the slot region (Blake et al., 1992;Li et al., 1993).The dynamic variations of radiation belt particles are the result of a complex competition between acceleration, transport, and loss mechanisms.The most important source process for inner belt electrons is inward radial transport from the outer belt (Cunningham et al., 2018;Selesnick, 2016) with cosmic ray albedo neutron decay (CRAND) contributing at the inner edge of the inner belt (Li et al., 2017;Xiang et al., 2019;Zhang et al., 2019).The energy-dependence of outer belt electrons penetrating inward can help quantify the populations of electrons transported to the inner belt during a certain event, which is important for determining source and loss processes of inner electrons.This study will focus on the mechanism responsible for the energy-dependent penetration of outer belt electrons into the low L region (L < 3.5).Our study provides Abstract Deep penetration of outer radiation belt electrons to low L (<3.5) has long been recognized as an energy-dependent phenomenon but with limited understanding.The Van Allen Probes measurements have clearly shown energy-dependent electron penetration during geomagnetically active times, with lower energy electrons penetrating to lower L.This study aims to improve our ability to model this phenomenon by quantitatively considering radial transport due to large-scale azimuthal electric fields (E-fields) as an energy-dependent convection term added to a radial diffusion Fokker-Planck equation.We use a modified Volland-Stern model to represent the enhanced convection field at lower L to match the observations of storm time values of E-field.We model 10-400 MeV/G electron phase space density with an energy-dependent radial diffusion coefficient and this convection term and show that the model reproduces the observed deep penetrations well, suggesting that time-variant azimuthal E-fields contribute preferentially to the deep penetration of lower-energy electrons.
Plain Language Summary Electrons trapped by the Earth's magnetic field gather in two regions known as the Van Allen radiation belts.It is well reported that electrons can be transported radially inward from the outer radiation belt during geomagnetically active times.More specifically, low energy (100 s of keV) electrons can be moved radially deeper than higher energy (∼1 MeV) electrons.Previous studies suggested that enhanced convection electric fields could contribute to the earthward transport of low energy (<200 keV) electrons.However, the mechanism which leads to different efficiencies of electron transport at different energies has not been quantified.This study expands the traditional radial diffusion model with an empirically determined convection term and shows that the net convection velocity increases for lower energy electrons.For the first time, we quantitatively modeled the energy-dependent penetration of radiation belt electrons in a wide energy range (10 s of keV to 2 MeV) in the presence of enhanced large-scale electric fields, during two geomagnetic storm events observed by the Van Allen Probes mission.MEI ET AL.In-situ observations have demonstrated the storm-time penetration of outer belt electrons.Baker et al. (2004) found that compression of the outer belt's inner edge is closely associated with the reduction of the plasmasphere, the cold, dense plasma region corotating with Earth.Li et al. (2006) reported the correlation between electron enhancements and the outer plasmasphere boundary using >1 MeV electron flux measurements from SAMPEX and CRRES.Many have studied the energy-dependence of outer belt electron penetration and flux enhancements, with finer energy and temporal resolution measurements provided by the Van Allen Probes mission (Mauk et al., 2013).Reeves et al. (2016) showed that lower energy electrons penetrate the inner zone more often than higher energy electrons.Khoo et al. (2018Khoo et al. ( , 2021) ) studied the relationship between the location of initial electron enhancement and the innermost plasmapause location during intense storms.Their results showed that initial enhancements occurred outside the innermost plasmapause, suggesting that enhanced convection is responsible for plasmasphere erosion and could contribute to lower energy electron enhancements.Mei et al. (2021) studied an energy-dependent linear relation between the upper boundary of the innermost slot region's L shell and the 15-hr-averaged K p index.Simulations have shown that enhanced convective electric fields are likely responsible for inward transporting <200 keV electrons to lower L (Korth et al., 1999;S. Liu et al., 2003;Thorne et al., 2007;Zhao et al., 2017).In this paper, we investigate the mechanism responsible for driving lower-energy outer belt electron penetration to lower L during geomagnetic storms.We quantify the net effect of the  ×  drift induced by time-variant large-scale azimuthal E-fields and consider this transport mechanism as a convection term in the radial diffusion equation for electrons at selected first adiabatic invariant,   , values.By modifying the Volland-Stern E-field model for high K p levels (  p ≥ 4 ), we enhance the convection E-field near Earth during storm times to match observations.Applying the modified Volland-Stern model to the convection term in 1-D diffusion-convection modeling of phase space density (PSD), we show the evolution of 10-400 MeV/G equatorial electrons during storm-time flux enhancement events.Our model achieves prediction efficiencies (PE) for lower   electrons in the low L region (L ∼ 3) of >∼0.9 for selected events.These results suggest the time-varying, large-scale azimuthal E-fields contributes to deeper penetration of low-energy electrons.

Quantification of the Energy-Dependent, E-Field-Induced Radial Transport
It is generally recognized that the azimuthal component of the convection E-field can cause electrons to move radially, but radial transport due to static E-fields will eventually negate this motion as an electron drifts around Earth.However, while the large-scale E-field enhances during storm-time, electrons with drift periods comparable to the timescales of the E-field enhancement   can be transported more efficiently in the radial direction.In general, the timescale refers to the magnitude of characteristic time variations of a field quantity Q, and can be defined as   =  ∕ (Roederer & Zhang, 2014); thus for E-field variations the timescale   can be defined as: Based on the K p -dependent Volland 10.1029/2022GL101921 3 of 10 of the particle.In addition, as the first adiabatic invariant   is conserved while the electron drifts, low   electrons could also be transported to lower L than high   electrons when they gain the same amount of kinetic energy due to large-scale E-field.Thus, higher energy electrons are less influenced by the time-varying E-field, and this radial transport mechanism is more important for lower energy electrons.Though azimuthal E-fields can cause electrons to radially drift inward or outward based on the azimuthal phase of an electron, the overall net effect on electron PSD is inward penetration if there is a positive radial gradient of radiation belt electron PSD.Since electron PSD at higher L for a fixed   is normally much greater than that at lower L, only a fraction of electrons moving inward will create a significant increase in flux (Califf et al., 2017).Together with the positive radial gradient of the PSD, another important factor affecting the inward penetration of electrons is the timescale of the convective E-field variation in relation to the drift periods of electrons.Its importance can be seen in the simulation results of Califf et al. (2017), and in particular their Figure 11, where E-field pulses of different duration can be seen to affect different μ-values of electrons with an MLT dependence.With respect to the dependence of the changes in PSD on local time, it is noted that Califf et al. (2017) used symmetric distributions of particles in MLT.The  ×  drift motion of electrons caused by large-scale E-fields can be described by convection processes (Aseev et al., 2016;Shprits et al., 2015).In this study, we confine the modeling to the radial dimension by providing an approximation of the net convection effect to simulate the contribution of this mechanism to electron penetration.

1-D Modeling of Radial Diffusion and Convection
To study the radial transport induced by storm-time, time-varying E-fields and investigate their contribution to the energy-dependent penetration, we expand the traditional form of the radial diffusion Fokker-Planck equation (Schulz & Lanzerotti, 1974) In Equation 3  are dependently determined by maintaining the continuity of  Liu at 400 MeV/G and K p = 6.The optimal values for the parameters are determined by searching for the best overall performance of the "radial diffusion-only"-modeled electron PSD compared to observed PSD over a wide range of   < 400 MeV/G (see Text S3 in Supporting Information S1).
For modeling with Equation 2, we set the source term S = 0 for our model, as local heating effects are not considered in this study since we focus on the energization of lower energy electrons by inward radial transport.The time step for computing the model is 3-min.The empirical models of electron lifetime due to chorus wave or hiss wave pitch angle scattering are applied to determine the loss term  −   (Gu et al., 2012;Zhu et al., 2021).Inside the empirical plasmapause location (Carpenter & Anderson, 1992)

Modification on the Volland-Stern E-Field Model
Enhanced convection E-fields can lead to the erosion of the plasmasphere, which could also cause electrons to penetrate to lower L. Motivated by the discrepancies between the Volland-Stern model (Maynard & Chen, 1975;Stern, 1975;Volland, 1973) and statistical observations of large-scale E-field during active times (Califf et al., 2014), we modify the Volland-Stern model at higher K p by increasing storm-time convection E-fields near Earth.Figure 2 In Equation 6,  Φ is the electric potential, R is the radial distance,   is the azimuthal angle from noon, A is given by Maynard and Chen (1975 is the radial distance in Earth radii where the   6.The above model is a time-varying convection electric field, that is parameterized according to 1-hr interpolated K p .

Results
A moderate storm on 8 June 2015 is studied to investigate whether convective radial transport resulted by large-scale E-field can explain the deep penetration of lower energy electrons.During the event, the minimum Dst index is −73 nT and the maximum K p is 6.
Figure 3 shows the comparison between the observed PSD for 20 MeV/G,   = 0.12  1∕2  electrons from RBSP A& B flux measurements (Blake et al., 2013) using the T89D model (Tsyganenko, 1989) (panel a), modeled PSD with different coefficient sets (panels b through e), the prediction efficiency (PE) for the various models with different coefficient sets for the presented period (panel f), and radial profiles of PSD during the initial enhancement (panel g).Comparisons at other   values are shown in the Supporting Information.Three groups of radial-diffusion-only modeling are conducted with the   -independent coefficients  & (Brautigam & Albert, 2000),  Ozeke (Ozeke et al., 2014), and the   -dependent coefficient  Liu−mod given by Equation 5.The test group is a diffusion-convection model with the same  Liu−mod and with the additional convection term where   is given by Equation 3, aiming to illustrate the relative contribution of the energy-dependent convection term   .Prediction efficiency is defined as: where   and   respectively denote the observed and modeled log (PSD),   is the mean of   .In this study, PE is calculated for a 1-day period during the PSD enhancement.Positive PE values suggest better prediction than the average value (e.g., Barker et al., 2005;Li et al., 2001).Black dotted lines in Figure 3a-3e show the innermost   Carpenter and Anderson (1992) empirical model.In Figure 3a, the observations show PSD enhancements by at least 2 orders of magnitude outside the plasmapause (which is   ∼ 3 during the event).Figure 3g-3h show that "radial diffusion-only"-modeling based on any of the  & ,  Ozke or  Liu−mod diffusion coefficients is not sufficient to create the observed PSD enhancement near the plasmapause, whereas the modeling results with both diffusive and convective radial transport using the combination of  Liu−mod and   provide an enhancement that is comparable to the observations.Furthermore, PE values of pure radial diffusion modeling with  Liu−mod drop to below 0.9 for    3.5 and eventually drop to negative values when approaching the plasmapause at Electrons with energy <200 keV move inward to the slot region and the radiation belts develop a "V" shaped structure as these lower energy outer belt electrons penetrate to lower L. The diffusion-only model with  Ozeke in the middle panel cannot reproduce the "V" shaped structure, and lower energy electrons cannot reach L ∼ 3-3.5 when only driven by radial diffusion, similar results are obtained with the  & and  Liu−mod (not shown herein).Diffusion-convection modeling with the   term, shown in the right panel, reproduces the energy-dependent electron penetration better and captures the slot region filling features for <200 keV electrons, including the "V" shaped structure.It is noted that Ripoll et al. (2016) reproduced the formation of "S" shaped structure during quiet time due to the post-storm energy-dependent electron decay.Our study suggests that enhanced convection plays an important role on the formation of "V" shaped structures, as those of Figure 4, and in the storm-time energy-dependent inward penetration and energization of low energy electrons.
To further examine the diffusion-convection model and the mechanism responsible for energy-dependent penetration, another moderate storm on 4 November 2014 is studied.During this event, the minimum Dst is −44 nT and maximum K p is 4+.Parameters for  Liu−mod and the modified Volland-Stern model are kept the same as those in the 8 June 2015 event.As the Figure S6 in Supporting Information S1 shows, PE for modeled lower   electron PSD is significantly improved to greater than 0.6 near the inner edge of the outer belt with combination of the   and  Liu−mod .Radial profiles of 10-60 MeV/G electron PSD with the convective term   better reproduces that of observations while diffusion-only modeling with   -independent   is not sufficient to move electrons inward to L < ∼3.The good comparison with the electron flux variations in Figure S7 in Supporting Information S1 further shows the contribution of enhanced convection on low energy electron penetrations.

Discussion and Conclusion
This study investigates the mechanism of energy-dependent outer belt electron penetration by conducting simulations over a broad energy range and extending the general radial diffusion model with an additional convection term determined by a time-variant large-scale E-field model.The results suggest that the convective transport effect due to time-variant large-scale E-fields is responsible for the deep inward penetration of lower (<500 keV) energy electrons.Higher energy electrons with shorter drift periods are less affected.In this study, the scenario where the large-scale E-field varies on shorter timescales than that of the electron drift period (   ≪  ), such as shock-induced E-field impulses, is not addressed.
Both events studied correspond to moderate storms with maximum K p ≤ 6.Previous studies (Liemohn & Jazowski, 2008;Menz et al., 2019) suggested that the performance of the Volland-Stern E-field model is better when K p < 7. The modified Volland-Stern model mimics the trend of large-scale E-fields at low L from statistics by Califf et al. (2014), which also concentrated on K p < 6.Based on the assumptions made, MEI ET AL.The results of this study can be summarized as follows: 1.The energy-dependence of outer radiation belt electron penetration depth can be explained by the relationship between the large-scale azimuthal E-field variation timescale and the drift period of electrons at different energies and radial locations.Higher energy electrons with much shorter drift periods are less affected by the convection E-field, while 10-100 s of keV electrons are moved deeper inward as a result of the combination of the convective and diffusive effects.

•
Convective radial transport of stormtime enhanced large-scale E-fields is an efficient inward transport mechanism of 10-100 s keV electrons • The energy-dependent electron penetration can be explained by the relation between the timescales of electron drift and large-scale E-fields • A radial diffusion-convection model is developed to reproduce the storm-time penetration of lower energy electrons to lower L Supporting Information:Supporting Information may be found in the online version of this article.radiation belts from the perspective of the energy spectrum and reveals the physical mechanisms underlying radiation belt dynamics.
Califf et al. (2017) conducted test particle simulations and showed that a large-scale electric field (E-field) with amplitudes of 1-2 mV/m can convectively transport hundreds of keV electrons and explain the observed enhancements in the slot region.It still remains an open question how to quantitatively explain and model the energy-dependent deep penetration of electrons.

Figure 1 .
Figure 1.Demonstration of a time-varying large-scale dawn-dusk E-field leading to the energy-dependent radial transport of electrons.(Left) The E-field induced radial transport is negligible when   ≫  .(Right) The E-field induced radial transport becomes more efficient when   ∼  .Dashed blue lines represent the large-scale E-field, and the dashed black circle around Earth is the drift orbit of trapped particles.The dashed red and green curves show the drift trajectory of high   and low   electrons in the presence of a time-varying, large-scale E-field, respectively.
shows the comparison between a statistical study of the dawn-dusk component of the E-field (   ) from Califf et al. (2014) and our modified Volland-Stern model.Statistics show that   during times of high K p can reach a local maximum at low L, especially in the dusk sector.Despite the discrepancies like the dawn-dusk asymmetry, we create a similar local maximum   to mimic the statistical results on the nightside at low L. The modification on the Volland-Stern model follows the piecewise functions:

Figure 2 .
Figure 2. (Left) Radial profiles of the dawn-dusk E-field according to Califf et al. (2014) in the frame corotating with Earth, as a function of L, for select K p values, in the dusk, midnight, dawn and noon regions, as marked.(Right) Radial profiles of the modified Volland-Stern model convection E-field as a function of L, for select K p values, and for the same local time regions as the left-hand side panels.
∼ 3 , while radial PSD values are constantly lower than the observed PSD by one order of magnitude in the region  ≅ 3 -3.5.These suggest that radial diffusion alone cannot provide sufficient electron penetration to lower L (  ≅ 3 -3.5)for low   electrons.When the convective radial transport is considered, PE performance is significantly improved especially for   ∼ 3 near the innermost edge of the observed enhancement.The results suggest that the large-scale E-fields play an important role to radially transporting low   electrons to lower L. To illustrate the energy-dependent electron penetration, we convert the modeled 10-400 MeV/G,   = 0.12  1∕2  electron PSD to differential flux with the T89D model and compare the modeled flux with electron flux measurements.In Figure 4, electron fluxes are displayed as a function of L and energy.The left column shows the observed 70° local pitch angle electron flux profiles during each half orbit pass, the middle column shows the modeled electron flux with   -independent  Ozeke at epochs between the corresponding timespan of observations, and the right column shows the modeled electron flux with   -dependent  Liu−mod and   terms.During this period the outer belt electron fluxes were significantly enhanced and electrons were transported to L ∼ 3-4.
convection model confined to the radial dimension for simplicity is suitable for studying radial transporting electrons at low L during moderate storms.As FigureS9in Supporting Information S1 shows, locally observed PSD suggest an MLT-dependent electron deep penetration at short timescales.A drift-resolved Fokker-Planck code, which would allow investigating the MLT-dependent PSD time-evolution, is beyond the scope of this study.
2. Time-varying large-scale azimuthal E-fields must be considered to model the deep penetration of lower   electrons.By introducing a convection term determined by a modified Volland-Stern E-model on top of the

Figure 4 .
Figure 4. Electron flux variations as a function of kinetic energy and L shell.(left column) Pass-averaged fluxes of  loc = 70 ° electrons observed by RBSP A on 8 June 2015; (middle column) Modeled electron fluxes for  = 0.12 1∕2  with  Ozeke ; (right column) Modeled electron fluxes with  Liu−mod and   .
-Stern E-field model, the lowest values for   are found for the highest rate of change in K p .For example, for a moderate storm in which K p changes from 3 to 6 within 3 hr, is typically ∼1 hr at  = 3.5 .As a comparison, the drift period for a 250 keV electron at  = 3.5 is ∼1 hr. Figure 1 is a schematic diagram conceptually showing how large-scale E-fields can lead to the energy-dependent radial displacement of electrons depending on the relationship between the electron drift period (   ) and   .When   ≪  , the E-field can be considered nearly static during an electron's drift cycle.Shown by the red dashed drift trajectory in Figure 1, radial displacement cannot accumulate over drift cycles and the net radial transport is negligible.However, as the green dashed arc shows, when   becomes comparable to   , the time-varying E-field leads to imbalanced inward or outward radial drift, and thus causes considerably larger net radial motion Writing -review & editing: Xinlin Li, Hong Zhao, Theodore Sarris, Lengying Khoo, Benjamin Hogan, Declan O'Brien, Sam Califf MEI ET AL.

,
net approximates the time-varying large-scale E-field which can result the equivalent drift velocity  ×net .
is the azimuthal component of large-scale E-field,   is the background local geomagnetic field.In this study,   and   are obtained from the Volland-Stern model and the static dipole magnetic field, respectively.  is the azimuthal angle from noon.
Parameter N is set equal to 2 in Volland-Stern model, but it is modified as a function of K p in this study to produce an enhanced E-field near Earth during active times.When  p ≤ 4 , we use the original Volland-Stern model.At more disturbed times when  p > 4 , the modified Volland-Stern model is used to enhance the convection E-field with larger N values according to Equation 10.1029/2022GL101921 6 of 10 local maximum dawn-dusk E-field component   is observed as a function of K p according to Califf et al. (2014).