Efficacy of Electric Field Models in Reproducing Observed Ring Current Ion Spectra During Two Geomagnetic Storms
This article is a companion to https://doi.org/10.1029/2019JA026859.
Abstract
We use the UNH-IMEF, Weimer 1996, https://doi.org/10.1029/96GL02255 and Volland-Stern electric field models along with a dipole magnetic field to calculate drift paths for particles that reach the Van Allen Probes' orbit for two inbound passes during two large geomagnetic storms. We compare the particle access in the models with the observed particle access using both realistic and enhanced solar wind model parameters. To test the accuracy of the drift paths, we estimate the H+ charge exchange loss along these drift paths. While increasing the strength of the model electric field drives particles further inward, improving agreement, energy-dependent cutoffs in the spectra do not agree, indicating that potential patterns for highly disturbed times are inaccurate. While none of the models were able to reproduce the observed features of the more dawnward pass during the 17 March 2013 storm, the UNH-IMEF model with enhanced inputs was able to adequately reproduce the access, charge exchange loss, and H+ particle pressure during the 17 March 2015 storm.
Key Points
- Storm time drift trajectories using the UNH-IMEF model are compared with results using Weimer and Volland-Stern models
- The electric field models are too weak in the inner magnetosphere, and particles do not convect in as far as observed without enhanced inputs
- Additional change to the potential patterns for highly disturbed times is necessary to reproduce observed energy cutoffs
1 Introduction
During geomagnetic storms, the large-scale dawn-to-dusk convection electric field is enhanced, pushing particles sunward in the equatorial magnetosphere. This process enhances the access of plasma sheet ions to the inner magnetosphere. Because transport due to large-scale magnetospheric convection conserves the first adiabatic invariant, the magnetic moment, the ions are energized as they drift earthward to stronger magnetic fields. This process contributes to the buildup of the storm time ring current, an enhancement of the ~20-200 keV ion population that creates a magnetic disturbance during geomagnetic storms. The peak of the ring current particle pressure during the main phase is generally inside L = 4 (De Michelis et al., 1999; Ebihara et al., 2002; Lui, 2003) and can be seen as low as L = 2.5 (Menz et al., 2017). Modeling the effects of magnetospheric convection has shown it to be a sufficient energization mechanism to explain observed ring current energization (Fok et al., 1995; Liemohn et al., 2001; Kozyra et al., 1998; Jordanova et al., 1998, 2001), and recent observational papers (Kistler et al., 2016; Menz et al., 2017) have shown that inward adiabatic convection of the plasma sheet population is a sufficient source for the inner magnetosphere storm time ring current population. Inductive electric fields, such as those caused by magnetic reconfiguration during substorms, cause particle injections that can impact the higher energy (>50–100 keV) ring current population, but these particles do not convect inside L = 4 where we observe the peak of the particle pressure. In a single event study, Gkioulidou et al. (2014) showed these injections penetrating down to L = 4 and estimated that they could account for 30% of the energy gain in the inner magnetosphere. While particle-wave interactions and radial diffusion do occur, Chen et al. (1994) and Jordanova and Miyoshi (2005) showed that their contributions are small during the main phase, although they can impact the higher energies (>80 keV) later in the storm.
The drift of ions due to the enhanced electric field, along with gradient and curvature drifts, leads to distinctive spectral features in the inner magnetosphere. These include a sharp upper energy cutoff that represents the open/closed drift path boundary and more complex upper and lower energy cutoffs at the inner edge of the convection region that lead to “nose structures” (Smith & Hoffman, 1974). The energies and location of these cutoffs depend on the form of the electric field (e.g., Angelopoulos et al., 2002; Ebihara et al., 2004; Ejiri et al., 1978, 1980) with the form of the magnetic field being less important in determining the energy spectrum (Kistler & Larson, 2000). Thus, comparing the details of the observed energy spectra, and in particular the energy cutoffs, with the predictions of the different electric field models can help to establish how well the empirical models are able to reproduce the storm time field.
A number of empirical electric field models have been developed. An early attempt at modeling the electric field in the inner magnetosphere was developed by Volland (1973) and Stern (1975). An analytic form of the electric potential was made to fit observations of the plasmapause and the ionospheric electric field. This resulted in an electric field that was weaker closer to the Earth. Maynard and Chen (1975) parameterized the strength of the Volland-Stern field based on the Kp index, a 3-hr index measuring geomagnetic activity. Further studies found that a shielding factor of 2 (e.g., Ejiri et al., 1978, 1980) and a rotation eastward by 2 hours in magnetic local time (MLT) (Kistler et al., 1989; Ebihara et al., 2004) best fit the observed features. Jordanova et al. (1999) and Ebihara and Ejiri (2000) introduced parametrization of the model based on solar wind conditions. Because of the analytical form of the Volland-Stern potential and its parameterization with geomagnetic indexes, it has been a popular choice for ring current models (Fok et al., 1993; Chen et al., 1994; Jordanova et al., 1996). These models have been relatively successful using a dipole magnetic field and a modified Volland-Stern electric field to reproduce the observed ring current. Liemohn and Jazowski (2008) simulated 90 storms using a Kp-dependent Volland-Stern electric field and found that the electric field did a poor job at Kp > 7 and for storms with minimum Dst < −75.
Rowland and Wygant (1998), using CRRES electric field data, performed an analysis of the dependence of the inner magnetosphere electric field on Kp, an indicator of geomagnetic activity. While the study was limited to the nightside and duskside, they found a peak in the electric field that intensified and moved earthward with increasing Kp. This was in contrast to the Volland-Stern electric field where the magnitude always decreases with decreasing radial distance. They did not create an empirical model with this data, but it was a clear indication that more work was necessary to understand the storm time electric field.
Weimer (1996) developed an empirical model for the ionospheric electric potential based on electric field measurements from the DE-2 satellite. A spherical harmonic fit was performed on the data to produce an ionospheric potential based on solar wind interplanetary magnetic field (IMF), dipole tilt angle, and solar wind velocity. These ionospheric potentials can then be mapped to the equatorial inner magnetosphere, providing an electric field. Kistler and Larson (2000) and Kistler et al. (1999) found that the Weimer 96 electric field better represented the losses and spectral features over a Volland-Stern electric field. Ferradas et al. (2018) used the Weimer 96 model combined with a dipole magnetic field to model the energy spectra changes from prestorm through the main and recovery phase and found good qualitative agreement between the observed spectral features as the inner edge of the plasma sheet transitioned from having multiple noses to having enhanced, single noses and then back to multiple noses. While the qualitative agreement was good, the exact energies and L values of the features were not identical.
Weimer revised their model to include substorm enhancement measured by the AL index (Weimer, 2001). Angelopoulos et al. (2002) simulated the same storm as Kistler et al. (1999) using the Volland-Stern and different versions of the Weimer model. They found that the best fit was obtained using a Weimer, 2001 electric field modified to incorporate inductive fields. However, none of the models were able to reproduce the L dependence of some features in the spectra.
Matsui et al. (2008, 2013) developed an empirical inner magnetosphere electric field model (UNH-IMEF) based on electric field measurements from the Cluster spacecraft. Inside of Cluster's perigee (L ~ 4), the model uses data from ground radar and low-altitude spacecraft in order to extend the coverage of the model down to L = 2. The model divides the electric field data into several ranges using the interplanetary electric field (IEF) as measured by ACE. Through interpolation and extrapolation, the model provides an electric field for any discrete IEF value. Jordanova et al. (2008) simulated two storms using a Kp-dependent Volland-Stern model and the initial version of the UNH-IMEF model and found that the Volland-Stern model better predicted the strength of the ring current due to a stronger electric field. However, they only analyzed the energy density spatial distribution and did not analyze in detail the predicted energy spectra or the penetration of different energies into the inner magnetosphere. Thus, a more detailed analysis of the updated model is important to assess this model.
The Van Allen Probes spacecraft provides a wealth of storm time ring current data that can be used to determine how well the existing empirical electric field models are able to match the observed main phase spectra. Menz et al. (2017) previously studied the 17 March 2013 storm and found signatures of convection (enhanced fluxes with sharp energy cutoffs) from the Van Allen Probes' apogee (L ~ 6) down the inner edge of the ring current at L = 2. They found that inward drift of time-varying O+ enhancements in the near-earth plasma sheet was the biggest contributor to the observed ring current. To show how the drift times from the source to different L values in the ring current could explain the observed spectra, a Weimer 96 electric field model using OMNI solar wind data from the period was used. While the model was able to show the trends, it was unable to reproduce the observed access to L < 3.5 during the storm. Thus, further work is needed to determine whether a different model or different parameterization of the model would improve the data/model comparison.
In this study we will use two storms to test how well particle tracing in different electric field models, combined with a dipole magnetic field, is able to reproduce the observed spectral features during the main phase. The UNH-IMEF, the Volland-Stern, and the Weimer 96 models will be tested and compared. For both storms, an inbound pass during the main phase when the spacecraft is on the dawnside will be simulated in detail. This local time region has proved difficult to model. The statistical study of Kistler et al. (1989) showed that the open/closed drift path boundary at this local time was consistently observed at higher energy than the models would predict, where there was reasonable agreement at other local times. As previous work has shown difficulty in reproducing storm time spectra with these models, versions in which the electric field is artificially enhanced will also be tested.
The spectra at the inner edge of the ring current during storms depend strongly on the drift paths, but even during the fast convection that occurs during storms, the inner edge can be affected by charge exchange losses. The drift paths for different ion species with the same energy per charge are the same, but for the energies of interest, <50 keV, H+ has a much shorter charge exchange lifetime than O+ and so is more likely to show charge exchange effects. Thus, we will perform our data/model comparison in two steps: We will first compare the spectra features observed in the O+ spectra, where minimal charge exchange effects are expected, with the energy cutoffs in particle access predicted by the drift modeling. We will then add charge exchange to the model and compare observed and predicted H+ spectra and spectral energies boundaries. Because the charge exchange loss depends on the drift time and path, this gives a second check on how well the electric fields are able to reproduce the particle transport.
2 Instrumentation/Methodology
The Van Allen Probes mission consists of two twin spacecraft flying in nearly identical orbits with perigee down to 600 km and apogee of ~5.8 RE., an inclination <10°, and a period of about 9 hr. HOPE (Funsten et al., 2013), part of the ECT instrument suite (Spence et al., 2013), measures H+, He+, O+, and electrons from ~1 up to 60 keV. The EFW instrument (Wygant et al., 2013) uses two wire booms in the spin plane to provide electric field and spacecraft potential data. In this study, we use O+ particle fluxes from HOPE instrument, 1–60 keV, as the Van Allen Probes move through the ring current to determine the ion access to the inner magnetosphere. We analyze the O+ distribution function to determine which energies have access to a given location from the plasma sheet. As particles move inward adiabatically, their distribution function is conserved at constant μ (the first adiabatic invariant, E⊥/B). Since there is a proportionally greater increase in O+ during the storm and since O+ has much lower charge exchange rates than H+ at energies below 45 keV, the O+ distribution function best indicates the access of the plasma sheet to the inner magnetosphere during storms assuming the source locations are the same. As low-energy (<10 keV) ions drift inward, they travel along equipotentials, moving earthward due to convection and eastward due to the corotation electric field. As particles travel to stronger magnetic fields, they energize adiabatically, increasing the energy-dependent gradient-curvature drift and transitioning from eastward to westward drift. This motion creates sharp boundaries in energy to where particles have access, and thus, we can identify the boundary between open and closed drift paths in the data by sharp drops in the distribution function.
Figure 1 shows the differential flux for ions at 90° pitch angle versus energy and time for H+ (panel a) and O+ (panel b) and particle pressures (panel c), calculated over the HOPE energy range using 90° (81–99°) differential flux (j) as 4π/3 ∑E
ΔE, where E is kinetic energy and m is mass, for Van Allen Probe A along with the Dst index, IEF (input into UNH-IMEF model), Kp (input for Volland-Stern model), and IMF angle and strength and solar wind velocity (inputs for Weimer model; panels c–i) during both storms. The storm onset is marked with a red line, and the modeled passes are highlighted in blue. Because the Van Allen Probes are generally close to the magnetic equator during these passes, we assume that the measured 90° pitch angle bin accurately represents the equatorial 90° particles used in the simulation.
During both storms, the IEF, a predictor of the strength of the inner magnetosphere electric field, increases and decreases significantly on the time scale of around an hour during the first few hours of the storm. This implies large changes in the inner magnetosphere electric field during this period. Shortly, after onset (red line), we see a sharp increase in both O+ and H+ fluxes, and in the following passes we see that the ring current particle pressure grows. This is the result of the increased plasma sheet source convecting into the inner magnetosphere. However, because ions take many hours to drift into the lowest L shells, we do not observe the full extent of the ring current penetration during these passes. Our chosen passes, near the end of the main phase, are both periods with relatively stable IEF value, predicting relatively stable, enhanced magnetospheric convection ideal for simulation. Furthermore, these passes are many hours after storm onset, allowing the ring current time to penetrate to low L shells.
The IEF value during the 2013 and 2015 passes is around 6 and 10 mV/m, respectively. Because the IEF is highly correlated with the inner magnetosphere electric field strength, we expect much stronger convection during the 2015 storm. Figure 2 shows the spacecraft trajectory for the chosen passes mapped to the equatorial plane. Here, we see that the trajectory for the 2013 storm is more eastward and sunward than the 2015 storm. Because the westward gradient-curvature drift becomes stronger with increasing energy, it is difficult for ions to reach low L shells eastward of the midnight. Kistler et al. (1989) showed a decrease in the energy of the expected open/closed drift boundary from midnight to dawn using a Volland-Stern model. Thus, we expect a higher open/closed boundary for the 2015 storm due to both a higher IEF value during the storm and a more duskward orbit.
3 Electric Field Models
Figure 2 shows the electric potential for different configurations of the electric field models in the inertial frame with the trajectory of the two simulated passes overplotted. We use a value of −92.4/R (kV) for the corotation potential, where R is the distance in earth radii. We have chosen a color bar to highlight the potential around the orbits. Panels a and b show the UNH-IMEF model electric potential for IEF values similar to those during the 2013 and 2015 storms, respectively (6 and 10 mV/m). Because of the high IEF values, the potentials here are based on extrapolation. Further increase in the IEF only results in an increase in the magnitude of the electric potential with no change to the overall shape. Panels c and d show the potentials for the Weimer 96 model using representative inputs for each storm. Here, we see the shape of the potential change between the two storms due to the differences in the interplanetary magnetic and electric fields. Like the UNH-IMEF model, the Weimer model shows a much stronger potential for 2015 storm (panel d) primarily due to an ~50% increase in IMF strength over the 2013 storm. The Weimer model's potential patterns also vary with the IMF angle. The potentials are strongest for an IMF angle of 180°, which corresponds to Bz purely southward. The IMF during the 2015 storm is closer to this maximum, a 13.5° offset from south compared with the 28° offset during the 2013 storm. However, while both IMF angles are close to 180°, they have different By components, with the >180° IMF angle for the 2013 storm denoting a negative By value, and the <180° IMF angle for the 2015 storm denoting a positive By value. Weimer (1995) found the duskward lobe to be stronger and more circular and the dawnward lobe to be weaker and more crescent in the northern hemisphere for positive By values. This trend reverses for negative By values, although the dawn cell is not nearly as enhanced for negative By as the dusk cell is for positive By. Thus, the 2015 potential is stronger not only due to the increase in the IMF but also due to having a more effective IMF angle. The potential's shape is affected further by the opposite directions of By, which changes the shape of the potential lobes. However, the By dependence is reversed in the southern hemisphere, and it is likely that using a model based on only one hemisphere overestimates this effect at the equator. Panel e shows a Volland-Stern potential with a shielding factor of 2 for Kp of 6.5 using the parameterization by Maynard and Chen (1975). Panel f shows a 2-hr offset Volland-Stern potential for a Kp of 8.5. Kistler et al. (1989) found that an eastward offset of 2 hr was needed to match observed features in the energy spectra.
The main purpose of this paper is to test how successful these models are in determining the drift trajectories during the storm. A drift trajectory that intersects the spacecraft has covered a wide range of positions and lasted for many hours. Comparing the in situ electric field during the inbound pass with the model field cannot show how well the electric field matches for the current storm over the drift trajectory because it is only a local measurement but can still give an indication of how well the model electric field compares with the actual storm time field. Figure 3 shows a comparison of the y-component of the electric field for the 2-min averaged MGSE VAP-A EFW data (black) with the Weimer 96 model (green), UNH-IMEF model (red), and Volland-Stern model with a 2-hr offset (dark blue) and no offset (light blue) as a function of L value during the storm time pass in the corotating frame. Because of the harmonic fit of the potential, the Weimer model can have oscillatory artifacts in the electric field at low L values. We plot the field with a dotted line where we believe the model is showing us these artifacts and not a prediction for the electric field. Because the models may underestimate the storm time electric field, we plot the electric field for the model with both the measured input (top) and an enhanced input (bottom) where we multiply the measured IEF values by a factor of 2 for the 17 March 2013 storm and a factor of 1.5 for the 17 March 2015 storm. These factors were found by gradually increasing the inputs when the model underestimated the observed ion access. For these storms, these factors represent the largest increase in the IEF while staying under the maximum value of 15 mV/m for the UNH-IMEF model. Our along-orbit comparison of the electric field shows that these increases are reasonable. For the Volland-Stern potential, we increase the Kp to a value of 8.5.
As expected, the measured electric field shows more structure in L than the model field that are based on fits and averages. For the 2013 storm (panels a and b), the Van Allen Probes EFW data show a wide enhancement of 1.5–2.0 mV/m along the orbit from L = 3.0–4.5 that exceeds the electric field predicted by any of the models. The EFW electric field steadily falls off below L = 3.0. Using the measured solar wind values as inputs (panel a), the UNH-IMEF and Weimer electric models predict an electric field of ~0.7 mV/m at L > 3.5. Compared with observations, both models greatly underestimate the field in this range. Both the Weimer and the UNH-IMEF fields drop off below L = 3.5; however, the Weimer field decreases about twice as fast as the UNH-IMEF field. This results in a stronger electric field for the UNH-IMEF from L = 2.7–3.5. The Volland-Stern field is relatively flat with a magnitude around 1 mV/m at L = 5 that slowly decreases to ~0.5 mV/m at L = 2. The offset potential is ~0.75 mV/m at L = 5 and decreases more slowly, ending up with a similar electric field strength at L = 2 as the nonrotated field. While all electric field models show agreement at L shells above 3.5, the Volland-Stern field is the only one that penetrates down to L = 2.5 as seen in observations. Increasing the solar wind inputs to the model (panel b) increases the model electric field but does not change their features. With the enhanced inputs, the models show agreement with the magnitude of the Van Allen Probe measurements at L > 3.5 but still underestimate the electric field at L = 2.5–3.5. For the enhanced Volland-Stern electric fields, the field with no rotation matches the magnitude and shape of the observed electric field at L = 2.5 and 3.5, while the rotated potential still underestimates the electric field along the orbit.
For the 2015 storm (panels c and d), the Van Allen Probes EFW data show an electric field just above 1 mV/m at L = 3.7, the highest L value for which there were data for this pass. The EFW field sharply peaks at just above 2 mV/m at L = 3.5 before falling to below 1 mV/m at L = 3.3. Below L = 3.0, the subtraction of the motional electric field (Vsc × B) for the EFW instrument does not work well. We have denoted this area with a transition from solid to dotted line and do not consider this data in our comparison. The Weimer model shows an electric field of 2 mV/m at its peak around L = 4.5, matching in magnitude but not location with the data. The UNH-IMEF field shows a peak at L = 3.6 of 1.5 mV/m, predicting the right location of the electric field peak along the orbit but with a lower magnitude and a less localized peak. At lower L shells the UNH-IMEF field shows a slow decrease below L = 3.5 down to around 1 mV/m at L = 3.0, overestimating the field at L = 3.0–3.3. The Weimer field decreases starting at L = 4.5 and fails to penetrate below L = 3.5. The Volland-Stern field shows little change in magnitude with L and underestimates the electric field at L > 3.3 while showing good agreement at L = 3.0–3.3. The offset Volland-Stern field agrees in magnitude at L = 3.5 but is slightly lower at lower L values and slightly higher at higher L values. Increasing the solar wind inputs (panel d) raises the UNH-IMEF field to ~2.2 mV/m at L = 3.5, slightly above the EFW peak. The enhanced inputs cause the UNH-IMEF field to overestimate the electric field at L < 3.5 by a significant amount. The enhanced Weimer field increases in magnitude but is still peaked too far out and does not penetrate to low L shells. The Volland-Stern field has almost no noticeable change, since the Kp index was 8 during this event and the magnitude of the field peaks at Kp = 8.5.
Comparing the electric field along the Van Allen probes orbit shows that the empirical models (Weimer and UNH-IMEF) estimate the electric field fairly well above L = 3.5 with enhanced inputs. In both storms, the Weimer model shows poor agreement below L = 3.5, falling off much faster than the data and other models. As expected, the measured field shows much more variation (either spatial or temporal) than the empirical models based on long-time averages. None of the models were able to reproduce the enhancement at L = 3.0 during the 2013 storm. The Volland-Stern gives reasonable electric field values for both storms but fails to capture the structure of the observed electric field.
4 Data-Model Comparison
The drifts in the prescribed electric and magnetic fields lead to particular features in the energy spectra. In particular, there is an energy cutoff above which ions do not have drift access to the inner magnetosphere. At the inner edge, there is an energy-dependent boundary below which the particles cannot convect. These access boundaries are quite clear in both the observations and the trajectory models and so can be used to test how well the models reproduce the convection electric field. For our first comparison, we compare the energy spectrum of O+. Because it has a relatively high charge exchange lifetime, we expect the observed O+ boundaries to better represent the drift boundaries. Figures 4 and 6 show the 90° pitch angle O+ distribution function versus energy spectrum as measured by HOPE compared with drift time versus energy spectrograms for the backward tracing of ions using the different electric field models. To determine the drift time spectrograms, we model the drift trajectory of an ion with a 90° pitch angle for each energy in the HOPE spectrogram at 1-min resolution along the Van Allen Probes' orbit. To determine the drift paths, we use the modeled potential, calculated with real-time inputs, combined with a dipole magnetic field to determine the drift velocity. Using a second-order Runge-Kutta method with a 30-s half step, we backward trace ions starting at the spacecraft observation point. We define the drift time of the ions as the time they take to reach the nightside plasma sheet, defined at L ≥ 6 and X < −5 on the nightside. If the drift does not reach the plasma sheet within 14 hr, it is shown as white. The 14-hr time frame covers the entire main phase of the 17 March 2013 storm. For the 2015 storm, which has a longer main phase, we assume ions with drifts longer than 14 hr have significant enough charge exchange loss that they do not have meaningful access. We plot the drift times for both realistic electric field input parameters along with the enhanced input parameters. Because the Kp is relatively constant during these periods, we found little difference between using a constant and dynamic Kp input. Thus, we choose to show only the results for a constant Kp value of 8.5, which was found to provide the most ion access. The first column shows the spectra for 1–50 keV. In the second column we plot the spectra in a reduced energy range (10–50 keV) to more clearly show the upper energy boundaries. Reference lines are plotted at 20 and 40 keV.
Figure 4 shows the data-model comparison for the 17 March 2013 storm. Panel a shows the 90° O+ distribution function versus energy spectra for Probe A. The Van Allen Probe data show a sharp dropoff in the distribution for energies above 40 keV starting below L = 3.8, a feature that often indicates the boundary between drift paths open to the plasma sheet and drift paths on closed circular trajectories. Ions with energies up to 40 keV have access down to L = 2.3. A nose structure (narrower energy range with access to lower L) centered around 20 keV has access down to L = 2. Panel b shows the particle drift times from the UNH-IMEF simulation using realistic IEF data as input. The white indicates regions where the particles are on closed drift paths or do not reach the plasma sheet in the 14-hr drift time. Here, we see that the highest energy with access is much lower than observed in the data. While the Van Allen Probes data showed that particles up to 40 keV have access, the model fails to provide access to particles >20 keV. The model shows a similar nose structure to the data, but the nose structure in the model is centered just below 10 keV, much lower in energy the 20 keV nose observed in the data. Panel c shows the results for the UNH-IMEF model simulation with the inputs enhanced by a factor of 2. This factor was necessary to match the measured electric field at L = 4 in the along-orbit electric field comparison, although it still underestimated the field at lower L shells. With the enhanced electric field, higher energy ions now have access to the lower Ls, with energies up to almost 40 keV reaching down to L = 4. However, at L shells below that, the high energy boundary drops off much faster than in the data. While the data show particles up to 40 keV at L = 2.5, the enhanced simulation shows access for particles only up to 20–25 keV. Furthermore, the enhanced electric field no longer produces the nose structure shown in the data and model with realistic inputs. The disagreement between the data and the enhanced IEF simulation is consistent with the electric field comparison, as the enhanced inputs matched well at L = 4 but not at lower L shells.
The Weimer model (Figure 4d) shows better agreement with the data at L > 3 compared with the UNH-IMEF model. The model shows full access to the HOPE energy range down to L = 4.3, where the open/closed drift boundary drops to ~35 keV. The boundary slowly drops down to 20 keV at L = 3. However, below L = 3 the Weimer model fails to provide significant access to particles at ring current energies. With enhanced inputs, the Weimer model shows even better agreement with the data, matching the drop of the open/closed drift boundary down to 40 keV at around L = 3.8. Below L = 3.2, however, the boundary drops off, and again, the model fails to provide significant access to ring current ions. The Volland-Stern models (panels f and g) show a steady decrease in the open/closed drift boundary down to L = 2. The 2-hr offset potential (panel g) raises the open/closed drift boundary by 5–10 keV, providing the highest open/closed drift boundary of any of the models. The model matches the drop in access around L = 3.5 but still underestimates the access at L < 3.
Figure 5 shows sample drift paths at three locations along the orbit for the three different electric field models, with realistic inputs used for the top row and the enhanced inputs for the UNH-IMEF and Weimer 96 and the rotated Volland-Stern in the bottom row. The initial energies and L values used for the backward tracing are marked as with diamonds symbols on the right side of Figure 4. The 15-keV ion at L = 2.6 (red trace) only has access for the UNH-IMEF and Volland-Stern fields, both realistic and enhanced versions. This is representative of the highest energy with access at this L shell. The drift paths show that these ions have direct access, with the energy-dependent westward gradient-curvature drift only starting to dominate the motion at the end of the orbit. Because the orbit is so eastward and dawnward at low L shells, the higher energy particles do not have access here due to the increased strength of their westward drift. The 40- and 50-keV ions at L = 3.3 and 4.3 (green/blue) have direct access for the Volland-Stern (both nonrotated and rotated) and enhanced Weimer models. Here particles take only a few hours to drift in. While the enhanced UNH-IMEF model was able to provide access for these ions, the drift paths are very different. Ions in the enhanced UNH-IMEF model must first drift westward around the dayside, reaching the spacecraft location after ~9 hr.
Figure 6 shows the data-model comparison for the 17 March 2015 storm. Panel a shows the O+ distribution function versus energy spectra for Probe A. The data show that energies above the HOPE energy range (>50 keV) have access down to L ~ 2.3, below which the access of the highest energies decreases down to 30 keV just below L = 2. Here, we define the boundary for access in the data as the transition from light blue to dark blue, which represents where the distribution function decreases off by at least a factor of 10, indicating significant charge exchange loss and long drifts. The data show a nose structure centered around 30 keV penetrating down to L ~ 1.85. Panels b and c show the particle access for the UNH-IMEF model using both realistic and enhanced IEF input. The realistic IEF input, panel b, shows the access for the full HOPE energy range down to L = 3.2, much further out than the L = 2.5 dropoff in the data. At L = 2.1, the model shows a drift boundary of ~25 keV while the data show access for ions up to ~40 keV. The model shows a nose structure centered around 10 keV that has access down to L = 1.7. The enhanced UNH-IMEF model, panel c, shows the energy dropoff at L ~ 2.7 further than the realistic IEF and closer to the L = 2.5 dropoff observed in the data. At L = 2.1, the enhanced model provides access for particles up to 40 keV, matching the observed access of the Van Allen Probes. However, the enhanced model shows particle access down to L = 1.55, much lower than observed in the data, with no nose structure. The Weimer model (panels d and e) matches the observed access at L > 3.5, but the open/closed drift boundary decreases fast at lower L shells. The Weimer model sees no appreciable increase in particle access with increased inputs and fails to provide access to ring current particles at L < 3. The Volland-Stern model (panel f) shows a decrease in the open/closed drift boundary starting at L = 3.5 down to just below 20 keV. The open/closed drift boundary decreases below the HOPE energy limit at L = 3.5, falling to 20 keV at L = 2.1. Rotating the potential (panel g) increases the open/closed drift boundary, moving the fall of the open/closed boundary below 50 keV down to L = 3.0 and creating a nose structure at ~15 keV. At L = 2.1, the offset Volland-Stern model shows an open/closed boundary just below 30 and 10 keV below the data.
For the 2015 storm, both the UNH-IMEF and the rotated Volland-Stern electric field models do a reasonable job recreating the observed access. While the enhanced UNH-IMEF model matches the access at L = 2.1, it also overestimates the access at low L shells. The offset Volland-Stern field, while underestimating the access, does a better job, creating the observed nose structure and L shell boundary at L = 2. However, if we look at the 3-hr drift boundary (transition from blue to green), the enhanced UNH-IMEF shows this boundary at L = 2.3, while the offset Volland-Stern shows the boundary at L = 3. Furthermore, the UNH-IMEF model shows this boundary to be highly energy dependent, with the boundary reaching the lowest L value around 10 keV. This boundary is less energy dependent in the UNH-IMEF model with enhanced inputs, and the Volland-Stern models show very little energy dependence for the boundary. Thus, while both models provide access to L = 2, they do so with different drift paths.
Figure 7 shows the drift path for a 40-keV ion at L = 2.6, 3.0, and 3.4 using the six variations of the electric field models. The energies and L value of these are marked as with diamonds symbols on the right side of Figure 6. Panels a and b show the drift paths for the realistic and enhanced UNH-IMEF model. While both provide access at all three L values, the enhanced model provides much faster drifts. At L = 2.6, the 40-keV ion must take a long drift path, drifting eastward around to the dayside before gradient drifting westward back to the spacecraft location for the realistic model. Enhancing the inputs provides more direct access, with a drift time around 3 hr rather than the ~11-hr drift using realistic inputs. The Weimer model is only able to provide access at L = 3.4 and only with enhanced inputs. For the Volland-Stern potential (panel e), the 40-keV ion only has access for L = 3.4. Rotating the potential (panel f) allows ions to reach down to L = 2.6 with a long drift path that reaches eastward to the dayside, similar to the UNH-IMEF model with realistic inputs. While the rotated Volland-Stern and both UNH-IMEF electric fields were able to provide access to the 40 keV at low L shells (L = 2.7), only the enhanced UNH-IMEF potential was able to provide fast access.
At low L shells, charge exchange losses are increased due to both the higher neutral hydrogen density closer earthward and the longer drift times particles take to reach the lower L shells (L < 3). Therefore, while the models show access at low L shells, if the drifts are too long, there will be significant flux lost due to this loss. Several times in our analysis, we observed that different models provide ion access with different drifts. These drifts have different losses associated with them. To better understand the “fitness” of the ion drift paths of our models, we simulate the effect of charge exchange along our modeled drift paths and compare this loss with the features observed in the data. For our charge exchange loss, we use a neutral H model (Hodges, 1994) along with species cross section interpolated from the values derived in Smith and Bewtra (1978) to calculate the net effect of charge exchange along our modeled drift paths.
Figures 8 and 9 show the H+ distribution function versus energy spectra (panel a) along with simulated H+ distribution versus energy spectra with and without the modeled charge exchange loss applied. To produce spectra for our models, we take a sample distribution function from the Van Allen Probes data further out in the simulated pass and move it in adiabatically, keeping the distribution function constant along lines of constant μ. We use the spectrum at L = 5 for the 2015 storm, but because of a large change in the distribution function from L = 5 to L = 3 for the 2013 storm, assumed to be a time-dependent change in the source, we use the spectra at L = 3.5 as input for the 2013 storm. We then apply the modeled charge exchange loss to this spectrum. Because charge exchange is modeled from the source region (L = 6), we use the model estimation to account for the loss already experienced in drifting from L = 6 to L = 3.5 for the 2013 storm. Because there is relatively little loss drifting to L = 5, no correction is needed for the 2015 storm.
For the 2013 storm, the Van Allen Probes data show steep dropoffs in distribution function for particles over 40 keV down to L = 2.6. This boundary moves down to above 25 keV at L = 2.3, where there is a boundary in L where particles at all energies experience a sharp dropoff in flux. The UNH-IMEF model (row b) shows significant charge exchange effects starting at L = 3. At higher L shells, we see fluxes start to fall due to charge exchange effects for the highest energy channel around 15 keV. Increasing the inputs results in faster drift times and moves the boundary inward to just above L = 2.5. The distribution function drops off for the enhanced UNH-IMEF model at just above 20 keV down to L ~ 2.6 where it gradually moves down below 10 keV by L ~ 2.3. While the Weimer model was not able to provide access below L = 3, where it does provide access, L = 3–3.5, there is more loss due to the longer drift times (Figure 6). With enhanced inputs, the Weimer model is able to provide access to ions up to 40 keV around L = 3.5. Here we see that there is less loss compared with the enhanced UNH-IMEF model, which also provided access for this energy range but had increased loss for energies above 20 keV. The Volland-Stern model shows a similar profile to the enhanced UNH-IMEF model, except with less loss at higher energies, better matching the data. Rotating the Volland-Stern potential (g) raises the energy where the distribution function drops off, with both potentials only showing significant loss in the highest energies that have access. The rotated potential shows particles drifting deeper into the magnetosphere before seeing a dropoff due to charge exchange, showing a similar boundary in L to the enhanced UNH-IMEF. It is clear that while none of the models show particularly good agreement with the data, the Volland-Stern and, where there is access, the Weimer model better match the data. Figure 5 showed the drift paths for these models having direct, fast access from the nightside while the UNH-IMEF model drift paths were long, drifting around the dayside. This shows that while the UNH-IMEF model does provide access for these higher energies, the drift paths provided are not appropriate.
The bottom of Figure 8 shows the calculated H+ particle pressure for E < 20 keV for the HOPE data and simulated spectra. Because the models fail to provide access for higher energy ions, we choose to compare the pressure for only energies below 20 keV. This allows for the differences in pressures to be dominated by the effects of charge exchange rather than the differences in particle access. For the UNH-IMEF and Weimer models, the enhanced inputs are shown with lines of greater thickness. The bottom-left panel shows the measured H+ pressure, black dashed line, along with the modeled pressures with no charge exchange loss for energies below 20 keV. Modeled particle pressures here are only a function of particles access. Pressures for the Volland-Stern and enhanced UNH-IMEF models start to outpace the measured pressure below L = 3.0 where the HOPE data start to experience charge exchange loss. As the models' open/closed drift boundary decrease below 20 keV at low L shells, we see that their pressures drop. Comparing the HOPE data to models with no loss shows the effect of charge exchange loss slowly increasing down to L = 2.5. By L = 2.3, charge exchange has effectively removed particle access. The bottom-right panel shows the same data with the modeled charge exchange loss applied. Here, none of the models show higher H+ pressure than the data. Both the offset Volland-Stern and enhanced UNH-IMEF models match the data well down to L = 2.7 but underestimate below due to the increasing charge exchange effects. This shows that ions in the data have much faster access in the data for L = 2.3–2.7 than provided by the models.
For the 2015 storm (Figure 9), the Van Allen Probes data show that the distribution brought inward down to L ~ 2.5 before significant loss is observed. At L = 2.5, we start to observe some loss for particles around 40 keV. At L = 2.3–2.5, there are signs of loss for 20-40 keV particles. A sharp boundary in L observed at L = 2.2, with fluxes dropping off significantly at lower L shells. A sharp energy boundary is observed extending from ~30 keV at L = 2.2 up to 50 keV at L = 2.4. The UNH-IMEF model (panel b) shows a similar dropoff in flux at all energies for L < 2.5 when charge exchange is included. The dropoff at energies >20 keV, however, starts at too low energy in the model, and particles below 20 keV show access to lower L shells than the data. Increasing the inputs to the UNH-IMEF model (panel c) allows the distribution to reach lower L shells and at higher energies. At L = 2.5, we see the transition from green to blue at 40 keV, matching the data. The enhanced model recreates the L boundary fairly well at 20–40 keV; however, at lower energies this boundary form extends even further inward than observation. Both Volland-Stern models (panels f and g) show similar profiles, with the rotated model showing the population penetrating lower in L and higher in energy than the nonrotated model. The Volland-Stern model shows signs of charge exchange loss starting at L = 3.5, much further out than either the data or UNH-IMEF model. This is due to the longer drift times provided by the Volland-Stern model compared with the UNH-IMEF model (Figure 6).
The bottom of Figure 9 shows the H+ particle pressure comparisons for the data and models. The bottom-left panel shows the measured H+ pressure, black dashed line, along with the modeled pressures with no charge exchange loss. Modeled particle pressures here are only a function of particles access. The UNH-IMEF (red) and offset Volland-Stern (dark blue) both agree with the data down to L = 3.0, suggesting that there is little charge exchange at L > 3. At lower L shells the offset Volland-Stern and realistic UNH-IMEF models agree well with the data. This is by coincident that the difference in access matches the charge exchange loss. The Volland-Stern model (blue) shows a similar profile, but with lower pressure due to the decreased access compared with the offset model. We found the enhanced UNH-IMEF model (thick red) best matched the observed access. We can infer from the difference between the model without loss and observation that there is an increasing effect of charge exchange in the HOPE data starting at L = 2.8 and the effects of charge exchange prevent any substantial ring current pressure to penetrate below L = 2.3. The bottom-right panel shows the model comparison taking into account charge exchange loss. The enhanced UNH-IMEF is the only model that produces a particle pressure comparable with the data. The enhanced UNH-IMEF model matches fairly well down to L = 2.5. The model shows the charge exchange loss to outpace the adiabatic energization around below L = 2.5, where the data show this effect at L = 2.3. The model overestimates the particles pressure below L = 2.2, likely due to the enhanced access to lower L shells in the model. Because of the similarity in access between the data and model, this result shows that the charge exchange losses are realistic assuming the source used is sufficiently accurate. The other models provide about half the measured particle pressure for L = 2.5–3.0. This is due to both an underestimation of particle access and increased charge exchange losses due to longer drift times for these models.
While both the Volland-Stern and the UNH-IMEF models provide particle access down to L = 2 for the 2015 storm, modeling charge exchange loss suggests that the UNH-IMEF electric field model provides more appropriate drifts. Comparing drift paths (Figures 7b and 7f) shows that ions in our Volland-Stern simulation take longer to drift in and spend more time at low L shells—The drift paths tend to send particles to low L shells eastward of the observation point from which they drift primarily azimuthally back to the spacecraft location. For example, Figures 7b and 7g show that a 40 keV in the enhanced UNH-IMEF simulation drifts azimuthally for just over an hour after drifting primarily radially, while the same particle in the offset Volland-Stern field drifts azimuthally for ~4 hr. The results for the 2015 storm are in contrast to the 2013 storm, where the Volland-Stern field showed faster access and better overall agreement.
5 Summary
We have looked at electric field data and particle access for two storms using three different electric field models. We have found for both storms that the Weimer electric field is too weak to reproduce the observed access for L < 3.5 and therefore is a poor choice for adequately strong storms where ions have access to these low L shells. Furthermore, we found that all the model electric fields were too weak during storm time conditions and better agreement was found by increasing the inputs to the models.
During the 2013 storm, all the models failed to capture the large electric field enhancement measured by the Van Allen Probes at L = 3.0. Both the Weimer and UNH-IMEF captured an enhancement at L = 4 but failed to capture the penetration of the electric field to lower L shells. The Volland-Stern electric field does not have a localized peak and always increases with increasing L. Due to the flatter shape of the Volland-Stern electric field along the orbit, it penetrated lower than the other models. None of the models were able to adequately capture the penetration of 20-40 keV particles down to L = 2 observed by the Van Allen Probes during the 2013 storm, most likely due to the underestimation of the electric field at low L shells in the models. Furthermore, our comparison showed that the drift paths of the energies that did have access were not always appropriate. At L = 3.5, the Weimer and Volland-Stern models were able to provide access to >20 keV with realistic charge exchange loss, but the UNH-IMEF model provided longer drift paths that traveled around the dayside, resulting in much more loss than observed in the data. At low L shells, for the energies that had access, the drift times were too long, resulting in more charge exchange loss than observed.
For the 2015 storm, the UNH-IMEF and offset Volland-Stern models were able to adequately reproduce the observed nose structure and access in L. Increasing the inputs to the UNH-IMEF model resulted in better agreement for 20-40 keV particles at low L shells but did not produce a nose and overestimated ion access at L < 2. The model with enhanced inputs, however, overestimated the electric field along the s/c path as well as the penetration of <20-keV particles. However, comparing charge exchange loss shows that the drift paths of the enhanced UNH-IMEF model best recreated the observed charge exchange loss, with the simulated pressure showing good agreement with the data for similar ion access. While the Volland-Stern model was able to recreate much of the access, the charge exchange comparison showed that the drift times were too long, resulting in much more charge exchange loss than observed in the data.
Our previous study of the 17 March 2013 storm (Menz et al., 2017) showed evidence that the large O+ ring current population that dominated the particle pressure was due to inward adiabatic convection of an observed O+ enhancement in the near-earth plasma sheet (L ~ 6). However, convection using these empirical models does not bring the higher energy particles in far enough, and so we do not expect to be able to reproduce the observed ring current pressure. Indeed, we found that all models predicted a pressure that is too low. However, our success in modeling the access for the 2015 storm allowed a simple comparison of the H+ particle pressure using a spectrum from further out in the orbit drifted inward adiabatically assuming charge exchange along the drift path. The enhanced UNH-IMEF model was able to sufficiently match the observed access and particle pressure.
Our calculations of inward transport including charge exchange assumed a constant source at the boundary. The H+ plasma sheet source spectrum was relatively constant during the main phase of these storms, so this assumption worked reasonably well for modeling the H+ spectra. However, because of the variation in the O+ source, this simple analysis is not sufficient to model the O+ contribution to the ring current. Using the convection results of this study, future work will model the O+ particle pressure using a more realistic, dynamic source.
Overall, we find that while inward convection combined with charge exchange loss explains quite well the features that are observed in the energy spectra, the specific energies and L values at which the transitions from open to closed drift paths occur are quite sensitive to the convection electric field. Because the electric field during storm times varies in both space and time during the storm, it is difficult for empirical models based on long-term averages to adequately replicate the storm time field. However, the fact that the observed energy cutoffs were not found in the simulation when increasing the magnitude of the electric field means that the shape of the electric field potentials must also be changing as geomagnetic activity increases. The particle pressure in the inner magnetosphere is also quite sensitive to the energies at which these energy cutoffs occur, and so a model that gives reasonable qualitative agreement can still overestimate or underestimate the pressure significantly. Thus, modeling the electric field in the inner magnetosphere remains a challenging problem for which improvement is necessary.
Acknowledgments
Work at UNH was supported by NASA under Grants NNX14AC03G and NNN06AA01C. Work at the University of Minnesota was supported by APL contract to UMN 922613 under NASA contract to APL NAS5-01072. HOPE data used in this paper were downloaded from http://www.rbsp-ect.lanl.gov/rbsp_ect.php. EFW data used in this paper were downloaded from http://www.space.umn.edu/rbspefw-data/. Solar wind plasma and IMF data and the Kp and Dst indices were obtained from http://omniweb.gsfc.nasa.gov.
