Rapid Acceleration Bursts in the Van Allen Radiation Belt

The fast Van Allen radiation belt electron dynamics during geomagnetic storms have not yet been fully explained, in part due to limitations of standard satellite missions in both orbit and the number of spacecraft. Here we overcome these limitations using measurements from the Global Positioning System (GPS) constellation during an acceleration event on 26 August 2018. We show that the acceleration of relativistic electrons occurs in two distinct bursts, each dominated by a different acceleration mechanism. The first burst enhances the radiation belt electrons by four orders of magnitude in 2 hr and is consistent with ULF‐wave radial diffusion. The second burst is likely caused by the local acceleration and delivers an order‐of‐magnitude increase in 20 min. This work demonstrates how distributed, operational measurements can be used to resolve phenomena not observable with previous capabilities, and that rapid energization of the radiation belt can occur much faster than previously reported.


Introduction
Electrons can be trapped and accelerated inside Earth's magnetosphere, creating the dynamic radiation belts (Friedel et al., 2002;Van Allen & Frank, 1959), where strong acceleration is often associated with geomagnetic storms (Murphy et al., 2020).The dynamics of the trapped relativistic electron population, with energies from tens to thousands of kilo-electronvolts (keV), are of practical interest because of their potentially harmful impact on space-based technology and astronauts (Baker et al., 1994;Eastwood et al., 2017;Lohmeyer & Cahoy, 2013;Welling, 2010).The radiation belt also provides a natural laboratory for studying the transport and acceleration of charged particles in magnetic fields (W.Li et al., 2020).These are typically driven by the interaction of electrons with waves of various types and frequencies, leading to the violation of one or more of the three adiabatic invariants and resulting in the observable evolution of the radiation belt content (e.g., Koskinen & Kilpua, 2022).However, the processes dominating these dynamics are still hotly debated (I.Mann et al., 2018;Shprits et al., 2018).
For example, during the main phase of a geomagnetic storm, radiation belt electrons can be lost outward into the solar wind due to the interaction with the ULF waves (see e.g., I. R. Mann et al., 2016;Thompson et al., 2020;Tu et al., 2013, and references therein), or precipitate downward into the atmosphere due to the interaction with the higher frequency waves like EMIC (e.g., Grach et al., 2022;Hendry et al., 2017;Usanova et al., 2014), hiss (e.g., Ma et al., 2021), or chorus (e.g., Chakraborty et al., 2022;L. Chen et al., 2021;Mourenas et al., 2022).Interestingly, the ULF waves also have the capability to modulate the intensity of electron precipitation (e.g., Shang et al., 2021;Watt et al., 2011).In addition to wave-driven electron losses, the charged particles can also be lost from the radiation belts due to field-line curvature scattering driven by fast changes in the geomagnetic field (see e.g., Tu et al., 2014;Y. C. Zhang et al., 2016).These processes lower the radiation belt content and are commonly referred to as the "radiation belt loss" or "dropout" (see also Turner et al., 2013).
Similarly, during a recovery phase of a storm, the newly trapped electrons can be transported inwards and thus accelerated by the ULF waves (e.g., Ozeke et al., 2020;Silva et al., 2022) or accelerated locally due to the interaction with hiss (e.g., J. Li et al., 2019) or chorus waves (e.g., Horne et al., 2005;Hua et al., 2022).Lower energy electrons (1-10s keV) are also often impacted by the presence of kinetic Alfven waves in the auroral regions of the magnetosphere (see, e.g., review by Lysak (2023), and references within).These processes are often referred to as radiation belt "acceleration" or "enhancement" (see, e.g., review by Millan and Baker (2012)).Importantly, both loss and acceleration processes affect a wide range of energies from a few keV to a few MeV.In this study, we focus on investigating fast radiation belt dynamics that affect multi-MeV electron populations.
From 2012 to 2019, radiation belt dynamics were continuously measured by NASA's flagship, twin-satellite, Van Allen Probes mission (formerly RBSP; Mauk et al., 2012).However, substantial radiation belt acceleration can occur on timescales much shorter than the half-orbit of Van Allen Probes (<4 hr) (e.g., I. R. Mann & Ozeke, 2016;Ozeke et al., 2017), limiting the capability of Van Allen Probes to capture the dynamics of the fastest events.For example, Olifer, Mann, Ozeke, Morley, and Louis (2021) showed that inward radial diffusion can increase radiation belt electron flux by more than two orders of magnitude over a ∼30-min period, thanks to a fortuitouslytimed conjunction between the two Van Allen Probes.Their analysis relied on serendipitous phasing of the acceleration and Van Allen Probes' orbits, showing that there is a need for continuous radiation belt monitoring on much shorter timescales to properly characterize fast radiation belt dynamics.This paper focuses on demonstrating such a capability using the recently released, publicly-available energetic particle data from the Global Positioning System (GPS) constellation (Morley et al., 2017).In particular, we analyze GPS electron flux data and convert the flux measurements to estimates of electron phase space density (PSD) as a function of three adiabatic invariants to investigate fast radiation belt dynamics during the August 2018 geomagnetic storm.By combining electron radiation measurements from 20 GPS satellites, we show that the 8hr-long acceleration during this event, as observed by the Van Allen Probes, actually consists of two distinct acceleration periods that last only 80 and 20 min, respectively.Due to their short nature, we refer to the observed acceleration periods as "acceleration bursts." To determine the dominant physical processes for each of the acceleration bursts, we assess ULF and chorus wave activities using in-situ and ground-based observations during this event.We also evaluate the timescales of the observed acceleration during the two acceleration bursts by deriving radial diffusion coefficients directly from the GPS PSD data.We compare GPS-driven coefficients with those obtained from various models highlighting the periods where acceleration rates are consistent with the ULF wave radial diffusion.Overall, this study shows that significant changes in the radiation belt population consistent with the diffusive processes can occur on much shorter timescales than previously thought and the only consistent way of resolving such dynamics is through the implementation of distributed multi-satellite missions like GPS.

Data and Methodology
In this paper, we rely on radiation belt electron measurements made by the twin-satellite Van Allen Probes mission, and the 20-satellite constellation of GPS satellites.In particular, we utilize differential directional electron flux measurements from the Magnetic Electron Ion Spectrometer (MagEIS; Blake et al., 2013) and the Relativistic Electron-Proton Telescope (REPT; Baker et al., 2012) onboard the Van Allen Probes.Meanwhile, we use omnidirectional differential flux data from the Combined X-Ray Dosimeter (CXD; Morley et al., 2016) onboard the GPS satellites.We supplement particle measurements with the in-situ and ground-based measurements of the plasma waves.In particular, we use the Plasma Wave Experiment (PWE; Kasahara et al., 2018) suite onboard Exploration of energization and Radiation in Geospace (ERG, also known as Arase; Miyoshi, Shinohara, Takashima, et al., 2018;Matsuoka, Teramoto, Nomura, et al., 2018) spacecraft to estimate wave dynamics in the very low frequency (VLF) band, in particular chorus.Finally, we use the ground-based magnetometer data, available through SuperMAG (Gjerloev, 2009(Gjerloev, , 2012)), to estimate the ULF wave activity across different latitudes and longitudes.
It is also important to note that to use only the highest quality of the GPS data, we remove potentially contaminated or low-quality CXD data.In particular, we use the same approach as described by Smirnov et al. (2020).

Estimating PSD for the GPS Satellites
Determining the dominant physical processes that drive the radiation belt acceleration generally requires analyzing the radiation belt content in terms of PSD as a function of three adiabatic invariants (Schulz & Lanzerotti, 1974): μ, the magnetic moment; K, associated with particle bounce motion; and L* (Roederer's generalized L, Roederer, 1970), associated with the drift motion.When expressed in terms of PSD as a function of these adiabatic invariants, any changes in the radiation belt population that are produced by non-adiabatic processes may be revealed with analysis of the radial PSD profiles (Y.Chen, Reeves, & Friedel, 2007).Electron PSD as the function of three adiabatic coordinates, f(μ, K, L*), can be estimated from the measurements of the differential directional electron flux, that is, flux as a function of both energy and pitch angle j(E, α).Equation 1 shows the relation between particle flux and PSD according to Liouville's theorem: where p is the relativistic momentum of an electron.The transformation from flux to PSD is followed by a straightforward coordinate change from f(E, α) to f(μ, K) and the estimation of the third adiabatic invariant L* for each of the locations of the spacecraft.For each spacecraft and measurement, we compute values of equatorial pitch angles that correspond to the desired value of K, α(K), as well as the values of particle energies that relate to the chosen μ and K values, E(μ, α(K), B).The transformation from (E, α) to (μ, K) coordinates is then performed by fitting f(E, α) = f(E(μ, α(K), B), α(K)) and mapping it onto the regular (μ, K) grid (Morley et al., 2013).In this study, we use a B-spline fit to j(E) for the Van Allen Probes data and the 4-component forward model described in Morley et al. (2016) for the GPS data (as provided in the original GPS data set).Using fits instead of linear interpolations reduces potential errors associated with the non-linearity of j and the (E, α) ↔ (μ, K) mapping (cf., Boyd et al., 2014;Y. Chen et al., 2005;Morley et al., 2013).In this paper, we investigate the electron dynamics for the first adiabatic invariant, μ, values of ∼500, ∼1,500, and ∼2,500 MeV/G at fixed second adiabatic invariant K = ∼0.1 R E G 0.5 .We also use the Tsyganenko and Sitnov (2005) magnetic field model to evaluate all three adiabatic invariants.These approaches for PSD calculation have been previously utilized by Morley et al. (2013) and Olifer, Mann, Ozeke, Morley, and Louis (2021) for detailed statistical and event-specific analysis of the radiation belt dynamics.
The Van Allen Probes mission measures differential directional flux, thus enabling the analysis of the PSD directly from the spacecraft measurements.However, as discussed earlier, the mission cannot provide continuous monitoring of a fixed region in L* at short timescales due to the availability of only two spacecraft and their relatively long orbital periods of ∼8 hr.To mitigate this limitation, we estimate the PSD using omnidirectional electron flux data from 20 GPS satellites and a model of pitch angle distribution (PAD) to acquire the directional flux.This enables continuous investigation of radiation belt dynamics over a wide range of adiabatic invariant coordinates with an unprecedented ∼20-min resolution.Here we show that this approach can resolve very fast radiation belt acceleration, revealing short timescale dynamics not observable using Van Allen Probes alone.
The CXD instrument is a wide-angle (approximately hemispheric) energetic charged particle detector that measures the radiation environment in medium-Earth orbit on board GPS satellites (Morley et al., 2016), with the boresight of the detector pointing towards the Earth.As the GPS satellites carry no magnetometer and CXD does not provide a directionally-resolved measurement, the publicly released data product provides omnidirectional electron flux data and not directional flux (Morley et al., 2017).Therefore, to estimate a directional electron flux at the GPS satellites for PSD calculation, we use the PAD model developed by Zhao et al. (2018).The model provides an estimate of the shape of the PAD, j(E,α eq ), at the GPS satellites as the function of equatorial pitch angle, α eq .The model is driven by the L-shell and MLT location of the spacecraft, as well as the geomagnetic activity as determined by the Dst index at the time.
To estimate directional electron flux, we use the Zhao et al. (2018) PAD model to first simulate the omnidirectional flux around the GPS satellites, assuming the conservation of the first adiabatic invariant and energy of a particle, and then normalize the functional form to match the omnidirectional flux from GPS flux measurements, J GPS (E), at different energies.The absolute value of the directional differential flux j(E,α eq ) needed for PSD calculation can be then estimated using Equation 2: where B and B eq are the magnetic field strengths at the satellite and the magnetic equator along the same magnetic field line respectively, obtained from the Tsyganenko and Sitnov (2005) magnetic field model.
) is the equatorial pitch angle that corresponds to the 90°pitch angle at the location of the satellite.In the case of the Zhao et al. (2018) model the above equation can be solved analytically.A detailed derivation of Equation 2 is presented in Supporting Information S1.A similar approach for estimating PSD at GPS satellites has been used by Kalliokoski et al. (2023)   In this study, we calculate electron PSD using differential directional electron flux measurements from the Van Allen Probes and its estimations from the GPS satellites using Zhao et al. (2018) pitch angle model during the 25-27 August 2018 geomagnetic storm.To ensure that the presented approach for estimating electron PSD from the omnidirectional GPS measurements is sufficiently accurate, we compare the estimates of the j(E, α eq ) electron flux (obtained from Equation 2) and PSD for the GPS data with those from the Van Allen Probes measurements.
Figure 1 shows that the evaluated PSD at GPS satellites is in good agreement with that of the Van Allen Probes for the relativistic particle population.However similar to the electron flux comparison described earlier, a relatively small systematic bias-of a factor of ∼1.79-exists for PSD.For an accurate comparison between the two data sets, we correct for this bias in the GPS PSD data set only such that it minimizes the magnitude of the symmetric signed percentage bias (Morley et al., 2018), and in this work we use the bias-corrected PSD from GPS, as illustrated in Figure 1c. Figure S1 in Supporting Information S1 also shows similar plots of the flux and PSD matching between GPS and the Van Allen Probes for the other two energy and first adiabatic invariant data sets that were used in this study.

Estimating Radial Diffusion Coefficients From Electrons PSD Data
In this study, we also investigate if the observed rates of acceleration are consistent with the acceleration rates provided by the ULF radial diffusion paradigm.These can be related to the radial diffusion coefficients D LL (see e.g., the discussions in Ozeke et al. (2018) and Olifer, Mann, Ozeke, Claudepierre, et al. ( 2021)) which can be estimated using the in-situ PSD spacecraft data.In particular, we follow the method described by Schulz and Lanzerotti (1974) for evaluating the D LL coefficients directly from satellite measurements.Equation 3shows the formula from Schulz and Lanzerotti (1974) (their equation 5.15) used to estimate the D LL coefficients during the two acceleration bursts.This equation is a direct consequence of the radial diffusion equation and assuming that the radial diffusion coefficient, D LL , and electron lifetimes, τ, do not change with time.
where F ≡ ln f; D n is the normalization of the radial diffusion coefficient that does not depend on L or time; n is the order of L dependence of the D LL coefficient.For this study, we use n = 10 (e.g., Lejosne, 2020;Osmane et al., 2023) and calculate electron lifetime due to interaction with EMIC waves using the X.Zhang et al. (2017) empirical model for the case of a strong EMIC driving (R EMIC = 0.2), which yields τ ≈ 0.5 days.Note that the resulting D LL coefficients do not depend strongly on the choice of either n or τ (Table S1 in Supporting Information S1).Also note that the second derivative term in Equation 3 may not always be evaluated using GPS data due to its poor L* coverage at the specified invariants, and the availability of only two data points.In those cases we set ∂ 2 F ∂L 2 = 0. Figure S2 in Supporting Information S1 shows that this assumption is mostly valid in the heart of the radiation belt (L* between 3.5 and 4) where the rate of changes in the PSD gradient, that is, the second derivative term ∂ 2 F ∂L 2 , is small in comparison with the other two terms under the integral in Equation 3. Nonetheless, because typically ∂ 2 F ∂L 2 < 0 in the Van Allen Probes data, the GPS-driven D LL coefficients may be underestimated by ∼20% at most.

Overview of the August 2018 Geomagnetic Storm
Figure 2 shows geomagnetic conditions and the radiation belt response during the 25-27 August 2018 geomagnetic storm.This strong storm is characterized by prolonged geomagnetic activity, during which the Dst index reaches a minimum of 174 nT (Figure 2a).As seen in Figures 2b and 2c, the storm contains a radiation belt loss period, depleting the relativistic electron flux by almost two orders of magnitude, from around 18 UT on August 25 to 6 UT on August 26.This is followed by rapid, intense acceleration of the belt leading to electron fluxes exceeding pre-storm levels within ∼6 hr.Due to the phasing of their orbits, Van Allen Probes (Figure 2b) only observe the pre-acceleration radiation belt at around 5:30 UT on August 26 and the post-acceleration belt at around 12 UT, completely missing the main portion of the acceleration.On the other hand, the electron flux data from the GPS satellites (Figure 2c) provide nearly continuous coverage of the radiation belt around L* of 4. These   August 26, but the temporal evolution is not resolved.On the other hand, the GPS PSD (Figure 2e) enables continuous examination of the dynamics within the heart of the radiation belt (3.5 ≤ L* ≤ 4.0) during the acceleration period.Note that a significant amount of GPS data at high L* cannot be converted to PSD(μ, K, L*) at the desired K because the selected K is not measured by the GPS satellites at those locations due to the orbital inclination.Nonetheless, Figure 2e presents the acceleration progression in greater detail.The GPS data reveal that the acceleration occurs not continuously, indeed, there is an initial change in PSD around 6-8 UT on August 26, where the color in Figure 2e goes from black to orange.After this initial acceleration, the PSD stays relatively constant for almost 4 hr, until the secondary acceleration at 12 UT where the color changes from orange to bright yellow.This is shown more clearly in Figure 3 where the evolution in PSD profiles is marked for select intervals.

Revealing Radiation Belt Acceleration Bursts With the GPS Constellation
Figure 3 shows detailed electron PSD evolution obtained from both Van Allen Probes and GPS missions for 26 August 2018.As described earlier, this event is characterized by a quick enhancement of the election population between two subsequent Van Allen Probe passes between ∼6 and ∼12 UT.Figures 3a and 3b show radial PSD profiles during this enhancement period.Due to the relatively slow orbital movement of the Van Allen Probes, we characterize each of the Van Allen Probes PSD values with the time of the measurement, as shown using a colormap, to highlight the spatio-temporal dependence of these measurements.On the other hand, the radial profiles from the GPS satellites are obtained through combining the PSD data from different satellites located at different L*, thus enabling a reconstruction of a near-instantaneous radial PSD profile.The times of these nearinstantaneous GPS profiles are shown on the right side of the plot.GPS data shown in Figures 3a and 3b have a spatial resolution of 0.1 L* and a temporal resolution of 20 min Figure 3c shows a temporal evolution of the GPS PSD in the heart of the radiation belt (between L* of 3.5 and 4.0) with the highlighted color-shaded regions from where the radial profiles were reconstructed in panels a and b at higher L* resolution.
Figure 3c also shows the uncertainty of the PSD calculated using the PSD matching technique between different pairs of GPS satellites, similar to the approaches used in prior studies (e.g., Y. Chen, Friedel, et al., 2007;Morley et al., 2013;Reeves et al., 2013).Figures 3a and 3b show the median size of the error bars across all data points obtained in the similar way separately for GPS and Van Allen Probes.In particular, the error bars are estimated as the median spread of the PSD difference factor calculated for all pairs of the GPS satellites or between the two Van Allen Probes during the times of their drift shell conjunctions.For the purposes of the PSD uncertainty calculation, we assume that two spacecraft are in conjunction if they are withing 0.1 L* and 5 min from each other.Due to their orbital separation, the Van Allen Probes only contain five conjunctions close to the apogee during the entire event (cf., Figure 2d).To enable the estimation of the PSD uncertainty at the Van Allen Probes in the heart of the radiation belt, we shift Van Allen Probe A measurements forward by 75 min to match the temporal and L* coverage of the Van Allen Probe B. This provides multiple close conjunction points between the two satellites across the full range of L*, enabling the estimation of the errors in the PSD data.Importantly, this method is only applicable during the pre-storm phase and the end of the recovery phase where the radiation belts do not change on the 75-min timescales.We do not apply this method for estimating PSD uncertainties at the Van Allen Probes during the main phase of the storm to avoid assigning temporal belt variation to the uncertainty estimation.Interestingly, this method yields a similar result to that reported by Reeves et al. (2013) with the variability of the PSD across the two Van Allen Probes being around a factor of ∼1.5.
Figure 3 shows that the acceleration during the August 2018 storm occurs in two distinct bursts.In Figures 3a and  3b, the first acceleration period is characterized by four PSD profiles between 6:40 and 8:00 UT.During these 80 min, the electron PSD is monotonically increased by almost four orders of magnitude.Importantly, the GPS data reveals a series of profiles with positive PSD gradients.These can be a characteristic of the acceleration driven by inward ULF-wave radial diffusion (e.g., Boyd et al., 2018;I. R. Mann et al., 2016;Reeves et al., 2013, to list a few) or a locally growing peak due to local acceleration observed across a narrow L* range (Olifer, Mann, Ozeke, Morley, & Louis, 2021).At the time of the first acceleration burst, both Van Allen Probes are outside the LCDS and do not measure the trapped population.After 8:00 UT the electron PSD remains largely unchanged until 11:50 UT.This is illustrated by near static PSD profiles from GPS (Figures 3a and 3b), a relatively flat portion of the GPS PSD time series (Figure 3c), as well as a conjunction of GPS data with the PSD measured by Van Allen Probe B at 11:30 UT.Indeed, Figure 3c demonstrates minimal variation in PSD between 8:00 and 11:50 UT, with most data points residing within error bars.Following these three hours, a very rapid increase in PSD is observed as the radiation belt electrons are accelerated by almost an order of magnitude over the next 20 min.This is highlighted by a sharp increase in GPS PSD by almost an order of magnitude from 11:50 to 12:10 UT in Figure 3.This increase in PSD also affects a wide range of L* values, as shown by the Van Allen Probe A pass later in the day (16-18 UT).Notably, Figures 3a and  3b also show that the radial GPS PSD profile at 12:40 UT (immediately after the second acceleration burst) is not monotonic, exhibiting a local peak.This is characteristic of the acceleration driven by local particle heating, often linked to VLF chorus waves (e.g., Boyd et al., 2018;I. R. Mann et al., 2016;Reeves et al., 2013, to list a few).
Figures S3 and S4 in Supporting Information S1 show the GPS and Van Allen Probe PSD for μ = 501 MeV/G and μ = 2,508 MeV/G, for fixed K = 0.1 R E /G 0.5 (in the same format as Figure 3).These electron populations correspond to ∼1 MeV and ∼3 MeV at L* = 3.6 respectively.Figure S4 in Supporting Information S1 shows that the acceleration of these populations is similar to the one described in Figure 3.However, the lower energy population for μ = 501 MeV/G does not show a prominent second acceleration burst.

Probing the Origin of the Acceleration Bursts
As described above, Figure 3 reveals characteristic monotonic PSD profiles during the first acceleration burst, and the peaked PSD profiles during the second acceleration burst.Such dynamics of the PSD profiles are commonly associated with ULF-wave radial diffusion and local acceleration respectively.To confirm the dominant physical mechanisms that causes these acceleration bursts, we investigate chorus and ULF wave dynamics during the acceleration phase of the August 2018 geomagnetic storm.Figure 4 shows the power spectral density of in-situ magnetic field fluctuations as measured by ERG satellite (panels a and b), as well as the integrated wave power in the Pc5 ULF band for four ground-based magnetometers in Wells Gray (Western Canada), Sept Iles (Eastern Canada), Lerwick (Northern UK), and Mekrijärvi (Eastern Finland) (Figures 4c-4f).
Figure 4a shows that there is an elevated wave power in the lower chorus band starting at around 11:30 UT, consistently with the timing of the second acceleration burst.Figure 4b also shows the waveform capture (WFC) data from the ERG satellite with the characteristic rising tone chorus elements clearly visible at 11:49, in the middle of the second acceleration burst.On the other hand, the time of the first acceleration burst (6:40 to 8:00) is not accompanied by the elevated chorus wave power as measured by ERG.Meanwhile, the ULF wave power is at its highest earlier in the recovery phase across all four magnetometers (Figures 4c-4f), slowly decaying by approximately an order of magnitude by the end of the first acceleration burst.There is a secondary enhancement of the ULF wave power later in the day, contemporaneous with the timing of the chorus wave activity observed by ERG.This behavior is similar to that observed by Dimitrakoudis and Mann (2019) where they reported a close correspondence between the timing of the enhanced chorus and ULF wave power across multiple events and locations.Interestingly, previous studies have shown that high amplitude ULF waves may modulate whistler wave growth (e.g., Shang et al., 2021;Watt et al., 2011).In our view, further investigation of the second acceleration burst that utilizes both particle and wave measurements from the Arase satellite will be interesting for testing if ULF waves modulate whistler activity in this event and how it may impact particle acceleration.

Discussion
The dynamics of the PSD as the function of L* (Figure 3), in concert with the observed chorus and ULF wave power (Figure 4) implies likely dominant acceleration processes active for each of the acceleration bursts.In particular, the first acceleration burst is likely consistent with the ULF wave inward radial diffusion, as it is characterized by consistently positive PSD gradients in L*, as well as the highest ULF wave power during this event.High temporally-resolved GPS PSD data reveal that ULF waves enhance electron population by almost four orders of magnitude over 80 min during the first acceleration burst.The second acceleration burst is likely linked to the local acceleration of electrons by chorus waves, active for only 20 min and enhancing the electron PSD by almost an order of magnitude.Therefore, the two acceleration bursts are likely governed by different dominant mechanisms active on different timescales and enhancing the belt by different fractions of the total population.

Assessing Very Fast ULF Wave Transport
Importantly, the 80-min acceleration by the ULF waves is often considered to be very quick and comparable to earlier reported cases of very fast inward transport (Olifer, Mann, Ozeke, Morley, & Louis, 2021).To investigate if the ULF wave acceleration during the first acceleration burst in this event is consistent with the existing radial diffusion paradigm, it is interesting to compare the observed timescales of the acceleration from the GPS data with those provided by various empirical and event-specific models.Figure 5 shows the obtained D LL from the GPS PSD data (see Section 2.2 for more detail) as well as those from various empirical and event-specific models and from ground-based observations during the two acceleration bursts.In particular, Figure 5a shows that comparison between the GPS-driven D LL coefficients at three different first adiabatic invariants with the Kp D LL parameterizations by Ozeke et al. (2014) and Lejosne (2020), as well as a more complex model that depends on solar wind speed, dynamic pressure, IMF B z , and SYM-H index by Murphy et al. (2023).Figure 5b shows the comparison of the GPS D LL with event-specific D LL coefficients provided by Lejosne (2020) (found in the supporting information to their paper) for the three electron energies, that correspond to the selected first adiabatic invariants (see Section 3 for more details).Finally, Figure 5c shows the comparison of the GPS D LL coefficients with the D E LL component derived from the network of ground-based magnetometers using Ozeke et al. (2009) approach for mapping of the horizontal magnetic field component into the inner magnetosphere using ground magnetometer data from Figures 4c-4f.
Overall, Figure 5 shows that the GPS-driven D LL coefficients are in strong agreement with the D LL coefficients obtained from various other sources during the first acceleration burst to within reported levels of D LL variability during storm times (Olifer et al., 2019;Sandhu et al., 2021).This result confirms that the ULF wave inward radial diffusion is capable of producing the observed four orders of magnitude enhancement over the 80-min period at the timescales consistent with the current state-of-the-art radial transport models.Meanwhile, the GPS diffusion coefficients are strongly overestimated compared to all other models during the second burst.Indeed, the observed set of much larger D LL coefficients than those predicted by the models represents that the ULF wave transport should be extremely fast, much faster than that predicted by the current state-of-the-art or the observed ULF wave dynamics, to produce such a short-lived but nonetheless significant acceleration in the radiation belt.This result shows that the second acceleration burst was dominated by a mechanism different from the ULF-wave transport and the chorus wave acceleration presents a viable option for it (see Section 3.3 for more detail).
Finally, it is interesting to discuss the energy dependence of the radial diffusion coefficients obtained from GPS and those from other sources.The GPS-driven D LL shows a clear structuring in μ, with lower μ (and thus lower energy) having larger D LL .This is consistent with the radial diffusion coefficients obtained from the ground-based ULF wave observations for this event (Figure 5c), as well as prior simulation studies (Z.Li et al., 2016Li et al., , 2017)).
Journal of Geophysical Research: Space Physics 10.1029/2024JA032544However, the energy dependence is directly opposite to the one in the Lejosne (2020) D LL model (Figure 5c).Lejosne (2020) model shows a weak dependence on energy with the lower energies having lower D LL coefficients.While probing the causes of this discrepancy between Lejosne (2020) description of the radial transport and other models is outside of the scope of this paper, the authors think that it may prove an interesting future study.Recent studies of ULF radial diffusion have also revealed the importance of taking into account the azimuthal evolution and diffusion of the radiation belt electron population (Lejosne & Albert, 2023;Osmane et al., 2023).Taking into account that GPS orbits are situated in six different MLT regions, it may be possible to test the theoretical framework of azimuthal diffusion proposed by Lejosne and Albert (2023) or the impact of m ≠ 1 waves proposed by Osmane et al. (2023) with the GPS PSD data and resulting diffusion coefficients.

Assessing Whistler Wave Acceleration
It is also important to investigate if the observed chorus waves can produce the acceleration of the multi-MeV electrons as observed during the second acceleration phase.In particular, it is valuable to investigate what electron energies and pitch angles are resonant with the observed chorus wave spectrum.Those are shown in Figure 6 for three different chorus wave frequencies.The resonance condition is obtained using Equation 4 taken from Allanson et al. (2019, their equation (5)): where ɛ = E/(m 0 c 2 ), E and α are the resonant kinetic energy and a pitch angle of electrons, ω and k are the angular frequency and angular wavenumber of a chorus wave, ω ce is the angular cyclotron frequency.kc/ω can be obtained from the chorus wave cold plasma dispersion relation.For the resonance curves shown in Figure 6, we assume background magnetic field strength of 200 nT and cold plasma density of 5 cm 3 , both consistent with Arase and Van Allen Probe measurements.
The observed elevated chorus wave power is confined to a lower portion of the lower chorus band as observed by Arase satellite at the time of the second acceleration buts (cf., Figure 4b).The main frequency band of the elevated wave activity is observed between ∼600 and ∼1,000 Hz with somewhat weaker wave power being present between ∼400 and ∼600 Hz.Such low frequencies are typically resonant with higher energy particles.Indeed, the resonance condition (4) for the observed chorus wave frequencies reveals that near equatorial electrons with predominantly high energies (∼> 1 MeV) are resonant with the observed waves at the time of the second acceleration burst (Figure 5).These electron energies are consistent with the first and second adiabatic invariants studied in this paper.Overall, this confirms that the chorus waves observed early in the second acceleration burst period are resonant with the electrons that were accelerated.However, a more detailed analysis of the type of wave-particle interaction that dominates this acceleration phase, that is, quasilinear or non-linear, would be of interest.

Conclusions
This study reveals how observations of the Van Allen radiation belt by the GPS constellation can simultaneously provide continuous monitoring, reveal very fast radiation belt dynamics, and enable identification of the dominant physical mechanisms driving them.Although the energetic charged particle sensors on GPS are operational space environment monitors, the large number (>20) on orbit brings unique scientific capabilities.This is clearly exemplified here using the August 2018 geomagnetic storm.The GPS satellite data reveal a clear separation of the acceleration into two distinctly different bursts, the first lasting ∼2 hr and the second lasting ∼20 min, while Van Allen Probes only measure their combined effect.Without constellation measurements the acceleration would not have been resolved into the two bursts, potentially leading to incorrect physical interpretation.
By evaluating wave dynamics during this event as well as calculating radial diffusion coefficients directly from the GPS PSD data in the heart of the radiation belt, we have shown that the first acceleration burst was consistent with the ULF-wave radial diffusion, while the second acceleration burst was not.Our analysis provides strong support for the conclusion that inward radial diffusion was a dominant driver of the first acceleration burst that lasted 2 hr.It increased the electron radiation belt population by almost four orders of magnitude.Meanwhile, we provide evidence to support our conclusion that the second acceleration burst was likely caused by local waveparticle interaction increasing relativistic electron PSD by an order of magnitude over 20 min.Resolving fast PSD dynamics across an extended interval has previously not been possible, leading to an incomplete understanding of rapid radiation belt acceleration.Even as new science missions aim to probe the radiation belts with highlycapable instruments (Blum et al., 2020;Miyoshi, Shinohara, Takashima, et al., 2018), we foresee a critical role for constellation measurements, such as from GPS, in the future of radiation belt science.4assuming background magnetic field strength of 200 nT and cold plasma density of 5 cm 3 .At the time of the second acceleration burs, the Arase satellite observes elevated chorus wave power between 400 and 1,100 Hz (cf., Figure 4).
with the pitch angle model developed by Y.Chen et al. (2014).Meanwhile,Staples et al. (2023) also appliedZhao et al. (2018) pitch angle model to estimate GPS PSD.Note that theZhao et al. (2018) PAD model has been derived based on the directional differential electron flux measurements made by the Van Allen Probes (electron energies of 30 keV to 5.2 MeV) for the inner and outer radiation belt (L = 1 6) across a wide range of Dst conditions.In this study, we apply this PAD model to the strongly relativistic electron fluxes (1-4 MeV) in the heart of the radiation belt (L = 4 5) which are both well represented by the model.However, the August 2018 geomagnetic storm studied here represents an intense geomagnetic storm with Dst = 174 nT which may not be well-captured by the statisticalZhao et al. (2018) PAD model due to the relatively quiet geomagnetic activity and limited intense geomagnetic storms measured in the Van Allen Probe era.Despite this, Figure1shows that theZhao et al. (2018) model performs well when applied to the August 2018 geomagnetic storm.The left panel of Figure1provides the comparison between the estimated differential directional electron flux at GPS and those explicitly measured by the Van Allen Probes during satellite conjunctions in L, MLT, and time.While the correlation between the two data sets is strong (R = 0.97), a small bias of a factor of 1.24 is present with the GPS electron fluxes consistently underestimated by theZhao et al. (2018) PAD model.

Figure 1 .
Figure 1.Comparison between estimated electron differential directional flux and phase space density (PSD) from Global Positioning System (GPS) satellite constellation and that of the Van Allen Probes (marked as RBSP) during the August 2018 geomagnetic storm.Van Allen Probes data is shown on the x-axis and the estimated GPS data is shown on the y-axis.Only measurements during the temporal and L* conjunction of the satellites are shown.Each panel also shows the Pearson correlation coefficient (R) and coefficient of determination (R 2 ) between the two data sets in log-space, mean square error (MSE), symmetric signed percentage bias (SSPB), and the 95% confidence interval for linear fit slope, m TS , obtained using the Theil-Sen robust linear regression.The red line shows the resulting Theil-Sen fit with the yellow region showing the 95% confidence interval of the fit.(left) comparison between differential directional electron flux estimates at GPS (using Equation 2 and Zhao et al. (2018) pitch angle distribution model) and Van Allen Probes measurements.(middle) comparison between PSD at the GPS and Van Allen Probes.(right) Same as middle panel but GPS data has been bias corrected.
show the progression of the acceleration during that time.It is important to note that the substantial undersampling of the acceleration phase by Van Allen Probes is also worsened by undefined L* due to the proximity to the last closed drift shell (LCDS).

Figure 2
Figure 2 also shows the PSD as the function of three adiabatic invariants, μ, K and L*, obtained from the Van Allen Probes and GPS satellites as per the description above.Figures2d and 2eshow PSD at μ = 1,466 MeV/G and K = 0.10 R E G 0.5 as a function of time and L*.This population corresponds to ∼2 MeV electrons in the heart of the radiation belt at L* = 3.6 during the beginning of the acceleration phase.Similar to the flux, the Van Allen Probes PSD data (Figure2d) show that the main portion of the acceleration occurs between 6 and 12 UT on

Figure 2 .
Figure 2. Geomagnetic indices and radiation belt response during the August 2018 geomagnetic storm.Panel (a) disturbance storm time index, Dst, (line plot, left y-axis) and Planetary K-index, Kp, (histogram plot, right y-axis).Panels (b and c) show 1.8 MeV electron flux measurements by the Van Allen Probes and Global Positioning System (GPS) spacecraft respectively with L* values calculated for local 90°pitch angle.Panels (d and e) show electron phase space density (PSD) for fixed first and second adiabatic invariants (μ = 1,466 MeV/G, K = 0.1 R E G 0.5 ) obtained from the Van Allen Probes and GPS constellation respectively with L* values calculated for K = 0.1 R E G 0.5 .Roederer (1970) L* parameter is calculated for the Tsyganenko and Sitnov (2005) magnetic field model.The black lines overlaid on panels (b-e) show the last closed drift shell (LCDS, cf., Albert et al., 2018), calculated using the LANLGeoMag (Henderson et al., 2018) library.Panel (e) shows median PSD across all satellites on a grid with a cell size of 10 min and 0.1 L*.

Figure 3 .
Figure 3. Electron phase space density (PSD) during the acceleration phase of the August 2018 geomagnetic storm.Panels (a and b), the electron PSD profiles obtained from the Van Allen Probes and Global Positioning System (GPS) data.Van Allen Probes PSD are shown with a scatter plot in both panels (panel a for Probe A and panel b for Probe B).Due to the orbital movement of the spacecraft, the Van Allen Probe PSD data are color-coded by UT on 26 August 2018.The PSD profiles evaluated from the GPS satellites are shown with line plots, duplicated in both panels, during the first (06:40-08:00 UT) and second (11:50-12:10 UT) acceleration bursts.The star marker represents the Van Allen Probe PSD during the conjunction with the GPS at 11:30 UT.Panel (c), time series of PSD evolution as observed by the GPS satellites between L* of 3.5 and 4.0.The shaded regions represent the times when the L* PSD profiles are plotted in the above panels.

Figure 4 .
Figure 4. Chorus and ULF wave power during the acceleration phase of the August 2018 geomagnetic storm.Panels (a and b) show frequency-time spectrograms of the magnetic field oscillations in the very low frequency (VLF) range as measured by the ERG satellite using the PWE instrument.Panel (a) shows the timing of the high-resolution WFC measurements from panel (b) with two red arrows.Panels (c through f) show integrated ULF wave power in the Pc5 band (0.55-7.78 mHz) from four different ground-based magnetometers.

Figure 5 .
Figure 5. Radial diffusion coefficients (D LL ) during the acceleration phase of the August 2018 geomagnetic storm.D LL derived from the Global Positioning System (GPS) phase space density (PSD) radial profiles are shown with horizontal lines for the first and second acceleration bursts.These were calculated separately for three values of the first adiabatic invariant, μ.We compare the GPS-driven D LL coefficients with those obtained from empirical models by Ozeke et al. (2014), Lejosne (2020), and Murphy et al. (2023) (panel a); with the event-specific and energy-dependant D LL data from Lejosne (2020) (panel b); as well as with the electric field component of event-specific radial diffusion coefficients, D E LL , estimated from the network of ground magnetometers using Ozeke et al. (2009) approach.

Figure 6 .
Figure6.Resonant electron energy and pitch angle for three characteristic chorus wave frequencies during the second acceleration burst.Resonant cures are calculated using Equation 4 assuming background magnetic field strength of 200 nT and cold plasma density of 5 cm 3 .At the time of the second acceleration burs, the Arase satellite observes elevated chorus wave power between 400 and 1,100 Hz (cf., Figure4).