Electron fluxes routinely increase by several orders of magnitude within days in the Earth’s outer radiation belt (Figure 1, the article by Reeves et al., 2013). This flux increase may or may not be preceded by a brief, large decrease (e.g., Blake et al., 1997). Broadly speaking, the radiation belt electron flux enhancements are coherent: Increases occur on similar timescales across the radiation belt energy spectrum (50 keV–10 MeV) and across the outer zone (equatorial radial distance, L, between ∼3 Re and 6.5 Re), regardless of altitude (e.g., Kanekal et al., 2001). Yet, the specific characteristics of these enhancements are variable. Figure 1 shows that the magnitude of MeV electron flux usually peaks at a variable location within the outer region (i.e., below geostationary orbit). The rise time for electron flux enhancements increases with energy (e.g., Blake et al., 1997). In addition, the frequency and the L-coverage for electron flux enhancements generally decrease with energy (e.g., Reeves et al., 2013; Zhao et al., 2016).

An association between the state of the Earth’s outer belt and the Sun was established in the early days of radiation belt science (Williams, 1966). It has remained a subject of research ever since (e.g., Hudson et al., 2008; Kellerman and Shprits, 2012; Kilpua et al., 2015; Zhao et al., 2019a; Ripoll et al., 2020). Connecting the Sun and solar wind properties with the state of the radiation belts is of prime importance for two main reasons. First, it provides observational constraints to radiation belt acceleration theories. Second, it constitutes the basis of radiation belt model developments, whether they are physics-based models (e.g., Xiang et al., 2021), empirical models such as AE9 and predecessors (Ginet et al., 2013), or machine learning models (e.g., Katsavrias et al., 2021a).

The most pronounced signature of the solar wind properties in the state of the Earth’s outer radiation belt is the correlation between solar wind speed and MeV electron flux magnitude (Paulikas and Blake, 1979; Reeves et al., 2011; Wing et al., 2016). The sign of the north-south component of the interplanetary magnetic field (IMF), B z , is also key (Blake et al., 1997). Most of the time, MeV electron flux enhancements occur when the speed of the solar wind is high (≳ 500 km/s) and the IMF B z is southward, i.e., during conditions that are associated with geomagnetic storm times (e.g., Baker et al., 2019). While these solar wind conditions are the most common conditions for MeV electron flux enhancements, not all of them are necessary. Significant MeV electron flux enhancements have also been reported during non-storm times (e.g., Schiller et al., 2014) and without a high-speed solar wind (e.g., Li X. et al., 2011). A sustained southward IMF is the only necessary condition for MeV electron flux enhancements at geosynchronous orbit according to Li X. et al. (2011). Yet, this necessary condition is not a sufficient condition to guarantee radiation belt enhancements. Indeed, even though geomagnetic storms are associated with a strong and sustained southward IMF, not all geomagnetic storms result in electron flux enhancements (Reeves et al., 2003). The most significant relativistic electron flux enhancements occur outside the plasmapause, in association with periods of prolonged substorm activity, as quantified by the AE index (Meredith et al., 2003). Moreover, MeV electron enhancements have been tied to High-Intensity Long-Duration Continuous AE Activity (HILDCAA) events (Tsurutani et al., 2006; Miyoshi and Kataoka, 2008; Hajra et al., 2015) and substorm clusters during geomagnetic disturbances (e.g., Rodger et al., 2022).

When electron flux enhancements occur during geomagnetic storms, the location of the peak in MeV electron flux enhancements during recovery phase is strongly correlated with the magnitude of the storm, as quantified by the Dst index (Tverskaya et al., 2003) or, equivalently, with the plasmapause location (O’Brien et al., 2003; Moya et al., 2017; Bruff et al., 2020). The equatorial pitch angle distribution of MeV electron flux enhancements at the center of the outer belt is most anisotropic (i.e., 90 ° peaked) within a day of the start of the recovery phase, and the degree of anisotropy increases with energy (e.g., Ozeke et al., 2022). The pitch-angle distribution of MeV electron flux becomes more isotropic in the week following the start of the recovery phase (Greeley et al., 2021).

One consequence of the relationship between the state of the Sun and the state of the Earth’s outer radiation belt is that periodicities of the Sun, of the solar activity, and of the Sun-Earth connection lead to periodicities in the intensity of the outer belt occurring on a variety of timescales. For instance, the 27-day periodicity of the electron flux enhancements (Williams, 1966) is associated with the 27-day recurrence of geomagnetic activity. The latter comes from the fact that long-lived solar wind features, such as high-speed streams, recur at Earth after every Sun rotation period of ∼27 days (e.g., Paulikas and Blake, 1979). The strong semiannual variations of MeV electron fluxes have been tied to the semiannual variation in the orientation between the Earth’s magnetic dipole axis and the Sun vector, and more precisely, to the Russell-McPherron effect (McPherron et al., 2009; Katsavrias et al., 2021b). MeV electron fluxes are also more intense during the declining phase of the solar cycle than during the ascending phase. This is due to the fact that the declining phase of the solar cycle is dominated by recurrent high-speed solar wind streams while the ascending phase is dominated by more sporadic coronal mass ejection events (Kanekal, 2006; Reeves et al., 2011), and radiation belts respond differently to storms driven by coronal mass ejections (CMEs) and storms driven by corotating interaction regions (CIRs) (e.g., Turner et al., 2019).

It takes a few hours down to a few minutes for the 50 keV to 5 MeV electrons of the outer belt to orbit around the Earth. During that time, these particles, trapped by the geomagnetic field, bounce 500 to 50,000 times from one hemisphere to the other while they gyrate 10⁵ to 10⁹ times around the magnetic field direction. In this context, it is convenient to describe the motion of radiation belt particles as the superposition of three quasi-periodic motions, each of them evolving on a very different timescale (e.g., Schulz and Lanzerotti, 1974):

1) A very fast motion of gyration around the magnetic field direction,

Each quasi-periodic motion is determined by the particle’s characteristics (charge, mass, kinetic energy, pitch angle) as well as by the characteristics of the magnetic and electric fields (magnitude, direction, as well as spatial and temporal variability of the fields).

The magnitude of each quasi-periodic motion is quantified by an adiabatic coordinate, that is, by a quantity that is a constant of motion under certain spatial and temporal conditions. In particular, an adiabatic coordinate remains constant as long as the time variations for the fields are negligible on the timescale of the corresponding quasi-periodic motion (e.g., Northrop, 1963). That is why the reformulation of trapped particle dynamics in terms of adiabatic coordinates allows for a simplified description of radiation belt dynamics (e.g., Roederer, 2014).

In the absence of significant time variations in the fields, trapped radiation belt particles remain at about the same average equatorial radial distance from the center of the planet. They move along closed surfaces called drift shells. Their kinetic energy is conserved on average. In other words, in the steady state, there is neither net acceleration nor net deceleration occurring in the radiation belts. Energy variation for the trapped radiation belt particles requires time variations of the electric and/or magnetic fields. Since the magnetic force does no work, it is the electric field that exchanges energy with the trapped particles. This electric field may be induced by magnetic field time variations, or it may be due to variations in the electric potential. It is usually assumed that there is no component of the electric field parallel to the magnetic field direction, a good approximation in the inner magnetosphere on timescales longer than the gyro-period. In the DC realm, the electrical conductivity is orders of magnitude greater in the parallel direction of the magnetic field than in the perpendicular direction of the magnetic field so that the parallel conductivity is often approximated to be infinitely high (e.g., Stern, 1977).

Electron flux enhancements are conventionally viewed as indicative of radiation belt acceleration because radiation belt spectra typically decrease with energy. Thus, the energization of a population of trapped particles is expected to manifest as a flux enhancement. The standard practice is to split the mechanisms driving radiation belt irreversible (i.e., non-adiabatic) acceleration into two categories, depending on the source location of the population that is accelerated:

• The non-adiabatic acceleration is local when the energized population is already present within the drift shell.

We focus below on the two most favored mechanisms for radiation belt energization, namely: 1) global acceleration via radial transport (Section 2.2.2.1) and 2) local acceleration via resonant interactions with chorus waves (Section 2.2.2.2). That said, many other mechanisms have been proposed over the years (see for instance the review by Friedel et al. (2002) for details).

A detailed and accurate modeling of radiation belt particle dynamics is nothing short of impossible: It would require a complete and highly accurate specification of the spatial and temporal variations of the electromagnetic fields on a multiplicity of spatio-temporal scales—from the drift-scale down to the gyro-scale. Particle-in-cell simulations allow for self-consistent interactions between particles and wave fields to be simulated, however computational requirements are high and only small spatial scales and time periods can be modelled this way (e.g., Camporeale, 2015; Allanson et al., 2019). Even when the fields are specified by numerical models (e.g., MHD fields), injecting test particles to simulate radiation belt dynamics remains cumbersome. This impossibility calls for necessary tradeoffs. A powerful way to reduce the number of variables to handle is the use of the adiabatic theory of magnetically trapped particles (Section 2.2.1). Adiabatic theory “provides correct answers only as long as we don’t look too close and are not expecting too detailed information” (Roederer and Zhang, 2014). To account for uncertainties in electromagnetic field dynamics, we leverage probability theory, in particular the Fokker-Planck formalism. This formalism accounts for uncertainty by assuming random changes in the variables, relating average characteristics of the electromagnetic fields to average properties of the radiation belt dynamics. It is these tools (Fokker-Planck equation and adiabatic invariant theory) that have been successfully combined for more than 25 years (Beutier and Boscher, 1995) to facilitate operational radiation belt modeling. In particular, these simplifications allow for radiation belt simulations over long time intervals (months to years) (e.g., Glauert et al., 2018).

Specifically, most physics-based radiation belt models consist of solving a Fokker-Planck equation reduced to a diffusion equation, with the objective of providing an approximate description for the time evolution of the radiation belts: ∂ f ∂ t = ∑ i , j ∂ ∂ J i ( D i , j ∂ f ∂ J j ) + S o u r c e s − L o s s e s (6) where f ( J 1 , J 2 , J 3 ) is the drift-averaged particle distribution function, J i are the action variables, proportional to the adiabatic coordinates, and D i , j are the drift-averaged diffusion coefficients (e.g., Schulz and Lanzerotti, 1974). The “ S o u r c e s ” and “ L o s s e s ” terms account for other non-diffusive processes affecting the distribution function. In practice, diffusion in terms of action variables is often reformulated in different coordinate systems. In particular, diffusion in terms of pitch angle, energy and L ∗ is often preferred. Thus, diverse reformulations of the same Eq. 6 exist.

Defining realistic boundary conditions and performing model-observation comparisons require relating the variables of Eq. 6 to measurable quantities. On one hand, it is straightforward to relate the trapped particle distribution in phase space (i.e., the phase space density, PSD) to experimental data: PSD is proportional to the directional differential flux, a measurable quantity (e.g., Roederer, 1970, p.93). On the other hand, defining the adiabatic coordinates cannot be done relying solely on experimental data. Indeed, since the adiabatic coordinates are:

1) μ = p ⊥ 2 / 2 m o B , where m o is the particle rest mass.

Quantifying the adiabatic coordinates of MeV populations associated with a PSD sample requires information on the instantaneous magnetic field topology along the full drift contour. This means working with a magnetic field model. In addition, the adiabatic coordinates of a measurement can be undefined under certain conditions, as in the case in the presence of open drift shells—where particles are lost before completing a full drift around the Earth. This is a spatial limit to the scope of the model and outer boundary specification (e.g., Albert et al., 2018).

Diffusion-driven radiation belt models solving Eq. 6 are thought to work best for very high energy particles (e.g., Fok, 2020). That said, they remain limited in several ways, as discussed below.

First, Eq. 6 assumes that radiation belt dynamics are mainly due to physical processes whose overall effects can be encapsulated by diffusion coefficients. In other words, according to Eq. 6, radiation belt dynamics are primarily due to many very small, uncorrelated, time-stationary field fluctuations, resulting in many very small ( Δ J i / J i ≪ 1 ) , uncorrelated, perturbations of the trapped particle dynamics, akin to random walks in phase space at all scales—from the drift-scale down to the gyro-scale. In this diffusive picture, the scattering of a population of particles with the same initial characteristics increases linearly with time in phase space. This concept is illustrated in Figure 4 in the case of pitch angle diffusion.

The postulate of a regime that is mainly diffusive also means that Eq. 6 is ill-suited at times when particle dynamics are coherent, in particular at times when significant particle injections occur, and at times when large amplitude waves result in non-diffusive regimes (e.g., nonlinear phase bunching, phase trapping) (e.g., Riley and Wolf, 1992; Albert, 2002; Bortnik et al., 2008; Ukhorskiy et al., 2009; Omura et al., 2015).

Second, Eq. 6 cannot resolve radiation belt dynamics on a timescale shorter than that of drift phase mixing. The dynamics of the PSD described by Eq. 6 are drift-averaged. As a result, all equation variables are independent of magnetic local time by design. This means, for instance, that radiation belt drift echoes (e.g., Lanzerotti et al., 1967) cannot be reproduced using Eq. 6. That is also why this framework cannot reproduce shock-injections during sudden storm commencements for instance. In this case, the modelling efforts favor test particle simulations (e.g., Li X. et al., 1993; Hudson et al., 1997: Kress et al., 2007: Hudson et al., 2017).

A first version of Eq. 6 focuses on the effects of 1) radial diffusion and 2) losses due to pitch angle scattering into the loss cone, meaning finite electron lifetimes: ∂ f ∂ t = L 2 ∂ ∂ L ( D L L L 2 ∂ f ∂ L ) − f τ   (7) where L stands for L ∗ , inversely proportional to the third adiabatic coordinate, D L L is the radial diffusion coefficient, and τ is the electron lifetime resulting from the combined effect of the pitch angle scattering induced by different waves.

The use of the master Eq. 7 to describe radiation belt dynamics constrains the range of possible time variations for the modeled PSD. Indeed, following Fick’s first law of diffusion, the net “current” of particles that flow through a unit area of drift shell per unit of time, i.e., the diffusion flux, is (e.g., Walt, 1994): " Current " = − D L L L 2 ∂ f ∂ L (8)

This means that radial diffusion acts to smooth the PSD radial profile. In other words, radial diffusion decreases peaks and increases valleys present in the PSD radial profile. An illustration is provided in Figure 5, in the case of a 1D diffusion equation in the absence of loss.

D L L contains all the information on the physical processes that drive cross drift shell motion, i.e., it quantifies the efficiency of the radial diffusion process, and it directly relates to the field dynamics. Yet, the magnitude of D L L alone is not enough to determine how much radial diffusion affects radiation belt dynamics. Indeed, it is the " C u r r e n t " quantity, i.e., − ( D L L / L 2 ) ( ∂ f / ∂ L ) (Eq. 8), that determines the manifestation of the diffusion process – that is, it is this quantity that drives the PSD time variations, ∂ f / ∂ t   (Eq. 7). Thus, if the PSD radial gradient, ∂ f / ∂ L , is significant, the effect of radial diffusion may appear significant, even if the magnitude of the radial diffusion coefficient D L L is relatively small (as is the case at very low L shells in the Earth’s inner radiation belt for instance). Conversely, if the PSD radial gradient is relatively small, radial diffusion may appear unimportant for the dynamics of this region of the belts, regardless of the magnitude of the radial diffusion coefficient. This reasoning applies to all diffusion modes. It demonstrates the importance of taking into account PSD gradients when comparing the effects of various diffusion processes (i.e., various waves) on the time evolution of the PSD. This also highlights the difficulty of directly relating measured wave power to PSD and/or flux variations (e.g., Simms et al., 2021).

Solving Eq. 7 for the PSD, f , requires characterizing the radial diffusion coefficient,   D L L , the electron lifetime, τ , and setting boundary conditions. Different studies choose different settings. The time-varying coefficients, D L L , are often provided by an empirical law for electromagnetic radial diffusion, such as defined by Brautigam and Albert (2000) for example based on a combination of in situ and ground-based measurements of time-varying magnetic fields parametrized by a geomagnetic activity index (Kp). More recent data sets have used both ground-based and in situ magnetic and electric field measurements to infer D _LL (Ozeke et al., 2014a; Liu et al., 2016; Ali et al., 2016; Sandhu et al., 2021). Radial diffusion coefficients can also be determined from solar wind measurements (e.g., Li X. et al., 2001; Lejosne, 2020) or MHD test-particle simulations (Tu et al., 2012; Li Z. et al., 2017). The electron lifetime is usually parameterized based on plasmapause location and magnetic activity (e.g., Orlova et al., 2016). In all cases, the solution of the standard diffusion equation (Eq. 7) displays much of the variability of the Earth’s outer belt on long timescales (months to years) (e.g., Li X. et al., 2001; Shprits et al., 2005; Chu et al., 2010; Ozeke et al., 2014b; Drozdov et al., 2015, 2017). In particular, it describes radiation belt dynamics well during geomagnetically quiet times (e.g., Selesnick et al., 1997; Su et al., 2015; Ripoll et al., 2019).

However, model-observation comparisons can also present notable discrepancies, in particular for MeV electron fluxes during the recovery phase of magnetic storms (e.g., Brautigam and Albert, 2000; Shprits et al., 2005; Shprits et al., 2007a; Ozeke et al., 2020). Specifically, the development of a peak in the PSD radial profile of the outer belt has been put forward as evidence of the effect of an additional local acceleration mechanism (e.g., Miyoshi et al., 2003; Chen et al., 2007; Reeves et al., 2013): It is contrary to what is expected from radiation belt dynamics driven primarily by radial diffusion (Figure 5). A growing local peak in the PSD radial profile appears to be a common feature of PSD enhancement events—at least for near-equatorial particles with a first adiabatic coordinate that corresponds to ∼1 MeV at L = 5. Indeed, based on four years of flux measurements from Time History and Events of Macroscale Interactions during Substorms (THEMIS) and Van Allen Probes converted into PSD, 70 out of 80 observed enhancement events presented a growing peak (Boyd et al., 2018). The geomagnetic conditions for a growing peak are variable: 38 out of 70 occurred during moderate or strong storms, while 32 occurred during small storm or non-storm times (i.e., with a Dst index no less than −50 nT). In all cases, the location of the local peak in the PSD radial profile was shown to be outside of the plasmasphere, about 1.25 Earth radius away from the plasmapause location on average.

When the effect of chorus waves on radiation belt dynamics is included as an additional source term in the Fokker-Planck equation (e.g., Tu et al., 2009; Xiang et al., 2021), or, more commonly as additional diffusion terms (e.g., Varotsou et al., 2005; Glauert et al., 2018), the quality of radiation belt modeling improves: Simulations yield a peak in the PSD radial profile, in reasonable agreement with observations (e.g., Shprits et al., 2008; Subbotin et al., 2010; Thorne et al., 2013; Ma et al., 2018; Wang and Shprits, 2019).

This apparent improvement in radiation belt modeling leads to the “two-step” picture for the acceleration to relativistic and ultra-relativistic energies in the outer radiation belt, in which both local and radial processes contribute to MeV electron production. This mechanism is well supported by both case studies (e.g., Jaynes et al., 2015; Zhao et al., 2018) and statistical analysis (e.g., Zhao et al., 2019b). It works as follows: First, the injection of source (tens of keV) and seed (hundreds of keV) electrons during substorms lead to whistler mode chorus wave generation and subsequent acceleration of the seed population to relativistic, and potentially ultra-relativistic (Allison et al., 2021), energies via local wave-particle interactions (e.g., Meredith et al., 2002), on a relatively rapid timescale. An illustration of this concept is provided in Figure 6 (Jaynes et al., 2015). Meanwhile, radial diffusion progressively redistributes the newly created MeV population, smoothing out the PSD radial profile and providing additional energy to the MeV particles transported inward.

In summary, the picture of electron radiation belt acceleration has evolved over time in response to measurements from new missions in the Earth’s inner magnetosphere, most notably thanks to the NASA Combined Release and Radiation Effects Satellite (CRRES) in the 1990s and most recently, the NASA THEMIS and the Van Allen Probes. In comparison, the magnetospheres of the outer planets are lacking data to differentiate between the leading processes for electron radiation belt acceleration, as summarized below.

First, the overall radiation belt dynamics result from concurrent processes that can influence each other and whose individual contributions are difficult to evaluate and time-varying (e.g., Tu et al., 2009; Xiang et al., 2021). Thus, any uncertainty in the magnitude of a source or loss process leads to other uncertainties in the magnitude of other processes.

In this context, it is also critical to quantify the losses that contribute to the “ L o s s es ” term in Eq. 6 to fully understand acceleration events. Losses can be created internally via wave-particle pitch-angle scattering or ULF wave effects, resulting in atmospheric precipitation, or at the outer boundary of the magnetosphere, a process known as magnetopause shadowing (Turner et al., 2012), resulting in losses to the interplanetary medium. Radiation belt electrons are susceptible to pitch-angle scattering by three main wave modes: broadband VLF hiss, electromagnetic ion cyclotron (EMIC) waves, and coherent VLF chorus (Thorne et al., 2005). Hiss losses are most relevant within the dense plasmasphere region where hiss can persist (Thorne et al., 1979), although this loss mechanism becomes less important during active times when the plasmapause location can move inward on short timescales (Goldstein et al., 2005). When this happens, the particle distribution that was within the plasmasphere is suddenly outside and susceptible to other loss or acceleration processes. Electron lifetimes within the plasmasphere have been estimated using both theoretical and observational techniques (Jaynes et al., 2014; Orlova et al., 2014; Claudepierre et al., 2020). Hiss-driven loss is considered to be a slower, steady loss rather than an impulsive event. On the other hand, EMIC waves can cause intense, sudden scattering that manifests as localized depletions, and are thought to be a primary loss factor of relativistic and ultra-relativistic electrons in the heart of the outer radiation belts (Drozdov et al., 2021b). VLF chorus waves also scatter outer belt electrons efficiently, particularly in the ring current energy range (Shprits et al., 2007b). Microbursts, trains of which may be created by quasi-periodic chorus waves typically seen in the outer radiation belt, can cause relativistic losses in concert with lower energy loss due to wave propagation to higher latitudes (Miyoshi et al., 2020). Relativistic losses can also be contributed by the phenomenon referred to as dusk-side relativistic electron precipitation (Comess et al., 2013), which are driven by both microburst events and non-microburst events. Microburst trains may be long-lasting, as evidenced by their connection to pulsating aurora which can be long-duration and widespread (Jones et al., 2013), and therefore may be a significant loss process for relativistic outer belt electrons. Finally, ULF waves have been implicated in energetic electron losses through a mechanism by which the radial oscillatory motion causes a lowering of the mirror point in a modulated manner (Brito et al., 2012). Taken together, these effects contribute to a net loss term in the characterization of the outer radiation belt system, and must be accounted for in order to accurately quantify the acceleration terms.

In addition, an accurate determination of the location of the last closed drift shell is an important parameter to include in Earth’s radiation belt modeling as it contributes to radiation belt losses during active times. Yet it requires assuming an instantaneous magnetic field topology, including the magnetopause location (e.g., Albert et al., 2018; Olifer et al., 2018; Staples et al., 2020), and the accuracy of such assumption is hard to quantify. In addition, diffusion coefficients require knowledge of instantaneous field variations and plasma conditions all along the trapped particles’ drift shell, including plasmapause location (e.g., Malaspina et al., 2016, 2020; Wang et al., 2020). For the energy diffusion coefficient, this means knowing the chorus wave spectral intensity, amplitude, and plasma density at all magnetic local times over the drift shell in real time (Thorne et al., 2013; Allison et al., 2021). For the radial diffusion coefficient, this means knowing instantaneous electric and magnetic field variations all along the drift contour (e.g., Lejosne and Kollmann, 2020). Thus, assumptions need to be made, and averaged conditions are usually preferred. As a result, diffusion coefficients are often parameterized in terms of magnetic activity indices, smoothing out estimated errors as well as natural variability (e.g., Watt et al., 2017). Yet, the need for “event-specific” diffusion coefficients is now well recognized (e.g., Tu et al., 2009) and efforts have been made to provide such information (Tu et al., 2012; Li Z. et al., 2017; Lejosne, 2020; Ozeke et al., 2020).

That said, converting measurements into inputs to the 3D Fokker-Planck equation means complying with the presupposed diffusion framework (Section 2.3), an increasingly complicated task as data resolution improves.

Most counter-arguments to local acceleration as the prevailing radiation belt acceleration mechanism challenge the interpretation of experimental data resulting in a peak in the PSD radial profile. These counter-arguments boil down to two main reasons.

The first is technical: Mapping measurements into phase space requires assuming a magnetic field model, whose real-time accuracy is difficult to quantify (see also the review by Green (2006) for methods to obtain PSD estimates). Let us also mention that the DC and low frequency electric fields can affect the dynamics of source and seed particles (tens to hundreds of keV): They can distort trapped particle drift shells, thereby modifying their third adiabatic coordinate L*. While this effect has been observed and studied for tens to hundreds of keV electrons in the Earth’s inner belt (e.g., Selesnick et al., 2016; Lejosne et al., 2021), drift shell distortion by large-scale electric fields is reasonably omitted when it comes to defining the adiabatic coordinates of MeV particles in the Earth’s outer radiation belt. Even when so, the conversion of experimental data into phase space density (PSD) parameterized by adiabatic coordinates remains a pitfall (e.g., Selesnick and Blake, 2000; Green and Kivelson, 2004). In particular, errors in magnetic field models can lead to the apparition of an artificial peak in the PSD radial profile, which vanishes when a realistic magnetic field model is used (e.g., Loridan et al., 2019). In addition, transient PSD peaks can also be spatio-temporal artifacts that disappear when leveraging multipoint measurements (e.g., Olifer et al., 2021). One way to test magnetic field model accuracy is to compare magnetic field model outputs and in-situ magnetic field measurements when available (e.g., Ozeke et al., 2019). In addition, the detection of a growing local peak requires observations during the acceleration process. Yet, the time resolution of in-situ measurements is constrained by spacecraft orbit period or revisit time.

The second reason is physical: Radial transport dynamics can also generate a local peak in the PSD radial profile (e.g., Ukhorskiy et al., 2006; Degeling et al., 2008), thereby further questioning the appropriateness of summarizing radial transport in terms of a diffusion process in the radiation belt master equation (Eq. 7) (e.g., Elkington et al., 1999; Kress et al., 2012, Figure 7).

In fact, a comparison between diffusion and particle drift descriptions of radial transport showed that the two modeling choices provide best agreement in the case of a series of sequential small storms and mediocre agreement during event analysis (Riley and Wolf, 1992). This is also why case events associated with fast radial transport (injection or drift resonance) are usually modeled by tracking test particles (i.e., guiding centers) drifting in analytical descriptions of the wave-associated electric field (e.g., Zong et al., 2017; Zong, 2022) or in MHD fields (e.g., Hudson et al., 2017). In contrast, summarizing local wave particle interactions in terms of diffusion in energy and pitch angle appears more reasonable (e.g., Tao et al., 2012), even though nonlinear effects occur in the presence of intense chorus waves, routinely measured in-situ (e.g., Zhang et al., 2019). In that context, alternative methods have been proposed to summarize the effect of chorus wave particle interactions on distribution functions (e.g., Furuya et al., 2008; Kubota and Omura, 2018; Artemyev et al., 2020).

While adjustments to the Fokker-Planck framework have been proposed to improve the description of trapped particle radial transport on timescales smaller than the drift period for the radiation belts (e.g., Bourdarie et al., 1997; Shprits et al., 2015) and for the ring current population (e.g., Fok et al., 2014; Jordanova et al., 2016), they also call for improved experimental knowledge of the electric and magnetic field variations driving radiation belt dynamics.

In summary, the appropriateness of our current master equation for modeling radiation belt dynamics has limitations, in particular when it comes to rendering the effects of radial transport on radiation belt dynamics on short time scales. In the absence of a modeling framework able to account for the effects of both diffusive and non-diffusive (i.e., coherent) radial transport, as well as for the effects of local acceleration (including non-linear regimes), it is not possible to quantify unequivocally the importance of local acceleration versus large-scale acceleration associated with radial transport. While the Fokker-Planck formalism has done well for long-term radiation belt modeling, it appears to be insufficient for definitive event analysis during active times. Thus, care must be taken when drawing conclusions on the physics at play solely based on PSD dynamics, and even more so on flux dynamics.