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One of the key assumptions of radiation belt modeling based on a three-dimensional Fokker-Planck equation is that trapped particle fluxes do not depend on the drift phase (i.e., the azimuthal angle, or magnetic local time, MLT). It is usually considered that MLT-dependent structures (such as particle injection signatures and subsequent drift echoes) are rapidly smoothed out by drift phase mixing. Yet, the characteristic times for radiation belt drift phase mixing are not well known.In this presentation, we show the existence of a naturally occurring phase mixing process in the presence of field fluctuations. This process complements the observational phase mixing due to the finite resolution of the measuring instrument.We present a first quantification for the characteristic time of natural phase mixing and we discuss the implications in terms of radiation belt modeling.

Electron resonant interaction with electromagnetic whistler-mode waves plays a crucial role for electron flux dynamics in planetary magnetospheres. One of the most intense types of whistler-mode waves consists of chorus waves generated via nonlinear resonant interaction with hot anisotropic electrons and propagating with time-varying (drifting) wave frequency. Electron nonlinear resonant interactions with such waves in a dipole magnetic field are well described analytically within the Hamiltonian approach under the approximation of monochromatic waves (of constant frequency). This paper aims to generalize this description to waves with drifting frequency. We show how frequency drift modifies two main nonlinear resonant effects: phase trapping and phase bunching. The obtained results contribute to the development of the Hamiltonian approach for wave–particle resonant interactions.

Proton flux measurements from the Proton Telescope instrument aboard the CRRES satellite are revisited, and used to drive a radial diffusion model of the inner proton belt at 1.1 ≤ L ≤ 1.65. Our model utilises a physics‐based evaluation of the cosmic ray albedo neutron decay (CRAND) source, and coulomb collisional loss is driven by a drift averaged density model combining results from the International Reference Ionosphere, NRLMSIS‐00 atmosphere and Radio Plasma Imager plasmasphere models, parameterised by solar activity and season. We drive our model using time‐averaged data at L = 1.65 to calculate steady state profiles of equatorial phase space density, and optimise our choice of radial diffusion coefficients based on four defining parameters to minimise the difference between model and data. This is first performed for a quiet period when the belt can be assumed to represent steady state. Additionally, we investigate fitting steady state solutions to time averages taken during active periods where the data exhibits limited deviation from steady state, demonstrated by CRRES measurements following the 24th March 1991 storm. We also discuss a way to make the optimisation process more reliable by excluding periods of variability in plasmaspheric density from any time average. Lastly, we compare our resultant diffusion coefficients to those derived via a similar process in previous work, and diffusion coefficients derived for electrons from ground and in situ observations. We find that higher diffusion coefficients are derived compared with previous work, and suggest more work is required to derive proton diffusion coefficients for different geomagnetic activity levels.

Multidimensional diffusion is central to radiation belt and ring current dynamics, but obtaining numerical solutions reliably is subject to difficulties related to “cross terms” in the diffusion equation. Eliminating them, by constructing new coordinates which diagonalize the diffusion matrix, has been found to be effective in two dimensions. Here this approach is reformulated to be both clearer and more robust, with analytical expressions replacing numerical solutions of differential equations. An approach to extending the method to three dimensions is presented and discussed.

The nonlinear resonant interaction of intense whistler-mode waves and energetic electrons in the Earth's radiation belts is traditionally described by theoretical models based on the consideration of slow–fast resonant systems. Such models reduce the electron dynamics around the resonance to the single pendulum equation that provides solutions for the electron nonlinear scattering (phase bunching) and phase trapping. Applicability of this approach is limited to not-too-small electron pitch-angles (i.e., sufficiently large electron magnetic moments), whereas model predictions contradict to the test particle results for small pitch-angle electrons. This study is focused on such field-aligned (small pitch-angle) electron resonances. We show that the nonlinear resonant interaction can be described by the slow–fast Hamiltonian system with the separatrix crossing. For the first cyclotron resonance, this interaction results in the electron pitch-angle increase for all resonant electrons, contrast to the pitch-angle decrease predicted by the pendulum equation for scattered electrons. We derive the threshold value of the magnetic moment of the transition to a new regime of the nonlinear resonant interaction. For field-aligned electrons, the proposed model provides the magnitude of magnetic moment changes in the nonlinear resonance. This model supplements existing models for not-too-small pitch-angles and contributes to the theory of the nonlinear resonant electron interaction with intense whistler-mode waves.

A new 3D diffusion code using a recently published layer method has been developed to analyze radiation belt electron dynamics. The code guarantees the positivity of the solution even when mixed diffusion terms are included. Unlike most of previous codes, our 3D code is developed directly in equatorial pitch angle (α0), momentum (p), and L-shell coordinates; this eliminates the need to transform back and forth between (α0,p) coordinates and adiabatic invariant coordinates. Using (α0,p,L) is also convenient for direct comparison with satellite data. The new code has been validated by various numerical tests, and we apply the 3D code to model the rapid electron flux enhancement following the geomagnetic storm on March 17, 2013, which is one of the GEM Focus Group challenge events. An event-specific global chorus wave model, an AL-dependent statistical plasmaspheric hiss wave model, and a recently published radial diffusion coefficient formula from Time History of Events and Macroscale Interactions during Substorms (THEMIS) statistics are used. The simulation results show good agreement with satellite observations in general, supporting the scenario that the rapid enhancement of radiation belt electron flux for this event results from an increased level of the seed population by radial diffusion, with subsequent acceleration by chorus waves. Our results prove that the layer method can be readily used to model global radiation belt dynamics in three dimensions.

Quasi-linear diffusion coefficients are considered for highly oblique whistler mode waves, which exhibit a singular “resonance cone” in cold plasma theory. The refractive index becomes both very large and rapidly varying as a function of wave parameters, making the diffusion coefficients difficult to calculate and to characterize. Since such waves have been repeatedly observed both outside and inside the plasmasphere, this problem has received renewed attention. Here the diffusion equations are analytically treated in the limit of large refractive index μ. It is shown that a common approximation to the refractive index allows the associated “normalization integral” to be evaluated in closed form and that this can be exploited in the numerical evaluation of the exact expression. The overall diffusion coefficient formulas for large μ are then reduced to a very simple form, and the remaining integral and sum over resonances are approximated analytically. These formulas are typically written for a modeled distribution of wave magnetic field intensity, but this may not be appropriate for highly oblique whistlers, which become quasi-electrostatic. Thus, the analysis is also presented in terms of wave electric field intensity. The final results depend strongly on the maximum μ (or μ∥) used to model the wave distribution, so realistic determination of these limiting values becomes paramount.

“Peculiar” or “butterfly” electron pitch angle distributions (PADs), with minima near 90°, have recently been observed in the inner radiation belt. These electrons are traditionally treated by pure pitch angle diffusion, driven by plasmaspheric hiss, lightning-generated whistlers, and VLF transmitter signals. Since this leads to monotonic PADs, energy diffusion by magnetosonic waves has been proposed to account for the observations. We show that the observed PADs arise readily from two-dimensional diffusion at L = 2, with or without magnetosonic waves. It is necessary to include cross diffusion, which accounts for the relationship between pitch angle and energy changes. The distribution of flux with energy is also in good agreement with observations between 200 keV and 1 MeV, dropping to very low levels at higher energy. Thus, at this location radial diffusion may be negligible at subrelativistic as well as ultrarelativistic energy.

An efficient and positivity-preserving layer method is introduced to solve the radiation belt diffusion equation and is applied to study the bounce resonance interaction between relativistic electrons and magnetosonic waves. The layer method with linear interpolation, denoted by LM-L (layer method-linear), requires the use of a large number of grid points to ensure accurate solutions. We introduce a monotonicity- and positivity-preserving cubic interpolation method to be used with the Milstein-Tretyakov layer method. The resulting method, called LM-MC (layer method-monotone cubic), can be used to solve the radiation belt diffusion equation with a much smaller number of grid points than LM-L, while still being able to preserve the positivity of the solution. We suggest that LM-MC can be used to study long-term dynamics of radiation belts. We then develop a 2D LM-MC code and use it to investigate the bounce resonance diffusion of radiation belt electrons by magnetosonic waves. Using a previously published magnetosonic wave model, we demonstrate that bounce resonance with magnetosonic waves is as important as gyro-resonance; both can cause several orders of magnitude increase of MeV electron fluxes within one day. We conclude that bounce resonance with magnetosonic waves should be taken into consideration together with gyro-resonance.

Predicting variations in the near-Earth space environment that can lead to spacecraft damage and failure is one example of “space weather” and a big space physics challenge. A project recently funded through the Los Alamos National Laboratory (LANL) Directed Research and Development (LDRD) program aims at developing a new capability to understand, model, and predict Space Hazards Induced near Earth by Large Dynamic Storms, the SHIELDS framework. The project goals are to understand the dynamics of the surface charging environment (SCE), the hot (keV) electrons representing the source and seed populations for the radiation belts, on both macro- and micro-scale. Important physics questions related to particle injection and acceleration associated with magnetospheric storms and substorms, as well as plasma waves, are investigated. These challenging problems are addressed using a team of world-class experts in the fields of space science and computational plasma physics, and state-of-the-art models and computational facilities. A full two-way coupling of physics-based models across multiple scales, including a global MHD (BATS-R-US) embedding a particle-in-cell (iPIC3D) and an inner magnetosphere (RAM-SCB) codes, is achieved. New data assimilation techniques employing in situ satellite data are developed; these provide an order of magnitude improvement in the accuracy in the simulation of the SCE. SHIELDS also includes a post-processing tool designed to calculate the surface charging for specific spacecraft geometry using the Curvilinear Particle-In-Cell (CPIC) code that can be used for reanalysis of satellite failures or for satellite design.

Wabs. Plasmaspheric hiss constitutes one of the main components of the waves, but lightning and VLF ground transmitters also make important contributions to the wave population. Benchmark models for each of these types of waves, pre-VAP but post-CRRES, were given by Abel and Thorne [1998]. Much work on refining these estimates has been done since then, using wave measurements from VAP as well as Cluster, THEMIS, and others. (The bulk of these observations have been equatorial, though informed by ray tracing.) The effects of very low frequency hiss, highly oblique whistlers, and magnetosonic (MS) waves have come to be recognized as significant. MS waves particularly come into consideration in discussions of butterfly pitch angle distributions. Of greater global significance is the presence or absence of MeV electrons at L

Chorus waves are typically observed as a series of discrete, narrowband rising or falling tone elements, as opposed to a continuous uniform band, as it is often modeled. The effects of this discreteness on the applicability of quasilinear theory to interactions between electrons and parallel-propagating chorus waves in a dipole field are investigated using test particle simulations. Previous work indicated that quasilinear theory might not be directly applicable, because chorus elements are coherent or quasi-coherent. Nonlinear processes such as phase trapping and bunching were demonstrated by modeling a chorus element using a single wave. Here we represent a chorus wave field with a series of coherent elements with subpacket structures using a previously developed method involving test particle simulations to explore the applicability of quasilinear theory. By comparing electron distribution functions from test particle simulations and quasilinear predictions, we demonstrate that, besides the wave amplitude, the discreteness of chorus waves also affects the applicability of quasilinear theory. When chorus elements are close to each other and the wave amplitude is small, quasilinear theory can well describe the evolution of the electron distribution. However, when chorus elements are widely separated in space and time, the discreteness of chorus might reduce the possibility of resonant interactions between electrons and chorus. Nonlinear effects of chorus waves on electrons are also shown using the current model. The method presented in this work should be helpful for investigating the applicability of quasilinear theory in general situations. Our results should be important to understanding and modeling electrons dynamics due to interactions with chorus.

This paper is devoted to the systematic study of electron lifetimes from narrowband wave-particle interactions within the plasmasphere. It relies on a new formulation of the bounce-averaged quasi-linear pitch angle diffusion coefficients parameterized by a single frequency, ω, and wave normal angle, θ. We first show that the diffusion coefficients scale with ω/Ωce, where Ωce is the equatorial electron gyrofrequency, and that maximal pitch angle diffusion occurs along the line α0 = π/2–θ, where α0 is the equatorial pitch angle. Lifetimes are computed for L shell values in the range [1.5, 3.5] and energies, E, in the range [0.1, 6] MeV as a function of frequency and wave normal angle. The maximal pitch angle associated with a given lifetime is also given, revealing the frequencies that are able to scatter nearly equatorial pitch angle particles. The lifetimes are relatively independent of frequency and wave normal angle after taking into consideration the scaling law, with a weak dependence on wave normal angle up to 60–70°, increasing to infinity as the wave normal angle approaches the resonance cone. We identify regions in the (L, E) plane in which a single wave type (hiss, VLF transmitters, or lightning-generated waves) is dominant relative to the others. We find that VLF waves dominate the lifetime for 0.2–0.4 MeV at L ~ 2 and for 0.5–0.8 MeV at L ~ 1.5, while hiss dominates the lifetime for 2–3 MeV at L = 3–3.5. The influence of lightning-generated waves is always mixed with the other two and cannot be easily differentiated. Limitations of the method for addressing effects due to restricted latitude or pitch angle domains are also discussed. Finally, for each (L, E) we search for the minimum lifetime and find that the optimal frequency that produces this lifetime increases as L diminishes. Restricting the search to very oblique waves, which could be emitted during the Demonstration and Science Experiments satellite mission, we find that the optimal frequency is always close to 0.16Ωce.

A 3-D model for solving the radiation belt diffusion equation in adiabatic invariant coordinates has been developed and tested. The model, named REM (for Radbelt Electron Model), obtains a probabilistic solution by solving a set of Itô stochastic differential equations that are mathematically equivalent to the diffusion equation. This method is capable of solving diffusion equations with a full 3-D diffusion tensor, including the radial-local cross diffusion components. The correct form of the boundary condition at equatorial pitch-angle α0 = 90° is also derived. The model is applied to a simulation of the October 2002 storm event. At α0 near 90°, our results are quantitatively consistent with GPS observations of phase-space density (PSD) increases, suggesting dominance of radial diffusion; at smaller α0, the observed PSD increases are overestimated by the model, possibly due to the α0-independent radial diffusion coefficients, or to insufficientelectron loss in the model, or both. Statistical analysis of the stochastic processes provides further insights into the diffusion processes, showing distinctive electron source distributions with and without local acceleration.

The effects of amplitude modulation on nonlinear interactions between a parallel propagating whistler wave and electrons in a dipole field are investigated in this work using a test particle code. Here we first use the test particle simulation to validate a previous single-wave nonlinear theory. Then we adopt a simple two-wave model to represent the recently observed amplitude modulation of a whistler wave field. By varying the frequency spacing between the two waves, we investigate the effects of different modulation frequencies on the nonlinear interactions. We demonstrate that when the resonance overlap condition is satisfied, the resulting change in the electron pitch angle and energy could be very different from what has been predicted by ideal single-wave nonlinear theories. Using a previously observed probability distribution of the subpacket modulation frequency of a chorus event, we obtain the probability distribution of different types of electron response. Our results indicate that the observed subpacket distribution produces particle responses in both non-overlapping and overlapping regimes. Our results suggest that the observed amplitude modulation should be considered when quantitatively treating interactions between electrons and recently observed large amplitude whistler waves or chorus waves.

[1] Observed signatures of electron precipitation from the inner radiation belt are shown to be consistent with the theory of resonant scattering by whistler-mode plasma waves, assuming the waves originate in VLF radio transmissions from the ground station NWC. The conclusion is based on a stochastic model of electron transport that includes pitch angle diffusion, radial diffusion, energy loss, and azimuthal drift. The wave scattering causes an increase in quasi-trapped electron intensity, forming the “wisp” signature, and a corresponding decrease in stably trapped intensity at low altitude. A smaller decrease at high altitude is expected to be obscured by inward radial diffusion. If NWC were shut down, the resulting increase in stably trapped electron intensity would be minimal.

The Earths inner and outer radiation belts, comprising energetic electrons and protons, pose a hazard to DoD spacecraft. Air ForceResearch Laboratory (AFRL) has an ongoing research effort to model and forecast the configurations of the belts, and to develop protective technologies for spacecraft. This final report summarizes these developments and research performed at AFRL during FY 20132015 to address them, funded by AFOSR grant 13RV08COR, Radiation Belt Dynamics.

The fluxes of energetic particles in the radiation belts are found to be strongly controlled by the solar wind conditions. In order to understand and predict the radiation particle intensities, we have developed a physics-based Radiation Belt Environment (RBE) model that considers the influences from the solar wind, ring current and plasmasphere. Recently, an improved calculation of wave-particle interactions has been incorporated. In particular, the model now includes cross diffusion in energy and pitch-angle. We find that the exclusion of cross diffusion could cause significant overestimation of electron flux enhancement during storm recovery. The RBE model is also connected to MHD fields so that the response of the radiation belts to fast variations in the global magnetosphere can be studied. We are able to reproduce the rapid flux increase during a substorm dipolarization on 4 September 2008. The timing is much shorter than the time scale of wave associated acceleration. Published by Elsevier Ltd.

Understanding the dynamics of the Earth’s radiation belts is important for modeling and forecasting the intensities of energetic electrons in space. Wave diffusion processes are known to be responsible for loss and acceleration of electrons in the radiation belts. Several recent studies indicate pitch angle and energy mixed-diffusion are also important when considering the total diffusive effects. In this study, a two-dimensional Fokker Planck equation is solved numerically using the Alternating Direction Implicit method. Mixed diffusion due to whistler-mode chorus waves tends to slow down the total diffusion in the energy-pitch angle space, particularly at smaller equatorial pitch angles. We then incorporate the electron energy and pitch angle mixed diffusions due to whistler-model chorus waves into the 4-dimensional Radiation Belt Environment (RBE) model and study the effect of mixed diffusion during a storm in October 2002. The 4-D simulation results show that energy and pitch angle mixed diffusion decrease the electron fluxes in the outer belt while electron fluxes in the slot region are enhanced (up to a factor of 2) during storm time.

We consider timescales on which particle distributions respond to pitch angle diffusion. On the longest timescale, the distribution decays at a single rate independent of equatorial pitch angle α 0 , even though the diffusion coefficient, and the distribution itself, may vary greatly with α 0 . We derive a simple integral expression to approximate this decay rate and show that it gives good agreement with the full expression. The roles of both the minimum and loss cone values of the diffusion coefficient are demonstrated and clarified.

Wave–particle interactions are described in the quasi-linear formulation by bounce-averaged diffusion coefficients for equatorial pitch angle α 0 and for momentum p , along with a mixed or cross diffusion coefficient D α 0 p . Because the cross terms complicate the associated time-dependent diffusion equation, and because D α 0 p has a somewhat unfamiliar character (e.g., it may be negative), it has frequently been omitted in numerical simulations of multidimensional diffusion. Generally, D α 0 p becomes increasingly important for small values of f pe / f ce , as does D pp , and for small widths of the wave frequency and wavenormal angle distributions. Here we use very simple models of diffusion to investigate the effect of the cross terms, and to demonstrate numerical problems associated with them.

International audience; Wave normal distributions of lower-band whistler-mode waves observed outside the plasmapause exhibit two peaks: one near the parallel direction and the other at very oblique angles. We analyze a number of conjunction events between the Van Allen Probes near the equatorial plane and Polar Orbiting Environmental Satellites (POES) at conjugate low altitudes, where lower-band whistler-mode wave amplitudes were inferred from the two-directional POES electron measurements over 30–100 keV, assuming that these waves were quasi-parallel. For conjunction events, the wave amplitudes inferred from the POES electron measurements were found to be overestimated as compared with the Van Allen Probes measurements primarily for oblique waves and quasi-parallel waves with small wave amplitudes (< ~20 pT) measured at low latitudes. This provides plausible experimental evidence of stronger pitch angle scattering loss caused by oblique waves than by quasi-parallel waves with the same magnetic wave amplitudes, as predicted by numerical calculations.

A recent Hamiltonian analysis of magnetized test particles acted on by a small monochromatic wave is compared to a quasi-linear formulation of pitch angle diffusion by a turbulent spectrum of whistler waves. The quasi-linear expression has previously been applied to anthropogenic VLF transmissions to conclude that they dominate pitch angle diffusion of inner radiation belt electrons. However, because these transmissions are very narrow in frequency range, the monochromatic approach might be more appropriate. It is shown analytically that the monochromatic limit of the quasi-linear pitch angle diffusion coefficient reduces to the same expression as the diffusive regime of the Hamiltonian treatment. Agreement of the Hamiltonian pitch angle diffusion coefficient with the quasi-linear values is verified numerically using realistic parameters for the spread in frequency and wave-normal angle. As an aside, a simple, alternate derivation of quasi-linear diffusion coefficients is given.

The interaction of magnetized, relativistic test particles with a monochromatic electromagnetic wave is analyzed, taking into account the passage through cyclotron resonance in the spatially varying background magnetic field in slab geometry. A resonance-averaged Hamiltonian is used to delineate two distinct regimes. In both cases, termed “adiabatic” and “nonadiabatic,” the first adiabatic invariant of the particle is broken during a resonant interaction, leading to change in energy and pitch angle. The adiabatic case is characterized by a limited range of resonant phase and a well-defined value of the change of the invariant. In the nonadiabatic case the phase at resonance ranges over 0 to 2π, and only the magnitude of the change of the invariant is determined; a phase factor gives the invariant change an effectively random sign. The appropriate regime is determined by a ratio of timescales which, in turn, depends on the particle and wave properties: adiabaticity is favored by large wave amplitudes and small parallel gradients of the geomagnetic field Bo. The long-term consequences of the two regimes are explored with a Monte Carlo simulation in the form of an iterated mapping. In a particular example, while some particles pitch angle scatter into the loss cone, energy is also removed from the distribution by particles in the adiabatic regime decaying in energy owing to repeated resonant interactions with the wave. Important potential magnetospheric applications include flux levels in the inner radiation belts, observations of “pancake” pitch angle distributions, and energization of storm time “killer” electrons.

Lifetimes and pitch angle distributions of radiation belt electrons injected into the slot region are determined from CRRES/MEA particle flux measurements. The dominant loss mechanism is presumed to be pitch angle scattering due to whistler waves, for which a theoretical formulation is available. The empirical lifetimes are compared to those from recent calculations, which rely on model whistler wave parameters. CRRES measurements of cyclotron-frequency electric fields are also presented and compared to the wave model.

The response of outer radiation belt relativistic electrons to the October 9, 1990, magnetic storm is analyzed in detail using a radial diffusion model and data from the Combined Release and Radiation Effects Satellite (CRRES) and the Los Alamos National Laboratory (LANL) geosynchronous satellite 1989-046. Electron measurements are expressed in terms of phase space density as a function of the three adiabatic invariants determined from CRRES magnetic field data and the Tsyganenko 1989 Kp-dependent magnetic field model. The radial diffusion model is implemented with a time-dependent radial diffusion coefficient parameterized by Kp, and a time-dependent outer boundary condition scaled by geosynchronous electron data. The results show that radial diffusion propagates outer boundary variations into the heart of the outer radiation belt, accounting for both significant decreases and increases in the 1 MeV electrons is inconsistent with the radial diffusion model given the parameter regime chosen for this study. Greatly enhanced whistler chorus waves observed by CRRES throughout the recovery phase suggest that a possible explanation for the inconsistency may be electron acceleration via wave-particle interaction.

The global evolution of an artificially injected relativistic electron beam is simulated and examined. The study focuses on injections originating in the upper ionosphere, magnetically mirroring above the lower atmosphere where significant energy loss occurs, and so a long-lived population arises in the inner magnetosphere from this particle source. This investigation is conducted by solving the bounce-averaged relativistic kinetic equation for the electron distribution function for various L shells. It is found that the beam quickly spreads in MLT due to differential drift rates, eventually morphing into a fairly uniform shell around the Earth. Wave interactions are comparable to collisional losses in reducing the beam content. It is also found that the beam total particle loss rate is a complicated function of L and, for the chosen conditions, the total beam particle counts are 73%, 77%, and 52% of the initial count at t=24 hours after injection for L=2, 3, and 4, respectively. The loss rates at this time are ∼1%/hour (of the remaining beam strength) and very slowly decreasing with time. These loss rates and other features of the beam evolution are discussed in detail. There are now four distinct stages recognized in the evolution of an injected relativistic beam: (1) the immediate loss of particles injected at pitch angles mapping to the lower thermosphere; (2) the initial loss of particles injected right next to the loss cone by collisional scattering; (3) the continuation of this collisional loss along with the spread of the beant to all local times by differential drift rates; and (4) the transformation of the beam into a fairly uniform shell covering all energies and all local times, with the loss rate mainly governed by wave scattering. Other stages may exist beyond the 1-day limit set on these simulations. While the study dwells on beam dynamics, it is also a general examination of the leading edge population next to the loss cone. This has implications for the physics of the naturally occurring radiation belt particles, as this region of phase space regulates the actual precipitation of these particles into the atmosphere. The applicability of this model for studying the natural radiation environment around the Earth is also pondered.