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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.

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.

“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.

[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.

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.

This paper considers the quasi-linear treatment of either electrons or protons cyclotron-resonant with either whistler or ion cyclotron waves. The dispersion relation used does not yield a simple, explicit expression for the resonant frequency ω. Detailed analysis is made of the conditions determining the resonant frequencies ω for a given wave normal angle θ; there may be up to three values of ω(θ). Criteria are derived which identify the frequencies and indicate when they are outside specified frequency cutoffs. From this analysis, much unnecessary computational effort to evaluate the diffusion coefficients may be avoided. For electron-whistler interactions in the radiation belts, the sensitivity of the local results to the parameters of the wave distribution is considered. It is shown how parameter-independent calculations may be used to analyze the dependence on the parameter values. This approach is extended to the bounce-averaged coefficients, which determine estimates of the pitch angle distribution and corresponding time to pitch angle scatter into the loss cone. Numerical results are obtained using recently reported parameters for whistler waves due to hiss, lightning, and VLF transmitters, as well as perturbations to these values.

Flux measurements made by CRRES of high-energy, equatorially mirroring protons in the range L ≤ 3 are observed to differ greatly from diffusive steady state profiles, both before the magnetic storm of March 24, 1991, and during two intervals following it. Time rates of change of the phase space density ƒ are obtained from time series of the observations and compared to those calculated from the time-dependent radial diffusion equation using profiles from the time averages of the observations. In general, good agreement is obtained for E ≤ 20 MeV, indicating that the outer zone protons were undergoing diffusive relaxation from highly disturbed configurations. Such a configuration is known to have been produced by the March 1991 storm, and we hypothesize that a similarly disruptive event occurred prior to the CRRES launch. For particle energy above 20 MeV or L > 3, limitations in the instrument sensitivity and integration time preclude a quantitative comparison.

The Proton Telescope (PROTEL) instrument on the CRRES satellite made measurements of omnidirectional differential proton flux in the energy range 1-100 MeV, with full pitch angle resolution. An overview of the equatorially mirroring particle data is presented for the entire CRRES mission. We then consider the equatorially mirroring particle distribution in terms of steady state radial diffusion theory. The outer zone, 1.7 ≤ L ≤ 3, is shown to have deviated drastically from steady state profiles both before and after the magnetic storm of March 24, 1991, which rearranged the outer zone but left the inner zone, L ≤ 1.7, largely unaffected. Time-averaged measurements are compared to theoretical steady state diffusion profiles of the inner zone, which are calculated using several different, previously introduced models of the plasmaspheric electron density and cosmic ray albedo neutron decay (CRAND). For each combination of models, a four-parameter (electrostatic plus magnetic) radial diffusion coefficient is sought which minimizes the difference between the measurements and calculated flux values. While the best fit values found are physically implausible, nearly equally good results are obtained for values only moderately adjusted from standard, consensus values.

Despite its well-known limitations, diffusion remains an important concept for modeling radiation belt energization and transport. Recent work at AFRL has applied radial diffusion to observations of: (a) 1 - 10 MeV equatorially mirroring protons at 1.2 2 MeV electrons at geosynchronous orbit associated with high speed solar wind streams. In general, encouraging results were obtained, but only after implementing: (1) non-steady state initial conditions; (2) time-dependent boundary conditions; (3) realistic magnetic field models; (4) tweaked, scaled, fit, activity-dependent, or enhanced diffusion coefficients. This paper reviews these various modeling efforts, including the modifications that they required, their successes and failures, and lessons learned.

The Electric Fields and Waves (EFW) Instruments on the two Radiation Belt Storm Probe (RBSP) spacecraft (recently renamed the Van Allen Probes) are designed to measure three dimensional quasi-static and low frequency electric fields and waves associated with the major mechanisms responsible for the acceleration of energetic charged particles in the inner magnetosphere of the Earth. For this measurement, the instrument uses two pairs of spherical double probe sensors at the ends of orthogonal centripetally deployed booms in the spin plane with tip-to-tip separations of 100 meters. The third component of the electric field is measured by two spherical sensors separated by ∼15 m, deployed at the ends of two stacer booms oppositely directed along the spin axis of the spacecraft. The instrument provides a continuous stream of measurements over the entire orbit of the low frequency electric field vector at 32 samples/s in a survey mode. This survey mode also includes measurements of spacecraft potential to provide information on thermal electron plasma variations and structure. Survey mode spectral information allows the continuous evaluation of the peak value and spectral power in electric, magnetic and density fluctuations from several Hz to 6.5 kHz. On-board cross-spectral data allows the calculation of field-aligned wave Poynting flux along the magnetic field. For higher frequency waveform information, two different programmable burst memories are used with nominal sampling rates of 512 samples/s and 16 k samples/s. The EFW burst modes provide targeted measurements over brief time intervals of 3-d electric fields, 3-d wave magnetic fields (from the EMFISIS magnetic search coil sensors), and spacecraft potential. In the burst modes all six sensor-spacecraft potential measurements are telemetered enabling interferometric timing of small-scale plasma structures. In the first burst mode, the instrument stores all or a substantial fraction of the high frequency measurements in a 32 gigabyte burst memory. The sub-intervals to be downloaded are uplinked by ground command after inspection of instrument survey data and other information available on the ground. The second burst mode involves autonomous storing and playback of data controlled by flight software algorithms, which assess the “highest quality” events on the basis of instrument measurements and information from other instruments available on orbit. The EFW instrument provides 3-d wave electric field signals with a frequency response up to 400 kHz to the EMFISIS instrument for analysis and telemetry (Kletzing et al. Space Sci. Rev. 2013)., United States. National Aeronautics and Space Administration (Contract NAS5-01072), Johns Hopkins University. Applied Physics Laboratory (Radiation Storm Belt Probes ECT Contract 967399), Johns Hopkins University. Applied Physics Laboratory (Radiation Storm Belt Probes EFW Contract 922613)

The refractive index μ for whistler mode plane waves in a magnetized electron–proton plasma is considered as a function of normalized wave frequency ω/Ωe at fixed density and angle of propagation. A factorization into an increasing function of frequency times a decreasing function of frequency is found, valid for any value of ωpe/Ωe, for ω between (ΩeΩi)1/2 and the smaller of Ωe and ωpe. This leads to lower and upper bounds on μ2 over any given frequency subinterval. The bounds can be directly applied to a quasilinear formulation of velocity-space diffusion of electrons in the Earth’s radiation belts, in a generalization of previously developed techniques to low density conditions found outside the plasma pause.

[1] The electron decay timescale (τ) is well known to be associated with radiation belt loss processes. Knowledge of τ is important for understanding pitch angle and radial diffusion mechanisms. Previous studies reported decay timescales from the inner belt to geosynchronous orbits; however, relatively few statistical studies have been focused on the region beyond the traditional outer belt. In this paper, a systematic calculation of electron decay times at 5 1 day. τ is examined as a function of energy, pitch angle, L shell, Kp, and AE during magnetically disturbed periods when Dst ≤ −50 nT. Results show that τ increases with increasing electron energy at L 6.6. This suggests radial transport as the dominant effect at L > 6.6. Additionally, τ decreases with increasing L-shell. This dependence has the strongest correlation and is seen in all energies and pitch angles. However,τ has no systematic dependence with pitch angle suggesting that pitch angle diffusion also plays a key role in the electron loss process. Based on our results, τ can be expressed as a function of energy and L, and coefficients are provided for a two-variable fit. Surprisingly,τ is slightly longer for higher activity cases at L < 6.6, which is inconsistent with the current radial or pitch angle diffusion models. Global effective decay times on the timescale of days place an upper bound on the true loss timescale.

[1] We explore Cluster and CRRES observations of plasmaspheric broadband hiss-like VLF emissions related with subauroral ion drifts/polarization streams (SAID/SAPS) to numerically simulate losses of energetic outer zone electrons due to wave-particle interactions. These emissions represent a distinctive subset of substorm/storm-related VLF activity in the region co-located with substorm injected energetic ions. Significant values of pitch-angle diffusion coefficients suggest that SAID/SAPS-related VLF waves could be responsible for the alteration of the outer radiation belt boundary during (sub)storms.

Properly treating wave-particle interactions is crucial to modeling and predicting the behavior of radiation belt electrons. The usual quasi-linear theory alone cannot capture the effects likely to be caused by nonlinear interactions with coherent waves, particularly chorus. Detailed analytical estimates of nonlinear particle motion in a specified wave have been developed, and and combined with detailed wave models can be used to formulate a combined diffusion-advection equation for the electron phase space density. Quasi-linear diffusion is recovered for small amplitude waves, but phase bunching and phase trapping, caused by larger amplitude waves, can also be included.

Relativistic (>l MeV) electron flux increases in the Earth's radiation belts are significantly underestimated by models that only include transport and loss processes, suggesting that some additional acceleration process is required. Here we use a new, threedimensional code that includes radial diffusion and quasi-linear pitch angle and energy diffusion due to chorus waves, including cross terms, to simulate the 9 October 1990 magnetic storm. The diffusion coefficients are activity dependent, and time-dependent boundary conditions are imposed on all six boundary faces, taken from fits to CRRES Medium Electrons A electron data. Although the main phase dropout is not fully captured, the persistent phase space density peaks observed during the recovery phase are well explained, but this requires both chorus wave acceleration and radial diffusion.

[1] The rapid loss of radiation belt electrons in the main phase of geomagnetic storms is believed to be aided by EMIC waves, and is usually analyzed with quasi-linear theory. However, even moderate EMIC wave intensities easily cause resonant electrons to respond nonlinearly, with drastically different results. We map out the region of nonlinear behavior with a single parameter, and show that both the direction and magnitude of scattering can be estimated by analytical expressions. The nonlinear interactions typically lead to advection toward large pitch angles, rather than diffusion toward the loss cone. This is expected to reduce the overall loss rate and greatly affect the distribution of trapped electrons.

[1] A new code using layer methods is presented to solve radiation belt diffusion equations and is used to explore effects of cross diffusion on electron fluxes. Previous results indicate that numerical problems arise when solving diffusion equations with cross diffusion when using simple finite difference methods. We show that layer methods, which are based on stochastic differential equations, are capable of solving diffusion equations with cross diffusion and are also generalizable to three dimensions. We run our layer code using two chorus wave models and a combined magnetosonic wave and hiss wave model (MH wave model). Both chorus and magnetosonic waves are capable of accelerating electrons to MeV levels in about a day. However, for the chorus wave models, omitting cross diffusion overestimates fluxes at high energies and small pitch angles, while for the MH wave model, ignoring cross diffusion overestimates fluxes at high energies and large pitch angles. These results show that cross diffusion is not ignorable and should be included when calculating radiation belt electron fluxes.

We show evidence that left-hand polarised electromagnetic ion cyclotron (EMIC) plasma waves can cause the loss of relativistic electrons into the atmosphere. Our unique set of ground and satellite observations shows coincident precipitation of ions with energies of tens of keY and of relativistic electrons into an isolated proton aurora. The coincident precipitation was produced by wave-particle interactions with EMIC waves near the plasmapause. The estimation of pitch angle diffusion coefficients supports that the observed EMIC waves caused coincident precipitation ofboth ions and relativistic electrons. This study clarifies that ions with energies of tens of ke V affect the evolution of relativistic electrons in the radiation belts via cyclotron resonance with EMIC waves, an effect that was first theoretically predicted in the early 1970's.

[1] Combined pitch angle and energy diffusion are key ingredients in current models of radiation belt electron dynamics. Bounce-averaged quasi-linear diffusion coefficients can be approximated with a recently developed approach based on the ranges of wave normal angle compatible with cyclotron and Landau resonance within a prescribed wave frequency band, as has been demonstrated for a model of chorus waves. The method casts nested, multiple integrals over wave normal angle as a single weighted average, which is further approximated by evaluation at only a few, carefully chosen points. Here, the method is shown to agree well with results for more recent models of chorus, electromagnetic ion cyclotron waves, and whistler mode hiss. Highly oblique magnetosonic waves are also considered, and a related approach is developed which is shown to give a good approximation for their diffusion rates.

[1] We study the effect of electromagnetic ion cyclotron (EMIC) wave scattering on radiation belt electrons during the large geomagnetic storm of 21 October 2001 with minimum Dst = −187 nT. We use our global physics-based model, which solves the kinetic equation for relativistic electrons and H+, O+, and He+ ions as a function of radial distance in the equatorial plane, magnetic local time, energy, and pitch angle. The model includes time-dependent convective transport and radial diffusion and all major loss processes and is coupled with a dynamic plasmasphere model. We calculate the excitation of EMIC waves self-consistently with the evolving plasma populations. Particle interactions with these waves are evaluated according to quasi-linear theory, using diffusion coefficients for a multicomponent plasma and including not only field-aligned but also oblique EMIC wave propagation. The pitch angle diffusion coefficients increase from 0° to ∼60° during specific storm conditions. Pitch angle scattering by EMIC waves causes significant loss of radiation belt electrons at E ≥ 1 MeV and precipitation into the atmosphere. However, the relativistic electron flux dropout during the main phase at large L ≥ 5 is due mostly to outward radial diffusion, driven by the flux decrease at geosynchronous orbit. We show first results from global simulations indicating significant relativistic electron precipitation within regions of enhanced EMIC instability, whose location varies with time but is predominantly in the afternoon-dusk sector. The precipitating electron fluxes are usually collocated with precipitating ion fluxes but occur at variable energy range and magnitude. The minimum resonant energy increases at low L and relativistic electrons at E ≤ 1 MeV do not precipitate at L < 3 during this storm.

Quasi-linear diffusion by cyclotron-resonant plasma waves is likely a key ingredient of the behavior of electrons in the Earth's radiation belts. Multidimensional dynamical simulations are under development, which require the diffusion coefficients to be evaluated quickly as well as accurately. The recently developed parallel propagation approximation replaces the integration over wavenormal distribution to a closed form expression, and can be quite accurate. However, it can also perform badly, especially for electron energy > or = 1 MeV. Here, the accuracy, justification, and limits of the approximation are explored, and an improved version is presented. It is based on a previously developed procedure for identifying wavenormal angles compatible with imposed cutoffs on the wave frequency distribution. Because it also requires evaluation at only a small number of points, it features computational efficiency comparable to the parallel propagation version, while preserving contributions from oblique waves and all relevant harmonic numbers. Detailed comparisons are presented using an established model of nightside chorus.

[1] Prior to 2003, there are two known cases where ultrarelativistic (≳10 MeV) electrons appeared in the Earth's inner zone radiation belts in association with high speed interplanetary shocks: the 24 March 1991 and the less well studied 21 February 1994 storms. During the March 1991 event electrons were injected well into the inner zone on a timescale of minutes, producing a new stably trapped radiation belt population that persisted for ∼10 years. More recently, at the end of solar cycle 23, a number of violent geomagnetic disturbances resulted in large variations in ultrarelativistic electrons in the inner zone, indicating that these events are less rare than previously thought. Here we present results from a numerical study of shock-induced transport and energization of outer zone electrons in the 1–7 MeV range, resulting in a newly formed 10–20 MeV electron belt near L ∼ 3. Test particle trajectories are followed in time-dependent fields from an MHD magnetospheric model simulation of the 29 October 2003 storm sudden commencement (SSC) driven by solar wind parameters measured at ACE. The newly formed belt is predominantly equatorially mirroring. This result is in part due to an SSC electric field pulse that is strongly peaked in the equatorial plane, preferentially accelerating equatorially mirroring particles. The timescale for subsequent pitch angle diffusion of the new belt, calculated using quasi-linear bounce-averaged diffusion coefficients, is in agreement with the observed delay in the appearance of peak fluxes at SAMPEX in low Earth orbit. We also present techniques for modeling radiation belt dynamics using test particle trajectories in MHD fields. Simulations are performed using code developed by the Center for Integrated Space Weather Modeling.

Following enhanced magnetic activity the fluxes of energetic electrons in the Earth's outer radiation belt gradually decay to quiet-time levels. We use CRRES observations to estimate the energetic electron loss timescales and to identify the principal loss mechanisms. Gradual loss of energetic electrons in the region 3.0 ≤ L ≤ 5.0 occurs during quiet periods (Kp 7), indicating that the decay takes place in the plasmasphere. We compute loss timescales for pitch-angle scattering by plasmaspheric hiss using the PADIE code with wave properties based on CRRES observations. The resulting timescales suggest that pitch angle scattering by plasmaspheric hiss propagating at small or intermediate wave normal angles is responsible for electron loss over a wide range of energies and L shells. The region where hiss dominates loss is energy-dependent, ranging from 3.5 ≤ L ≤ 5.0 at 214 keV to 3.0 ≤ L ≤ 4.0 at 1.09 MeV. Plasmaspheric hiss at large wave normal angles does not contribute significantly to the loss rates. At E = 1.09 MeV the loss timescales are overestimated by a factor of ∼5 for 4.5 ≤ L ≤ 5.0. We suggest that resonant wave-particle interactions with EMIC waves, which become important at MeV energies for larger L (L > ∼4.5), may play a significant role in this region.

[1] Combined Release and Radiation Effects Satellite (CRRES) Electric Field Instrument (EFI) data are used to determine the electric field power spectral density as a function of L and Kp over the frequency range 0.2 to 15.9 mHz. The power at each frequency is fit to the function P(L, Kp) = a Lb exp(cKp). Assuming a purely electrostatic field and making several other assumptions regarding the azimuthal dependence of the field fluctuations, a Kp-dependent radial diffusion coefficient DLLE is computed from the power spectra. The model average DLLE for high activity (Kp = 6) are between 1 to 2 orders of magnitude larger than that for low activity (Kp = 1), dependent upon L and first invariant.

[1] Electron acceleration inside the Earth's magnetosphere is required to explain increases in the ∼MeV radiation belt electron flux during magnetically disturbed periods. Recent studies show that electron acceleration by whistler mode chorus waves becomes most efficient just outside the plasmapause, near L = 4.5, where peaks in the electron phase space density are observed. We present CRRES data on the spatial distribution of chorus emissions during active conditions. The wave data are used to calculate the pitch angle and energy diffusion rates in three magnetic local time (MLT) sectors and to obtain a timescale for acceleration. We show that chorus emissions in the prenoon sector accelerate electrons most efficiently at latitudes above 15° for equatorial pitch angles between 20° and 60°. As electrons drift around the Earth, they are scattered to large pitch angles and further accelerated by chorus on the nightside in the equatorial region. The timescale to accelerate electrons by whistler mode chorus and increase the flux at 1 MeV by an order of magnitude is approximately 1 day, in agreement with satellite observations during the recovery phase of storms. During wave acceleration the electrons undergo many drift orbits and the resulting pitch angle distributions are energy-dependent. Chorus scattering should produce pitch angle distributions that are either flat-topped or butterfly-shaped. The results provide strong support for the wave acceleration theory.

Current research has reemphasized the importance of cyclotron resonant wave-particle interactions for radiation belt electrons. Whistler mode hiss, chorus, and EMIC waves can act in combination to cause acceleration and loss of radiation belt electrons at greater rates than previously appreciated. These processes can be described by quasi-linear theory, but calculating quasi-linear diffusion coefficients is computationally demanding. Recent advances have been made in computing bounce averaged quasi-linear pitch angle, energy, and mixed diffusion coefficients for hiss and EMIC in the high density plasmasphere; this paper outlines generalization of these techniques for chorus waves, prevalent in the low density region outside the plasmasphere. These coefficients are associated with a two-dimensional diffusion equation whose numerical solution by finite differencing methods requires care, for reasons having to do with the relation between the mixed and other diffusion coefficients, as discussed.

[1] A numerical simulation of the low-altitude electron radiation belt is described. It includes dependences on the electrons' bounce and drift phases, equatorial pitch angle, and kinetic energy in the range of ∼1 to several MeV at L = 3.5. Physical processes in addition to the adiabatic electron motion are pitch angle diffusion and backscattering from a realistic model atmosphere. Quasi-linear diffusion coefficients are calculated from a model of the whistler mode plasmaspheric hiss wave intensity. Comparisons of the simulation results with electron data from a low-altitude satellite show that the model accounts for the main features of the electron spatial distribution during selected periods of differing geomagnetic activity.

[1] First-order estimates indicate that the lifetime of energetic (a few MeV) electrons in the inner radiation belts (e.g., near L = 2) may be significantly reduced by in situ injection of whistler mode waves at radiated power levels of a few kW at frequencies of a few kHz. Our estimates are based on previously published results concerning the effect on the electron lifetimes of VLF signals from ground-based VLF transmitters operating in the 17–23 kHz range. Waves at lower frequencies (a few kHz) can drive diffusion rates that are higher by a factor of as much as ∼30 and can also be efficiently stored in the magnetospheric cavity, resulting in additional effective enhancement of wave power density of a factor of ∼16. This wave power enhancement is also expected to enhance the MeV electron diffusion rates.

A spectral description of a model of convective turbulence in the E region of the equatorial ionosphere is considered. It is demonstrated that three complex modes are sufficient to generate a stochastic steady state, in a manner analogous to similar models of the Euler equation and of drift waves. Complex terms in the nonlinear coupling coefficient are retained that are usually dropped on the basis of large kL, where L is the density gradient scale length. In addition, some parallel damping is included. Parameters that lead to a stochastic steady state are then found by explicitly solving for the wave vectors that reproduce growth rates and nonlinear coupling terms from a known stochastic case of a drift wave model.

The refractive index for a constant-frequency whistler mode wave in an electron-proton plasma is considered as a function of position, through the local gyrofrequencies Ωe,i. The full cold plasma dispersion relation is used. The wave frequency can take any value up to the smaller of Ωe and the plasma frequency ωpe, but ωpe is allowed to take any fixed value, as is the wavenormal angle. It is rigorously established that the refractive index is a decreasing function of Ωe. One application of this is to finding locations of Landau and cyclotron resonances, to evaluate the effects of whistler mode waves on radiation belt electrons.

Simulations of the evolution of a relativistic electron beam injected into the Earth's magnetosphere are examined. It is found that the lifetime of beam particles strongly depends on the initial pitch angle distribution and that lifetimes similar to those found for the radiation belts are obtained for nearly equatorially mirroring injections. It is concluded that the results of our previous study on beam propagation [Khazanov et al, 1999], which only considered upper ionospheric injections, are consistent with other relativistic electron studies and should be regarded as an examination of the loss cone edge population, whether naturally occurring or artificially injected into near-Earth space.