Simulating the Earth’s radiation belts: internal acceleration and continuous losses to the magnetopause

In the Earth's radiation belts the flux of relativistic electrons is highly variable, sometimes changing by orders of magnitude within a few hours. Since energetic electrons can damage satellites it is important to understand the processes driving these changes and, ultimately, to develop forecasts of the energetic electron population. One approach is to use 3-dimensional diffusion models, based on a Fokker-Planck equation. Here we describe a model where the phase-space density is set to zero at the outer L* boundary, simulating losses to the magnetopause, using recently published chorus diffusion coefficients for 1.5 ≤ L* ≤ 10. The value of the phase-space density on the minimum energy boundary is determined from a recently published, solar wind dependent, statistical model. Our simulations show that an outer radiation belt can be created by local acceleration of electrons from a very soft energy spectrum without the need for a source of electrons from inward radial transport. The location in L* of the peaks in flux for these steady state simulations is energy dependent and moves Earthward with increasing energy. Comparisons between the model and data from the CRRES satellite are shown; flux drop-outs are reproduced in the model by the increased outward radial diffusion that occurs during storms. Including the inward movement of the magnetopause in the model has little additional effect on the results. Finally, the location of the low energy boundary is shown to be important for accurate modelling of observations

Particle-in-cell experiments examine electron diffusion by whistler-mode waves: 1. Benchmarking with a cold plasma

Using a particle-in-cell code, we study the diffusive response of electrons due to wave particle interactions with whistler-mode waves. The relatively simple configuration of field-aligned waves in a cold plasma is used in order to benchmark our novel method, and to compare with previous works that used a different modelling technique. In this boundary-value problem, incoherent whistler-mode waves are excited at the domain boundary, and then propagate through the ambient plasma. Electron diffusion characteristics are directly extracted from particle data across all available energy and pitch-angle space. The ‘nature’ of the diffusive response is itself a function of energy and pitch-angle, such that the rate of diffusion is not always constant in time. However, after an initial transient phase, the rate of diffusion tends to a constant, in a manner that is consistent with the assumptions of quasilinear diffusion theory. This work establishes a framework for future investigations on the nature of diffusion due to whistler-mode wave-particle interactions, using particle-in-cell numerical codes with driven waves as boundary value problems

Variability of quasilinear diffusion coefficients for plasmaspheric hiss

In the Outer Radiation Belt, the acceleration and loss of high-energy electrons is largely controlled by wave-particle interactions. Quasilinear diffusion coefficients are an efficient way to capture the small-scale physics of wave-particle interactions due to magnetospheric wave modes such as plasmaspheric hiss. The strength of quasilinear diffusion coefficients as a function of energy and pitch-angle depends on both wave parameters and plasma parameters such as ambient magnetic field strength, plasma number density and composition. For plasmaspheric hiss in the magnetosphere, observations indicate large variations in the wave intensity and wavenormal angle, but less is known about the simultaneous variability of the magnetic field and number density. We use in-situ measurements from the Van Allen Probe mission to demonstrate the variability of selected factors that control the size and shape of pitch-angle diffusion coefficients: wave intensity, magnetic field strength and electron number density. We then compare with the variability of diffusion coefficients calculated individually from co-located and simultaneous groups of measurements. We show that the distribution of the plasmaspheric hiss diffusion coefficients is highly non-Gaussian with large variance, and that the distributions themselves vary strongly across the three phase-space bins studied. In most bins studied, the plasmaspheric hiss diffusion coefficients tend to increase with geomagnetic activity, but our results indicate that new approaches that include natural variability may yield improved parameterizations. We suggest methods like stochastic parameterization of wave-particle interactions could use variability information to improve modelling of the Outer Radiation Belt

Quasi-linear simulations of inner radiation belt electron pitch angle and energy distributions

“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

Effects of VLF transmitter waves on the inner belt and slot region

Signals from very low frequency (VLF) transmitters can leak from the Earth‐ionosphere wave guide into the inner magnetosphere, where they propagate in the whistler mode and contribute to electron dynamics in the inner radiation belt and slot region. Observations show that the waves from each VLF transmitter are highly localized, peaking on the nightside in the vicinity of the transmitter. In this study we use ∼5 years of Van Allen Probes observations to construct global statistical models of the bounce‐averaged pitch angle diffusion coefficients for each individual VLF transmitter, as a function of L*, magnetic local time (MLT), and geographic longitude. We construct a 1‐D pitch angle diffusion model with implicit longitude and MLT dependence to show that VLF transmitter waves weakly scatter electrons into the drift loss cone. We find that global averages of the wave power, determined by averaging the wave power over MLT and longitude, capture the long‐term dynamics of the loss process, despite the highly localized nature of the waves in space. We use our new model to assess the role of VLF transmitter waves, hiss waves, and Coulomb collisions on electron loss in the inner radiation belt and slot region. At moderate relativistic energies, E∼500 keV, waves from VLF transmitters reduce electron lifetimes by an order of magnitude or more, down to the order of 200 days near the outer edge of the inner radiation belt. However, VLF transmitter waves are ineffective at removing multi–megaelectron volt electrons from either the inner radiation belt or slot region

Electron losses from the radiation belts caused by EMIC waves

Electromagnetic Ion Cyclotron (EMIC) waves cause electron loss in the radiation belts by resonating with high energy electrons at energies greater than about 500 keV. However, their effectiveness has not been fully quantified. Here we determine the effectiveness of EMIC waves by using wave data from the fluxgate magnetometer on CRRES to calculate bounce averaged pitch angle and energy diffusion rates for L* =3.5 - 7 for five levels of Kp between 12 - 18 MLT. To determine the electron loss EMIC diffusion rates were included in the BAS Radiation Belt Model together with whistler mode chorus, plasmaspheric hiss and radial diffusion. By simulating a 100 day period in 1990 we show that EMIC waves caused a significant reduction in the electron flux for energies greater than 2 MeV but only for pitch angles lower than about 60°.The simulations show that the distribution of electrons left behind in space looks like a pancake distribution. Since EMIC waves cannot remove electrons at all pitch angles even at 30 MeV, our results suggest that EMIC waves are unlikely to set an upper limit on the energy of the flux of radiation belt electrons

On the importance of gradients in the low energy electron phase space density for relativistic electron acceleration

Observations of the electron radiation belts have shown links between increases in the low-energy seed population and enhancements in the >1-MeV flux. During active times, low-energy electrons are introduced to the radiation belt region before being accelerated to higher energies via a range of mechanisms. The impact of variations in the seed population on the 1-MeV flux level were explored using the British Antarctic Survey Radiation Belt Model. We find that, for a period from the 21 April to 9 May 2013, the increase in the low-energy electron flux was vital to recreate the observed 1-MeV flux enhancement on the 1 May but was less important for the 1-MeV enhancement on the 27 April 2013. To better understand the relationships between the different energy populations, a series of idealized experiments with the 2-D British Antarctic Survey Radiation Belt Model were performed, which highlight a careful balance between losses and acceleration from chorus waves. Seed population enhancements alter this balance by increasing the phase space density gradient, and consequently, the rate of energy diffusion, allowing acceleration to surpass loss. Additionally, we demonstrate that even with the same chorus diffusion coefficients and the same low-energy boundary condition, the flux of ∼500-keV to 1-MeV electrons increased when starting with a hard spectrum but decreased for a soft initial spectrum. This suggests that initial energy gradients in the phase space density were important to determine whether >500-keV electrons were enhanced due to chorus wave acceleratio

A 30-year simulation of the outer electron radiation belt

As society becomes more reliant on satellite technology, it is becoming increasingly important to understand the radiation environment throughout the Van Allen radiation belts. Historically most satellites have operated in low Earth orbit or geostationary orbit (GEO), but there are now over 100 satellites in medium Earth orbit (MEO). Additionally, satellites using electric orbit raising to reach GEO may spend hundreds of days on orbits that pass through the heart of the radiation belts. There is little long‐term data on the high‐energy electron flux, responsible for internal charging in satellites, available for MEO. Here we simulate the electron flux between the outer edge of the inner belt and GEO for 30 years. We present a method that converts the >2‐MeV flux measured at GEO by the Geostationary Operational Environmental Satellites spacecraft into a differential flux spectrum to provide an outer boundary condition. The resulting simulation is validated using independent measurements made by the Galileo In‐Orbit Validation Element‐B spacecraft; correlation coefficients are in the range 0.72 to 0.88, and skill scores are between 0.6 and 0.8 for a range of L∗ and energies. The results show a clear solar cycle variation and filling of the slot region during active conditions and that the worst case spectrum overlaps that derived independently for the limiting extreme event. The simulation provides a resource that can be used by satellite designers to understand the MEO environment, by space insurers to help resolve the cause of anomalies and by satellite operators to plan for the environmental extremes

A new diffusion matrix for whistler mode chorus waves

Global models of the Van Allen radiation belts usually include resonant wave-particle interactions as a diffusion process, but there is a large uncertainty over the diffusion rates. Here we present a new diffusion matrix for whistler mode chorus waves that can be used in such models. Data from seven satellites are used to construct 3,536 power spectra for upper and lower band chorus for 1.5 ≤ L∗ ≤ 10, MLT, magnetic latitude 0o ≤ |λm| ≤ 60o and five levels of Kp. Five density models are also constructed from the data. Gaussian functions are fitted to the spectra and capture typically 90% of the wave power. The frequency maxima of the power spectra vary with L∗ and are typically lower than that used previously. Lower band chorus diffusion increases with geomagnetic activity and is largest between 21:00 and 09:00 MLT. Energy diffusion extends to a few MeV at large pitch angles > 60o and at high energies exceeds pitch angle diffusion at the loss cone. Most electron diffusion occurs close to the geomagnetic equator (< 12o). Pitch angle diffusion rates for lower band chorus increase with L∗ and are significant at L∗ = 8 even for low levels of geomagnetic activitywhile upper band chorus is restricted to mainly L∗ < 6. The combined drift and bounce averaged diffusion rates for upper and lower band chorus extend from a few keV near the loss cone up to several MeV at large pitch angles indicating loss at low energies and net acceleration at high energies

The magnetic local time distribution of energetic electrons in the radiation belt region

Using 14 years of electron flux data from the National Oceanic and Atmospheric Administration Polar Operational Environmental Satellites, a statistical study of the magnetic local time (MLT) distribution of the electron population is performed across a range of activity levels, defined by AE, AE*, Kp, solar wind velocity (Vsw), and VswBz. Three electron energies (>30, >100, and >300 keV) are considered. Dawn-dusk flux asymmetries larger than order of magnitude were observed for >30 and >100 keV electrons. For >300 keV electrons, dawn-dusk asymmetries were primarily due to a decrease in the average duskside flux beyond L* ∼ 4.5 that arose with increasing activity. For the >30 keV population, substorm injections enhance the dawnside flux, which may not reach the duskside as the electrons can be on open drift paths and lost to the magnetopause. The asymmetries in the >300 keV population are attributed to the combination of magnetopause shadowing and >300 keV electron injections by large electric fields. We suggest that 3-D radiation belt models could set the minimum energy boundary (Emin) to 30 keV or above at L* ∼ 6 during periods of low activity. However, for more moderate conditions, Emin should be larger than 100 keV and, for very extreme activities, ∼300 keV. Our observations show the extent that in situ electron flux readings may vary during active periods due to the MLT of the satellite and highlight the importance of 4-D radiation belt models to fully understand radiation belt processes