J. M. Albert

Energetic Outer Zone Electron Loss Timescales During Low Geomagnetic Activity

Following enhanced magnetic activity the fluxes of energetic electrons in the Earths 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.

Three-Dimensional Diffusion Simulation of Outer Radiation Belt Electrons During the 9 October 1990 Magnetic Storm

Abstract : Relativistic (>l MeV) electron flux increases in the Earths 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.

Multidimensional quasi-linear diffusion of radiation belt electrons

Abstract : We consider diffusion of outer zone radiation belt electrons by chorus waves. Quasi-linear diffusion coefficients valid outside the plasmasphere have only been calculated recently, and indicate that the energy and cross diffusion rates can be comparable to that for pitch angle diffusion. Proper solution of the diffusion equation for phase space density must therefore be based on the full diffusion tensor, but this has been plagued by numerical problems associated with the large and rapidly varying cross terms. To circumvent this, techniques are developed for transforming to variables in which the cross diffusion term vanishes. A model calculation shows significant diffusion of phase space density at L = 4.5 from 0.2 MeV up to a few MeV in less than a day.

Simple approximations of quasi‐linear diffusion coefficients

Abstract : Quasi-linear diffusion by cyclotron-resonant plasma waves is likely a key ingredient of the behavior of electrons in the Earths 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.

Stochastic modeling of multidimensional diffusion in the radiation belts

Abstract : A new code for solving radiation belt diffusion equations has been developed and applied to the 2-D bounce-averaged energy pitch angle quasi-linear diffusion equation. The code uses Monte Carlo methods to solve Ito stochastic differential equations (SDEs) which are mathematically equivalent to radiation belt diffusion equations. We show that our SDE code solves the diffusion equation with off-diagonal diffusion coefficients in contrast to standard finite difference codes which are generally unstable when off-diagonal diffusion coefficients are included. Our results are in excellent agreement with previous results. We have also investigated effects of assuming purely parallel propagating electromagnetic waves when calculating the diffusion coefficients and find that this assumption leads to errors of more than an order of magnitude in flux at some equatorial pitch angles for the specific chorus wave model we use. Further work is needed to investigate the sensitivity of our results to the wave model parameters. Generalization of the method to 3-D is straight-forward, thus making this model a very promising new way to investigate the relative roles of pitch angle, energy, and radial diffusion in radiation belt dynamics.

Comparison of quasilinear diffusion coefficients for parallel propagating whistler mode waves with test particle simulations

[1] We present a comparison between the classical quasilinear diffusion coefficients and those calculated using a general test particle code. The trajectories of a large number of electrons are followed as they traverse a numerically‐ constructed, broadband, small‐amplitude wave field, using a general relativistic test particle code. The change in each electron’s pitch angle and energy is shown to be stochastic and the resulting diffusion of the entire population is found to be in excellent agreement with quasilinear theory. We also demonstrate that the diffusion coefficients presented by Summers, derived specifically for parallel propagating waves, are a factor of two larger than the test particle results if the power spectral density is one‐sided (w > 0). Our results demonstrate the general validity of using quasilinear theory to describe the effects of broadband small amplitude waves on radiation belt electrons. Citation: Tao, X., J. Bortnik, J. M. Albert, K. Liu, and R. M. Thorne (2011), Comparison of quasilinear diffusion coefficients for parallel propagating whistler mode waves with test particle simulations, Geophys. Res. Lett., 38, L06105,

Evidence of stronger pitch angle scattering loss caused by oblique whistler‐mode waves as compared with quasi‐parallel waves

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.

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.

Electron lifetimes from narrowband wave‐particle interactions within the plasmasphere

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.

Three‐dimensional stochastic modeling of radiation belts in adiabatic invariant coordinates

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 Ito 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.