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 wave normal 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 colocated 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 modeling of the outer radiation belt. Plain Language SummaryThe electrons in Earth's radiation belts exist in a highly rarefied part of space where collisions between particles is very rare. The only way in which the energy or direction of the trapped high-energy electrons can be changed is through interactions with electromagnetic waves. The efficacy of the interaction is a function of the energy and direction of travel of the electrons. In physics-based models of the radiation belts, the efficacy of the wave-particle interactions is captured in diffusion coefficients. These functions are constructed from information about the amplitude and frequency properties of the waves in the interaction, the magnetic field strength, ion composition, and density of the local plasma. We build up collections of observations of these properties from multiple passes of one of the NASA Van Allen probes through the same three small regions of space. The observations display significant temporal variability. We report on the statistical distributions of wave intensity, magnetic field strength and plasma number density and investigate the statistical distribution of the resulting diffusion coefficient. We find that the diffusion coefficients are highly variable and suggest that, by borrowing methods from other branches of geophysics...

Physics-based radiation belt models of electron behavior often focus on the wave-particle interactions that accelerate and scatter particles or contribute to radial diffusion. These models make considerable use of quasilinear theory to describe the wave-particle interactions (e.g., Lyons et al., 1972; Ripoll et al., 2020) and can be used to study the flux of high-energy electrons on a range of time scales, from single storms (e.g.,

Many modern outer radiation belt models simulate the long‐time behavior of high‐energy electrons by solving a three‐dimensional Fokker‐Planck equation for the drift‐ and bounce‐averaged electron phase space density that includes radial, pitch‐angle, and energy diffusion. Radial diffusion is an important process, often characterized by a deterministic diffusion coefficient. One widely used parameterization is based on the median of statistical ultralow frequency (ULF) wave power for a particular geomagnetic index Kp. We perform idealized numerical ensemble experiments on radial diffusion, introducing temporal and spatial variability to the diffusion coefficient through stochastic parameterization, constrained by statistical properties of its underlying observations. Our results demonstrate the sensitivity of radial diffusion over a long time period to the full distribution of the radial diffusion coefficient, highlighting that information is lost when only using median ULF wave power. When temporal variability is included, ensembles exhibit greater diffusion with more rapidly varying diffusion coefficients, larger variance of the diffusion coefficients and for distributions with heavier tails. When we introduce spatial variability, the variance in the set of all ensemble solutions increases with larger spatial scales of variability. Our results demonstrate that the variability of diffusion affects the temporal evolution of phase space density in the outer radiation belt. We discuss the need to identify important temporal and length scales to constrain variability in diffusion models. We suggest that the application of stochastic parameterization techniques in the diffusion equation may allow the inclusion of natural variability and uncertainty in modeling of wave‐particle interactions in the inner magnetosphere.

Kinetic wave-particle interactions in Earth’s outer radiation belt energize and scatter high-energy electrons, playing an important role in the dynamic variation of the extent and intensity of the outer belt. It is possible to model the effects of wave-particle interactions across long length and time scales using quasi-linear theory, leading to a Fokker-Planck equation to describe the effects of the waves on the high energy electrons. This powerful theory renders the efficacy of the wave-particle interaction in a diffusion coefficient that varies with energy or momentum and pitch angle. In this article we determine how the Fokker-Planck equation responds to the temporal variation of the quasi-linear diffusion coefficient in the case of pitch-angle diffusion due to plasmaspheric hiss. Guided by in-situ observations of how hiss wave activity and local number density change in time, we use stochastic parameterisation to describe the temporal evolution of hiss diffusion coefficients in ensemble numerical experiments. These experiments are informed by observations from three different example locations in near-Earth space, and a comparison of the results indicates that local differences in the distribution of diffusion coefficients can result in material differences to the ensemble solutions. We demonstrate that ensemble solutions of the Fokker-Planck equation depend both upon the timescale of variability (varied between minutes and hours), and the shape of the distribution of diffusion coefficients. Based upon theoretical construction of the diffusion coefficients and the results presented here, we argue that there is a useful maximum averaging timescale that should be used to construct a diffusion coefficient from observations, and that this timescale is likely less than the orbital period of most inner magnetospheric missions. We discuss time and length scales of wave-particle interactions relative to the drift velocity of high-energy electrons and confirm that arithmetic drift-averaging is can be appropriate in some cases. We show that in some locations, rare but large values of the diffusion coefficient occur during periods of relatively low number density. Ensemble solutions are sensitive to the presence of these rare values, supporting the need for accurate cold plasma density models in radiation belt descriptions.

The operational and research-focused modeling of high-energy electron fluxes in Earth's radiation belts is based upon the physics of electron motion in the Earth's magnetic field. High-energy electrons are trapped in the approximately dipolar magnetic field and execute three motions: very fast gyromotion (with periods of less than 1 ms), fast bounce motion between hemispheres (with periods of ∼1 s) and drift motion around the planet (with periods of minutes). Models of radiation belt dynamics use a coordinate system based upon these motions, which can be described using a system of adiabatic invariants μ, J, and L*. This means that slow, reversible changes to the energy and path of electrons due to slow changes in the magnetic field are automatically taken into account in the model, since the computational grids themselves are based upon the invariant. It therefore becomes very important to be able to map between real space, and energy space, to the values of these adiabatic invariants at every stage in model development. The creation of initial conditions, boundary conditions (e.g., Glauert et al., 2018), diffusion matrices (e.g., Horne et al., 2018),