Accounting for Variability in ULF Wave Radial Diffusion Models

Abstract

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.

The Implications of Temporal Variability in Wave‐Particle Interactions in Earth's Radiation Belts

“…Both this paper and Thompson et al (2020) highlight that the distribution and variability time scales of diffusion coefficients are important for the evolution of electron phase space density due to diffusion. In other words, key details of the microphysical wave-particle interaction are important for accurate modeling of the macroscale radiation belt system, and the evolution of phase space density is not solely reliant on the average properties of the diffusion coefficients.…”

“…Other evidence suggests it is important to understand the underlying distribution of diffusion coefficients. Idealized numerical experiments using a radial diffusion equation (Thompson et al., 2020) noted that the amount of diffusion depends upon the nature of the underlying distribution of diffusion coefficients. Those distributions with statistically heavier tails experienced greater diffusion, even when the distributions of diffusion coefficients had the same statistical average value.…”

ULF Wave Driven Radial Diffusion During Geomagnetic Storms: A Statistical Analysis of Van Allen Probes Observations

“…The L * value is calculated using the International Radiation Belt Environment Modeling (IRBEM) code (https://sourceforge.net/projects/irbem/) with the Tsyganenko and Sitnov (2005) magnetic field model for an equatorially trapped particle. It is noted that multiple methods for calculating L * values exist, and this is a source of additional variability to the statistical analysis (Thompson et al., 2020). It is also stressed that in a distorted and non‐dipolar magnetic field configuration, the L value and L * values are not directly comparable.…”

“…It is noted that multiple methods for calculating L* values exist, and this is a source of additional variability to the statistical analysis (Thompson et al, 2020). It is also stressed that in a distorted and non-dipolar magnetic field configuration, the L value and L* values are not directly comparable.…”

Determining the Global Scale Size of Chorus Waves in the Magnetosphere

“…Numerical experiments demonstrate that the solution of Fokker-Planck equation for the diffusion model is sensitively dependent on the temporal variability of the diffusion coefficient (Thompson et al, 2020;Watt et al, 2021), which will be affected by the spatial and temporal variations of the wave and plasma parameter. Watt et al (2021) performed multiple numerical experiments of bounce-averaged diffusion, and indicated that rapid variations (with timescales of ∼2 min) of whistler-mode wave diffusion coefficients resulted in solutions of the Fokker-Planck equations that were similar to results obtained from an averaged diffusion coefficient.…”