ULF Wave Transport of Relativistic Electrons in the Van Allen Belts: Criteria for Transition to Radial Diffusion

Relativistic electrons in the radiation belts can be transported as a result of wave-particle interactions (WPI) with ultra-low frequency (ULF) waves. Such WPI are often assumed to be diffusive, parametric models for the radial diffusion coefficient often being used to assess the rates of radial transport. However, these WPI transition from initially coherent interactions to the diffusive regime over a finite time, this time depending on the ULF wave power spectral density, and local resonance conditions. Further, in the real system on the timescales of a single storm, interactions with finite discrete modes may be more realistic. Here, we use a particle-tracing model to simulate the dynamics of outer radiation belt electrons in the presence of a finite number of discrete frequency modes. We characterize the point of the onset of diffusion as a transition from separate discrete interactions in terms of wave parameters by using the "two-thirds" overlap criterion (Lichtenberg & Lieberman, 1992, ), a comparison between the distance between, and the widths of, the electron's primary resonant islands in phase space. Further, we find the particle decorrelation time in our model system with typical parameters to be on the timescale of hours, which only afterward can the system be modeled by one-dimensional radial diffusion. Direct comparison of particle transport rates in our model with previous analytic diffusion coefficient formulations show good agreement at times beyond the decorrelation time. These results are critical for determining the time periods and conditions under which ULF wave radial diffusion theory can be applied. The dynamics of Earth's outer Van Allen radiation belt electrons have up to now been almost exclusively modeled using statistical methods. However, such approaches may not be valid for all scenarios. In this work, we defined a criterion separating the regimes where the dynamics of the outer radiation belt electrons can and cannot be modeled statistically, and in particular using a model based around the concepts of diffusion where averaging over many individual interactions leads to an assessment for the overall behavior of a set, or ensemble, of electrons. We use a test particle-tracing model to assess the actual dynamics of particle ensembles when perturbed by a type of plasma waves with ultra-low frequency in space. We showed that there is a distinctive qualitative and quantitative difference between diffusive and the more coherent regimes and identified their point of transition. We further verified that once the system has evolved beyond our derived transition criteria it does indeed match the common statistical predictions, verifying the applicability of a diffusion model after that time. Significantly, however, at earlier times the more correlated system behaves differently and may be characterized by a much faster and coherent transport. We derive analytic expressions required for the transition from coherent to diffusive transport applied to ultra-low frequency (ULF) wave-particle interactions We characterize the behavior using an equivalent dynamical system model, highlighting the importance of the particle decorrelation time For decorrelation times longer than typical ULF wavetrains, particle transport rates cannot be estimated under a radial diffusion paradigm