On the Transition from Coherent to Diffusive ULF Wave Transport of Relativistic Electrons in the Van Allen Belts

Relativistic electrons trapped in the terrestrial Van Allen belts can be transported through drift-resonant interactions with ultra-low frequency (ULF) plasma waves. Interactions with a single mode can produce signatures of a coherent resonant process, and which transitions towards the diffusive paradigm as the wave frequency spectra and/or the azimuthal wavenumber spectra become increasingly broadband. Whilst the transport of electrons through interactions with ULF waves is often modeled using statistical approaches which assume diffusive dynamics caused by the perturbed electromagnetic fields, in the real system on the timescales of particle transport during a single storm a limited number of discrete mode interactions may be more realistic. Here we use a single particle-tracing approach to the dynamics of ensembles of relativistic electrons, and simulate the trajectories and particle dynamics of the electrons in the presence of a finite number of discrete frequency modes. We focus on interactions with Alfvenic disturbances which are assumed to be locally standing field line resonances, and examine how the ensemble responds to varying numbers of discrete modes which span a finite frequency range. We will attempt to characterize the point of onset of diffusion as a transition from separate discrete interactions using the two-thirds resonance island overlap rule found in statistical theory (e.g., Lichtenberg and Lieberman, 1992) which involves the comparison between the distance and widths of the electron's primary resonant islands in its phase space. With a range of simulations, we directly compare the transport in our numerical model to that expected from finite numbers of resonance islands. We further derive a criterion in terms of wave frequency separation and wave amplitude which predicts the transition to diffusive behaviour and test it with our simulations. Finally, we compare the rates of diffusion produced in the ensemble particle dynamics with analytic predictions from a number of prior analyses and approaches. These results are critically important for determining the conditions under which ULF wave radial diffusion theory can be applied in radiation belt models.