Approximate analytical formulation of radial diffusion and whistler‐induced losses from a preexisting flux peak in the plasmasphere

Abstract

Modeling the spatiotemporal evolution of relativistic electron fluxes trapped in the Earth's radiation belts in the presence of radial diffusion coupled with wave‐induced losses should address one important question: how deep can relativistic electrons penetrate into the inner magnetosphere? However, a full modeling requires extensive numerical simulations solving the comprehensive quasi‐linear equations describing pitch angle and radial diffusion of the electron distribution, making it rather difficult to perform parametric studies of the flux behavior. Here we consider the particular situation where a localized flux peak (or storage ring) has been produced at low L < 4 during a period of strong disturbances, through a combination of chorus‐induced energy diffusion (or direct injection) at low L together with enhanced wave‐induced losses and outward radial transport at higher L. Assuming that radial diffusion can be further described as the spatial broadening within the plasmasphere of this preexisting flux peak, simple approximate analytical solutions for the distribution of trapped relativistic electrons are derived. Such a simplified formalism provides a convenient means for easily determining whether radial diffusion actually prevails over atmospheric losses at any particular time for given electron energy E and location L. It is further used to infer favorable conditions for relativistic electron access to the inner belt, providing an explanation for the relative scarcity of such a feat under most circumstances. Comparisons with electron flux measurements on board the Van Allen Probes show a reasonable agreement between a few weeks and 4 months after the formation of a flux peak.

Scaling laws for the inner structure of the radiation belts

“…For the sake of simplicity, we shall use the corresponding electric radial diffusion coefficient D LL ≈ 10 −7+0.46Kp L 6 day −1 derived from a statistics of 15 years of ULF wave data at L = 2-6 for Kp < 5 by Ozeke et al [2014]. The corresponding D LL is weakly dependent on E and 0 [Ozeke et al, 2014] when considering the main electron population at 0 > 50 ∘ Mourenas et al, 2015], in agreement with a D LL model based on in situ Van Allen Probes statistics [Ali et al, 2016]. We further assume that electron dynamics is dictated by radial diffusion and hiss and EMIC-induced losses acting nearly independently of each other [e.g., Schulz and Lanzerotti, 1974;Mourenas et al, 2015].…”

“…The corresponding D LL is weakly dependent on E and 0 [Ozeke et al, 2014] when considering the main electron population at 0 > 50 ∘ Mourenas et al, 2015], in agreement with a D LL model based on in situ Van Allen Probes statistics [Ali et al, 2016]. We further assume that electron dynamics is dictated by radial diffusion and hiss and EMIC-induced losses acting nearly independently of each other [e.g., Schulz and Lanzerotti, 1974;Mourenas et al, 2015]. Then the quantity = p 2 ∕(2m e B) (≃ M = sin 2 0 , the first adiabatic invariant) is nearly conserved by both radial diffusion and pitch angle diffusion [Walt, 1970;Schulz and Lanzerotti, 1974], corresponding to inward electron trajectories in (E, L) space given by p(t)∕p(t = 0) = [L(t = 0)∕L(t)] 3∕2 .…”

Diffusive Transport of Several Hundred keV Electrons in the Earth's Slot Region

“…Several important factors involved in wave‐particle interactions could be coupled. For example, the long‐term radial diffusion of energetic electrons is commonly accompanied by a flux decay feature (Baker et al, ; Ma et al, ; Mourenas et al, ). The electron flux evolution via wave‐particle interactions between a variety of wave modes can be reasonably described as a summation of diffusion processes using quasi‐linear theory (Kennel & Engelmann, ; Lyons, , ) and has been recently simulated using radiation belt diffusion models (Albert & Young, ; Glauert & Horne, ; Glauert et al, ; Ma et al, ; Li, Ma, Thorne, et al, ; Shprits et al, ; Tu et al, ; Xiao et al, ).…”