Approximate analytical solutions for the trapped electron distribution due to quasi‐linear diffusion by whistler mode waves

Abstract

The distribution of trapped energetic electrons inside the Earth's radiation belts is the focus of intense studies aiming at better describing the evolution of the space environment in the presence of various disturbances induced by the solar wind or by an enhanced lightning activity. Such studies are usually performed by means of comparisons with full numerical simulations solving the Fokker‐Planck quasi‐linear diffusion equation for the particle distribution function. Here we present for the first time approximate but realistic analytical solutions for the electron distribution, which are shown to be in good agreement with exact numerical solutions in situations where resonant scattering of energetic electrons by whistler mode hiss, lightning‐generated or chorus waves, is the dominant process. Quiet time distributions are well recovered, as well as the evolution of energized relativistic electron distributions during disturbed geomagnetic conditions. It is further shown that careful comparisons between the analytical solutions and measured distributions may allow to infer important bounce‐ and drift‐averaged wave characteristics (such as wave amplitude). It could also help to improve the global understanding of underlying physical phenomena.

Electron Flux Enhancements at L = 4.2 Observed by Global Positioning System Satellites: Relationship With Solar Wind and Geomagnetic Activity

“…Intense chorus waves may also be present at other MLTs at L ∼ 4.2 during this event, but the orbital coverage of the Van Allen Probes did not allow us to confirm this. Such high chorus amplitudes are similar to chorus amplitudes measured during the strong storm on 17 March 2013 at L ∼ 4.25, when ∼100 pT chorus waves, present over 8–10 hr in MLT, were measured and were sufficient to account for the observed increase of 1‐ to 5‐MeV electron fluxes over a 10‐hr period through chorus‐induced energization alone, provided that the local electron plasma density was sufficiently small, <13 cm −3 (W. Li et al, ; Mourenas et al, ). During our event on 28 September 2016, the density inferred from spacecraft potential measurements of the Van Allen Probes at L = 4.2–4.6 was roughly ∼10–20 cm −3 .…”

“…At 3–5 MeV, fluxes increased by a factor of 5–10 over 8–9 hr. Such characteristic behaviors are expected in the case of quasi‐linear energization by chorus waves from an initially narrow energy distribution, leading to a progressive energy broadening of this distribution (Horne et al, ; Mourenas et al, ; Thorne et al, ). Then, K p was around 4–5 and A E ∼ 1,000–1,500 nT, S Y M − H decreased from −30 to −49 nT before recovering, and P dyn remained stable at ∼3.5–4.…”

Empirical model of lower band chorus wave distribution in the outer radiation belt

“…Therefore, additionally to a general increase of the wave amplitude (and a corresponding global intensification of electron scattering) when K p grows, the simultaneous evolution of the realistic g ( θ ) distribution with K p modifies the profile of the pitch angle diffusion rate as a function of α 0 . This new effect may in turn result in different pitch angle distribution shapes for the electrons [ Mourenas et al , ].…”

Properties of Intense Field‐Aligned Lower‐Band Chorus Waves: Implications for Nonlinear Wave‐Particle Interactions

“…Once the distribution of wave intensity in geometrical space ( L ‐shell, MLT, and magnetic latitude) and the distribution of wave propagation angles are known, the local quasi‐linear diffusion coefficients can be calculated (Kennel & Engelmann, ; Lerche, ) and averaged over the 3‐D spatial domain of the radiation belts (e.g., Albert, ; Artemyev, Agapitov, et al, ; Horne et al, ; Glauert & Horne, ; Mourenas, Artemyev, Agapitov & Krasnoselskikh, ; Shprits & Ni, ). Such diffusion coefficients (expressed as functions of L ‐shell, MLT, and geomagnetic activity/solar wind conditions) are central to state‐of‐the‐art diffusion models describing radiation belt dynamics—that is, relativistic electron acceleration (e.g., Horne et al, ; Li et al, ; Mourenas et al, ; Thorne et al, ), electron precipitation (e.g., Ni et al, ; Thorne et al, ), and electron transport to lower L ‐shells (together with radial diffusion, e.g., Ma et al, ). Because a wide range of phenomena observed in the radiation belts can be described relatively successfully by quasi‐linear diffusion models (see also Albert et al, ; Drozdov et al, ; Glauert et al, ; Ma et al, ; Su et al, ), most past investigations of chorus waves have focused on the aforementioned wave parameters/characteristics needed to evaluate the quasi‐linear diffusion coefficients.…”