Drift phase resolved diffusive radiation belt model: 1. Theoretical framework

Abstract

Most physics-based models provide a coarse three-dimensional representation of radiation belt dynamics at low time resolution, of the order of a few drift periods. The description of the effect of trapped particle transport on radiation belt intensity is based on the random phase approximation, and it is in one dimension only: the third adiabatic invariant coordinate, akin to a phase-averaged radial distance. This means that these radiation belt models do not resolve the drift phase or, equivalently, the magnetic local time. Yet, in situ measurements suggest that radiation belt intensity frequently depends on magnetic local time, at least transiently, such as during active times. To include processes generating azimuthal variations in trapped particle fluxes and to quantify their relative importance in radiation belt energization, an improvement in the spatiotemporal resolution of the radiation belt models is required. The objective of this study is to pave the way for a new generation of diffusive radiation belt models capable of retaining drift phase information. Specifically, we highlight a two-dimensional equation for the effects of trapped particle transport on radiation belt intensity. With a theoretical framework that goes beyond the radial diffusion paradigm, the effects of trapped particle bulk motion, as well as diffusion, are quantified in terms of Euler potentials, α,β, quantities akin to the radial and azimuthal directions. This work provides the theoretical foundations underlying the drift phase resolved transport equation for radiation belt dynamics. It also brings forward the concept of azimuthal diffusion as a phase-mixing agent.

The challenge to understand the zoo of particle transport regimes during resonant wave-particle interactions for given survey-mode wave spectra

“…Very interestingly, Lemons (2012) indicated that, nonetheless, a new form of the Fokker-Planck equation may be required in such a regime during early decorrelation times (in the case of pitchangle dynamics only), despite the fact that the quasilinear conditions of the electromagnetic perturbations are satisfied (in their case it was magnetic perturbations only). We leave investigations of this nature for our system to future work, noting that related (analogous) phenomena have been recently described in Lejosne and Albert (2023) for the case of radial transport.…”

“…A grand challenge problem in radiation belt science is the understanding and parameterisation of the relative contributions of quasilinear and nonlinear wave-particle interactions respectively (e.g., see Bortnik et al (2008a); Omura et al (2008); Albert et al (2012); Tao et al (2012b); Allanson et al (2021); Artemyev et al (2021b); Gan et al (2022); Allanson et al (2023)). Informed by spacecraft observations (e.g., Agapitov et al, 2015;Foster et al, 2017;Kurita et al, 2018;Mozer et al, 2018;Shumko et al, 2018;Zhang et al, 2019;Tsai et al, 2022;, many theoretical/analytical and numerical advances have been made in recent years (see discussions and context in e.g., Vainchtein et al (2018); Mourenas et al (2018); Lukin et al (2021); Allanson et al (2022); Albert et al (2022b); Bortnik et al (2022); Frantsuzov et al (2023); Artemyev et al (2022) in the context of local wave-particle interactions; Lejosne (2019); Desai et al (2021); Osmane and Lejosne (2021); Camporeale et al (2022); Lejosne et al (2022); Osmane et al (2023); Lejosne and Albert (2023) considering radial transport; as well as Brizard and Chan (2022) for a combined and self-consistent formalism). Recent works have also pioneered the importance of including inherent 'sub-grid' variability of particle dynamics in radiation belt modelling (Watt et al, 2017;Watt et al, 2019;Ross et al, 2020;Ross et al, 2021;…”

Electron Resonant Interaction With Coherent ULF Waves: Hamiltonian Approach

“…Such motion introduces an additional time‐scale into the system, the bounce frequency $${\omega }_{b}\gg \omega ,\dot{\varphi }$$ (Schulz & Lanzerotti, 1974). Such two‐frequency system can be described using the same approach developed for resonant slow‐fast Hamiltonian systems (Neishtadt & Vasiliev, 2006), but several new effects should be taken into account (see also discussion in Brizard & Chan, 2022; Lejosne & Albert, 2023): Latitudinal distribution of ULF wavefield may be quite different for different $$\mathfrak{m}$$ and wave polarization (e.g., Degeling et al., 2010; Degeling et al., 2011; Sarris et al., 2022), and this may significantly complicate any theoretical description of wave‐particle resonant interactions. Non‐azimuthal ULF wave polarization (Wright & Elsden, 2016) also may bring additional effects on electron nonlinear resonances (see Li et al., 2021, for results of investigation of electron dynamics in toroidal ULF waves). Although the longitudinal invariant, J ‖ = (2 π ) −1 ∮ p ‖ dr ‖ , can be introduced and will be constant in the absence of bounce‐resonances, the new Hamiltonian $${\mathcal{H}}_{\Vert }={J}_{\Vert }(\alpha ,\beta )$$ will not have an analytical form for the dipole field (Roederer, 1970).…”

“…Although we have included the effect of MLT dependence of ULF wave amplitude on electron resonant dynamics in some test particle simulations (see Figures 7 and 8), this effect has not been fully investigated in our study. The importance of the dependence of radial transport provided by electron scattering by ULF waves on MLT has been recently reevaluated and new theoretical framework for this effect has been proposed (see Lejosne & Albert, 2023; Lejosne et al., 2023). This framework is based on adaptation of the Euler potential and Hamiltonian approach, and thus can be merged with our model of the nonlinear electron interaction with ULF waves.…”

“…The principal difference between the electron diffusive resonant scattering by broadband waves and the resonant interaction (linear and nonlinear) with coherent intense waves is that the former requires strong gradients of electron phase space density to provide electron acceleration via the drift resonance (see discussion in Hartinger et al., 2020; Sarris et al., 2021), whereas the latter may create such gradients (e.g., Komar et al., 2017; Li et al., 2021, 2022). Therefore, investigation of electron resonant interactions with coherent intense ULF waves, especially nonlinear aspects of such interactions, may supplement the standard theory of diffusive radial transport and acceleration (see discussion Lejosne & Albert, 2023; Osmane et al., 2023). Our study aims to provide the first attempt to develop a Hamiltonian model of the nonlinear resonant interaction of ULF waves and electrons, by adopting already well developed Hamiltonian models of nonlinear cyclotron resonances (Albert et al., 2013; Artemyev et al., 2018; Shklyar & Matsumoto, 2009).…”