On application of stochastic differential equations for simulation of nonlinear wave–particle resonant interactions

Abstract

Long-term simulations of energetic electron fluxes in many space plasma systems require accounting for two groups of processes with well separated time-scales: a microphysics of electron resonant scattering by electromagnetic waves and a macrophysics of electron adiabatic heating/transport by mesoscale plasma flows. Examples of such systems are Earth's radiation belts and Earth's bow shock, where ion-scale plasma injections and cross-shock electric fields determine a general electron energization, whereas electron scattering by waves relaxes anisotropy of electron distributions and produces small populations of high-energy electrons. The application of stochastic differential equations is a promising approach for including effects of resonant wave–particle interaction into codes tracing electrons in models of large-scale electromagnetic fields. This study proposes and verifies such equations for the system with non-diffusive wave–particle interactions, i.e., the system with nonlinear effects of phase trapping and bunching. We consider electron resonances with intense electrostatic whistler-mode waves often observed in the Earth's radiation belts. We demonstrate that nonlinear resonant effects can be described by stochastic differential equations with the non-Gaussian probability distribution of random variations of electron energies.

On the Role of Whistler‐Mode Waves in Electron Interaction With Dipolarizing Flux Bundles

“…For the proposed mapping technique, there are two possible generalizations allowing us to include electron diffusion by low amplitude whistlers. First, the electron phase space trajectory equations can be supplemented by stochastic differential terms (Lukin, Artemyev, & Petrukovich, 2021; Tao et al., 2008) modeling the random electron pitch‐angle/energy changes due to scattering by waves. Second, the scattering can be added directly into the map (see, e.g., the description of diffusion by the Chirikov map; Chirikov, 1979; Lichtenberg & Lieberman, 1983) and more sophisticated maps (Benkadda et al., 1996; Khazanov et al., 2014)).…”

“…Thus, electron interactions with waves are mostly modeled by solving the diffusion equation, which is mainly applicable in a strong background magnetic field, such as that of the near‐Earth magnetotail (Ni et al., 2016) and inner magnetosphere (see reviews by Li & Hudson, 2019; Shprits et al., 2008; Thorne et al., 2021 and references therein). Merging these two approaches (test particle simulations to follow adiabatic heating and solving the diffusion equation to track the effect of waves on the electron distributions) has been proposed, mostly for applications to the inner magnetosphere (e.g., Elkington et al., 2019; Lukin, Artemyev, & Petrukovich, 2021; Michael et al., 2021), but it has yet to be implemented.…”

On the Incorporation of Nonlinear Resonant Wave‐Particle Interactions Into Radiation Belt Models

“…(2019); Lukin et al. (2021) and numerically calculate F (Δ I x ) distributions for a wide range of B w / B 0 and β parameters.…”

“…Since such phase gain δζ ∼ ωτ ≫ 1 for τ about a fraction of the bounce period, ζ can be taken as a random number with a distribution such that the sin ζ distribution repeats the properties of 𝐴𝐴 Δ𝐼𝐼𝑥𝑥 = 𝐼𝐼 (𝑛𝑛+1) 𝑥𝑥 − 𝐼𝐼 (𝑛𝑛) 𝑥𝑥 distributions. Although the ΔI x distribution can be evaluated numerically (Artemyev et al, 2019;Itin et al, 2000;Lukin et al, 2021), this is computationally very expensive for a realistic multi-parameter system. Moreover, we are mostly interested in describing the phase averaged properties of I x changes and, thus, we can reduce the map Equation 3to…”

Differentiating Between the Leading Processes for Electron Radiation Belt Acceleration

“…In the absence of a modeling framework able to account for 1) the effects of both diffusive and non-diffusive (i.e., rapid, significant and coherent) radial transport, as well as 2) the effects of local acceleration (including nonlinear effects), it is not possible to quantify the importance of local vs. radial acceleration unequivocally. Given current computational advances, time may have come to go beyond a purely diffusion-driven model, towards a more realistic modeling framework (e.g., Artemyev et al, 2021;Lukin et al, 2021;Allanson et al, 2022). That said, improved radiation belt modeling would also require improved knowledge of the characteristics of trapped particle interactions with VLF and ULF waves-via experimental determination of the correlation decay time for instance (e.g., Ukhorskiy and Sitnov, 2013).…”