Understanding cold electron impact on parallel-propagating whistler chorus waves via moment-based quasilinear theory

TL;DR

This work addresses how cold electrons influence wave-particle interactions in Earth's magnetosphere by introducing secondary drift-driven instabilities of field-aligned whistler chorus. It develops a moment-based quasilinear theory to quantify energy exchange from the primary parallel-propagating whistler wave to cold electrons via oblique electrostatic whistlers and Bernstein-like modes, and validates the theory against fully kinetic PIC simulations. The study finds persistent secondary instabilities across broad ranges of cold-electron densities and temperatures, with oblique modes typically delivering the dominant damping to the primary wave and heating the cold population, often driving substantial reductions in magnetic energy density by the primary wave (up to around 95% damping in some cases). These results offer a computationally efficient framework for predicting energy partitioning in the inner magnetosphere and provide potential explanations for why certain high-amplitude wave modes are not observed simultaneously, with implications for radiation belt dynamics and magnetospheric monitoring.

Abstract

Earth's magnetosphere hosts a wide range of collisionless particle populations that interact through various wave-particle processes. Among these, cold electrons, with energies below 100eV, often dominate the plasma density but remain poorly characterized due to measurement challenges such as spacecraft charging and photoelectron contamination. Understanding the contribution of these cold populations to wave-particle interaction is of significant interest. Recent kinetic simulations identified a secondary drift-driven instability in which parallel-propagating whistler-mode chorus waves excite oblique electrostatic whistler waves near the resonance cone and Bernstein-mode turbulence. These secondary modes enable a new channel of energy transfer from the parallel-propagating whistler wave to the cold electrons. In this work, we develop a moment-based quasilinear theory of the secondary instabilities to quantify such energy exchange. Our results show that these secondary instabilities persist for a wide range of parameters and, in many cases, lead to nearly complete damping of the primary wave. Such secondary instability might limit the amplitude of parallel-propagating whistler waves in Earth's magnetosphere and might explain why high-amplitude oblique whistler or electron Bernstein waves are rarely observed simultaneously with high-amplitude field-aligned whistler waves in the inner magnetosphere.

Figures (11)

• Figure 1: Primary wave magnetic energy density $|B_{W}|^2$ and the cold electron average bulk velocity $\vec{U}_{c}(t) \coloneqq \int \mathrm{d}^3 v \int \mathrm{d}^3 x \vec{v} f_{c}(\vec{x}, \vec{v}, t)$ from the 2D3V PIC simulation. The primary wave magnetic energy density is proportional to the perpendicular cold electron drift until the onset of electrostatic turbulence, which transfers energy to cold electrons.
• Figure 2: Frequency and growth rate for the PIC simulation parameters, shown for (a) oblique and (b) perpendicular modes. The oblique whistler has a frequency close to the primary field-aligned whistler, $\omega_{0} \approx 0.5 |\Omega_{ce}|$, near the resonance cone, while the perpendicular modes correspond to short-wavelength perturbations. Subfigures (c) and (d) show the cold electron density spectra in $(k_{\perp}, k_{|})$ from the PIC simulation at $t|\Omega_{ce}| = 850$ and $t|\Omega_{ce}| = 1000$.
• Figure 3: Comparison of PIC and QLT results: (a) cold electron heating and (b) primary whistler wave damping, i.e. $\Delta |B_{W}|^2 = |B_{W}(t)|^2/|B_{W}(t|\Omega_{ce}|=700)|^2$. The moment-based QLT results qualitatively resemble the PIC results, capturing the overall trends accurately; however, QLT slightly underestimates the cold electron heating (resonant interactions) in the perpendicular direction and slightly overestimates the whistler wave damping due to overestimation of non-resonant interactions.
• Figure 4: The (normalized) cold electron equilibrium distribution function in the co-drifting frame $F_{0c}(\vec{v}', \theta_{v}=90^{\circ}, t)$ from the PIC simulation (dotted) and QLT simulations (solid) at times: (a) $t|\Omega_{ce}| = 800$ and (b) $t|\Omega_{ce}| = 1300$. Overall, the QLT and PIC results show good agreement, and the distribution function remains approximately bi-Maxwellian throughout the simulation, validating a fundamental assumption of the moment-based QLT approach.
• Figure 5: Growth rate of the secondary instabilities as a function of cold electron density $n_{c}/n_{e}$ and wavenumber amplitude $|\vec{k}| d_e$. Subfigures (a/b) show results from the full dispersion relation in Eq. \ref{['f-theta-equation']}, while subfigures (c/d) show results from the corresponding approximations in Eq. \ref{['dispersion_relation_ecdi']} and Eq. \ref{['approx-dispersion-relation']}. The approximate solutions show good qualitative agreement with the full dispersion solver. The results in subfigures (a/b) indicate that the growth rate is the largest when $0.3 \lesssim n_{c}/n_{e} \lesssim 0.6$.