Whistler waves play a critical role in the dynamics of the Earth’s magnetosphere since wave-particle interactions with whistlers play a dual role of both accelerating and scattering of energetic electrons. The corresponding diffusion coefficients depend on the global distribution of spectral power and polarization of the waves, making understanding of these properties essential for successful modeling of ring current or radiation belt dynamics (Thorne, 2010). As a consequence, there exists a significant interest in understanding of the basic physical processes affecting evolution of the magnetospheric waves in general and of whistler waves in particular.

Statistical studies show that whistler waves in the Earth’s magnetosphere appear either as “chorus”, i.e., discrete emissions appearing typically in two distinct frequency bands in a region outside the plasmasphere (Burtis and Helliwell, 1969; Tsurutani and Smith, 1974; Li et al., 2013), or “hiss”, i.e., broadband emissions found predominantly in the plasmasphere (Dunckel and Helliwell, 1969; Russell et al., 1969; Thorne et al., 1973; Hartley et al., 2018) and in plasmaspheric plumes (Chan and Holzer, 1976). The chorus waves are associated with local energization of energetic particles (Summers et al., 1998; Meredith et al., 2002; Meredith et al., 2003), as well as electron precipitation in the form of diffuse (Swift, 1981; Ni et al., 2014) and pulsating aurora (Nishimura et al., 2010; Kasahara et al., 2018), or microbursts (Oliven and Gurnett, 1968; Breneman et al., 2017). Chorus waves observed in the Earth’s magnetosphere can reach very large amplitudes, e.g., (Cattell et al., 2008; Wilson et al., 2011; Tyler et al., 2019b; Tyler et al., 2019a). For small magnetic latitudes, lower-band chorus waves are predominantly field-aligned (Hayakawa et al., 1984; Haque et al., 2010; Li et al., 2011; Agapitov et al., 2012; Li et al., 2013; Teng et al., 2019), which is consistent with their excitation via a cyclotron instability of anisotropic electrons with energies in the range of several to tens of keV (Kennel and Petschek, 1966; Gary and Wang, 1996; Gary et al., 2011; Fu et al., 2014). In addition to the field-aligned chorus waves, waves near the resonance cone angle are often present (Santolík et al., 2009; Agapitov et al., 2012; Li et al., 2013; Li et al., 2016; Teng et al., 2019). The oblique waves play an important role in scattering of energetic electrons (Mourenas et al., 2014; Artemyev et al., 2015). In general, the whistler waves become more oblique as they propagate to high latitudes. They may further experience significant radial propagation, which complicates identification of their sources (Chen et al., 2013). Nevertheless, it is likely that the oblique chorus at small magnetic latitudes is generated by a mechanism distinct from that producing field-aligned chorus (Artemyev et al., 2015). Analyses of specific events often find plateau/beam-like features in the parallel electron energy distribution with typical energy from ∼ 100 eV (Li et al., 2016) to several keV (Agapitov et al., 2015). This led to suggestions that oblique modes are generated by a linear instability associated either with temperature anisotropy of ∼ keV electrons plus a plateau/beam distribution at lower energies or with a beam-plasma instability driven by low-energy (≲ 1 keV) beams (Mourenas et al., 2015; Artemyev et al., 2016; Li et al., 2016). An alternative mechanism was proposed by Fu et al. (2017), who suggested that oblique chorus waves are generated by wave-wave nonlinear coupling from the interaction of quasi-field-aligned chorus waves in the lower band and mildly oblique upper-band chorus waves.

Since chorus waves are typically observed in the regions adjacent to the plasmasphere, the density of cold plasma populations in these regions is often significant (here “cold” is loosely defined to mean energies below approximately 100 eV.) In fact, in cases where the total plasma density can be measured by spacecraft (e.g., from the frequency of the upper-hybrid line), it is often found that cold plasma populations dominate the total density. The focus of this paper is on a recently identified nonlinear instability experienced by whistler waves in the presence of cold plasma populations (Roytershteyn and Delzanno, 2021). Specifically, kinetic simulations and theory were used to show that in the presence of cold electron populations, quasi-field-aligned whistler waves of sufficiently large amplitude experience a parametric nonlinear instability. This process leads to damping of quasi-field-aligned waves, heating of the cold plasma, and generation of oblique electrostatic modes, including oblique whistlers waves near the resonance cone. The latter in particular are of significant interest since the instability in question may provide a new mechanism of generating oblique chorus that is complementary to the earlier proposed models. In this paper, we extend the analysis of the instability to the case of a multi-component distribution of low-energy electrons, focusing specifically on the generation of the oblique whistlers. Inclusion of such additional electron populations is motivated by the need to make a closer contact with observations that often show presence of “warm” electron populations with energies in the range of ∼ 100 eV, which can also strongly interact with the oblique chorus waves (e.g., Agapitov et al., 2015; Li et al., 2016; Li et al., 2019). Using theoretical analysis and kinetic particle-in-cell (PIC) simulations, we show that the previously identified instability producing oblique whistler waves is robust in the sense that is remains operational in the presence of such warm electron populations. Further, it may produce observed features of the electron distributions, such as formation of a plateau or a weak beam in the resonant range of energies.

Properties of cold plasma populations, especially of the electrons, are hard to measure because their typical energies are comparable to the spacecraft potential and/or the energy of the photo-electrons emitted by the spacecraft (see, e.g., a review in Delzanno et al., 2021). Photo-electrons, in particular, strongly confound the measurements of ambient cold electrons for spacecraft in sunlight. As a result, the energy distribution of the cold populations remains poorly constrained by observations even if the total plasma density could sometimes be estimated from wave measurements. In the present work, we treat the temperature of the cold populations as an unknown and vary it parametrically.

In what follows, we use the following notations: the background magnetic field B ₀ is along z in a local Cartesian (x, y, z) system. Perpendicular and parallel directions are defined with respect to B ₀. The plasma frequency for species s is defined by ω p , s 2 = 4 π e 2 n s / m s , with n _s the density of species s, e is the absolute value of the electron charge, and m _s is the species mass. Index s = eC, eW, eH, i labels cold (eC), warm (eW) and hot (eH) electrons and ions (i). Here “hot” refers to an anisotropic population with energies in the range of several keV and above. The electron plasma frequency computed with the total electron density n ₀ = n _0eC + n _0eW + n _0eH is denoted by ω _pe and the total electron inertial length is d _e = c/ω _pe with c the speed of light. The cyclotron frequency is Ω _c,s = eB ₀/m _s c. The species thermal velocity is v t , s = T s / m s , where T _s is the temperature (which could be anisotropic). The inertial length is d _s = c/ω _p,s . We will interchangeably use the terms “parent,” “driver,” or “primary” wave to refer to quasi field-aligned (whistler) wave that is the source of energy for the nonlinear interactions considered.

The specific nonlinear mechanisms that are the focus of this investigation originate from relative drifts between cold electrons and ions transverse to the background magnetic field. In principle, it is well known that transverse drifts, when present, may produce instabilities that play an important role in many settings in laboratory and space plasmas (see, e.g., a review of instabilities in the shock foot by Muschietti and Lembège (2017)). In the cases discussed here, the drifts are induced by the oscillating electric field of the parent whistler wave. The key observations are: i) whistlers are often present in the region where density is dominated by cold plasma populations and ii) for large, but realistic amplitude of whistlers (i.e., comparable to the observed values), the drifts can be significant in relation to the thermal speed of the cold electron component.

The kinetic theory to study the instabilities of quasi-parallel whistler waves caused by relative transverse drift was presented by Roytershteyn and Delzanno (2021). The key features of the theory are: 1) the underlying equilibrium corresponds to a time-dependent circularly-polarized flow of cold plasma populations induced by the primary whistler wave; 2) The dispersion relation for a given wavevector k couples several frequencies, separated by harmonics of the driver frequency ω ₀; 3) For typical parameters of magnetospheric plasmas, the dominant instabilities are either quasi-perpendicular, short-wavelength modes related to Electron Cyclotron Drift Instability (Forslund et al., 1970; Forslund et al., 1972), or oblique short-wavelength whistler modes near the resonance cone.

Specifically, the electrostatic dispersion relation, generalized here to account for multiple electron species, reads: k 2 + ∑ s = e C , e W ω p e , s 2 v t , s 2 ∑ n = − ∞ + ∞ 1 + ω 2 𝜐 t , s k z Z ξ s , n Γ n λ s ϕ ̃ k , ω = ω p i 2 2 v t i 2 ∑ p , m = − ∞ + ∞ J p a J m a Z ′ ω + p ω 0 2 k v t i ϕ ̃ k , ω + p − m ω 0 . (1)

Expression (1) is written in the electron co-moving (rotating) frame of reference. Here ω is the frequency, k = | k | is the magnitude of the wavevector k = {0, k _y , k _z }, a = k _y V ₀/ω ₀, Γ n ( λ ) = e − λ I n λ , λ s = k y 2 v t , s 2 / Ω c e 2 , ξ s , n = ( ω − n Ω c e ) / ( 2 v t , s k z ) , Z is the plasma dispersion function, J _m (I _n ) is the (modified) Bessel function of first kind, and ϕ ̃ ( k , ω ) is the Fourier component of the electrostatic potential in that frame, while V ₀ and ω ₀ are the amplitude and frequency of the equilibrium electron flow induced by a primary whistler wave. Eq. 1 is derived under assumption that the spatial variation of the driving quasi-parallel whistler mode can be neglected. This is rigorously justified under condition k _z ≫ k _z0, where k _z0 is characteristic wavenumber of the driving mode. We note that in deriving Eq. 1, we assumed that the cold and warm populations are isotropic. Accounting for anisotropy is straighforward: corresponding expressions describing electron response (in non-rotating frame) are given in many textbooks, e.g., Stix (1992).

As is emphasized above, expression (1) couples the response of the system at a given k and ω with that at the same k but at at different values of ω, Doppler-shifted by the frequency of the driver ω ₀. Thus, relation (1) constitutes a system of equations for ω and sidebands ω ± pω ₀, where p is integer. However, when |ω + pω ₀|≫ kv _ti the ion contribution (terms on the right-hand side of Eq. 1) can be neglected. In this case, system (1) describes the usual electron waves in a hot plasma, expressed however in the rotating frame. Instabilities may appear near intersections of a specific hot-plasma electron wave with the Doppler-shifted ion response, i.e., when |ω + pω ₀|≲ kv _ti . Finally, the transformation between Fourier components of any quantity A in the co-moving frame A ̃ and those in the stationary (ion) frame A ̂ is given by A ̃ ( ω , k ) = ∑ m J m ( a ) A ̂ ( ω + m ω 0 , k ) (Kaw and Lee, 1973). Since observations are made in the stationary frame, this implies that a mode with a single frequency ω in the co-moving frame may be observed as a series of sidebands.

A variety of useful approximations to Eq. 1 could be readily obtained. First, we ignore coupling to the side-bands and focus on perturbations in the vicinity of the resonance cone ω ≈ Ω _ce cos θ to obtain a simplified dispersion relation (c.f. Eq. 7 of Roytershteyn and Delzanno (2021)): k 2 + ∑ s = e C , e W ω p e , s 2 v t , s 2 ∑ n = − ∞ + ∞ 1 + ω 2 𝜐 t , s k z Z ξ e , n Γ n λ = ω p i 2 2 𝜐 t i 2 | J 1 a | 2 Z ′ ω − ω 0 2 k 𝜐 t i (2)

A justification of the assumptions leading to Eq. 2 was discussed by Roytershteyn and Delzanno (2021).

A further insight into the nature of the instability generating oblique whistler modes can be obtained by by considering a configuration where the warm population is absent and using cold plasma approximation for the left-hand side, |Ω _ce /(k _y v _te )|≫ 1 and |Ω _ce /(k _z 𝜐 _te )|≫ 1. The system (1) then reduces to k y 2 1 − ω p e 2 ω 2 − Ω c e 2 + k z 2 1 − ω p e 2 ω 2 − ω p i 2 2 𝜐 t i 2 J − 1 2 a Z ′ ζ = 0 , with ζ = ( ω − ω 0 ) / ( 2 𝜐 t i ) , and a = k _y |V ₀|/ω ₀. Expanding the Z function in the small argument limit, Z ′ ∼ − 2 i π ζ ⁡ exp ( − ζ 2 ) − 2 , and assuming γ/ω _r ≪ 1, where γ = Im ω and ω _r = Reω, yields an analytic expression for the instability growth rate: γ ω r = π 2 m e m i J − 1 2 a 𝜐 t i 2 ω 0 − ω r 2 k 𝜐 t i k y 2 ω r 2 ω r 2 − Ω c e 2 2 + k z 2 ω r 2 − 1 where ω _r is the solution of k y 2 1 − ω p e 2 ω r 2 − Ω c e 2 + k z 2 1 − ω p e 2 ω r 2 + ω p i 2 𝜐 t i 2 J − 1 2 a = 0 . (3)

The nature of the instability now becomes evident. When there is no flow, the terms involving J ₋₁ are zero and there is no coupling between electrons and ions. In that case there is no instability and Eq. 3 describes whistler waves near the resonance cone angle, cos θ _R ≈ ω _r /Ω _ce . The presence of the flow leads to coupling with the ions and to a drift instability when ω _r < ω ₀ which leads to the generation of the oblique whistler waves near the resonance cone.

In order to understand the influence of the low-energy part of the electron energy spectrum on the instability, we turn to the form of the dispersion relation given by Eq. 2, which retains the resonance terms. An example of its solution is presented in Figure 1. Here, we assumed parameters approximately corresponding to simulation case D discussed in Section 3 below: ω _pe /Ω _ce = 4, n _0eW /n ₀ = 0.2, n _0eC /n ₀ = 0.75, T _eC = T _i = 5eV, T _eW = 100eV, ω ₀ = 0.3Ω _ce , V ₀ ≈ 0.8v _teC , and m _i /m _e = 1836. We observe that in the presence of warm and cold populations, there are typically two distinct peaks in the growth rate, one at relatively low values of k and the other at relatively high values of k. The nature of these two peaks could be elucidated by varying the parameters of the background plasma. For example, increasing the temperature of the cold populations by a factor of two strongly affects the peak at higher values of k, but leaves the peak at low values of k virtually unchanged (see curve labeled T _eC × 2 in Figure 1). In contrast, increasing the temperature of the warm population affects both peaks, significantly reducing the growth rate at the low values of k (curve labeled T _eW × 2). Examination of the case where the warm population is absent (curve labeled n _eW = 0) suggests that the peaks are formed due to suppression of the growth rate in the range of wavenumbers where the warm population is resonant with the mode |ξ _eW,0| = 1.

Overall, the results of the linear analysis illustrate strong dependence of the instability properties on the energy spectrum of the electrons in the range below a few hundred eV. Further, it may be anticipated that development of the instability may influence distribution functions in this part of the energy spectrum, leading, for example, to formation of plateau and/or beam features in the distribution due to resonances. This point is illustrated further in the following section using PIC simulations.

We also note that the oblique instability near the whistler resonance cone persists (for a fixed driver amplitude) for relatively large values of electron temperature. For the parameters considered in the example above, that included cases where the cold population was absent and only the warm population with T _eW = 100eV was included, as shown in Figure 1. While the corresponding growth rate becomes lower with increasing temperature of the background populations, this behavior is in contrast to the strong temperature dependence of quasi-perpendicular ECDI-type modes. The latter generally appear when the magnitude of the transverse drift V ₀ is comparable to the thermal speed of the low-energy populations 𝜐 _te . The oblique instability of the whistler type clearly exist in the regime V ₀ < 𝜐 _te . This observation suggests that the oblique chorus generated by the mechanism presented here could often accompany large-amplitude quasi-field aligned chorus waves in the Earth’s magnetosphere.

Quasi-field-aligned whistler waves experience a nonlinear instability due to transverse currents when they propagate through a background plasma of sufficiently low energy. One of the most interesting effects associated with the instability is the generation of oblique electrostatic waves, including highly oblique whistler waves near the resonance cone. These modes are of great interest due to their possible connection with oblique whistler-mode chorus waves in the Earth’s magnetosphere that are routinely observed by spacecraft and are known to play an important role in wave-particle interactions. The results presented in this paper extend earlier analysis of Roytershteyn and Delzanno (2021) to the case where low-energy background consists of both a cold electron population with characteristic energy in the range of a few eV and a warm component with characteristic energy in range of ∼ 100 eV. This is motivated by spacecraft observations in the Earth’s magnetosphere where oblique chorus waves were shown to interact resonantly with the warm electrons.

In the presented analysis the low-energy electrons are modeled by including two (cold + warm) Maxwellian populations. The main conclusions are: 1) the instability producing oblique electrostatic waves is sensitive to the shape of the electron distribution at low energies. This is manifested by the sensitivity of both the wavenumbers corresponding to the maximum growth rate and the growth rate magnitude to the temperatures and the relative density of the two low-energy populations. Typically, two distinct peaks in the growth rate are present for the model considered: a peak at relatively low wavenumbers that is most sensitive to the properties of the warm population and a peak at relatively high wavenumbers that is most sensitive to the properties of the cold population. The sensitivity of these processes to the low-energy electron distribution function reinforces the need for the development of robust techniques for in-situ measurements in this energy range (Delzanno et al., 2021; Maldonado et al., 2023); 2) overall, the instability producing oblique whistler waves near the resonance cone persists even in the cases where the temperature of the cold populations is relatively high. For a reference case with (high) amplitude of the parent field-aligned whistler wave δB/B ₀ ∼ 0.01, the maximum growth rate of the oblique whistler mode was γ/Ω _ce ≈ 4 × 10⁻³ when the background consisted only of the warm component with temperature equal to 100 eV. 3) PIC simulations show that the development of the instability leads to heating of the low-energy electrons and formation of characteristic plateau and beam features in the parallel electron distribution function in the range of energies resonant with the instability. The characteristic energy of the resonant particles changes from 50 eV to 1 keV for the cases considered depending on the parameters of the background plasma and the driver. The plateau/beam features have been previously detected in spacecraft observations of oblique chorus waves. However, they have been previously attributed to external sources and have been proposed to be one of the mechanisms producing the oblique chorus (Li et al., 2016). In the present scenario, the causality link is reversed and the oblique whistler waves generated by the instability of the parent quasi-field-aligned whistler are the reason for formation of the plateau and the beam. Finally, 4) the relative energization of the background populations in the simulations, quantified by the relative change in energy between the cold and the warm populations, have been shown to scale linearly with the ratio of the corresponding peak values of the linear growth rates. We note that formation of the plateau in the parallel electron distribution functions has been proposed by Li et al. (2019) as the mechanism responsible for two-banded structure of the chorus. In that scenario, plateau is formed due to interaction with a slightly oblique primary whistler waves (in our terminology) and is consequently located at higher energies compared to the case discussed here, where the plateau is formed by interaction with highly oblique secondary whistlers.

In the presented analysis, the effect of drift-type instabilities associated with the presence of cold plasma has been to a large degree isolated by imposing geometrical constraints on the simulations. For example, the simulations were mostly 2D, were performed in relatively small domains with periodic boundary conditions, and considered only uniform background magnetic field and plasma conditions. Relaxing these constraints and changing the configuration towards a more realistic one (e.g., a dipolar magnetic field with non-uniform density) will introduce other important nonlinear phenomena into the problem, such as frequency chirping and the formation of rising or falling chorus elements (Omura et al., 2012), nonlinear scattering of whistlers (Ganguli et al., 2010), parametric interaction of whistler waves with electrostatic modes (Boswell and Giles, 1977; Umeda et al., 2017), and several others, e.g., (Artemyev et al., 2016; Demekhov et al., 2017; Agapitov et al., 2018; Vasko et al., 2018). Understanding of the relative significance of the processes discussed in this paper in the presence of other phenomena is an important future direction of this work.