Linear Vlasov–Maxwell theory is a framework for the description of small-amplitude plasma waves in kinetic plasmas. The starting point for the derivation of the hot-plasma dispersion relation in linear Vlasov–Maxwell theory is the Vlasov equation, ∂ f j ∂ t + v ⋅ ∂ f j ∂ x + q j m j E + 1 c v × B ⋅ ∂ f j ∂ v = 0 , (1) combined with Maxwell’s equations, ∇ ⋅ E = 4 π ϱ , (2) ∇ ⋅ B = 0 , (3) ∇ × E = − 1 c ∂ B ∂ t , (4) and ∇ × B = 4 π c j + 1 c ∂ E ∂ t . (5) In this coupled set of equations, f _j ( x , v , t) is the velocity distribution function of species j, E is the electric field, B is the magnetic field, q _j and m _j are the charge and the mass of a particle of species j, ϱ is the charge density, j is the current density, and c is the speed of light. Self-consistency demands that ϱ = ∑ j q j ∫ f j d 3 v (6) and j = ∑ j q j ∫ v f j d 3 v , (7) showing that Eqs. 1–5 represent a complicated, coupled set of integro-differential equations in six-dimensional phase space and time. Linear Vlasov–Maxwell theory simplifies this set of equations by linearisation so that f j x , v , t = f 0 j v + δ f j x , v , t , (8) E ( x , t) = δ E ( x , t), and B ( x , t) = B ₀ + δ B ( x , t), where the subscript 0 indicates a background quantity and δ indicates a fluctuating quantity that averages to zero over time and space. Moreover, we make the assumption that all fluctuating quantities behave like plane waves, ∝ e ^{i k ⋅ x −iωt} , where k is the wave vector and ω is the wave frequency. As described in the literature (e.g., Stix, 1992), the application of these assumptions and Landau’s procedure for the analytic continuation around poles in the complex plane lead to the dispersion relation in the form det D = 0 , (9) where D = ϵ x x − n z 2 ϵ x y ϵ x z + n x n z ϵ y x ϵ y y − n x 2 − n z 2 ϵ y z ϵ z x + n z n x ϵ z y ϵ z z − n x 2 , (10) is the dispersion tensor, ϵ is the plasma susceptibility tensor, and n = k c/ω. In this convention, the reference frame is chosen so that k _y = 0. The entries of the 3 × 3 matrix D depend on the plasma background properties (q _j , m _j , f _{0j} , and B ₀) and of the wave properties ( k and ω). Numerous numerical tools exist that solve Eq. 9, often assuming closed expressions for f _0j such as Maxwellian or bi-Maxwellian distributions (Roennmark, 1982; Klein et al., 2012; Verscharen and Chandran, 2018).

The standard approach for finding the dispersion relation (corresponding to an initial-value problem) involves the determination of a complex ω that solves Eq. 9 for given plasma background properties at fixed k . In general, these solutions are complex-valued. We define the complex ω that solves Eq. 9 for given background parameters and k as ω k = ω k r + i γ k , (11) where ω _kr = Re (ω _k ) is the real wave frequency and γ _k = Im (ω _k ) is the growth/damping rate at wave vector k . The fluctuation amplitudes of solutions with γ _k < 0 exponentially decrease with time, while the fluctuation amplitudes of solutions with γ _k > 0 exponentially increase with time. Therefore, we refer to solutions with γ _k < 0 as damped waves and to solutions with γ _k > 0 as instabilities. In linear theory, the damping rate γ _k is generally a function of k that possesses a global maximum at fixed plasma background properties. We refer to the maximum growth rate γ _m as the maximum γ _k over all k for a given instability and given plasma background properties.

Due to the kinetic (microphysical) nature of these instabilities, we also find the term micro-instabilities for these solutions in the literature. Peter Gary pioneered the application of linear Vlasov–Maxwell theory to the study of micro-instabilities in space plasmas.

If γ _k ≠ 0, the energy density of the electromagnetic fluctuations changes over time. This process exchanges energy between the electromagnetic field and the plasma particles, either in the form of a particle energy loss (γ _k > 0) or gain (γ _k < 0) in order to conserve the total energy. This fundamental concept helps us understand the evolution of the velocity distribution function under the action of micro-instabilities.

Resonant micro-instabilities are a family of micro-instabilities in which the energy exchange occurs via resonant wave–particle interactions between the unstable waves and the plasma particles.

Quasi-linear theory is a mathematical framework to describe the evolution of f _0j (Vedenov et al., 1961; Drummond and Pines, 1964; Kennel and Engelmann, 1966; Rowlands et al., 1966). It requires that the amplitude of the resonant waves and their damping rates are small (|δf _j |≪ f _0j in Eq. 8, and |γ _m|≪|ω _kr| in Eq. 11 at the unstable k ), so that the timescale of the evolution of f _0j is much greater than the period of the resonant wave 1/ω _kr.

Under the assumptions of quasi-linear theory, the background distribution is gyrotropic; i.e., cylindrically symmetric around B ₀. Therefore, it is helpful to work in cylindrical coordinates in velocity space, so that v is represented by the velocity component v _⊥ perpendicular to B ₀, the velocity component v _∥ parallel to B ₀, and the azimuthal angle ϕ _v . Likewise, we express k with its cylindrical coordinates k _⊥, k _∥, and ϕ _k .

The slow, quasi-linear evolution of f _0j over time due to resonant wave–particle interactions is given by the equation (Stix, 1992) ∂ f 0 j ∂ t = lim V → ∞ ∑ n = − ∞ + ∞ q j 2 8 π 2 m j 2 ∫ 1 v ⊥ V G ̂ v ⊥ δ ω k r − k ∥ v ∥ − n Ω j ψ k j , n 2 G ̂ f 0 j d 3 k , (12) where G ̂ ≡ 1 − k ∥ v ∥ ω k r ∂ ∂ v ⊥ + k ∥ v ⊥ ω k r ∂ ∂ v ∥ , (13) ψ k j , n ≡ 1 2 E k , r e i ϕ k J n + 1 ξ j + E k , l e − i ϕ k J n − 1 ξ j + v ∥ v ⊥ E k z J n ξ j , (14) Ω _j ≡ q _j B ₀/m _j c is the cyclotron frequency ¹ , ξ _j = k _⊥ v _⊥/Ω _j is the argument of the Bessel function J _m of order m, and n is an integer that marks the order of the resonance. We refer to the resonance with n = 0 as the Landau resonance and to all other resonances with n ≠ 0 as cyclotron resonances. The left and right circularly polarised components of the electric field are given by E k , l ≡ E k x + i E k y / 2 and E k , r ≡ E k x − i E k y / 2 , where we use the Fourier transformation of the electric field in the convention E k k , t = ∫ V δ E x , t e − i k ⋅ x d 3 x (15) over the spatial volume V. We define the sense of the polarisation of a given wave mode in terms of E _k,l, E _k,r, and E _kz .

Due to the δ-function in Eq. 12, only particles fulfilling the resonance condition ω k r = k ∥ v ∥ + n Ω j (16) participate in the resonant wave–particle interactions associated with a given n. In the case of Landau-resonant interactions, a resonant particle travels along B ₀ with the parallel phase speed of the resonant wave, v _∥ = ω _kr/k _∥. This resonant particle experiences a constant parallel wave electric field E _∥≡ δ E ⋅ B ₀/B ₀. In the case of cyclotron-resonant interactions, a resonant particle travels along B ₀ at a speed such that the Doppler-shifted wave frequency in the particle’s frame of reference is an integer multiple of the particle’s cyclotron frequency. Such a particle experiences a perpendicular wave electric field E _⊥ ≡ δ E − E _∥ B ₀/B ₀ that does not average to zero over multiple cyclotron periods of the particle. This description captures the fundamental nature of resonant wave–particle interactions. We note, however, that Eq. 12 includes more subtleties such as higher-order and anomalous cyclotron resonances as well as transit-time damping, which lie outside the scope of this review.

Figure 2 illustrates an example for a solution to the linear Vlasov–Maxwell dispersion relation from Eq. 9 and the resonance conditions from quasi-linear theory in Eq. 16. The black curve shows a solution of the Vlasov–Maxwell dispersion relation from Eq. 9 in terms of ω _kr for the fast-magnetosonic/whistler wave as a function of k _∥. The dashed lines describe the resonance conditions from Eq. 16 for n = −1 (blue), n = +1 (red), and n = 0 (green). Their slopes correspond to different values of v _∥. At any intersection between a line representing a resonance condition and the plot of the dispersion relation, Eq. 16 is fulfilled.

Eq. 12 represents a diffusion equation in velocity space. The operator G ̂ dictates the direction of the diffusive flux when the resonance condition is fulfilled and ψ k j , n ≠ 0 . The diffusive flux of resonant particles through velocity space is locally tangent to semicircles in velocity space of the form v ∥ − v ph 2 + v ⊥ 2 = constant , (17) where v _ph ≡ ω _kr/k _∥ is the field-parallel phase speed of the resonant waves. According to Eq. 17, quasi-linear diffusion conserves particle kinetic energy in the reference frame that moves with the velocity v _ph B ₀/B ₀. The description of Eq. 17 leaves us with an ambiguity in the direction of the diffusive flux of resonant particles (clockwise or counter-clockwise in velocity space). This ambiguity is resolved by the requirement that Eq. 12 demands, like other diffusion processes, a diffusive flux from larger values of f _0j to smaller values of f _0j .

Our cylindrical coordinate system is aligned with B ₀, so that waves with v _ph > 0 propagate in the direction of B ₀. The direction of propagation of wave solutions with respect to B ₀ can be reversed mathematically in two ways: either by changing the sign of ω _kr or by changing the sign of k _∥. Although this choice does not affect the physics described by the wave theory, it has an impact on the polarisation and thus the applicable resonance condition. We implicitly assume that ω _kr ≥ 0 throughout this work, so that a reversal of the direction of propagation corresponds to changing the sign of k _∥ in our convention ² .

Figure 3 illustrates the quasi-linear diffusion in velocity space. The black semi-circles represent Eq. 17. Particles at v _∥ = v _ph fulfill the Landau-resonance condition with n = 0 according to Eq. 16. If G ̂ f 0 j > 0 at v _∥ = v _ph, the direction of the diffusive flux of Landau-resonant particles is as indicated by the red arrow marked “Landau”. For Landau-resonant particles, G ̂ f 0 j = ( v ⊥ / v ph ) ( ∂ f 0 j / ∂ v ∥ ) .

Assuming that particles with v _∥ = v _res > v _ph fulfill a cyclotron resonance condition according to Eq. 16 in this example, the diffusive flux of these cyclotron-resonant particles is as indicated by the red arrow marked “cyclotron” as long as G ̂ f 0 j > 0 at v _∥ = v _res.

As the resonant particles diffuse through velocity space, their v _∥ changes. If the particles interact with waves with k _⊥ = 0 on only a single branch of the dispersion relation, then at each value of v _∥ the particles typically resonate with waves at a single value of k _∥. This resonant value of k _∥, which can be regarded as a function of v _∥, corresponds to a unique parallel phase velocity v _ph, and over time the particles trace out a one-dimensional diffusion contour in velocity space that is locally tangent to the semicircles defined by Eq. 17. For dispersive waves, v _ph varies with k _∥, and hence with v _∥ of the resonant particles, and thus the full diffusion contour is not semi-circular, because the centre of the locally tangent semi-circle evolves during the diffusion process.

The quasi-linear evolution according to the concepts outlined so far generally leads to a change of the integrated particle kinetic energy of f _0j (i.e., the second velocity moment of f _0j ). If the average kinetic energy m j ( v ⊥ 2 + v ∥ 2 ) / 2 of the resonant particles decreases in the quasi-linear diffusion process, this energy is transferred into the resonant waves, leading to growth of the wave amplitude and thus instability. If the average energy of the resonant particles increases, this energy is taken from the resonant waves, corresponding to wave damping. Our graphical representation in Figure 3 allows us to evaluate the energy behaviour by comparing the direction of the diffusive flux of resonant particles with semi-circles around the origin (green-dashed in Figure 3). These semi-circles represent isocontours of constant ( v ⊥ 2 + v ∥ 2 ) . If the direction of the diffusive flux locally crosses these semi-circles from larger to smaller ( v ⊥ 2 + v ∥ 2 ) , the process corresponds to an instability. If it crosses these semi-circles in the other direction, the process corresponds to damping. In the specific example shown in Figure 3, both the marked Landau-resonant and the marked cyclotron-resonant particles contribute to an instability of the resonant wave at phase speed v _ph. The question of damping/instability thus ultimately simplifies to an investigation of the relative alignments between the semi-circles in velocity space defined by Eq. 17, the isocontours of f _0j , and the isocontours of ( v ⊥ 2 + v ∥ 2 ) at the speed that fulfills Eq. 16 for resonant wave–particle interactions.

The propagation direction and the polarisation of the waves under consideration have a strong impact on the quasi-linear diffusion process. If the waves are purely parallel-propagating (i.e., k _⊥ = 0), then ξ _j = k _⊥ v _⊥/Ω _j = 0 in Eq. 14. The Bessel functions have the property J m 0 = 1 if  m = 0 , 0 otherwise . (18) This property simplifies Eq. 12 considerably for parallel-propagating waves. According to Eq. 14, parallel-propagating waves only have ψ k j , n ≠ 0 if n = +1, n = −1, or n = 0. If a parallel-propagating wave is purely left-circularly polarised (i.e., if E _k,r = E _kz = 0), only the cyclotron resonance with n = +1 contributes to the sum in Eq. 12. If a parallel-propagating wave is purely right-circularly polarised (i.e., if E _k,l = E _kz = 0), only the cyclotron resonance with n = −1 contributes to the sum in Eq. 12. Lastly, if a parallel-propagating wave is purely longitudinal (i.e., if E _k,r = E _k,l = 0), only the Landau resonance with n = 0 contributes to the sum in Eq. 12. This simplification of the quasi-linear diffusion equations is particularly useful since many instabilities have maximum growth for k _⊥ = 0, in which case they exhibit these pure polarisation properties according to linear Vlasov–Maxwell theory. For example, the parallel-propagating fast-magnetosonic/whistler wave is purely right-circularly polarised. The parallel-propagating Alfvén/ion-cyclotron wave is purely left-circularly polarised, and the parallel-propagating Langmuir wave is purely longitudinal. We note that these definitions only apply when ω _kr > 0 according to our convention. For waves with oblique wave vectors, in general, all n must be considered and the polarisation is typically “mixed” with contributions from non-zero E _k,r, E _k,l, and E _kz . Nevertheless, it is often useful to consider that the Bessel-function contributions J _m (ξ _j ) in Eq. 12 are greater for m = 0 than for other m when ξ _j is moderately small, which is often the case for the majority of the resonant particles.

In addition to its mathematical rigour, the quasi-linear-diffusion framework provides us with a visual aid to understand the physics of resonant micro-instabilities. It can be shown that the description of wave damping and instability in terms of quasi-linear diffusion is consistent with its description in terms of γ _k from solutions of linear Vlasov–Maxwell theory as far as the assumptions of both frameworks are fulfilled (Kennel and Engelmann, 1966; Kennel and Wong, 1967; see also Chandran et al., 2010).

The instabilities discussed in this review occur on a variety of length scales, which are often related to the characteristic plasma scales of the system (Verscharen et al., 2019b). We define the inertial length of species j as d j ≡ c ω p j = v A j Ω j = m j c 2 4 π n j q j 2 , (19) where w p j ≡ 4 π n j q j 2 / m j is the plasma frequency of species j, v A j ≡ B 0 / 4 π n j m j is the Alfvén speed of species j and n _j is the background number density of species j. We define the gyro-radius of species j as ρ j ≡ w ⊥ j Ω j = 2 k B T ⊥ j m j c 2 q j 2 B 0 2 , (20) where w ⊥ j ≡ 2 k B T ⊥ j / m j is the perpendicular thermal speed of species j and k _B is the Boltzmann constant. For electrostatic instabilities, the Debye length λ j ≡ k B T ∥ j 4 π n j q j 2 (21) of species j defines an important spatial reference scale. Lastly, we define the following dimensionless ratios of kinetic to magnetic pressure: β j ≡ 8 π n j k B T j B 0 2 , β ⊥ j ≡ 8 π n j k B T ⊥ j B 0 2 , and β ∥ j ≡ 8 π n j k B T ∥ j B 0 2 , (22) where T _j is the scalar temperature of species j, which we use in the case of isotropic plasmas when T _j = T _⊥j = T _∥j .

The electron whistler anisotropy instability is driven when T _⊥e > T _∥e (Kennel and Petschek, 1966; Scharer and Trivelpiece, 1967; Gary and Karimabadi, 2006; Lazar et al., 2022). It is an instability of the fast-magnetosonic/whistler wave with Ω_p ≪ ω _kr < |Ω_e| and k ≲ 1/d _e at maximum growth. The instability has maximum growth when k _⊥ = 0.

Figure 4 describes the quasi-linear evolution of f _0e under the action of the electron whistler anisotropy instability. The initial electron distribution is elongated (i.e., anisotropic) in the direction perpendicular to the background magnetic field. Electrons with v _∥ > 0 resonate with fast-magnetosonic/whistler waves at parallel phase speed −|v _ph| (i.e., propagating oppositely to the direction of B ₀) through the n = +1 cyclotron resonance according to Eq. 16. Electrons with v _∥ < 0 resonate with waves at parallel phase speed +|v _ph| through the n = −1 cyclotron resonance according to Eq. 16.

The relative alignment between the isocontours of f _0e at the value of v _∥ that fulfills Eq. 16 and the black semi-circles around ±|v _ph| in Figure 4 guarantees that the quasi-linear diffusion is locally directed tangent to the blue arrows. Therefore, v _⊥ of the resonant electrons decreases while their |v _∥| increases. Overall, this process leads to a reduction of ( v ⊥ 2 + v ∥ 2 ) and thus of the energy of the resonant electrons. This energy is transferred into the resonant fast-magnetosonic/whistler waves in both directions of propagation. As in the case of the propagating electron firehose instability presented in Section 3.2, the overall temperature anisotropy of the distribution decreases in this process until the distribution function relaxes to a quasi-linear plateau in the part of velocity space occupied by resonant electrons. If f _0e is asymmetric around v _∥ = 0, the energy in the unstable counter-propagating waves can be imbalanced.

Kennel and Petschek (1966) give a necessary condition for instability of the electron whistler anisotropy instability as T ⊥ e T ∥ e − 1 > 1 | Ω e | ω k r − 1 . (23)

The unstable mode follows the approximate dispersion relation (Gary, 1993) ω k r Ω p ≃ k ∥ 2 d p 2 1 + T ⊥ e T ∥ e − 1 β ∥ e 2 . (24) When T _∥e ∼ T _∥p and β _∥e ∼ 1 as in the solar wind, protons are unlikely to undergo a significant resonant interaction with parallel-propagating fast-magnetosonic/whistler waves. Therefore, this instability does not compete with proton-resonant damping.

The necessary relative alignment between the isocontours of f _0e at the value of v _∥ that fulfills Eq. 16 and the semi-circles around ±|v _ph| as shown in Figure 4 can also be fulfilled in bi-κ electron distributions (Lazar et al., 2011, 2013; Shaaban et al., 2021). Likewise, the instability criteria can also be fulfilled in plasmas consisting of anisotropic core and halo populations (Gary et al., 2012; Lazar et al., 2018a). In these cases, anisotropic halo electrons resonate with parallel-propagating fast-magnetosonic/whistler waves through the same mechanism as the electron core (Lazar et al., 2015). If the core is isotropic, the halo driving competes with cyclotron-resonant core damping.

The electron whistler anisotropy instability is believed to be responsible for the sporadic generation of parallel-propagating fast-magnetosonic/whistler waves that are intermittently observed in the solar wind (Tong et al., 2019a; Jagarlamudi et al., 2020; Vasko et al., 2020). Observations show that the solar wind very rarely exhibits plasma conditions above the instability threshold though (Štverák et al., 2008).

A review of early simulation work of the electron whistler anisotropy instability with quasi-linear context is given by Cuperman (1981). Particle-in-cell simulations reveal that this instability changes its properties in the low-β _∥e regime, in which the wave at maximum growth is predominantly oblique and electrostatic, so that Landau-resonant processes become important (Gary et al., 2011). Kinetic simulations of the electron whistler anisotropy instability agree reasonably well with quasi-linear predictions in terms of the behaviour of the instability at saturation (Kim et al., 2017). In particular, these simulations indicate the occurrence of weakly resonant wave–particle interactions. While these numerical simulations start with bi-Maxwellian electron distributions, more recently, the impact of more realistic electron distribution functions has been explored. For instance, Lazar et al. (2022) perform particle-in-cell simulations with bi-κ electron distributions with different β _∥e. In the explored cases, the presence of suprathermal electrons leads to higher growth rates and oscillation amplitudes than in the bi-Maxwellian case.

The propagating electron firehose instability is driven when T _∥e > T _⊥e (Hollweg and Völk, 1970; Pilipp and Völk, 1971; Li and Habbal, 2000). It corresponds to an instability of left-hand polarised fast-magnetosonic/whistler modes that undergo a significant change in their dispersion relation under the relevant unstable plasma conditions. In the following discussion, we focus on the parallel-propagating case, in which k _⊥ = 0.

When T _⊥e = T _∥e, the fast-magnetosonic/whistler branch of the dispersion relation is right-circularly polarised. However, when T _⊥e/T _∥e is sufficiently small, the fast-magnetosonic/whistler wave becomes left-circularly polarised as |k _∥| increases from small to large values. These left-circularly polarised fast-magnetosonic/whistler waves can interact with electrons when Eq. 16 is satisfied for the n = +1 resonance, and such interactions can drive the wave unstable. When unstable, this mode satisfies Ω_p < ω _kr ≪|Ω_e| and 1/d _p < k _∥ < 1/d _e (Micera et al., 2020a).

Figure 5 describes the quasi-linear evolution of f _0e under the action of the propagating electron firehose instability. The initial electron distribution is elongated (i.e., anisotropic) in the direction parallel to the background magnetic field. Electrons with v _∥ > |v _ph| resonate with waves at + |v _ph| through the n = +1 cyclotron resonance according to Eq. 16. Electrons with v _∥ < −|v _ph| resonate with waves propagating in the anti-parallel direction to B ₀ (i.e., with a phase speed − |v _ph|) through the n = −1 cyclotron resonance according to Eq. 16.

The relative alignment between the isocontours of f _0e at the value of v _∥ that fulfills Eq. 16 and the black semi-circles around ±|v _ph| in Figure 5 guarantees that the quasi-linear diffusion is locally directed tangent to the blue arrows. Therefore, v _⊥ of the resonant electrons increases while their |v _∥| decreases. Overall, this process leads to a reduction of ( v ⊥ 2 + v ∥ 2 ) and thus of the kinetic energy of the resonant electrons. This energy is transferred into the resonant fast-magnetosonic/whistler waves in both directions of propagation. As in the case of the electron whistler anisotropy instability discussed in Section 3.1, the overall temperature anisotropy of the distribution decreases in this process until the distribution function relaxes to a quasi-linear plateau in the part of velocity space occupied by resonant electrons. If f _0e is asymmetric around v _∥ = 0, the energy in the unstable counter-propagating waves can be imbalanced.

In addition to anisotropic core electrons, also suprathermal electron populations such as an anisotropic halo with a bi-κ distribution can drive the propagating electron firehose instability (Lazar et al., 2017; Shaaban et al., 2021).

Since this instability is left-hand polarised, cyclotron-resonant proton damping counteracts the driving by cyclotron-resonant electrons. Due to its high frequency compared to Ω_p, the instability’s growth rate depends only weakly on T _∥e/T _∥p and T _⊥p/T _∥p (Hollweg and Völk, 1970; Gary and Madland, 1985). The non-propagating firehose instability discussed in Section 5.2 often has a lower threshold than the propagating firehose instability under most solar-wind conditions (Paesold and Benz, 1999; Li and Habbal, 2000; Gary and Nishimura, 2003).

One-dimensional, relativistic particle-in-cell simulations of the propagating electron firehose instability underline its possible role as a temperature-isotropisation mechanism in solar-flare plasmas (Paesold and Benz, 1999; Messmer, 2002). Simulations with both anisotropic protons and electrons reveal that the concurrent presence of a proton and electron temperature anisotropy can increase the growth rate of the propagating proton firehose instability compared to plasmas with isotropic electrons (Micera et al., 2020a). Quasi-parallel and exactly parallel electron firehose modes become dominant after the saturation of oblique modes with higher growth rates (see also Section 5.2; Camporeale and Burgess, 2008; Innocenti et al., 2019a, the latter study is conducted within an expanding-box framework).

The electron/ion-acoustic instability is an example of an electrostatic instability driven by an asymmetry in the electron distribution function. It is driven by the Landau-resonance of electrons with the ion-acoustic mode (Fried and Gould, 1961). In a magnetised plasma, it has maximum growth when k _⊥ = 0.

The electron/ion-acoustic instability can be driven if there is a non-zero current parallel to B ₀ in a plasma with a single proton and a single electron component. While the ion-acoustic wave is strongly Landau-damped in plasmas with T _e ≈ T _p, a sufficiently large relative drift between the protons and the electrons leads to an instability. The dispersion relation of the ion-acoustic wave is given by (Gary, 1993; Verscharen et al., 2017) ω k r ≈ k ∥ c s = k ∥ 3 k B T ∥ p + k B T ∥ e m p , (25) where c _s is the ion-acoustic speed. The protons provide the wave inertia, while the proton and electron pressures provide the restoring force. In their wave evolution, protons behave like a one-dimensional adiabatic fluid, while the electrons are isothermal according to Eq. 25. When the instability is only weakly unstable (i.e., small γ _m > 0), it is a long-wavelength electrostatic instability with k ≪ 1/λ _p. With increasing γ _m, the unstable wave-vector space increases to 0 < k ≲ 1/λ _p (Gary, 1993).

The quasi-linear evolution of f _0e under the action of the electron/ion-acoustic instability corresponds to the case shown in Figure 6. If it is driven by the current between a single proton and a single electron component, the red strahl population in Figure 6 can be ignored. Due to the relative drift between the core population and the protons, Landau-resonant electrons diffuse towards smaller v _∥, leading to a reduction in the kinetic energy of the resonant electrons. This process drives the resonant ion-acoustic waves unstable at the expense of the relative drift between protons and electrons.

For the case of a single drifting electron component, Gary (1993) gives an expression for the growth rate of the electron/ion-acoustic instability in a plasma with Maxwellian distribution functions under the assumption that T _e ≫ T _p: γ k = π ω k r 3 2 | k ∥ | 3 m p k B T e k ∥ U c − ω k r w e exp − ω k r / k ∥ − U c 2 w e 2 , (26) where U _c is the bulk velocity of the single electron component in the proton reference frame. Combining Eqs 25 and 26 shows that γ _k > 0 if U _c > v _ph, where v _ph ≈ c _s. As shown in Figure 6, the transition from U _c < v _ph to U _c > v _ph marks the transition from a diffusion that raises v _∥ (the blue arrow would be pointing to the right in this case) of the Landau-resonant electrons to a diffusion that lowers their v _∥ (blue arrow pointing to the left as shown). The growth rate according to Eq. 26 has a strong dependence on the electron temperature.

If the electron beam is very fast or the electrons are cold (i.e., U _c ≳ w _e), the dispersion relation of this unstable mode changes significantly from Eq. 25. In this cold-plasma regime, the instability transitions into the classic Buneman electron/ion two-stream instability (Buneman, 1959). It corresponds to the P = 0 mode in cold-plasma theory (Stix, 1992) with ω _kr ∼ ω _pe and maximum growth at k _∥≃ ω _pe/U _c when U _c ≫ w _e.

In the solar wind, the persistent occurrence of sufficiently strong field-aligned currents to drive the electron/ion-acoustic instability via this mechanism is unlikely. For reference, the most intense current densities in the solar wind at 1 au are typically ∼ 5 n A / m 2 (Podesta and Roytershteyn, 2017), and the corresponding net drift between ions and electrons is very small (Vasko et al., 2022). However, a two-component electron configuration as shown in Figure 6 consisting of a core and strahl population enables the same instability mechanism. In an electron–proton plasma with core and strahl populations, the system is free from parallel currents if n s U s + n c U c = n p U p , (27) which is typically the case in the solar wind (Feldman et al., 1975; Salem et al., 2021). In this configuration, the Landau-resonant interaction between unstable core electrons and ion-acoustic waves leads to a reduction of v _∥ of the resonant electrons. As this corresponds to a reduction in U _c, the current-balance requirement from quasi-neutrality then also leads to a reduction in |U _s| (see also Schroeder et al., 2021). Therefore, the electron/ion-acoustic instability is a candidate for the (indirect) regulation of the strahl heat flux in the solar wind (Gary, 1978). As shown in Figure 6, this instability does not scatter strahl electrons into the halo, although such a behaviour is found in the solar wind (Štverák et al., 2009).

Ion-acoustic waves have been observed in the solar wind (Gurnett and Anderson, 1977; Kurth et al., 1979; Gurnett, 1991; Píša et al., 2021). They often occur in sporadic bursts and at times when T _∥e > T _∥p (Mozer et al., 2021b). Near the Sun, the condition that T _∥e ≫ T _∥p can be satisfied in low-speed solar wind. At small heliocentric distances, the proton temperature remains correlated with the wind speed, but the electron temperature is anti-correlated with the wind speed, most likely due to the initial conditions in the corona (Halekas et al., 2020; Maksimovic et al., 2020). The resulting conditions in slow-speed near-Sun solar wind thus favour the growth of the ion-acoustic wave. Indeed, Parker Solar Probe observes ion-acoustic waves under these conditions (Mozer et al., 2022). The loose correlation between ion-acoustic waves with periods of enhanced electron temperatures suggests that these waves may heat the core electrons. However, since high electron temperature itself favours the growth of the waves, the causality remains unclear.

Near 1 au, other instabilities often have lower thresholds than the electron/ion-acoustic instability though (Gary, 1978; Lemons et al., 1979). A direct stability analysis of measured electron distributions from Helios identifies a case that is unstable against the electron/ion-acoustic instability (Dum et al., 1980). Strong ion-acoustic wave bursts are also found near magnetic switchbacks (Mozer et al., 2021a). The exact generation mechanism of these waves is unclear, as an ion/ion-acoustic instability is also a possible candidate for the generation of these waves (Mozer et al., 2020; Graham et al., 2021).

The current-driven electron/ion-acoustic instability is studied numerically in the context of laser-heated laboratory plasmas (Detering et al., 2005).

The kinetic Alfvén heat-flux instability is driven by the same instability mechanism as the electron/ion-acoustic instability shown in Figure 6. Also in this instability, Landau-resonant core electrons diffuse towards smaller v _∥. The unstable wave mode in this case is the highly-oblique kinetic Alfvén wave, which is an electromagnetic plasma mode. It corresponds to the small-wavelength extension of the Alfvén wave in highly oblique propagation (i.e., k _⊥ ρ _p ≳ 1 and k _⊥≫ k _∥). Its dispersion relation is given by (Howes et al., 2006) ω k r ≈ k ∥ v A k ⊥ ρ p β p + 2 1 + T e / T p . (28)

In a Maxwellian electron–proton plasma, kinetic Alfvén waves undergo electron-Landau damping. With the introduction of a core–strahl configuration with sufficiently large U _c > 0, the plasma can achieve ∂f _0e/∂v _∥ > 0 at v _∥ = v _ph, in which case the kinetic Alfvén wave is driven unstable. The quasi-neutrality condition in Eq. 27 enforces a simultaneous reduction of |U _s| when the instability reduces U _c like in the case of the electron/ion-acoustic instability. The kinetic Alfvén heat-flux instability has maximum growth at k _⊥≲ 1/ρ _p (Gary et al., 1975b).

Under typical solar-wind conditions with β _j ∼ 1, the kinetic Alfvén heat-flux instability has a significantly greater threshold than the parallel whistler heat-flux instability (Section 4.6; Gary et al., 1975a; Gary et al., 1975b). In addition, this instability does not explain the observed scattering of strahl electrons into the halo population (Štverák et al., 2009) since only core electrons diffuse in velocity space towards smaller v _∥ (Verscharen et al., 2019a).

Kinetic Alfvén waves are often observed in the solar wind (Leamon et al., 1998; Bale et al., 2005; Chen et al., 2010; Salem et al., 2012; Šafránková et al., 2019; Roberts et al., 2020). However, their presence is generally neither attributed to ion-driven nor to electron-driven instabilities. Instead, they are interpreted as the small-wavelength extension of the Alfvénic cascade of solar-wind turbulence (Howes et al., 2006; Schekochihin et al., 2009).

The kinetic Alfvén heat-flux instability has not been investigated extensively via numerical simulations. In contrast, kinetic Alfvén waves more generally have been the subject of intense study. For example, Gary and Nishimura (2004) compare linear theory and particle-in-cell simulations (albeit employing a low proton-to-electron mass ratio) of kinetic Alfvén waves to quantify the associated electron heating. Particle-in-cell and gyrokinetic simulations of kinetic Alfvén turbulence in the solar wind are used to investigate ion and electron heating (Howes et al., 2008, 2011; Parashar et al., 2015; Hughes et al., 2017; Grošelj et al., 2018; Cerri et al., 2019). Furthermore, kinetic Alfvén waves are routinely generated in particle-in-cell simulations of magnetic reconnection in conjunction with Hall physics in the diffusion region (Rogers et al., 2001; Shay et al., 2011).

The Langmuir instability and the electron-beam instability are two examples of beam-driven electrostatic instabilities (also called “high-frequency electron/electron instabilities”; Gary, 1985b). They are both driven by Landau-resonant electrons, which requires that ∂f _0e/∂v _∥ > 0 at v _∥ = v _ph when v _ph > 0. This configuration corresponds to a bump-on-tail distribution. Both instabilities have maximum growth when k _⊥ = 0.

In the relevant high-frequency range and assuming only a small modification to the real-part of the dispersion relation from any electron beam components, there are two solutions to the dispersion relation that become unstable. The Langmuir wave follows the dispersion relation ω k r ≈ ω p e 2 + 3 2 k ∥ 2 w c 2 . (29) If an electron-beam (strahl) component with bulk velocity U _s is present, the plasma also supports an electron-beam mode with (Gary, 1978) ω k r ≈ k ∥ U s . (30)

The criterion for distinguishing which of the two modes becomes unstable when ∂f _0e/∂v _∥ > 0 at v _∥ = v _ph depends on the speed, temperature, and relative density of the beam (O’Neil and Malmberg, 1968). Under unstable conditions, the Langmuir wave is the relevant mode if (Gary, 1993) 2 U s w s 3 n s n c < 1 , (31) and the electron-beam mode is the relevant mode if 2 U s w s 3 n s n c ≳ 1 . (32)

Both the Langmuir and the electron-beam instability have high frequencies (compared to Ω_p) and wave numbers k _∥≪ 1/λ _e, often even k _∥≲ 1/λ _p, at maximum growth (Gary, 1993).

The quasi-linear evolution of f _0e under the action of the Langmuir instability or of the electron-beam instability is shown in Figure 7. Landau-resonant strahl electrons diffuse towards smaller v _∥ in this configuration, while the core bulk velocity increases in order to guarantee current balance according to Eq. 27. This process leads to a flattening of the distribution function around v _∥ = v _ph. If the dispersion relation in Eq. 30 were fulfilled exactly and the strahl distribution were symmetric around v _∥ = U _s, then ∂f _0e/∂v _∥ = 0 for Landau-resonant electrons at v _∥ = v _ph. In this case, the instability would not act. This illustrates the importance of subtle modifications to the dispersion relation due to the beam component itself so that U _s is slightly greater than v _ph in order to create an unstable configuration.

Under certain parameter combinations, especially at large beam speeds, the Langmuir mode and the electron-beam mode couple in their dispersion relation (Gary, 1993). Since the electron-beam mode’s phase speed is approximately equal to the parallel speed of the resonant electrons, it typically grows over a wide range of frequencies, which is important in the Earth’s foreshock, where bump-on-tail configurations occur frequently (Fuselier et al., 1985). However, this behaviour changes when the modes couple because then the dispersion relation becomes more complex.

In the cold-plasma limit (i.e., for very fast and cold beams with U _s ≫ w _s and U _s ≫ w _c), the electron-beam instability corresponds to the classical electron two-stream instability. In this limit, the instability has ω _kr ≃ ω _pe and a maximum growth rate of (Gary, 1993) γ m ≃ 3 2 n s 2 n c 1 / 3 ω pc . (33)

In low-β _c conditions, the Langmuir and electron-beam instabilities can have lower thresholds than other beam-driven instabilities. However, they both require a bump-on-tail configuration in order to be driven. The electron strahl does not generally generate such a non-monotonic v _∥-dependence of f _0e. However, observations in the Earth’s foreshock find evidence for the Langmuir instability when tenuous and fast electron beams are present, and for the electron-beam instability when denser and slower electron beams are present (Etcheto and Faucheux, 1984; Lacombe et al., 1985; Onsager and Holzworth, 1990).

Langmuir waves are often observed in the solar wind at different heliospheric distances (Kennel et al., 1980). They frequently occur at the same time as narrow-band electromagnetic waves identified as whistler waves, potentially suggesting a common origin (Jagarlamudi et al., 2021).

Two-dimensional, electromagnetic particle-in-cell simulations of a core-strahl electron configuration reveal that the electrostatic electron-beam instability also develops fluctuations in the perpendicular electric-field component, which scatter strahl electrons towards greater v _⊥ (Gary and Saito, 2007). A Fokker–Planck model of wave–particle interactions between an electron beam and the Langmuir instability suggests a similar process (Pavan et al., 2013).

In low-β _j plasmas, highly oblique electrostatic ion-cyclotron waves exist (D’Angelo and Motley, 1962; Stix, 1992). These modes occur in bands between the harmonics of the proton gyrofrequency (Gary, 1993): m Ω p < ω k r < m + 1 Ω p , (34) where m ≥ 1 is the integer harmonic order of the electrostatic ion-cyclotron wave.

In a plasma consisting of a single electron and a single proton population, electrostatic ion-cyclotron waves can become unstable if there is a sufficiently large current given by a difference in the bulk speeds of the electrons and the protons parallel to B ₀ (Drummond and Rosenbluth, 1962; Kindel and Kennel, 1971). Strong Landau-resonant interactions between the electrons and the harmonics of the ion-cyclotron wave are responsible for the driving of this instability. The wave number at maximum growth typically fulfills k _⊥∼ 1/ρ _p.

The quasi-linear evolution of f _0e is similar to the process described in Figure 7, but where the strahl population is the only electron population. Since electrostatic ion-cyclotron waves are highly dispersive, v _ph depends strongly on v _∥ of the resonant electrons.

The ion-cyclotron wave with m = 1 has the lowest threshold. With increasing U _c, harmonics with higher m become unstable as well. Cyclotron-resonant interactions with protons compete with the Landau-resonant electron driving. Therefore, the instability threshold depends strongly on T _p (Gary, 1993). For T _e/T _p ≲ 10, the electron/ion-cyclotron instability has a lower threshold than the current-driven electron/ion-acoustic instability. At T _e/T _p ≳ 50, the unstable ion-cyclotron branch of the dispersion relation merges with the unstable ion-acoustic branch, so that this instability loses its identity.

As in the case of other current-driven instabilities, we expect that the introduction of a core-strahl configuration can also create the conditions necessary for the electron/ion-cyclotron instability in a plasma fulfilling Eq. 27. In this case, the instability mechanism requires ∂f _0e/∂v _∥ > 0 at v _∥ = v _ph and would be the same as shown in Figure 7. However, we are not aware of detailed studies of the conditions necessary for a core-strahl electron system to drive electrostatic ion-cyclotron waves unstable.

In order to overcome cyclotron-resonant proton damping, this instability is only relevant in plasma environments with β _∥p ≪ 1. Therefore, the electron/ion-cyclotron instability is mostly thought to occur within low-β _j environments such as the auroral ionosphere (Ashour-Abdalla and Thorne, 1978; Bergmann, 1984). Driven by electrons and damped by protons, the unstable ion-cyclotron waves lead to efficient proton heating in this environment (Okuda and Ashour-Abdalla, 1983; Ashour-Abdalla and Okuda, 1984).

A Fokker–Planck model of the current-driven electron/ion-cyclotron instability shows the presence of quasi-linear cyclotron-resonant diffusion effects on the proton distribution function, leading to its flattening in the resonance region (Harvey, 1975). These simulation results also confirm that the conditions for this instability to be excited are not commonly satisfied in typical solar-wind conditions.

In a plasma consisting of protons and two electron populations, an additional electrostatic mode with properties similar to the ion-acoustic wave from Section 4.1 emerges (Watanabe and Taniuti, 1977). This mode is called the electron-acoustic wave. In order for this mode to have a small damping rate, it is required that the two electron components have comparable densities but that one of the electron components is much hotter than the other (Gary, 1987). We identify the hotter component with a possible halo population in the solar wind.

If n _h ∼ n _c, T _h ≫ T _c, and U _h = U _c = 0, the electron-acoustic wave has the dispersion relation (Watanabe and Taniuti, 1977; Gary, 1987) ω k r ≈ ω p c 1 + 3 k ∥ 2 λ c 2 1 + 1 / k ∥ 2 λ h 2 . (35) The cold electron component provides the wave inertia, while the high mobility of the hot electrons provides the restoring force of the electron-acoustic wave. At long wavelengths, v _ph is approximately proportional to T h .

With the introduction of a sufficient relative drift speed between the core and halo populations, the electron-acoustic mode becomes unstable through the Landau-resonant interaction between halo electrons and the electron-acoustic mode. In this case, the mode still fulfills Eq. 35, but in the frame of the core electrons, which provide the wave inertia. The wave number of the electron/electron-acoustic instability at maximum growth typically fulfills 1/λ _h < k _∥ < 1/λ _c. It has maximum growth when k _⊥ = 0.

If the electron-acoustic mode is moderately unstable (i.e., |γ _m|≪ ω _kr), its growth rate is given by (Gary, 1993) γ k ≃ ω pc 2 | k ∥ | 3 λ h 2 π k ∥ U h ′ − ω k r ′ w h exp − U h ′ − ω k r ′ / k ∥ 2 w h 2 (36) where U h ′ and ω k r ′ are the bulk velocity of the halo population and the wave frequency in the reference frame of the core population: U h ′ = U h − U c and ω k r ′ = ω k r − k ∥ U c . Eq. 36 shows that, as for all Landau-resonant instabilities, the electron/electron-acoustic instability requires U _h > v _ph when v _ph > 0 so that ∂f _0e/∂v _∥ > 0 at the value of v _∥ of the resonant electrons.

The quasi-linear evolution of f _0e under the action of the electron/electron-acoustic instability is shown in Figure 8. Landau-resonant halo electrons diffuse towards smaller v _∥ in this configuration.

The electron/ion-acoustic instability from Section 4.1 has a lower threshold than the electron/electron-acoustic instability unless T _h ≫ T _c (Gary, 1987). In addition, a substantial n _h ∼ n _c is required. Both conditions are not generally fulfilled in the solar wind. However, cusp hiss fluctuations in the magnetospheric context are attributed to electron-acoustic fluctuations (Thomsen et al., 1983; Marsch, 1985). Electron-acoustic waves, typically with nonlinearly steepened wave forms, are reported in the Earth’s inner magnetosphere, where the density of hot electrons can be greater than the density of cold electrons during hot plasma injections from the magnetotail (Vasko et al., 2017; Dillard et al., 2018). In the near-Sun environment, the electron/electron-acoustic instability may be relevant for the regulation of the solar-wind heat flux (Sun et al., 2021).

Fully electromagnetic particle-in-cell simulations of the electron/electron-acoustic instability show that heating of the cold core component quenches the instability due to a breakdown of its requirement that T _h ≫ T _c (Lin et al., 1985). This appears to be the dominant nonlinear saturation mechanism for the electron/electron-acoustic instability (Gary, 1993).

The whistler heat-flux instability is a cyclotron-resonant instability of the electromagnetic fast-magnetosonic/whistler wave (Gary et al., 1975a, Gary et al., 1975b; Schwartz, 1980). It has maximum growth at k _⊥ = 0. Quasi-parallel fast-magnetosonic/whistler waves follow the approximate cold-plasma dispersion relation (Verscharen et al., 2019b) ω k r Ω p ≈ k ∥ 2 d p 2 2 1 + 4 k ∥ 2 d p 2 + 1 , (37) when ω _kr ≪|Ω_e|. The cyclotron-resonant electrons interact with the right-hand circularly polarised electric field of the fast-magnetosonic/whistler wave through the n = −1 resonance. According to Eq. 16, only electrons with v _∥ < 0 can fulfill the resonance condition when v _ph > 0. In a plasma consisting of protons and two isotropic electron populations, the driving electron population must have a bulk velocity greater than v _ph. Therefore, this instability is typically driven by a hot beam population like the halo. At maximum growth, Ω_p ≪ ω _kr ≪|Ω_e| and 1/d _p ≪ k _∥ ≲ 1/d _e.

Figure 9 shows the quasi-linear evolution of f _0e in the whistler heat-flux instability. In this case, 0 < v _ph < U _h, and halo electrons with v _∥ < 0 resonate with the whistler wave. This setup guarantees that the flux of diffusing electrons in velocity space is directed as shown by the green arrow in Figure 9 as particles diffuse towards smaller values of f _0e (see also Shaaban et al., 2019c). The resonant electrons diffuse towards smaller v _⊥ but larger |v _∥|, while their ( v ⊥ 2 + v ∥ 2 ) decreases. This decreasing kinetic energy is transferred into the resonant fast-magnetosonic/whistler waves. Increasing the halo density and temperature brings a larger number of electrons into resonance with the wave and thus leads to an increase in the growth rate. This instability is not a good candidate to explain the regulation of the strahl heat flux since the strahl does not provide a sufficient number of electrons at v _∥ < 0 when U _s > 0 (Verscharen et al., 2019a).

With increasing halo speed, ω _kr decreases compared to the traditional dispersion relation in Eq. 37. Figure 7 of Gary (1985a) compares the thresholds of the whistler heat-flux instability with the thresholds of the electron/ion-acoustic instability (see Section 4.1) and the electron-beam instability (see Section 4.3). Under typical solar-wind conditions, the whistler heat-flux instability has the lowest threshold of these instabilities. Only at large n _h/n _c, large T _h/T _c, and small β _j , the other instabilities can compete with the whistler heat-flux instability.

The instability mechanism of the whistler heat-flux instability is similar to the instability mechanism of the electron whistler anisotropy instability presented in Section 3.1. The introduction of a halo anisotropy modifies the shape of f _0e in the velocity space occupied by resonant electrons. Consequently, the threshold of the whistler heat-flux instability decreases with increasing halo anisotropy T _⊥h/T _∥h > 0 (in general, the instability is sensitive to the shape of the halo distribution; see Abraham-Shrauner and Feldman, 1977; Dum et al., 1980).

The cyclotron-resonant halo driving competes with the cyclotron-resonant core damping of the fast-magnetosonic/whistler wave in this instability. Introducing a core anisotropy with T _⊥c/T _∥c > 0 lowers the cyclotron-resonant core damping though and thus raises the growth rate. Treatments of the whistler heat-flux instability in bi-Maxwellian and κ-distributed plasmas confirm this picture (Shaaban et al., 2018; Sarfraz and Yoon, 2020). Quasi-linear models of the whistler heat-flux and electron whistler anisotropy instability driven by a combination of heat flux and anisotropy are also available (Shaaban et al., 2019b; Vasko et al., 2020).

The thresholds of the whistler heat-flux instability have often been compared with the observed heat flux in the solar wind (Gary and Feldman, 1977; Gary et al., 1999a; Tong et al., 2019b). This instability is likely to operate near 1 au (Tong et al., 2019b), but it appears unlikely to prove important near the Sun, where the halo is an almost negligible component of the distribution (Halekas et al., 2020, 2021b; Abraham et al., 2022). Conditions relevant for the driving of this instability also occur near interplanetary shocks and in the Earth’s foreshock (Wilson et al., 2009, 2013; Page et al., 2021). Quasi-parallel whistler waves are observed in solar-wind intervals with strong heat flux, supporting the links between core-halo heat flux and the whistler heat-flux instability (Lacombe et al., 2014; Stansby et al., 2016; Tong et al., 2019a; Jagarlamudi et al., 2020). The contributions of core and halo anisotropies to the growth of the instability are confirmed observationally through the presence of a clear positive correlation between the occurrence of whistler waves and core/halo anisotropies (Jagarlamudi et al., 2020). Like the relative halo density, the core and halo anisotropies decrease with decreasing heliocentric distance, which is consistent with the observed lack of whistler waves at heliocentric distances ≲ 0.13 au (Cattell et al., 2022).

Particle-in-cell simulations of the whistler heat-flux instability reveal that beam scattering and core heating occur simultaneously due to cyclotron-resonant wave–particle interactions (López et al., 2019b). The quasi-linear pitch-angle diffusion of the resonant electrons as shown in Figure 9 saturates quickly so that a significant heat-flux regulation is not expected. Numerical simulations confirm this expectation under typical solar-wind conditions (Kuzichev et al., 2019). Therefore, despite a significant amount of research into the action of the whistler heat-flux instability, its contribution to the observed heat-flux regulation in the solar wind is still not fully understood (for observational constraints, see Feldman et al., 1976a,b).

When the fast-magnetosonic/whistler wave is not exactly parallel in propagation, it develops a non-zero left-circularly polarised component in its electric-field fluctuations. Unlike in the case of the whistler heat-flux instability presented in Section 4.6, this allows electrons with v _∥ > 0 to resonate according to Eq. 16 with fast-magnetosonic/whistler waves with v _ph > 0 via the n = +1 cyclotron resonance (Verscharen et al., 2019a), provided v _∥ is sufficiently large. This enables strahl electrons with v _∥ > 0 to drive the oblique whistler wave unstable (Vasko et al., 2019). At maximum growth, Ω_p ≪ ω _kr ≲ |Ω_e| and 1/d _p ≪ k _∥ ≲ 1/d _e.

The quasi-linear evolution of f _0e in the oblique fast-magnetosonic/whistler instability is shown in Figure 10. If the strahl distribution is isotropic, the resonant electrons diffuse towards larger v _⊥ and smaller v _∥ given that 0 < v _ph < U _s. In this case, ( v ⊥ 2 + v ∥ 2 ) of the resonant strahl electrons decreases, corresponding to a loss of kinetic energy and thus a transfer of energy into the growing fast-magnetosonic/whistler waves.

Due to its obliqueness, the fast-magnetosonic/whistler wave also possesses fluctuations in the electric-field component E _∥ parallel to B ₀. Therefore, the cyclotron-resonant driving by resonant strahl electrons competes not just with cyclotron-resonant damping by core electrons via n = −1 but also with Landau-resonant damping by core electrons via n = 0. These competitions between driving and damping define two regimes of the oblique fast-magnetosonic/whistler instability: a high-β _∥c regime and a low-β _∥c regime.

In the high-β _∥c regime (i.e., when w _∥c ≳ v _Ae/2), the competition between cyclotron-resonant strahl driving and Landau-resonant core damping determines the instability threshold. The oblique fast-magnetosonic/whistler wave is unstable in this case if (Verscharen et al., 2019a) U s ≳ 2 n c n s T ∥ s T ∥ c v A e 2 w ∥ c 2 1 + cos ⁡ θ 1 − cos ⁡ θ cos ⁡ θ 1 / 4 , (38) where θ is the angle between the wave vector and the background magnetic field.

In the low-β _∥c regime (i.e., when w _∥c ≲ v _Ae/2), the competition between cyclotron-resonant strahl driving, cyclotron-resonant core damping, and Landau-resonant core damping determines the instability threshold. The oblique fast-magnetosonic/whistler wave is unstable in this case if U s ≳ 3 w ∥ c . (39)

In the low-β _∥c regime, ω _kr ≈ |Ω_e|/2, k _∥ ≈ |Ω_e|/2w _∥c, and θ = 60° at maximum growth.

Eqs 38 and 39 have been tested successfully against numerical solutions to the linear Vlasov–Maxwell dispersion relation for typical solar-wind parameters (Verscharen et al., 2019a).

Statistical comparisons of instability thresholds with electron measurements in the solar wind from Wind (Verscharen et al., 2019a) and from Parker Solar Probe (Halekas et al., 2021b) show that the strahl parameters are limited by Eq. 38 to the stable parameter space. However, a recent analysis of Parker Solar Probe and Helios data suggests that the strahl very rarely reaches the threshold in the inner heliosphere, so that the importance of this instability is now put into question (Jeong et al., 2022a). This finding is consistent with the observed lack of fast-magnetosonic/whistler waves in Parker Solar Probe data at heliocentric distances ≲ 0.13 au (Cattell et al., 2022). Moreover, the majority of the fast-magnetosonic/whistler waves observed farther away from the Sun have a quasi-parallel direction of propagation with respect to the magnetic field (Kretzschmar et al., 2021). Therefore, other mechanisms than the self-induced scattering of strahl electrons by the oblique fast-magnetosonic/whistler instability may thus be needed to explain the observed scattering of strahl electrons into the halo population (e.g., the interaction with pre-existing fast-magnetosonic/whistler waves; Vocks et al., 2005; Vocks and Mann, 2009; Pierrard et al., 2011; Jagarlamudi et al., 2021; Cattell and Vo, 2021; Cattell et al., 2021). Moreover, Bernstein and ion-acoustic waves become more dominant than fast-magnetosonic/whistler waves in the very inner heliosphere, suggesting a transition into an electrostatic regime which could affect the electron distributions near the Sun (Mozer et al., 2021b; Malaspina et al., 2021). The observed Bernstein and ion-acoustic waves occur in very specific types of solar wind: Bernstein waves occur predominantly in quiet wind with magnetic field close to the ideal Parker spiral, while ion-acoustic waves occur predominantly in slow solar wind. This correlation underlines the importance of the careful investigation of the electron distribution’s evolution as a function of wind parameters.

Numerical evaluations of the quasi-linear diffusion equation confirm that the oblique fast-magnetosonic/whistler instability scatters resonant strahl electrons as shown in Figure 10 (Jeong et al., 2020; Sun et al., 2021). Numerical particle-in-cell simulations also confirm this evolution under conditions consistent with observed solar-wind parameters (Micera et al., 2020b).

At very large angles of propagation ( k ∥ 2 / k 2 < m e / m p ) , the fast-magnetosonic/whistler wave solution of the Vlasov–Maxwell dispersion relation is also known as the lower-hybrid mode. Its wave frequency asymptotes to the lower-hybrid frequency (Verdon et al., 2009) ω k r ≈ ω LH ≡ ω pp 1 + ω pe 2 Ω e 2 (40) in the low-β _j limit. Landau-resonant core damping strongly suppresses the lower-hybrid mode (Lakhina, 1979), so that it becomes most relevant when β _c ≪ 1, in which case the mode becomes largely electrostatic (Marsch and Chang, 1983).

In this low-β _c case, the lower-hybrid wave can be driven unstable by strahl electrons via the n = +1 cyclotron resonance like the oblique fast-magnetosonic/whistler instability (see Section 4.7; Omelchenko et al., 1994; Krafft and Volokitin, 2003; Shevchenko and Galinsky, 2010). At maximum growth, the lower-hybrid fan instability has ω _kr ≈ ω _LH and k ∥ ≈ ω LH + | Ω e | / U s .

The instability mechanism for the lower-hybrid fan instability is the same as the mechanism driving the oblique fast-magnetosonic/whistler instability shown in Figure 10. Since the cyclotron-resonant diffusion leads to a fan-like widening of the strahl component in the perpendicular direction, this instability received the name lower-hybrid fan instability.

Numerical Hamiltonian simulations of the lower-hybrid fan instability confirm the importance of both cyclotron-resonant and Landau-resonant interactions between electrons and waves (Krafft et al., 2005; Krafft and Volokitin, 2006). In its nonlinear stage, the lower-hybrid fan instability is prone to strong wave trapping that is not captured by quasi-linear theory.

Up until this point, we have discussed the canonical examples of electron instabilities driven by anisotropy, drifts, or beams in the electron distribution function. However, other deviations from thermodynamic equilibrium are also able to drive instabilities if the deformation of the distribution is sufficiently strong in the range of resonant velocities (Dum et al., 1980).

One example of such a deformation of f _0e is the sunward deficit in the electron distribution (Halekas et al., 2020; Abraham et al., 2022). As expected in exospheric models of the solar wind (Lemaire and Scherer, 1971; Pierrard and Lemaire, 1996; Maksimovic et al., 2001), the interplanetary potential reflects electrons that leave the Sun with a kinetic energy below a cut-off value that depends on the potential. These reflected electrons return towards the Sun and form part of the sunward half of the electron distribution in the inner heliosphere. Electrons above the cut-off energy do not return. If collisions and other scattering mechanisms are neglected, a sharp cut-off is thus expected on the sunward side of the electron distribution, marking the separation between the reflected and the (missing) electrons that have escaped the potential. This cut-off has been observed in the form of a sunward deficit in the electron distribution function in data from Parker Solar Probe (Berčič et al., 2021a; Halekas et al., 2021a).

The sunward deficit can create conditions in which electrons near the cut-off resonantly interact with parallel fast-magnetosonic/whistler waves such that they lose their kinetic energy and drive the wave unstable (Berčič et al., 2021b). This interaction leads to the electron-deficit whistler instability. The instability at maximum growth has Ω_p ≪ ω _kr ≪|Ω_e| and k _⊥ = 0. The wave number at maximum growth depends on the parallel velocity of the deficit in velocity space according to Eq. 16. The properties of this instability are still under study, but Berčič et al. (2021b) suggest that the wave number at maximum growth is typically ∼ 1 / d e .

The quasi-linear evolution of f _0e in the electron-deficit whistler instability is shown in Figure 11. The deficit is located at v _∥ < 0, where it modifies the relative alignment between the pitch-angle gradients of f _0e and the direction of the diffusive flux of resonant electrons (locally tangent to the black semi-circles). Resonant electrons diffuse towards smaller v _⊥ and larger |v _∥|. The parallel fast-magnetosonic/whistler wave with phase speed v _ph grows at the expense of the energy of the diffusing electrons. This quasi-linear process fills up the electron deficit.

High-cadence and high-resolution measurements of the electron distribution function from Solar Orbiter show pronounced deficits at times when pronounced amplitudes of quasi-parallel fast-magnetosonic/whistler waves are seen (Berčič et al., 2021b). This observation suggests the sporadic occurrence of the electron-deficit whistler instability in the solar wind. Since the sunward electron deficit is more pronounced near the Sun, a systematic study of the this instability would be worthwhile in data from Parker Solar Probe and Solar Orbiter at small heliocentric distances.

The observed deficit, rather than forming a sharp cutoff at a specific v _∥ as shown in Figure 11, also extends to larger pitch angles, encompassing locally mirroring electrons with largely perpendicular velocities. This observation could indicate that the operation of the instability has already resulted in diffusion from perpendicular to parallel velocities, or it could indicate that those regions of phase space are also unpopulated as suggested by Halekas et al. (2021a).

Simulations of the electron-deficit whistler instability are not available yet.

If the plasma consists of one electron population and multiple ion populations, relative drifts between the electrons and the ion populations can drive instabilities. The current balance in a system consisting of an electron component and two proton components demands n e U e = n pc U pc + n pb U pb , (41) where the subscript e refers to the single electron species, pc to the proton core, and pb to the proton beam. If such a proton beam–core configuration with U _pc ≠ U _pb exists, then U _e ≠ U _pc; i.e., there is a non-zero drift between the proton core and the electrons. If the proton beam is sufficiently dense and fast in the proton-core frame, the fast-magnetosonic/whistler wave can be driven through cyclotron-resonant wave–particle interactions with the electrons (Akimoto et al., 1987). The mechanism is similar to the one shown in Figure 9 for the whistler heat-flux instability, where the population marked as the halo now corresponds to the only electron population. For isotropic electrons, this instability still requires that U _e > v _ph. The separation between U _e and v _ph depends on the proton-beam and proton-core properties. The same mechanism is also potentially able to drive other instabilities presented in Section 4.

The mirror-mode instability is an example for a non-resonant, anisotropy-driven instability. It corresponds to the oblique non-propagating slow mode with ω _kr = 0 (Chandrasekhar et al., 1958; Barnes, 1966; Basu and Coppi, 1984; Verscharen et al., 2017). Due to the polarisation of the non-propagating slow mode, the mirror-mode instability exhibits a significant component of magnetic-field fluctuations δB _∥≡ δ B ⋅ B ₀/B ₀ parallel to B ₀. The fluctuations in δB _∥ are anti-correlated with the fluctuations in δn _e. Trapping of slow (v _∥ ≈ 0) particles through the mirror force plays an important role in the nonlinear evolution of the mirror-mode instability (Southwood and Kivelson, 1993). However, particle trapping is not captured by our quasi-linear framework, and our requirement that ω _kr ≫ γ _K is violated in the mirror-mode instability.

Unlike the resonant instabilities discussed in Sections 3 and 4, the mirror-mode instability is less sensitive to the shape of the distribution function in a defined narrow part of velocity space. Instead, its stability depends on the total pressure anisotropy of the system. This point is also illustrated by the analytical threshold for the mirror-mode instability which depends on the pressure contributions of both species in an electron–proton plasma: the mirror mode is unstable if (for a gyrokinetic derivation, see Verscharen et al., 2019a) β ⊥ p T ⊥ p T ∥ p − 1 + β ⊥ e T ⊥ e T ∥ e − 1 > 1 . (42) This type of analysis has also been extended to the case in which there are more than two particle species present (Hall, 1979; Hellinger, 2007; Chen et al., 2016).

Eq. 42 illustrates that the mirror-mode instability can be driven unstable by both ions and electrons (see also Migliuolo, 1986). The wave number at maximum growth shows an interesting transition between ion scales (k ≲ 1/d _p) and electron scales (k ≲ 1/d _e) depending on the species with the dominant anisotropy (Hellinger and Štverák, 2018). The saturation of the mirror-mode instability happens via a fluid-level rearrangement of the plasma that reduces the overall pressure anisotropy (Kivelson and Southwood, 1996; Rincon et al., 2015; Riquelme et al., 2015).

Linear theory predicts that, for the same β _∥e and T _⊥e/T _∥e in a bi-Maxwellian plasma, the electron whistler anisotropy instability discussed in Section 3.1 generally has higher growth rates and lower thresholds than the oblique electron mirror-mode instability (Gary and Karimabadi, 2006). However, two-dimensional particle-in-cell simulations show that both instabilities compete, and the oblique electron mirror-mode instability often becomes dominant in the nonlinear phase after the parallel electron whistler anisotropy instability saturates (Hellinger and Štverák, 2018). Hellinger and Štverák (2018) also note that the importance of the mirror-mode instability potentially increases in more realistic three-dimensional simulations due to the higher degrees of freedom in such a system ³ .

The non-propagating electron firehose instability is another non-resonant instability driven by anisotropy. It corresponds to a non-propagating (i.e., ω _kr = 0) solution of the oblique Alfvén-wave branch (Li and Habbal, 2000). If the plasma pressure anisotropy with p _∥ > p _⊥ is sufficiently large, the magnetic tension is unable to provide a sufficient restoring force for the propagation of the Alfvén wave anymore, and the mode becomes aperiodic (Squire et al., 2016). In this context, we define the total pressures p _⊥≡∑ _j n _j k _B T _⊥j and p _∥≡∑ _j n _j k _B T _∥j . Like in the case of the mirror-mode instability, the instability threshold of the non-propagating firehose instability depends on both electron and proton pressure contributions. In an electron–proton plasma, the non-propagating firehose mode is unstable if (for a gyrokinetic derivation, see Verscharen et al., 2019a) β ∥ p − β ⊥ p + β ∥ e − β ⊥ e > 2 . (43) If multiple species are present, this condition can be extended to (Kunz et al., 2015; Chen et al., 2016) ∑ j β ∥ j − β ⊥ j + 8 π m j n j U j 2 B 0 2 > 2 , (44) which also accounts for the contributions of relative drifts to the total parallel pressure (Gary et al., 1975b). Therefore, the non-propagating firehose instability can also be driven unstable in the presence of isotropic or anisotropic beam populations. At maximum growth, the non-propagating electron firehose instability has k ∼ 1/d _e.

The non-propagating electron firehose has a significantly lower threshold than the propagating electron firehose instability under typical solar-wind conditions (Gary and Nishimura, 2003; Lazar et al., 2022). The presence of suprathermal electron populations with properties consistent with observed solar-wind conditions lowers the threshold of the non-propagating firehose instability even further (Shaaban et al., 2019a). In the expanding solar wind, conservation of the first adiabatic invariant (magnetic moment) in a decreasing magnetic field naturally increases T _∥e/T _⊥e and drives the distribution toward the firehose instability thresholds (Innocenti et al., 2020). If this process happens faster than Coulomb collisions can moderate the anisotropy, then the non-propagating firehose instability can be triggered.

The non-propagating electron firehose instability thresholds constrain slow wind (but not fast wind) core electrons to the stable parameter space in electron measurements from Helios, Cluster, and Ulysses (Štverák et al., 2008). Electrons measured by Parker Solar Probe during encounters 1 to 9 are stable and far from the thresholds of the non-propagating electron firehose instability (Cattell et al., 2022).

Particle-in-cell simulations of the non-propagating electron firehose instability show that nonlinear wave–wave interactions play an important role during the nonlinear stage of the instability (Camporeale and Burgess, 2008; Hellinger et al., 2014). Highly oblique fluctuations grow initially, which then couple to modes with less oblique wave vectors. The interplay of the various modes leads to a situation in which the plasma “bounces” around the marginal stability threshold in parameter space. An anisotropy in suprathermal electron populations relaxes more quickly than an anisotropy in thermal electron populations during the instability’s nonlinear evolution (López et al., 2019a), which is also consistent with the observation of lower halo anisotropies (Štverák et al., 2008). Fully kinetic, expanding-box simulations demonstrate that the non-propagating electron firehose instability arises self-consistently in the expanding solar wind due to the conservation of the magnetic moment (Innocenti et al., 2019a). After onset, the firehose instability and its nonlinear evolution compete against the ongoing expansion to drive the system between stability and instability: also in expanding-box simulations, the electrons bounce around the marginal stability threshold in parameter space.