With the first in-situ explorations of solar wind plasmas in interplanetary space, the kinetic (micro- and mesoscopic) properties of plasma populations became of great interest (Olbert et al., 1968; Vasyliunas, 1968; Gary, 1993). These properties are mainly revealed by the fluxes and velocity distributions of particles measured in-situ, which suggest that, locally, particle plasma populations (electrons, protons, or minor ions) are not in thermal equilibrium. The non-equilibrium features frequently reported by the observations are kinetic anisotropies, such as temperature anisotropy or beam populations (Marsch, 2006), phase space density gradients (Page et al., 2021), as well as suprathermal populations (e.g., halo, beaming populations), with energies up to a few keVs (Olbert et al., 1968; Vasyliunas, 1968; Collier et al., 1996; Maksimovic et al., 2005; Štverák et al., 2008; Mason and Gloeckler, 2012; Tong et al., 2019a). By comparison to the low-energy, or core populations (up to a few tens of eV), which are well reproduced by the standard Maxwellian models, suprathermal populations are less dense (but hotter), and require a more laborious approach based on the Kappa (κ–power-law) distributions (Pierrard and Lazar, 2010; Viñas et al., 2015; López et al., 2021). These are probably the main reasons why suprathermal populations have been ignored for decades in the analysis of anisotropic temperatures and related wave excitations, usually invoking the dominance of the core population and the simplicity of standard Maxwellian models, see Ref. (Gary, 1993; Yoon, 2017). and references therein. However, suprathermal populations (e.g., electron halo) are much hotter and more anisotropic than quasi-thermal core (Maksimovic et al., 2000; Štverák et al., 2008; Pierrard et al., 2016; Lazar et al., 2020), suggesting a significant implication of suprathermals in the kinetic excitations of wave fluctuations and instabilities. Moreover, in recent years important progresses have been made, both theoretically and numerically, by developing kinetic approaches relying on the anisotropic Kappa distribution functions. These models become thus able to describe the observed suprathermal populations (Pierrard and Lazar, 2010; Lazar et al., 2012), and also their contributions to kinetic effects and instabilities of plasma particles (Shaaban et al., 2019a; Lazar et al., 2019; López et al., 2019; López et al., 2021).

In the present review we compile recent results consistently and rigorously describing the effects of suprathermal populations on electromagnetic (EM) excitations induced by the temperature anisotropy of major plasma species in solar wind plasma, i.e., electrons and protons. The present review is largely confined to the works carried out by the present authors, and is not meant to cover the broader aspects regarding the fundamental properties and origins of suprathermal charged particle distributions. Particularly interesting are the EM cyclotron and firehose excitations, known as plasma micro-instabilities, or kinetic instabilities, which enhance the small-scale magnetic fluctuations observed in the solar wind (Bale et al., 2009; He et al., 2011; Wilson et al., 2013; Lion et al., 2016; Wicks et al., 2016; Roberts et al., 2017). These instabilities and the resulting enhanced fluctuations should play a major role in self-regulating the properties of solar wind plasma, especially in the absence of particle-particle collisions whose frequency decreases significantly with increasing heliocentric distance (Štverák et al., 2008; Bale et al., 2009; Klein et al., 2018). Also, compared to low-energy (core) populations, suprathermal particles are less affected by collisions (Maksimovic et al., 2005), but should be more susceptible to kinetic instabilities. Moreover, the free energy of suprathermal particles is expected to stimulate kinetic instabilities, and increase the level of magnetic fluctuations. Despite these expectations, a large number of dispersion and stability analyses predict the opposite, especially an inhibition of these instabilities, for many conditions specific to the solar wind (Lazar and Poedts, 2009; Mace and Sydora, 2010; Lazar et al., 2011; Lazar et al., 2013). Such contradictory outcomes led to further investigations (Lazar et al., 2015; Lazar et al., 2016), which showed that these studies did not use the original Kappa distribution, but another simplified form assuming the corresponding kinetic temperature (as given by the second order moment) independent of the power exponent κ (Hau et al., 2009; Hellberg et al., 2009; Livadiotis and McComas, 2012). Empirically introduced by Olbert (Olbert et al., 1968) and Vasyliunas (Vasyliunas, 1968), the original Kappa model (see also section 2 below) enables realistic interpretation of suprathermal populations, and produces reliable results (Lazar et al., 2015), which show rather a stimulation of the temperature anisotropy instabilities and an enhancement of the resulting EM fluctuations in the presence of suprathermal populations (Thorne and Summers, 1991; Lazar et al., 2015; Lazar et al., 2016; Shaaban et al., 2019a; Lazar et al., 2019; López et al., 2019).

Section 2 presents the essence of such a realistic interpretation of suprathermal populations, which enhance the high-energy tails of the anisotropic bi-Kappa distribution, and are highlighted by contrast with the bi-Maxwellian (quasi-thermal) core. In Sections 3 and 4 we discuss the most important instabilities triggered by the temperature anisotropy of electrons and protons, the dominant species in the solar wind. Selected are the results from advanced quasi-linear (QL) theories and from numerical simulations, those able to show the extended evolution of the excited fluctuations, as well as their action back on the particles leading to the relaxation of their temperature anisotropy. The last section (Section 5) brings together the main conclusions and a series of perspectives of the present review study.

The modeling of suprathermal populations and their implications is often based on the Kappa distribution functions, introduced to reproduce the observed electron distributions with energies of up to a few eVs (Olbert et al., 1968; Vasyliunas, 1968). In such a case, the Kappa power-law is applied as a global (fitting) model, that incorporates both the low-energy electron core and the suprathermal population enhancing the high-energy tails of the observed distribution. Not only electrons, but also protons and minor ions observed in the solar wind exhibit such velocity distributions (Collier et al., 1996; Mason and Gloeckler, 2012).

The anisotropic model used to describe populations with anisotropic temperatures (i.e., A = T _⊥/T _∥ ≠ 1) is the bi-Kappa distribution (Summers and Thorne, 1991) f κ v ⊥ , v ∥ = 1 π 3 / 2 θ ⊥ 2 θ ∥ Γ κ + 1 κ 3 / 2 Γ κ − 1 / 2 1 + v ⊥ 2 κ θ ⊥ 2 + v ∥ 2 κ θ ∥ 2 − κ − 1 , (1) normalized to unity ∫d ³ vf _κ = 1, and written in terms of the normalization velocity parameters θ _∥,⊥ (where ∥, ⊥ indicate the components defining the anisotropy with respect to the uniform magnetic field lines). From the second-order moments of Kappa distribution we find the velocity parameters directly related to the corresponding components of the (Kappa) kinetic temperature T κ , ∥ , ⊥ = κ κ − 3 / 2 m θ ∥ , ⊥ 2 2 k B , (2) for particle populations of mass m (and k _B is the Boltzmann constant). These kinetic temperatures are positively defined only for a power exponent κ > 3/2.

Widely applied theoretically (Lazar and Poedts, 2009; Mace and Sydora, 2010; Lazar et al., 2011; Lazar et al., 2013; Shaaban et al., 2019a; Lazar et al., 2019; López et al., 2019), the bi-Kappa model has also been used in more recent and refined observations, as a partial fit reproducing only the suprathermal components of the observed distributions, e.g., halo or strahl populations, while the low-energy core was described by a more standard bi-Maxwellian distribution (Maksimovic et al., 2005; Štverák et al., 2008; Tong et al., 2019a; Wilson III et al., 2019; Scherer et al., 2020). Such a dual or even multi-component description using Kappa models may offer valuable insights distinguishing between various components, e.g., core, halo and, eventually, beaming (or strahl) populations, their properties and origin. In such a case the theory becomes not only complicated, due to the multitude of parameters involved, but also limited to the specific case defined by the set of parameters considered in the approach (Lazar et al., 2014; Shaaban et al., 2016; Lazar et al., 2018a; Lazar et al., 2018b; Shaaban et al., 2019b).

For plasma instabilities driven by the temperature anisotropy of the observed (gyrotropic) distributions relevant are the most unstable cases combining both the core and suprathermal halo populations with similar anisotropies, either both with A = T _⊥/T _∥ > 1, or both with A = T _⊥/T _∥ < 1 (Lazar et al., 2018a; Lazar et al., 2018b; Shaaban et al., 2019b). For instance, the electron plasma states with a dual core–halo composition, showing reduced relative drifts but similar (correlated-like) anisotropies, are often revealed by the solar wind observations (Štverák et al., 2008; Pierrard et al., 2016; Wilson III et al., 2019), and can be described by a global (bi-)Kappa (Lazar et al., 2017a), as also suggested in the early theories and observations (Pierrard and Lazar, 2010) ¹ . Such a simplified approach cannot provide details about the components like core, halo or additional beams, but has a particular advantage in the analysis of suprathermal halo populations, facilitating the understanding of their implications in the excitation of anisotropic temperature instabilities. Thus, the suprathermals can be emphasized and even quantified based on a comparison between the observed (global) Kappa distribution with enhanced high-energy tails (red line in Figure 1) and the Maxwellian low-energy core in the absence of suprathermal population (e.g., dashed blue line in Figure 1) (Lazar et al., 2015; Lazar et al., 2016).

Such a contrasting analysis becomes straightforward, since the quasi-thermal core (subscript c) is reproduced with a good approximation by the Maxwellian limit (of lower temperature) of the Kappa distribution, i.e., f κ → ∞ v ⊥ , v ∥ = f M v ⊥ , v ∥ (see also Figure 1) (Lazar et al., 2015; Lazar et al., 2016), which reads f c v ⊥ , v ∥ ≃ f M v ⊥ , v ∥ = 1 π 3 / 2 θ ⊥ 2 θ ∥ exp − v ⊥ 2 θ ⊥ 2 − v ∥ 2 θ ∥ 2 . (3) Here the normalization parameters θ _⊥,∥ are thermal velocities related to the corresponding components of the core temperature T c , ∥ , ⊥ ≃ m θ ∥ , ⊥ 2 2 k B < T κ , ∥ , ⊥ . (4) This implies an (approximate) relationship between temperatures (Lazar et al., 2015) T κ , ∥ , ⊥ ≃ κ κ − 3 / 2 T c , ∥ , ⊥ > T c , ∥ , ⊥ , (5) and between the corresponding plasma beta parameters, for instance, the parallel components β κ , ∥ ≃ κ κ − 3 / 2 β c , ∥ > β c , ∥ , (6) commonly invoked in the dispersion analysis. The plasma beta is defined (kinetically) as β = 8 π n k B T / B 0 2 , where n is the number density of plasma particles of temperature T, and B ₀ is the strength of the uniform magnetic field. In the analysis below, we will specify the values for the plasma beta corresponding to the (bi-)Maxwellian limit, i.e., β _c,∥ = β _∥, and for the power exponent κ. It becomes also clear that this bi-Maxwellian limit considerably facilitates the comparison with the bi-Kappa distribution, both in analytical and numerical computations, providing results of a general validity (Lazar et al., 2015; Lazar et al., 2016; Shaaban et al., 2021b). In this case one can conclude on the implications of suprathermal populations without resorting to a laborious statistical study of the results obtained for each core–halo combination.

If we reiterate those studies which invoke another comparison of the Kappa distribution with a Maxwellian limit of the same (kinetic) temperature (Hellberg et al., 2009; Livadiotis and McComas, 2012), i.e., T _M = T _κ , their results do not have the same relevance, and cannot highlight suprathermal populations and their implications (Hau et al., 2009; Lazar et al., 2015; Lazar et al., 2016) (a graphical comparison of a bi-Kappa with both bi-Maxwellian limits can be found in (Lazar et al., 2016)). Many of the early analyses of temperature anisotropy instabilities were also affected by such a simplified approach (Lazar and Poedts, 2009; Mace and Sydora, 2010; Lazar et al., 2011). However, more recent studies based on a realistic contrast between a (bi-)Kappa with its (bi-)Maxwellian core, as illustrated in Figure 1, have reached concurring conclusions providing a consistent interpretation of the suprathermal populations and their contribution to these instabilities (Lazar et al., 2016; Shaaban et al., 2019a; López et al., 2019). In the following we will revisit these more reliable studies, paying particular attention to the results obtained with advanced QL theories and numerical simulations.

In this paper we have reviewed a number of recent results on the instabilities induced by the temperature anisotropy of plasma particles, under conditions typical for the solar wind and planetary environments, e.g., in Refs. (Lazar et al., 2019; López et al., 2019; Shaaban et al., 2021b; Shaaban et al., 2021a). The interest is to outline and show how these instabilities are influenced by suprathermal populations (with energies up to 1 keV), which are often ignored despite their ubiquity in space plasmas (Gary, 1993; Yoon, 2017). The recent studies we have referred to have the merit of using a rigorous modeling and interpretation of suprathermal populations based on the (anisotropic) Kappa distribution functions, see Section 2 (Lazar et al., 2015; Lazar et al., 2016). What motivated our analysis even more was the fact that all these recent results showed a systematic stimulation of the instabilities induced by temperature anisotropy, whether they are induced by different species (i.e., electrons, protons), or are instabilities of different nature (e.g., cyclotron or firehose) (Lazar et al., 2015; Lazar et al., 2016; Shaaban et al., 2019a; Lazar et al., 2019; López et al., 2019). Therefore, our intention was not only to revisit but also to test these results, as well as theoretical approaches (i.e., linear and QL) and numerical simulations, which were re-applied for the same or slightly modified set-ups, but keeping a parameterization specific to space plasmas.

By formulating these conclusions, we can indeed confirm that suprathermal populations, both electrons and protons, have a systematic stimulating effect not only on cyclotron instabilities, i.e., whistler and EMIC, but also on firehose instabilities, both branches of periodic (or parallel) and aperiodic (or oblique) firehose modes. The unstable fluctuations, quantified by their, e.g., wave power or energy density, can be markedly enhanced by the suprathermal populations, especially when the low-energy (core) populations are initially in low plasma beta conditions. Indeed, comparisons between low and high beta conditions, clearly show that suprathermal populations contribute with an additional kinetic (free) energy, producing similar results as those obtained by increasing the plasma beta parameter. From now on, in-situ measurements can explain the increased levels of these specific fluctuations not only by association with temperature anisotropy (Bale et al., 2009) but also by the presence of suprathermal populations. On the other hand, the resulting enhanced fluctuations should act back on the anisotropic particles, and the results from long runs indicate stronger variations of their parameters in the presence of suprathermals, also leading to a deeper relaxation of temperature anisotropy.

The same stimulating effect is also reflected by the anisotropy thresholds of these instabilities, which can be significantly lowered by the suprathermals. Suggestive examples are shown in Figure 9, in a diagram of proton anisotropy (A _p ) vs. proton (parallel) beta parameter (β _p∥), not only by the thresholds of proton firehose (PFH) instability (A _p < 1), but also by the thresholds of EM ion-cyclotron (EMIC) instability in the upper part (A _p > 1) of this diagram (Shaaban et al., 2021a; Shaaban et al., 2021b). The anisotropy thresholds of electron instabilities are not shown here, but undergo the same influence in the presence of suprathermal electrons (Lazar et al., 2017b; Shaaban et al., 2019a; Lazar et al., 2019). This is another consequence of the presence of suprathermal populations, which can indirectly contribute to the accumulation of quasi-stable states in between these thresholds, near and along the isotropy condition A ≃ 1, as shown by the observations, not only for protons (Kasper et al., 2002; Bale et al., 2009) but also for electrons (Štverák et al., 2008). Other instabilities induced by beams or drifting populations (Verscharen et al., 2019; López et al., 2020; Micera et al., 2020; Schroeder et al., 2021), may show less systematic behavior when modeled by Kappa distributions (Shaaban et al., 2018a; Shaaban et al., 2020). Present resultss and recent advances in modeling the dispersion and stability of these anisotropic populations (López et al., 2021) should motivate future investigations to decode even more complex spectra of wave instabilities, as triggered by the interplay of various kinetic anisotropies of solar wind plasma populations, e.g., temperature anisotropy and relative drifts (Shaaban et al., 2018b; Shaaban and Lazar, 2020; Vasko et al., 2020), but also density gradients in phase space (Page et al., 2021).