their values are widely scattered between 0.5 and 2 (see Figure 1.15). Generally, Ac < 1 and Ah < 1, but instances with Ac > 1 and Ah < 1, and vice versa, are observed. The latter case, Ac < 1 and Ah > 1, then is much more common.
Figure 1.16 Typical electron velocity distribution function (VDF) observed by WIND at 1 AU in the low‐speed (left) and in the high‐speed (right) solar wind. Bottom panels: The plane of velocities (normalized to the thermal velocity) parallel and perpendicular to the interplanetary magnetic field. Top panels: Parallel (solid black line) and perpendicular (dashed blue line) cross section of the observed VDF. The dashed‐dotted red line represents the Maxwellian distribution that well fits the core of observed VDF.
(Source: From Pierrard, 2011. © 2011, Springer Nature.)
The maximum range of values for temperature anisotropies is constrained by the onset of plasma instabilities. A > > 1 values are unstable toward the generation of whistler waves, and A < < 1 values are subject to the firehose instability (Gary 1993). A > > 1
This picture is similar for solar wind proton distributions. They frequently show temperature anisotropies T⊥ > T∥ plus an anti‐sunward beam (see Marsch, 2012, for a review based on Helios data covering solar distances between 0.3 AU and 1 AU). The temperature anisotropy, A, shows a strong correlation with the observed ion cyclotron wave power, which indicates that the distributions are shaped by wave–particle interactions. As for electrons, the temperature anisotropies are constrained by thresholds for firehose and mirror instabilities, but are hardly observed near these limits. Observations suggest that temperature anisotropy instabilities may play a role in controlling the level of temperature anisotropy.
1.5.2. Wave–Particle Interactions, Kinetic Instabilities, and Collisions
The low density of the solar wind plasma could lead to the expectation that it could be described as collisionless; that is, Coulomb collisions are negligible. Nevertheless, Coulomb collisions do play a role in constraining temperature anisotropies (Pierrard et al., 2016; Pierrard et al., 2011; C. Salem et al., 2003; Štverák et al., 2008)
Figure 1.17 Kinetic shells formed in the electron VDF by resonant electron–whistler interaction (solid lines), and isolines of an undisturbed Maxwellian VDF (dotted lines). Shown are two plots for different values of the electron Alfvén speed, vA,e = 0.5 c (left, (a)), and vA,e = 0.05 c (right, (b)).
(Source: From Vocks & Mann, 2003. © 2003, IOP Publishing.)
Non‐Maxwellian particle distributions can become unstable toward numerous plasma instabilities (Štverák et al., 2008). Quasi‐linear theory (Kennel & Engelmann, 1966) is a useful tool to study the effect of whistler waves on the evolution of solar wind electron distributions. It considers nonlinear effects between plasma waves and the fluctuations they cause in the plasma, and therefore diffusion terms are included in the Boltzmann–Vlasov equation. Solar wind models including the effects of whistler wave turbulence are capable of reproducing the observed characteristics of solar wind electrons (Pierrard et al., 2011; Vocks, 2012).
While whistler wave turbulence leads to waves propagating in all directions relative to the background magnetic field (Chen et al., 2010; Gary & Smith, 2009), even solar wind models that consider only the interaction between electrons and whistlers propagating parallel to the magnetic field result in electron distributions with a core, halo, and strahl (Vocks et al., 2005).
The quasi‐linear interaction between parallel propagating whistlers and electrons leads to pitch‐angle scattering in the reference frame of the waves, which leads to the formation of “kinetic shells” (Isenberg et al., 2001); see Figure 1.17. In interplanetary space, whistler wave phase speeds are not larger than electron thermal speeds, so there is little difference between the wave frame and the plasma rest frame. So whistler turbulence tends to lead to more pitch‐angle diffusion. The observation of finite strahl widths already implies that some pitch‐angle scattering mechanism must be active in the solar wind, because otherwise the conservation of magnetic momentum in the expanding magnetic field geometry of the solar corona and wind would focus the electrons into an extremely narrow beam.
If the whistler wave phase speed is much higher than electron thermal speeds, as is the case in the solar corona, then the kinetic shell formation not only leads to pitch‐angle diffusion, but also deforms the distribution toward higher temperature anisotropies, T⊥ > T∥. The diffusion process brings electrons from a phase‐space location with low v∥, v⊥ = 0 to a position with high v⊥, v∥ = 0. In the plasma frame, this corresponds to a gain of kinetic energy. Therefore, this mechanism is capable of producing a suprathermal population out of an initially Maxwellian electron distribution (Vocks et al., 2008), thus forming a halo component with a power‐law distribution.
1.5.3. Suprathermal Particles
Kinetic effects induce strong distortions of the VDFs in the thermal regime, but also by forming suprathermal populations that include electron strahl, non‐thermal ion beams, and heavy ion differential streaming. In this chapter, we focus on the origin and important effects of suprathermal electrons and direct the reader to the review by Marsch et al. (2009) for a discussion of suprathermal ions.
As already mentioned, solar wind electron distributions show distinct particle populations; a thermal core; an extended suprathermal halo; the asymmetric, anti‐sunward strahl; and an isotropic super‐halo at higher energies (Lin, 1998). The thermal core can be described by a bi‐Maxwellian distribution, but this does not work so well for the halo (Pilipp et al., 1987) component. Core and halo can be much better fitted by kappa distributions (Maksimovic et al., 1997), that show power‐law suprathermal tails ∝v −2( κ + 1). The κ parameter can reach values as low as 2. Note that κ → ∞ corresponds to a Maxwellian, and for κ → 1.5, the thermal energy, that is, the temperature, diverges.
An even better fit to observed electron data can be achieved if the core is additionally fitted by a (bi‐)Maxwellian (Pierrard et al., 2016). The density of the halo is less than that of the core, but due to the higher halo temperature, their thermal energies are comparable (Maksimovic et al., 2000). A coronal origin of these suprathermal electrons is possible (Pierrard et al., 1999; Vocks et al., 2005), but whistler wave turbulence in the solar wind also plays a role in shaping these distributions (Yoon et al., 2015).
Pierrard et al. (2016) have studied the evolution of the suprathermal halo through the heliosphere between 0.3 AU and 4 AU. Temperature anisotropies were considered by applying bi‐kappa fits to the electron
