To develop better kinetic models requires substantial numerical and algebraic efforts. The starting point again is, as for any kinetic model, the full Boltzmann Equation (9). Since this equation depends on time and on three spatial and three velocity coordinates, the numerical effort in solving it is considerable, and therefore simplifications have to be made. For example, in the solar corona and solar wind, all characteristic time scales are small compared to the ion (and of course even more so the electron) gyroperiod. Thus, it is reasonable to assume a gyrotropic VDF. This reduces the number of velocity coordinates from three to two, instead of the full vector we may just consider the components and (with respect to the local magnetic field direction).
Concerning the electrons, some authors (Maksimovic et al., 1997a) have proposed a solar wind model that relies entirely on suprathermal electrons (represented by a kappa velocity distribution) in the corona to accelerate the fast solar wind ions through the ambipolar electric field. However, their model does not explain two major features of coronal holes, namely hot ions and cold electrons, and makes perhaps unrealistic assumptions at the coronal base. Pierrard et al. (1999) developed a test-particle model of electrons, including Coulomb collisions via the Fokker–Planck operator, and solved their kinetic equations according to boundary conditions posed at 1 AU, in order to construct the corresponding consistent VDF in the corona. They conclude that to match the in situ observations suprathermal electrons have to present in the corona.
Some progress in kinetic modelling of solar wind electrons has also been made by Lie-Svendsen et al. (1997) and Lie-Svendsen and Leer (2000), who were able to partly reproduce the core-strahl structure of the electrons, by using only the Fokker–Planck equation for Coulomb collisions (also including the effects of protons) while starting with Maxwellian electrons at the coronal base. They used a test particle approach, in which test electrons were injected into a prescribed background solar wind, the properties of which were calculated by means of the usual fluid equations. Yet, their model does not give the basic core-halo shape of the velocity distributions as measured in situ. Their electron pitch-angle distributions look realistic, yet show an agreement only for the strahl but not the halo (Pilipp et al., 1987a,b).
Lie-Svendsen and Leer (2000) found that the velocity filtration effect (see again Subsection 4.3) was rather small and not capable of producing sizable electron beams. The drift speed and heat flux were solely carried by the tail that is able to escape from the potential. These results essentially confirm the standard empirical (and exospheric) picture already described by Feldman et al. (1975) and later by Pilipp et al. (1987a,b). Some of their model results are presented in Figure 26, which shows on the left side three isocontours of the VDF, and on the right side cuts through these VDF, which illustrate the formation of electron collisional run-away tails and the missing of electrons on the Sunward side (left) at speeds below the escape speed (11,000 km s–1 in this case). Remarkably, the electron pitch-angle distributions resemble very much the measured ones from Helios (see Pilipp et al., 1987a), in particular the narrow electron strahl also occurs at energies beyond about 100 eV. These main kinetic features arise from self-collisions alone. Adding collisions with protons only leads to stronger isotropization (Lie-Svendsen and Leer, 2000).
Kinetic solutions of the Fokker–Planck equation were also obtained by Pierrard et al. (1999). Their model mainly differs from the one of Lie-Svendsen and Leer (2000) in the boundary VDF in the corona and the background electron distributions, which were determined from in situ WIND measurements without the artifacts introduced by an exospheric-type cut off, but the conclusions obtained are essentially the same. The core-halo electron VDF may therefore be produced by Coulomb collisions and the large-scale electric and gravitational forces (in this context see again the exospheric model in Subsection 3.3).
What role whistler-mode waves have to play in such a scenario (see again the clear evidence for wave effects provided in Subsection 6.8) has to remain an open issue. However, we want to mention the work of Chen et al. (2003), who numerically modelled the halo and core electrons by a two-fluid model, and found that the drift between them was not effectively enough regulated by Coulomb collisions. To reconcile their model with the observations, they concluded that enhanced friction by wave-particle interaction is required, beyond about say, from where on it becomes the dominant factor limiting the skewness.
Concerning solar wind protons, Tam and Chang (1999) first investigated the kinetic effects of wave-particle interactions by using a global hybrid model, which follows the evolution of the particle VDFs along an inhomogeneous field line. By considering diffusion in the wave field, the ambipolar interplanetary electric field and Coulomb collisions, the model corresponds to solving a simplified and approximate variant of the basic Equations (9, 10). This model could qualitatively account for the bulk proton acceleration, preferential heating of heavy ions, as well as double-beam formation (by a mechanism similar to the collisional runaway in the model of Livi and Marsch, 1987). It thus represented an instructive global evolutionary study of the solar wind that took into account these kinetic effects. However, quantitatively the details of the VDFs are poorly reproduced when being compared with the observations shown previously in Subsection 2.3.
The authors extended their study (Tam and Chang, 2001) and also considered cyclotron-resonant heating of the electrons, which did not change qualitatively the features obtained in their previous solar wind model. However, the electron heating increased the electric field, which is to say the electron partial pressure gradient, and thereby enhanced the terminal wind velocity. In a further kinetic study, Tam and Chang (2002) have compared the effects on the solar wind velocity of wave-proton interactions with those of suprathermal electrons. Besides Coulomb self-collisions, they considered no other process affecting the shape of the electron VDFs.
Therefore, the essential kinetic effect on the suprathermal electrons was velocity filtration, which arises from weak Coulomb collisions together with the globally kinetic nature of the solar wind. Their model results showed that in the presence of cyclotron-resonant heating of protons and alpha particles, the electron velocity filtration is relatively insignificant for the acceleration of the fast solar wind. They concluded, however, that when there are wave-cyclotron resonances that also affect the suprathermal electrons strongly, then kinetic effects on the suprathermals may no longer be negligible.
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