Feldman et al. (1974, 1996) argued that proton beams originally might stem from proton injections into the nascent solar wind at the base of the expanding solar corona. On the contrary, Livi and Marsch (1987) suggested that the solar wind proton double beams may be generated and shaped by coronal and interplanetary Coulomb collisions, which are insufficient to prevent proton runaway or pitch-angle focusing in the mirror field configuration. Montgomery et al. (1976), Daughton and Gary (1998) and Daughton et al. (1999) provided arguments from instability calculations and direct numerical simulations that the velocity of the proton beam is regulated by electromagnetic instabilities driven by the beam kinetic energy. Dum et al. (1980) found evidence for wave growth from the measured distributions, and so did Leubner and Viñas (1986) for some selected double-peaked proton VDFs.
A basic hypothesis in collisionless plasma theory is that wave-particle scattering by enhanced fluctuations stemming from a kinetic instability should constrain the source of free energy that drives the unstable wave. This idea implies that the stability threshold derived from linear theory should place observable bounds on the parameters that characterise a growing mode. Goldstein et al. (2000) found that the proton measurements obtained from the plasma instrument on Ulysses support this notion, as they indicated that linear plasma instabilities constrain the relative streaming of the two proton components. Their results are illustrated in the right frame of Figure 19. The four lines shown represent the threshold conditions for two proton-proton instabilities as calculated in Daughton and Gary (1998). The upper and lower solid lines represent the thresholds of the magnetosonic instability at and , respectively, whereas the upper and lower dashed lines display the thresholds of the Alfvén instability at and , respectively. Apparently, the proton beams in the solar wind as measured by Ulysses are practically stable.
In addition to linear beam instabilities, Tu et al. (2002, 2003) suggested that the proton beams (running faster than the core) could also be shaped by quasi-linear diffusion caused by cyclotron waves, which follow a second branch of the dispersion relation owing its existence to the ubiquitous alpha particles in the solar wind. Whether this branch is populated by waves in the real solar wind remains unclear.
More recently, Leubner (2004a) newly interpreted proton beam distribution functions as a natural equilibrium state in generalised thermo-statistics. Leubner (2004b) also fitted the VDFs by two superposed kappa functions for the proton core and beam, and discussed fundamental issues of such kappa distributions for interplanetary protons. In these papers the core-beam distributions are theoretically derived from a non-extensive entropy generalization and then tested on twelve measured Helios VDFs, where the core-beam separation scale is found to obey a condition of maximal entropy.
Tu et al. (2004) again analysed the Helios data with respect to the proton beam instability. Their statistical results from a large data set are shown in Figure 19, giving in the left frame a scatter plot of the measured beam drifts versus the core . The crosses represent the observed beam drift speed along the magnetic field, which is normalised to the local Alfvén speed, i.e., the ratio . From this plot we can see that the two parameters are surprisingly well correlated. The correlation coefficient between and is 0.82. By using a linear fit to represent the data, one finds a simple empirical relation, which may (including the standard deviations) be written as:
Although the stability boundary for the Alfvén I instability has been parameterised in terms of drift speed and plasma beta in Equation (61), it is worth mentioning that this instability is actually a good deal more complicated than can be represented in a two parameter fit. Equation (61) was derived assuming a bi-Maxwellian, with isotropic beam, isotropic core, isotropic electrons, and beam temperature equal to core temperature. In reality, the growth rate of the Alfvén I instability has a significant dependence on all four of these additional complications as shown in Figures 3 to 6 of Daughton and Gary (1998). So the true stability boundary (assuming a bi-Maxwellian) is a function of six parameters and thus too complicated to fit in a simple relation.
There are only 10 data points, indicated by diamonds in the right upper corner of Figure 19, which correspond to VDFs that are unstable against the Alfvén I instability, according to the results presented by and Daughton et al. (1999) and Daughton and Gary (1998). Their equation (1) was used to calculate the theoretical instability-threshold value of . If this is smaller than the observed value, , for a measured VDF, the corresponding data points are indicated by diamonds.
Marsch and Livi (1987) some time ago analysed solar wind ion beams and found observational evidence for marginal stability of many proton beams having , values which one hardly finds in Figure 19. Marsch and Livi (1987) were mostly looking for right hand polarised magnetosonic waves driven unstable by resonant protons at the high-energy flanks of the beam, and found about 20% of their fast beams to be weakly unstable. They concluded that this instability was important in regulating the proton beam and heat flux. Clear single examples for this were found before by Dum et al. (1980) from a stability analysis based on the full measured VDFs, as well as on several modelled VDFs by Leubner and Viñas (1986). According to the data in Figure 19, one comes to a more moderate conclusion, although on the basis of a different, and more restrictively and by visual inspection selected, data set of about 600 proton beams in fast solar wind of which one is shown in Figure 18. It turns out that most of the low-density (), high-speed proton beams obtained in the previous study by Marsch and Livi (1987) were perhaps spurious, and their identification severely suffered from low, unreliable counting statistics.
Voitenko and Goossens (2002b) studied kinetic excitation of high-frequency ion-cyclotron kinetic Alfvén waves (ICKAWs) by ion beams produced by magnetic reconnection in the solar corona. Plasma outflowing from a reconnection site may set up a neutralised proton beam, providing free energy for wave excitation. High growth rates of the order of were found for typical plasma conditions in the low corona. These ICKAWs can undergo Cerenkov resonances with both super- and sub-Alfvénic particles. The waves were found to be damped mainly by wave-particle interactions, with ions at the cyclotron resonance and electrons at the Landau resonance. Therefore, ICKAWs can heat all plasma species in the corona, and may also give rise to anisotropic ion heating.
As we have discussed, ion beams are permanently present in the solar wind, and their stability has been investigated for quite some time (see, e.g., the review of Gary, 1991). Some of the more recent studies within the framework of linear theory were carried out by Gomberoff and Elgueta (1991), Gnavi et al. (1996), Gomberoff and Astudillo (1998), Gomberoff and Astudillo (1999), Gomberoff et al. (2000). The non-linear behaviour of circularly polarised electromagnetic waves and parametric instabilities in a plasma with ion beams were studied by many authors (Hollweg et al., 1993; Gomberoff et al., 1994; Jayanti and Hollweg, 1993a,b, 1994; Gomberoff, 2000; Gomberoff et al., 2001, 2002, 2003). The influence of ion kinetics on the non-linear behaviour of the waves was also investigated by means of the drift-kinetic approach (Inhester, 1990), hybrid simulation (Daughton et al., 1999; Araneda et al., 2002), and quasi-linear theory (Tu et al., 2002; Tu and Marsch, 2002). A simulation study of the role of ion kinetics in low frequency wave-train evolution was carried out by Vasquez (1995). It is only recently that it was realised that in the presence of an ambient large-amplitude Alfvén wave the nature and threshold of the proton beam instability can be substantially modified.
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