Origin and regulation of proton beams

6.5 Origin and regulation of proton beams

In addition to the anisotropic core discussed above, a secondary proton component was often observed as a salient feature of the VDFs in fast solar wind. The beams were already found in the early era of space in situ explorations (Feldman et al., 1973 , 1974

; Goodrich and Lazarus, 1976 ; Marsch et al., 1981 , 1982c ,b ; Feldman et al., 1993 ; Goldstein et al., 2000

), and were extensively described in the review by Feldman and Marsch (1997 ). In Figure 3

several beams from Helios are shown, and another typical example is shown in Figure 18

. These double-beam distributions carry important information about the kinetic state of the solar wind plasma and on the interplanetary dynamic processes. However, the origin (either in the corona and/or interplanetary medium) of the proton beams and their spatial and temporal evolution has not yet been fully understood.

Figure 18:

Left: This panel shows 2-D contours of a typical proton beam VDF, based on a 3-D interpolation of the data obtained by Helios 2 at 09:08:33 on day 70 in 1976 after Tu et al. (2004

). The horizontal axis gives the velocity component parallel ( $v∥$ ) and the vertical perpendicular ( $vx$ ) to the magnetic field. The solid, dashed and dotted curves, respectively, show the contours of the VDF relative to its maximum value which is located at $v∥ = 0$ and $vx = 0$ . The respective values of the contours from the center correspond to $0.8$ , $0.6$ , $0.4$ , $0.2$ , $0.1$ , $0.032$ , $0.01$ , $0.0032$ , and $0.001$ of the maximum. Right: This panel shows a 1-D reduced VDF, $ΔF (v∥)$ , obtained by integration along the vertical $vx$ -direction indicated in the left panel. The VDF is normalised to its maximum value. The horizontal axis gives the velocity component $v ∥$ in $km s−1$ .

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 $β∥c = 0.2$ and $1.0$ , respectively, whereas the upper and lower dashed lines display the thresholds of the Alfvén instability at $β∥c = 1.0$ and $0.2$ , 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.

Figure 19:

Left: The normalised beam drift speed is plotted versus the plasma beta after Tu et al. (2004

). Each cross point represents a single Helios plasma measurement of the proton beam drift speed plotted against the core plasma beta, $β ∥c$ . The dash-dot line shows the result of a linear least-squares fit to the logarithm of the observed data points. The dotted line and the dashed line show the threshold of the Alfvén I instability after Daughton and Gary (1998

) with a constant ratio of the proton beam density, $nb$ , to the electron density, $ne$ . The values of their ratio, $nb ∕ne$ , are $0.05$ (upper line) and $0.2$ (lower line). The maximum instability growth rate at the threshold is $γ ∕Ω = 0.01 m p$ . The diamonds show the few data points for which the VDFs are found to be unstable. Right: The normalised proton-proton relative drift speed is shown versus the relative beam density. Individual data points as measured by Ulysses are given after Goldstein et al. (2000 ). The four lines represent threshold conditions for two proton-proton instabilities as shown in Daughton and Gary (1998

). The upper and lower solid lines represent the thresholds of the magnetosonic instability at $β = 0.2 ∥c$ and $1.0$ , respectively, whereas the upper and lower dashed lines display the thresholds of the Alfvén instability at $β∥c = 1.0$ and $0.2$ , respectively.

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 $β ∥c$ . The crosses represent the observed beam drift speed along the magnetic field, which is normalised to the local Alfvén speed, i.e., the ratio $vd∕VA$ . From this plot we can see that the two parameters are surprisingly well correlated. The correlation coefficient between $lg(β∥c)$ and $lg (vd∕VA )$ 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:

This relation is plotted in Figure 19

as the dashed-dotted line, which fits the data points well, especially for the plasma beta range: $β = 0.1– 0.6 ∥c$ . There are two additional curves in Figure 19

. The dotted line and the dashed line refer to theoretical results. They give the threshold values of the Alfvén I instability according to Daughton and Gary (1998

). Their threshold curve was regular enough that they could fit it by the simple relation:

where $0.06 Δ1 ∼= 1.65β ∥c$ , and $Δ2 ∼= 5.1 + 1.9β∥c$ . This theoretical curves is shown for a constant relative beam density, having the values: $nb∕ne = 0.05$ , respectively $0.2$ . From Figure 19

we can see that the majority of the data points correspond to relative beam densities ranging between $0.05$ and $0.2$ . This plot clearly indicates that most of the measured proton beam distributions that qualified for our data set are stable against the oblique electromagnetic Alfvén I beam instability, since we see that the majority of the data points are distributed below (outside of) the regions delineated by the dashed and dotted lines.

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 $vd∕VA$ . If this is smaller than the observed value, $vd∕VA$ , 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 $vd∕VA ≥ 2$ , 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 ( $nb∕ne ≤ 0.05$ ), high-speed $(vd∕VA ≥ 2)$ 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 $4 −1 γ ≈ 10 s$ 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.