As Figure 20 demonstrates, that in the presence of a large-amplitude Alfvén-cyclotron wave the power of the waves corresponding to the linear instability strongly decreases (see the fading of the left grey bar with the relative amplitude , rising from 0.0 to 0.2). There is low wave activity in the third quadrant, except for the power at , which may be due to a parametric decay. This region would be expected to be stable in a linear system with . These properties show that there is a stabilization of the linear instability (Gomberoff, 2003), due to the presence of the large amplitude wave that is visible as dark bar in the lower frame (b) at . The beam drift is large, , and the beam density sizable, , with . Finally, note that the beam-modified linear dispersion relation is nicely outlined by the two curves crossing at the origin. On these lines the fluctuation level clearly appears to be enhanced.
These simulation findings are particularly important in the light of the results presented in the previous section, in which the observed proton double beams in the fast solar wind were found to be largely stable. However, in the standard linear stability analyses (as discussed in the previous section) the presence of the large-amplitude Alfvénic turbulence, which is ubiquitous in the fast solar wind (Tu and Marsch, 1995), was not accounted for. But Kaghashvili et al. (2004) showed that the relative streaming between proton components decelerated among non-linear low-frequency Alfvén waves. Some of their results are shown in Figure 21, which gives the normalised proton beam drift versus time in gyroperiods for various ambient Alfvén wave amplitudes.
The evolution of streaming minor ions in the presence of large-amplitude Alfvén waves is similar to weak proton beam deceleration. Kaghashvili et al. (2003) showed that minor ion deceleration is associated with the development of a compressional wave component. But they did not relate this with instability, and their simulation results show that minor ion deceleration only begins after an onset time. Therefore, minor ion deceleration may be related with an effective beam instability.
To summarise, Gomberoff (2003), Gomberoff et al. (2003), Araneda and Gomberoff (2004) and Kaghashvili et al. (2004) all have clearly shown that finite-amplitude Alfvén waves have a parametric (often stabilizing) effect on parallel-propagating magnetosonic waves that are generated by a beam instability. Similarly, we may conclude that the obliquely-propagating proton-proton Alfvén cyclotron waves generated by a beam instability are altered by finite-amplitude Alfvén waves. To verify all these theoretical findings by an analysis of measured wave and particle data is an important future task.
Dubinin et al. (2005) analysed the non-linear evolution of differential ion streaming with the non-linear multifluid MHD equations and showed that the cold ion beam-plasma system possesses an equilibrium with a remnant of differential streaming. The plasma may, through the non-linear action of the hydromagnetic waves, attain such a dynamic equilibrium state. It is shown that at zero plasma beta the differential speed between the alphas and protons can range between and , and for a proton double beam between and . The waves involved are either the Alfvén or magnetosonic modes, depending upon the beam speed, whereby Alfvén waves enable larger differential speeds.
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