Effects of wave couplings on linear beam instabilities

6.6 Effects of wave couplings on linear beam instabilities

A non-uniform background plasma can strongly change the linear properties of common plasma instabilities. For example, Gomberoff (2003

) showed that in the presence of large-amplitude Alfvén-cyclotron waves, the beam-driven instability of linear right hand polarised waves could be stabilised. Recently, Araneda and Gomberoff (2004

) demonstrated by direct numerical simulation with a one-dimensional hybrid code (Winske and Leroy, 1984 ) that, if the non-linear wave amplitude exceeded a certain threshold value, the linear instability was completely stabilised. We present some results from these simulations in Figure 20

, which shows the spectral density of the waves in the frequency-wavevector plane, such that the dispersion curves become clearly recognizable. The wave power is given in grey coding.

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 $A$ , rising from $0.0$ to $0.2$ ). There is low wave activity in the third quadrant, except for the power at $k∥VA ∕Ωp = − 0.35$ , which may be due to a parametric decay. This region would be expected to be stable in a linear system with $A = 0$ . 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 $k∥VA∕Ωp = 0.4$ . The beam drift is large, $U = 2.1 VA$ , and the beam density sizable, $nb ∕ne = 0.15$ , with $ne = nb + np$ . 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.

Figure 20:

Power spectrum arranged according to the dispersion relation of transverse magnetic field fluctuations propagating along the field, for the case of zero background-wave amplitude, $A = 0$ , on the upper (a) and large amplitude, $A = 0.3$ , on the lower panel (b). The bottom gray codes correspond to the logarithms of the wave power. A strong beam is present, but the original beam instability (a) is highly suppressed in the presence of the ambient wave (b) (after Araneda and Gomberoff, 2004

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.

Figure 21:

Left: Time evolution of the average differential streaming speed $Upp$ for a relative beam density $nb = 0.25$ . Initially, the main and beam protons are isotropic and have the same temperatures. Curves for five values of the starting speed are plotted, each for the case without waves (dotted lines) and with initial waves (solid lines). Only the fast initial beams show significant deceleration in the presence of waves. Right: Time evolution (extending to $3000$ cyclotron periods) of $Upp$ for a strong beam with $nb = 0.5$ and starting speed $1.57 VA$ . The case without waves is given by the dotted line, and the cases with waves are indicated by solid lines for different relative wave amplitudes, $A$ . All cases show speeds saturating well below $VA$ . The maximum deceleration rate (for $A = 0.0,0.1,0.25$ , and $0.5$ ) is $1.13$ , $1.31$ , $2.04$ , and $2.53$ , respectively, in units of $10−3V Ω A p$ (after Kaghashvili et al., 2004

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 $0.8$ and $1.5VA$ , and for a proton double beam between $0.3$ and $1.5V A$ . The waves involved are either the Alfvén or magnetosonic modes, depending upon the beam speed, whereby Alfvén waves enable larger differential speeds.