5.2 Local production of Alfv´enic turbulence at high latitude

5.2 Local production of Alfvénic turbulence at high latitude

An interesting solution to the radial behavior of the minority modes might be represented by local generation mechanisms, like parametric decay (Malara et al., 2001

; Del Zanna et al., 2001

), which might saturate and be inhibited beyond $2.5 AU$ .

Parametric instability has been studied in a variety of situations depending on the value of the plasma $b$ (among others Sagdeev and Galeev, 1969 ; Goldstein, 1978 ; Hoshino and Goldstein, 1989 ; Malara and Velli, 1996 ). Malara et al. (2000 ) and Del Zanna et al. (2001 ) recently studied the non-linear growth of parametric decay of a broadband Alfvén wave, and showed that the final state strongly depends on the value of the plasma $b$ (thermal to magnetic pressure ratio). For $b < 1$ the instability completely destroys the initial Alfvénic correlation. For $b ~ 1$ (a value close to solar wind conditions) the instability is not able to go beyond some limit in the disruption of the initial correlation between velocity and magnetic field fluctuations, and the final state is $sc ~ 0.5$ as observed in the solar wind (see Section 4.2).

These authors solved numerically the fully compressible, non-linear MHD equations in a one-dimensional configuration using a pseudo-spectral numerical code. The simulation starts with a non-monochromatic, large amplitude Alfvén wave polarized on the $yz$ plane, propagating in a uniform background magnetic field. Successively, the instability was triggered by adding some noise of the order $10-6$ to the initial density level.

During the first part of the evolution of the instability the amplitude of unstable modes is small and, consequently, non-linear couplings are negligible. A subsequent exponential growth, predicted by the linear theory, increases the level of both $- e$ and density compressive fluctuations. During the second part of the development of the instability, non-linear couplings are not longer disregardable and their effect is firstly to slow down the exponential growth of unstable modes and then to saturate the instability to a level that depends on the value of the plasma $b$ .

Spectra of $± e$ are shown in Figure 61 for different times during the development of the instability. At the beginning the spectrum of the mother-wave is peaked at $k = 10$ , and before the instability saturation $(t < 35)$ the back-scattered $e-$ and the density fluctuations $er$ are peaked at $k = 1$ and $k = 11$ , respectively. After saturation, as the run goes on, the spectrum of $e-$ approaches that of $e+$ towards a common final state characterized by a Kolmogorov-like spectrum and $e+$ slightly larger than $- e$ .

Figure 61:

Spectra of $+ e$ (thick line), $- e$ (dashed line), and $r e$ (thin line) are shown for 6 different times during the development of the instability. For $t > 50$ a typical Kolmogorov slope appears. These results refer to $b = 1$ (figure adopted from Malara et al., 2001

The behavior of outward and inward modes, density and magnetic magnitude variances and the normalized cross-helicity $sc$ is summarized in the left column of Figure 62

. The evolution of $sc$ , when the instability reaches saturation, can be qualitatively compared with Ulysses observations (courtesy of B. Bavassano) in the right panel of the same figure, which shows a similar trend.

Figure 62:

Top left panel: time evolution of $+ e$ (solid line) and $- e$ (dashed line). Middle left panel: density (solid line) and magnetic magnitude (dashed line) variances. Bottom left panel: normalized cross helicity $sc$ . Right panel: Ulysses observations of $sc$ radial evolution within the polar wind (left column is from Malara et al., 2001

, right panel is a courtesy of B. Bavassano).

Obviously, making this comparison, one has to take into account that this model has strong limitations like the presence of a peak in $+ e$ not observed in real polar turbulence. Another limitation, partly due to dissipation that has to be included in the model, is that the spectra obtained at the end of the instability growth are steeper than those observed in the solar wind. Finally, a further limitation is represented by the fact that this code is 1D.

In addition, Umeki and Terasawa (1992 ) studying the non-linear evolution of a large-amplitude incoherent Alfvén wave via 1D magnetohydrodynamic simulations, reported that while in a low beta plasma ( $b ~~ 0.2$ ) the growth of backscattered Alfvén waves, which are opposite in helicity and propagation direction from the original Alfvén waves, could be clearly detected, in a high beta plasma ( $b ~~ 2$ ) there was no production of backscattered Alfvén waves. Consequently, although numerical results obtained by Malara et al. (2001 ) are very encouraging, the high beta plasma ( $b ~~ 2$ ), characteristic of fast polar wind at solar minimum, plays against a relevant role of parametric instability in developing solar wind turbulence as observed by Ulysses. However, these simulations do remain an important step forward towards the understanding of turbulent evolution in the polar wind until other mechanisms will be found to be active enough to justify the observations shown in Figure 55 .