The picture of the evolution of incompressible MHD turbulence, which comes out is rather nice but solar wind turbulence displays a more complicated behavior. In particular, as we have reported above, observations seems to point out that solar wind evolves in the opposite way. The correlation is high near the Sun, at larger radial distances, from to the correlation is progressively lower, while the level in fluctuations of mass density and magnetic field intensity increases. What is more difficult to understand is why correlation is progressively destroyed in the solar wind, while the natural evolution of MHD is towards a state of maximal normalized cross-helicity. A possible solution can be found in the fact that solar wind is neither incompressible nor statistically homogeneous, and some efforts to tentatively take into account more sophisticated effects have been made.
A mechanism, responsible for the radial evolution of turbulence, was suggested by Roberts and Goldstein (1988), Goldstein et al. (1989), and Roberts et al. (1991, 1992) and was based on velocity shear generation. The suggestion to adopt such a mechanism came from a detailed analysis made by Roberts et al. (1987a,b) of Helios and Voyager interplanetary observations of the radial evolution of the normalized cross-helicity at different time scales. Moreover, Voyager’s observations showed that plasma regions, which had not experienced dynamical interactions with neighboring plasma, kept the Alfvénic character of the fluctuations at distances as far as (Roberts et al., 1987b). In particular, the vicinity of Helios trajectory to the interplanetary current sheet, characterized by low velocity flow, suggested Roberts et al. (1991) to include in his simulations a narrow low speed flow surrounded by two high speed flows. The idea was to mimic the slow, equatorial solar wind between north and south fast polar wind. Magnetic field profile and velocity shear were reconstructed using the lowest Fourier modes as shown in Figure 59. An initial population of purely outward propagating Alfvénic fluctuations () was added at large and was characterized by a spectral slope of . No inward modes were present in the same range. Results of Figure 59 show that the time evolution of spectrum is quite rapid at the beginning, towards a steeper spectrum, and slows down successively. At the same time, modes are created by the generation mechanism at higher and higher but, along a Kolmogorov-type slope .
The above simulations by Roberts et al. (1991) were successively implemented with a compressive pseudo-spectral code (Ghosh and Matthaeus, 1990) which provided evidence that, during this turbulence evolution, clear correlations between magnetic field magnitude and density fluctuations, and between and density fluctuations should arise. However, such a clear correlation, by-product of the non-linear evolution, was not found in solar wind data (Marsch and Tu, 1993b; Bruno et al., 1996). Moreover, their results did not show the flattening of spectrum at higher frequency, as observed by Helios (Tu et al., 1989b). As a consequence, velocity shear alone cannot explain the whole phenomenon, other mechanisms must also play a relevant role in the evolution of interplanetary turbulence.
Compressible numerical simulations have been performed by Veltri et al. (1992) and Malara et al. (1996, 2000) which invoked the interactions between small scale waves and large scale magnetic field gradients and the parametric instability, as characteristic effects to reduce correlations. In a compressible, statistically inhomogeneous medium such as the heliosphere, there are many processes which tend to destroy the natural evolution toward a maximal correlation, typical of standard MHD. In such a medium an Alfvén wave is subject to parametric decay instability (Viñas and Goldstein, 1991; Del Zanna et al., 2001; Del Zanna, 2001), which means that the mother wave decays in two modes: i) a compressive mode that dissipates energy because of the steepening effect, and ii) a backscattered Alfvénic mode with lower amplitude and frequency. Malara et al. (1996) showed that in a compressible medium, the correlation between the velocity and the magnetic field fluctuations is reduced because of the generation of the backward propagating Alfvénic fluctuations, and of a compressive component of turbulence, characterized by density fluctuations and magnetic intensity fluctuations .
From a technical point of view it is worthwhile to remark that, when a large scale field which varies on a narrow region is introduced (typically a -like field), periodic boundaries conditions should be used with some care. Roberts et al. (1991, 1992) used a double shear layer, while Malara et al. (1992) introduced an interesting numerical technique based on both the glue between two simulation boxes and a Chebyshev expansion, to maintain a single shear layer, say non periodic boundary conditions, and an increased resolution where the shear layer exists.
Grappin et al. (1992) observed that the solar wind expansion increases the lengths normal to the radial direction, thus producing an effect similar to a kind of inverse energy cascade. This effect perhaps should be able to compete with the turbulent cascade which transfers energy to small scales, thus stopping the non-linear interactions. In absence of non-linear interactions, the natural tendency towards an increase of is stopped.
A numerical model treating the evolution of and , including parametric decay of , was presented by Marsch and Tu (1993a). The parametric decay source term was added in order to reproduce the decreasing cross-helicity observed during the wind expansion. As a matter of fact, the cascade process, when spectrum equations for both and are included and solved self-consistently, can only steepen the spectra at high frequency. Results from this model, shown in Figure 60, partially reproduce the observed evolution of the normalized cross-helicity. While the radial evolution of is correctly reproduced, the behavior of shows an over-production of inward modes between and probably due to an overestimation of the strength of the pump-wave. However, the model is applied to the situation observed by Helios at where a rather flat spectrum already exists.
© Max Planck Society and the author(s)