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5.1 Local production of Alfvénic turbulence in the ecliptic

The discovery of the strong correlation between velocity and magnetic field fluctuations has represented the motivation for some MHD numerical simulations, aimed to confirm the conjecture by Dobrowolny et al. (1980bJump To The Next Citation Point). The high level of correlation seems to be due to a kind of self-organization (dynamical alignment) of MHD turbulence, generated by the natural evolution of MHD towards the strongest attractive fixed point of equations (Ting et al., 1986Jump To The Next Citation PointCarbone and Veltri, 19871992Jump To The Next Citation Point). Numerical simulations (Carbone and Veltri, 1992Ting et al., 1986) confirmed this conjecture, say MHD turbulence spontaneously can tends towards a state were correlation increases, that is, the quantity sc = 2Hc/E, where Hc is the cross-helicity and E the total energy of the flow (see Appendix 13.1), tends to be maximal.

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 1 to 10 AU 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. (1991Jump To The Next Citation Point1992Jump To The Next Citation Point) and was based on velocity shear generation. The suggestion to adopt such a mechanism came from a detailed analysis made by Roberts et al. (1987aJump To The Next Citation Point,bJump To The Next Citation Point) of Helios and Voyager interplanetary observations of the radial evolution of the normalized cross-helicity sc 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 8 AU (Roberts et al., 1987bJump To The Next Citation Point). In particular, the vicinity of Helios trajectory to the interplanetary current sheet, characterized by low velocity flow, suggested Roberts et al. (1991Jump To The Next Citation Point) 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 6 lowest ± Z Fourier modes as shown in Figure 59View Image. An initial population of purely outward propagating Alfvénic fluctuations (+ z) was added at large k and was characterized by a spectral slope of k -1. No inward modes were present in the same range. Results of Figure 59View Image show that the time evolution of z+ spectrum is quite rapid at the beginning, towards a steeper spectrum, and slows down successively. At the same time, z- modes are created by the generation mechanism at higher and higher k but, along a Kolmogorov-type slope - 5/3 k.

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Figure 59: Time evolution of the power density spectra of z+ and z- showing the turbulent evolution of the spectra due to velocity shear generation (adopted from Roberts et al., 1991Jump To The Next Citation Point).
These results, although obtained from simulations performed using 2D incompressible spectral and pseudo-spectral codes, with fairly small Reynolds number of Re -~ 200, were similar to the spectral evolution observed in the solar wind (Marsch and Tu, 1990a). Moreover, spatial averages across the simulation box revealed a strong cross-helicity depletion right across the slow wind, representing the heliospheric current sheet. However, magnetic field inversions and even relatively small velocity shears would largely affect an initially high Alfvénic flow (Roberts et al., 1992Jump To The Next Citation Point). However, Bavassano and Bruno (1992) studied an interaction region, repeatedly observed between 0.3 and 0.9 AU, characterized by a large velocity shear and previously thought to be a good candidate for shear generation (Bavassano and Bruno, 1989Jump To The Next Citation Point). They concluded that, even in the hypothesis of a very fast growth of the instability, inward modes would not have had enough time to fill up the whole region as observed by Helios 2.

The above simulations by Roberts et al. (1991Jump To The Next Citation Point) 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 - z 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, 1993bJump To The Next Citation PointBruno et al., 1996Jump To The Next Citation Point). Moreover, their results did not show the flattening of e-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. (1996Jump To The Next Citation Point2000Jump To The Next Citation Point) 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, 1991Del Zanna et al., 2001Jump To The Next Citation PointDel 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 dr /= 0 and magnetic intensity fluctuations d|B |/= 0.

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 tanh-like field), periodic boundaries conditions should be used with some care. Roberts et al. (1991Jump To The Next Citation Point1992Jump To The Next Citation Point) 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 sc is stopped.

A numerical model treating the evolution of e+and e-, including parametric decay of e+, was presented by Marsch and Tu (1993aJump To The Next Citation Point). 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 e+and e-are included and solved self-consistently, can only steepen the spectra at high frequency. Results from this model, shown in Figure 60View Image, partially reproduce the observed evolution of the normalized cross-helicity. While the radial evolution of e+is correctly reproduced, the behavior of - eshows an over-production of inward modes between 0.6 and 0.8 AU probably due to an overestimation of the strength of the pump-wave. However, the model is applied to the situation observed by Helios at 0.3 AU where a rather flat e-spectrum already exists.

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Figure 60: Radial evolution of + eand - espectra obtained from the Marsch and Tu (1993aJump To The Next Citation Point) model, in which a parametric decay source term was added to the Tu’s model (Tu et al., 1984Jump To The Next Citation Point) that was, in turn, extended by including both spectrum equations for e+and e -and solved them self-consistently (adopted from Marsch and Tu, 1993aJump To The Next Citation Point, © 1993 American Geophysical Union, reproduced by permission of American Geophysical Union).

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