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). 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., 1986; Carbone and Veltri, 1987, 1992). Numerical simulations
(Carbone and Veltri, 1992; Ting et al., 1986) confirmed this conjecture, say MHD turbulence
spontaneously can tends towards a state were correlation increases, that is, the quantity 
,
where 
is the cross-helicity and 
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 
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
.
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, 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., 1992). However, Bavassano and
Bruno (1992) studied an interaction region, repeatedly observed between 
and 
, characterized
by a large velocity shear and previously thought to be a good candidate for shear generation (Bavassano
and Bruno, 1989). 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. (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.
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http://www.livingreviews.org/lrsp-2005-4 |
© Max Planck Society and the author(s)
Problems/comments to |
and 
showing the turbulent
evolution of the spectra due to velocity shear generation (adopted from 
and 
spectra obtained from the 
and 
and solved
them self-consistently (adopted from