9.1 Radial evolution of intermittency in the ecliptic

Marsch and Liu (1993

) investigated for the first time solar wind scaling properties in the inner heliosphere. These authors provided some insights on the different intermittent character of slow and fast wind, on the radial evolution of intermittency, and on the different scaling characterizing the three components of velocity. In particular, they found that fast streams were less intermittent than slow streams and the observed intermittency showed a weak tendency to increase with heliocentric distance. They also concluded that the Alfvénic turbulence observed in fast streams starts from the Sun as self-similar but then, during the expansion, decorrelates becoming more multifractal. This evolution was not seen in the slow wind, supporting the idea that turbulence in fast wind is mainly made of Alfvén waves and convected structures (Tu and Marsch, 1993 ), as already inferred by looking at the radial evolution of the level of cross-helicity in the solar wind (Bruno and Bavassano, 1991 ).

Bruno et al. (2003a ) investigated the radial evolution of intermittency in the inner heliosphere, using the behavior of the flatness of the PDF of magnetic field and velocity fluctuations as a function of scale. As a matter of fact, probability distribution functions of fluctuating fields affected by intermittency become more and more peaked at smaller and smaller scales. Since the peakedness of a distribution is measured by its flatness factor, they studied the behavior of this parameter at different scales to estimate the degree of intermittency of their time series, as suggested by Frisch (1995 ).

In order to study intermittency they computed the following estimator of the flatness factor $F$ :

where $t$ is the scale of interest and $Sp = < |V (t + t) - V (t)|p > t$ is the structure function of order $p$ of the generic function $V(t)$ . They considered a given function to be intermittent if the factor $F$ increased when considering smaller and smaller scales or, equivalently, higher and higher frequencies.

In particular, vector field, like velocity and magnetic field, encompasses two distinct contributions, a compressive one due to intensity fluctuations that can be expressed as $d| B(t, t)|= |B(t + t )| - |B(t)|$ , and a directional one due to changes in the vector orientation $V~ s um ----------------------2- dB(t, t) = i=x,y,z(Bi(t + t)- Bi(t))$ . Obviously, relation $dB(t, t)$ takes into account also compressive contributions, and the expression $dB(t, t) > |d| B(t, t)||$ is always true.

Figure 94:

Flatness $F$ vs. time scale $t$ relative to magnetic field fluctuations. The left column (panels A and C) refers to slow wind and the right column (panels B and D) refers to fast wind. The upper panels refer to compressive fluctuations and the lower panels refer to directional fluctuations. Vertical bars represent errors associated with each value of $F$ . The three different symbols in each panel refer to different heliocentric distances as reported in the legend (adopted from Bruno et al., 2003b

Figure 95:

Flatness $F$ vs. time scale $t$ relative to wind velocity fluctuations. In the same format of Figure 94

panels A and C refer to slow wind and panels B and D refer to fast wind. The upper panels refer to compressive fluctuations and the lower panels refer to directional fluctuations. Vertical bars represent errors associated with each value of $F$ (adopted from Bruno et al., 2003b

Looking at Figures 94

and 95

, taken from the work of Bruno et al. (2003a

), the following conclusions can be drawn:

Magnetic field fluctuations are more intermittent than velocity fluctuations.
Compressive fluctuations are more intermittent than directional fluctuations.
Slow wind intermittency does not show appreciable radial dependence.
Fast wind intermittency, for both magnetic field and velocity, clearly increases with distance.
Magnetic and velocity fluctuations have a rather Gaussian behavior at large scales, as expected, regardless of type of wind or heliocentric distance.

Moreover, they also found that the intermittency of the components rotated into the mean field reference system (see Appendix 15.1) showed that the most intermittent component of the magnetic field is the one along the mean field, while the other two show a similar level of intermittency within the associated uncertainties. This different behavior is then enhanced for larger heliocentric distances. These results agree with conclusions drawn by Marsch and Tu (1994 ) who, analyzing fast and slow wind at $0.3 AU$ , found that the PDFs of the fluctuations of transverse components of both velocity and magnetic fields, constructed for different time scales, were appreciably more Gaussian-like than fluctuations observed for the radial component, which resulted to be more and more spiky for smaller and smaller scales. However, this difference between radial and transverse components seemed to vanish with increasing heliocentric distance, and Tu et al. (1996 ) could not establish a clear radial trend or anisotropy. These results might be reconciled with conclusions by Bruno et al. (2003b ) if the analysis by Tu et al. (1996 ) was repeated in the mean field reference system. The reason is that components normal to the mean field direction are more influenced by Alfvénic fluctuations and, as a consequence, their fluctuations are more stochastic and less intermittent. This effect largely reduces during the radial excursion mainly because the Solar Ecliptic (SE) reference system is not the most appropriate one for studying magnetic field fluctuations, and a cross-talking between different components is artificially introduced. As a matter of fact, the presence of the large scale spiral magnetic field breaks the spatial symmetry introducing a preferential direction parallel to the mean field. The same Bruno et al. (2003b ) showed that it was not possible to find a clear radial trend unless magnetic field data were rotated into this more natural reference system.

On the other hand, it looks more difficult to reconcile the radial evolution of intermittency found by Bruno et al. (2003b ) and Marsch and Liu (1993 ) in fast wind with conclusions drawn by Tu et al. (1996 ), who stated that “Neither a clear radial evolution nor a clear anisotropy can be established. The value of P1 in high-speed and low-speed wind are not prominent different.”. However, it is very likely that the conclusions given above are related with how to deal with the flat slope of the spectrum in fast wind near $0.3 AU$ . Tu et al. (1996 ) concluded, indeed: “It should be pointed out that the extended model cannot be used to analyze the intermittency of such fluctuations which have a flat spectrum. If the index of the power spectrum is near or less than unity $...$ P1 would be $0.5$ . However, this does not mean there is no intermittency. The model simply cannot be used in this case, because the structure function(1) does not represent the effects of intermittency adequately for those fluctuations which have a flat spectrum and reveal no clear scaling behavior”.

Bruno et al. (2003a ) concluded that the two major ingredients of interplanetary MHD fluctuations are compressive fluctuations due to a sort of underlying, coherent structure convected by the wind, and stochastic Alfvénic fluctuations propagating in the wind. Depending on the type of solar wind sample and on the heliocentric distance, the observed scaling properties would change accordingly. In particular, the same authors suggested that, as the radial distance increases, convected, coherent structures of the wind assume a more relevant role since the Alfvénic component of the fluctuations is depleted. This would be reflected in the increased intermittent character of the fluctuations. The coherent nature of the convected structures would contribute to increase intermittency while the stochastic character of the Alfvénic fluctuations would contribute to decrease it. This interpretation would also justify why compressive fluctuations are always more intermittent than directional fluctuations. As a matter of fact, coherent structures would contribute to the intermittency of compressive fluctuations and, at the same time, would also produce intermittency in directional fluctuations. However, since directional fluctuations are greatly influenced by Alfvénic stochastic fluctuations, their intermittency will be more or less reduced depending on the amplitude of the Alfvén waves with respect to the amplitude of compressive fluctuations.

The radial dependence of the intermittency behavior of solar wind fluctuations stimulated Bruno et al. (1999b ) to reconsider previous investigations on fluctuations anisotropy reported in Section 3.1.3. These authors studied magnetic field and velocity fluctuations anisotropy for the same corotating, high velocity stream observed by Bavassano et al. (1982a ) within the framework of the dynamics of non-linear systems. Using the Local Intermittency Measure (Farge et al., 1990 ; Farge, 1992 ), Bruno et al. (1999b ) were able to justify the controversy between results by Klein et al. (1991 ) in the outer heliosphere and Bavassano et al. (1982a ) in the inner heliosphere. Exploiting the possibility offered by this technique to locate in space and time those events which produce intermittency, these authors were able to remove intermittent events and perform again the anisotropy analysis. They found that intermittency strongly affected the radial dependence of magnetic fluctuations while it was less effective on velocity fluctuations. In particular, after intermittency removal, the average level of anisotropy decreased for both magnetic and velocity field at all distances. Although magnetic fluctuations remained more anisotropic than their kinetic counterpart, the radial dependence was eliminated. On the other hand, the velocity field anisotropy showed that intermittency, although altering the anisotropic level of the fluctuations, does not markedly change its radial trend.