Despite the above considerations, Yaglom’s law results surprisingly verified in some solar wind samples. Results of the occurrence of Yaglom’s law in the ecliptic plane, has been reported by MacBride et al. (2008, 2010) and Smith et al. (2009) and, independently, in the polar wind by Sorriso-Valvo et al. (2007). It is worthwhile to note that, the occurrence of Yaglom’s law in polar wind, where fluctuations are Alfvénic, represents a double surprising feature because, according to the usual phenomenology of MHD turbulence, a nonlinear energy cascade should be absent for Alfénic turbulence.
In a first attempt to evaluate phenomenologically the value of the energy dissipation rate, MacBride et al. (2008) analyzed the data from ACE to evaluate the occurrence of both the Kolmogorov’s 4/5-law and their MHD analog (39). Although some words of caution related to spikes in wind speed, magnetic field strength caused by shocks and other imposed heliospheric structures that constitute inhomogeneities in the data, authors found that both relations are more or less verified in solar wind turbulence. They found a distribution for the energy dissipation rate, defined in the above paper as , with an average of about .
In order to avoid variations of the solar activity and ecliptic disturbances (like slow wind sources, coronal mass ejections, ecliptic current sheet, and so on), and mainly mixing between fast and slow wind, Sorriso-Valvo et al. (2007) used high speed polar wind data measured by the Ulysses spacecraft. In particular, authors analyze the first seven months of 1996, when the heliocentric distance slowly increased from 3 AU to 4 AU, while the heliolatitude decreased from about to . The third-order mixed structure functions have been obtained using 10-days moving averages, during which the fields can be considered as stationary. A linear scaling law, like the one shown in Figure 92, has been observed in a significant fraction of samples in the examined period, with a linear range spanning more than two decades. The linear law generally extends from few minutes up to 1 day or more, and is present in about 20 periods of a few days in the 7 months considered. This probably reflects different regimes of driving of the turbulence by the Sun itself, and it is certainly an indication of the nonstationarity of the energy injection process. According to the formal definition of inertial range in the usual fluid flows, authors attribute to the range where Yaglom’s law appear the role of inertial range in the solar wind turbulence (Sorriso-Valvo et al., 2007). This range extends on scales larger than the usual range of scales where a Kolmogorov relation has been observed, say up to about few hours (cf. Figure 25).
Several other periods are found where the linear scaling range is reduced and, in particular, the sign of is observed to be either positive or negative. In some other periods the linear scaling law is observed either for or rather than for both quantities. It is worth noting that in a large fraction of cases the sign switches from negative to positive (or viceversa) at scales of about 1 day, roughly indicating the scale where the small scale Alfvénic correlations between velocity and magnetic fields are lost. This should indicate that the nature of fluctuations changes across the break. The values of the pseudo-energies dissipation rates has been found to be of the order of magnitude about few hundreds of J/Kg s, higher than that found in usual fluid flows which result of the order of .
The occurrence of Yaglom’s law in solar wind turbulence has been evidenced by a systematic study by MacBride et al. (2010), which, using ACE data, found a reasonable linear scaling for the mixed third-order structure functions, from about 64 s. to several hours at 1 AU in the ecliptic plane. Assuming that the third-order mixed structure function is perpendicular to the mean field, or assuming that this function varies only with the component of the scale that is perpendicular to the mean field, and is cylindrically symmetric, the Yaglom’s law would reduce to a 2D state. On the other hand, if the third-order function is parallel to the mean field or varies only with the component of the scale that is parallel to the mean field, the Yaglom’slaw would reduce to a 1D-like case. In both cases the result will depend on the angle between the average magnetic field and the flow direction. In both cases the energy cascade rate varies in the range (see MacBride et al., 2010, for further details).
Quite interestingly, Smith et al. (2009) found that the pseudo-energy cascade rates derived from Yaglom’s scaling law reveal a strong dependence on the amount of cross-helicity. In particular, they showed that when the correlation between velocity and magnetic fluctuations are higher than about 0.75, the third-order moment of the outward-propagating component, as well as of the total energy and cross-helicity are negative. As already made by Sorriso-Valvo et al. (2007), they attribute this phenomenon to a kind of inverse cascade, namely a back-transfer of energy from small to large scales within the inertial range of the dominant component. We should point out that experimental values of energy transfer rate in the incompressive case, estimated with different techniques from different data sets (Vasquez et al., 2007; MacBride et al., 2010), are only partially in agreement with that obtained by Sorriso-Valvo et al. (2007). However, the different nature of wind (ecliptic vs. polar, fast vs. slow, at different radial distances from the Sun) makes such a comparison only indicative.
As far as the scaling law (46) is concerned, Carbone et al. (2009a) found that a linear scaling for as defined in (46), appears almost in all Ulysses dataset. In particular, the linear scaling for is verified even when there is no scaling at all for (39). In particular, it has been observed (Carbone et al., 2009a) that a linear scaling for appears in about half the whole signal, while displays scaling on about a quarter of the sample. The linear scaling law generally extends on about two decades, from a few minutes up to one day or more, as shown in Figure 93. At variance to the incompressible case, the two fluxes coexist in a large number of cases. The pseudo-energies dissipation rates so obtained are considerably larger than the relative values obtained in the incompressible case. In fact it has been found that on average . This result shows that the nonlinear energy cascade in solar wind turbulence is considerably enhanced by density fluctuations, despite their small amplitude within the Alfvénic polar turbulence. Note that the new variables are built by coupling the Elsässer fields with the density, before computing the scale-dependent increments. Moreover, the third-order moments are very sensitive to intense field fluctuations, that could arise when density fluctuations are correlated with velocity and magnetic field. Similar results, but with a considerably smaller effect, were found in numerical simulations of compressive MHD (Mac Low and Klessen, 2004).
Finally, it is worth reporting that the presence of Yaglom’s law in solar wind turbulence is an interesting theoretical topic, because this is the first real experimental evidence that the solar wind turbulence, at least at large-scales, can be described within the magnetohydrodynamic model. In fact, Yaglom’s law is an exact law derived from MHD equations and, let us say once more, their occurrence in a medium like the solar wind is a welcomed surprise. By the way, the presence of the law in the polar wind solves the paradox of the presence of Alfvénic turbulence as first pointed out by Dobrowolny et al. (1980a). Of course, the presence of Yaglom’s law generates some controversial questions about data selection, reliability and a brief discussion on the extension of the inertial range. The interested reader can find some questions and relative answers in Physical Review Letters (Forman et al., 2010; Sorriso-Valvo et al., 2010a).