8.2 Probability distribution functions

As said in Section 7.2 the statistics of turbulent flows can be characterized by the PDF of field differences over varying scales. At large scales PDFs are Gaussian, while tails become higher than Gaussian (actually, PDFs decay as $exp[- dZ ±] l$ ) at smaller scales.

Marsch and Tu (1994 ) started to investigate the behavior of PDFs of fluctuations against scales and they found that PDFs are rather spiky at small scales and quite Gaussian at large scales. The same behavior have been obtained by Sorriso-Valvo et al. (1999 , 2001 ) who investigated Helios 2 data for both velocity and magnetic field.

In order to make a quantitative analysis of the energy cascade leading to the scaling dependence of PDFs just described, the distributions obtained in the solar wind have been fitted (Sorriso-Valvo et al., 1999 ) by using the log-normal ansatz

The width of the log-normal distribution of $s$ is given by $2 V~ -----2- c (l) = <(ds) >$ , while $s0$ is the most probable value of $s$ .

Table 5:

The values of the parameters $s0$ , $m$ , and $g$ , in the fit of $c2(t)$ (see Equation (45

) as a kernel for the scaling behavior of PDFs. FW and SW refer to fast and slow wind, respectively, as obtained from the Helios 2 spacecraft, by collecting in a single dataset all periods.







parameter	B field (SW)	V fiele (SW)	B field (FW)	V field (FW)









$s 0$	$0.90 ± 0.05$	$0.95 ± 0.05$	$0.85 ± 0.05$	$0.90 ± 0.05$
$m$	$0.75 ± 0.03$	$0.38 ± 0.02$	$0.90 ± 0.03$	$0.54 ± 0.03$
$g$	$0.18 ± 0.03$	$0.20 ± 0.04$	$0.19 ± 0.02$	$0.44 ± 0.05$

The Equation (42 ) has been fitted to the experimental PDFs of both velocity and magnetic intensity, and the corresponding values of the parameter $c$ have been recovered. In Figures 84 and 85 the solid lines show the curves relative to the fit. It can be seen that the scaling behavior of PDFs, in all cases, is very well described by Equation (42 ). At every scale $r$ , we get a single value for the width $2 c (r)$ , which can be approximated by a power law $2 -g c (r) = mr$ for $r < 1 h$ , as it can be seen in Figure 86 . The values of parameters $m$ and $g$ obtained in the fit, along with the values of $s0$ , are reported in Table 5. The fits have been obtained in the range of scales $t < 0.72 h$ for the magnetic field, and $t < 1.44 h$ for the velocity field. The analysis of PDFs shows once more that magnetic field is more intermittent than the velocity field.

Figure 84:

Normalized PDFs of fluctuations of the longitudinal velocity field at four different scales $t$ . Solid lines represent the fit made by using the log-normal model (adopted from Sorriso-Valvo et al., 1999

Figure 85:

We show the normalized PDFs of fluctuations of the magnetic field magnitude at four different scales $t$ as indicated in the different panels. Solid lines represent the fit made by using the log-normal model (adopted from Sorriso-Valvo et al., 1999

Figure 86:

Scaling laws of the parameter $c2(t)$ as a function of the scales $t$ , obtained by the fits of the PDFs of both velocity and magnetic variables (see Figures 84

and 85

The same analysis has been repeated by Forman and Burlaga (2003

). These authors used $64 s$ averages of radial solar wind speed reported by the SWEPAM instrument on the ACE spacecraft, increments have been calculated over a range of lag times from $64 s$ to several days. From the PDF obtained through the Equation (45

) authors calculated the structure functions and compared the free parameters of the model with the scaling exponents of the structure functions. Then a fit on the scaling exponents allows to calculate the values of $2 c$ and $s0$ . Once these parameters have been calculated, the whole PDF is evaluated. The same authors found that the PDFs do not precisely fit the data, at least for large values of the moment order. Interesting enough, Forman and Burlaga (2003 ) investigated the behavior of PDFs when different kernels $Gc(s)$ , derived from different cascade models, are taken into account in Equation (42

). They discussed the physical content of each model, concluding that a cascade model derived from lognormal or log-Lévy theories¹⁰ , modified by self-organized criticality proposed by Schertzer et al. (1997 ), seems to avoid all problems present in other cascade models.