List Of Footnotes

1		This concept will be explained better in the next sections.
2		A fluid particle is defined as an infinitesimal portion of fluid which moves with the local velocity. As usual in fluid dynamics, infinitesimal means small with respect to large scale, but large enough with respect to molecular scales.
3		The translation of the original paper by Kolmogorov (1941 ) can be found in the book by Hunt et al. (1991 ).
4		These authors were the first ones to use physical technologies and methodologies to investigate turbulent flows from an experimental point of view. Before them, experimental studies on turbulence were motivated mainly by engineering aspects.
5		We can use a different definition for the third invariant $H(t)$ , for example a quantity positive defined, without the term $n (- 1)$ and with $a = 2$ . This can be identified as the surrogate of the square of the vector potential, thus investigating a kind of 2D MHD. In this case, we obtain a shell model with $c = 2$ , $a = 5/4$ , and $c = -1/3$ . However, this model does not reproduce the inverse cascade of the square of magnetic potential observed in the true 2D MHD equations.
6		We have already defined fluctuations of a field as the difference between the field itself and its average value. This quantity has been defined as $dy$ . Here, the differences $dyl$ of the field separated by a distance $l$ represents characteristic fluctuations at the scale $l$ , say characteristic fluctuations of the field across specific structures (eddies) that are present at that scale. The reader can realize the difference between both definitions.
7		To be precise, it is worth remarking that there are no convincing arguments to identify as inertial range the intermediate range of frequencies where the observed spectral properties are typical of fully developed turbulence. From a theoretical point of view, here the association “intermediate range” $-~$ “inertial range” is somewhat arbitrary as it can be inferred from the short discussion given in Section 2.10.
8		Since the solar wind moves at supersonic speed $Vsw$ , the usual Taylor’s hypothesis is verified, and we can get information on spatial ( $l$ ) scaling laws by using time differences $r = l/Vsw$ .
9		It is worthwhile to remark that neither the fluid relation (26 ) nor its MHD counterpart (27 ) are satisfied in the solar wind. Namely there is not any extended range of scales, from which we can derive scaling exponents, where the above relations which formally define the inertial range are verified. Here we are in a situation similar to a low-Reynolds number fluid flow.
10		The log-Lévy model is a modification of the lognormal model. In such case, the central limit theorem is used to derive the limit distribution of an infinite sum of stochastic variables by relaxing the hypothesis of finite variance usually used. The resulting limit function is a Lévy function.