## 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 (Kolmogorov, 1991) of the original paper by Kolmogorov (1941) can also 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 , for example a quantity positive defined, without the term and with . 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 , , and . 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 . Here, the differences of the field separated by a distance represents characteristic fluctuations at the scale , 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 again 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. Really an operative definition of inertial range of turbulence is the range of scales where relation (41) (for fluid flows) or (40) (for MHD flows) is verified. | |

8 | Since the solar wind moves at supersonic speed , the usual Taylor’s hypothesis is verified, and we can get information on spatial scaling laws by using time differences . | |

9 | Note that, according to the occurrence of the Yaglom’s law, that is a third-order moment is different from zero, the fluctuations at a given scale in the inertial range must present some non-Gaussian features. From this point of view the calculation of structure functions with the absolute value is unappropriate because in this way we risk to cancel out non-Gaussian features. Namely we symmetrize the probability density functions of fluctuations. However, in general, the number of points at disposal is much lower than required for a robust estimate of odd structure functions, even in usual fluid flows. Then, as usually, we will obtain structure functions by taking the absolute value, even if some care must be taken in certain conclusions which can be found in literature. | |

10 | The lognormal model is derived by using a multiplicative process, where random variable generates the cascade. Then, according to the Central Limit Theorem, the process converges to a lognormal distribution of finite variance. 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 random variables by relaxing the hypothesis of finite variance usually used. The resulting limit function is a Lévy function. | |

11 | For a discussion on non-turbulent mechanism of solar wind heating cf. Tu and Marsch (1995a). | |

12 | Of course, this is based on classical turbulence. As said before, in the solar wind the dissipative term is unknown, even if it might happens at very small kinetic scales. | |

13 | It is worthwhile to remark that a turbulent fluid flows is out of equilibrium, say the cascade requires the injection of energy (input) and a dissipation mechanism (output), usually lying on well separated scales, along with a transfer of energy. Without input and output, the nonlinear term of equations works like an energy redistribution mechanism towards an equilibrium in the wave vectors space. This generates an equilibrium energy spectrum which should in general be the same as that obtained when the cascade is at work (cf., e.g., Frisch et al., 1975). However, even if the turbulent spectra could be anticipated by looking at the equilibrium spectra, the physical mechanisms are different. Of course, this should also be the case for the Hall MHD. |