Most of these observations are consistent with the general understanding that, for increasing distance from the Sun, the turbulent power-law spectrum expands toward larger scales (Bavassano et al. 1982; Bruno & Carbone 2013; Chen et al. 2020). However, spectral properties may be misleading, as it is not possible to unequivocally ascribe the Kolmogorov-like power-law scaling to the presence of a turbulent cascade. For example, it is universally observed that turbulence is associated with intermittency (Kolmogorov 1962), as is routinely observed in solar wind measurements (Sorriso-Valvo et al. 1999; Bruno & Carbone 2013). A standard way for characterizing the intermittency of a field ϕ (ϕ being, for example, a velocity or magnetic field component) is by means of the scale-dependent increments Δϕ = ϕ(t + Δt) − ϕ(t), which account for the presence of gradients on a timescale Δt (Anselmet et al. 1984). Intermittency is related to the scale-dependent shape of the probability distribution function of the increments Δϕ (Castaing et al. 1990). This, additionally, implies the existence of nonvanishing odd moments. In particular, a scaling law can be derived for the third-order moment directly from the dynamical MHD equations, as the conservation law of the appropriate inviscid invariants (de Karman & Howarth 1938; Danaila et al. 2001). Such a relation, known in the MHD description as the Politano–Pouquet (PP) law (Politano & Pouquet 1998; Carbone et al. 2009), establishes that under the hypothesis of homogeneity, stationarity, local isotropy, and incompressibility, in the turbulent inertial range the mixed third-order moment of the increments of velocity (v) and magnetic field (in velocity units, 




b
=
B

/



4
π
ρ




, with B the magnetic field vector and ρ the plasma mass density) is a linear function of the scale Δt. Moreover, the proportionality coefficient is related to the mean energy transfer rate of the turbulent cascade. This can be written as
1







Y
(
Δ
t
)


≡


〈
Δ


v


R


(
∣
Δ
v


∣


2


+
∣
Δ
b


∣


2


)








−
2
Δ


b


R


(
Δ
v
·
Δ
b
)
〉
=



4


3



ε


V


sw


Δ
t
.





Here Δv and Δb are scale-dependent vector increments of the plasma velocity and magnetic field, as defined above; ΔvR = vR(t + Δt) − vR(t) and ΔbR = bR(t + Δt) − bR(t) are velocity and magnetic field longitudinal increments measured in the sampling direction R; ε is the mean energy transfer rate; and brackets indicate ensemble average. Note that the solar wind speed Vsw is used for switching between space scales, ℓ, and timescales, Δt, through the Taylor hypothesis, ℓ = VswΔt (Taylor 1938). This also results in the reversal of the sign in the left-hand side of Equation (1) with respect to the traditional formulation in terms of spatial increments. The PP law is a fundamental relation for MHD turbulence, since it describes the energy cascade and ultimately allows us to estimate the energy that will be dissipated at small scales. It is particularly relevant for solar wind turbulence, where the collisionless processes responsible for removing the energy at the bottom of the nonlinear cascade are not yet fully understood (Chen et al. 2019; Sorriso-Valvo et al. 2019; Matthaeus et al. 2020; Smith & Vasquez 2021). The linear relation (1) has been observed in various regions of the heliosphere, for different conditions of the space plasma, confirming the turbulent nature of their dynamics and providing an estimate of the energy transfer rate (MacBride et al. 2005; Sorriso-Valvo et al. 2007; Marino et al. 2008; Smith et al. 2009; Stawarz et al. 2010; Coburn et al. 2012; Bandyopadhyay et al. 2018; Hadid et al. 2018; Andrés et al. 2019; Bandyopadhyay et al. 2020; Sorriso-Valvo et al. 2021).
Citation: Castaing et al. 1990
Citation function: Background; Background