SELF-CONSISTENT CASTAING DISTRIBUTION OF SOLAR WIND TURBULENT FLUCTUATIONS

The interplanetary space is permeated by the solar wind, a rarefied, magnetized plasma continuously expanding from the solar corona. The solar wind blows radially away from the Sun and extends up to about 100 AU, at supersonic and super-Alfvénic speed. Measurements collected by spacecraft instruments during the past decades have evidenced that low-frequency fluctuations have power-law spectra (Coleman 1968). This supports the study of fluctuations in the framework of magnetohydrodynamic (MHD) turbulence (Biskamp 1997; Bruno & Carbone 2013). The MHD nature of the turbulent fluctuations has recently been confirmed by more detailed analysis of the linear scaling of the mixed third-order moment (Politano & Pouquet 1998; MacBride et al. 2005, 2008; Sorriso-Valvo et al. 2007, 2010; Marino et al. 2008, 2011, 2012; Carbone et al. 2009). One of the most interesting properties of solar wind turbulence is the intermittent character of the fluctuations (Frisch 1995) of fields such as velocity, magnetic field, or the Elsasser fields. Intermittency is related to the non-homogeneous generation of energetic structures in the flow, owing to the nonlinear transfer of energy across the scales as observed in geophysical flows (Marino et al. 2013a, 2014, 2015; Pouquet & Marino 2013) and heliospheric plasmas (Bruno & Carbone 2013), whose efficiency can be correlated with levels of cross-helicity (Smith et al. 2009) or self-generated kinetic helicity (Marino et al. 2013b). Being ubiquitous in turbulence, intermittency plays a relevant role in the statistical description of the field fluctuations. The main manifestation of intermittency in fully developed turbulence is the scale-dependent variation of the statistical properties of the field increments, customarily defined as ${\rm{\Delta }}{\psi }_{{\ell }}(x)=\psi (x+{\ell })-\psi (x)$ for a unidimensional generic field $\psi (x)$ at the scale ℓ. In particular, many studies have focused on scaling properties of the probability distribution functions (PDFs) of the field fluctuations, $P({\rm{\Delta }}{\psi }_{{\ell }}$), and of the structure functions, defined as the moments of the distribution function of the field fluctuations ${S}_{p}({\rm{\Delta }}{\psi }_{{\ell }})=\langle {\rm{\Delta }}{\psi }_{{\ell }}{(x)}^{p}\rangle$ (here $\langle \cdot \rangle$ indicates an ensemble average). The intermittent (i.e., non-homogeneous) concentration of turbulent energy on small-scale structures, as the energy is transferred through the scales, results in the enhancement of the PDF tails, indicating that large-amplitude fluctuations are more and more probable at smaller and smaller scale. Correspondingly, the structure functions' scaling exponents deviate from the linear prescription valid for scale-invariant PDFs (Frisch 1995). Such deviation has been studied since early works on turbulence. Methods based on wavelet transform (Farge 1992; Onorato et al. 2000) or threshold techniques (Greco et al. 2009) have been developed for the identification, description, and characterization of the intermittent structures. On the other hand, models for intermittency try to reproduce the shape of the PDFs (Castaing et al. 1990; Benzi et al. 1991; Frisch 1995; Frisch & Sornette 1997) or the structure functions' anomaly (Meneveau & Sreenivasan 1987; She & Lévêque 1994), allowing the quantitative evaluation of intermittency.

In solar wind plasma, intermittency has been extensively characterized in recent years (Tu & Marsch 1995; Marsch & Tu 1997; Sorriso-Valvo et al. 1999, 2001; Hnat et al. 2003; Leubner & Vörös 2005a; Bruno et al. 2007a; Kiyani et al. 2009; Macek & Wawrzaszek 2009; Ragot 2013). As a result of intermittent processes, small-scale structures, such as thin current sheets or tangential discontinuities, have been identified in solar wind (Veltri & Mangeney 1999; Bruno et al. 2004, 2007b; Greco et al. 2009). Their dependence on parameters such as solar activity level, heliocentric distance, heliolatitude, and wind speed has been also pointed out (Pagel & Balogh 2002; Bruno et al. 2003; Chapman et al. 2008; Nicol et al. 2008; Yordanova et al. 2009).

One of the models for the description of the increment PDFs was introduced by Castaing (Castaing et al. 1990) and successfully applied in several contexts (Sorriso-Valvo et al. 1999, 2000, 2001; Padhye et al. 2001; Hnat et al. 2003; Pagel & Balogh 2003; Stepanova et al. 2003; Carbone et al. 2004; Leubner & Vörös 2005a, 2005b). Within the multifractal framework (Frisch 1995), for each scale, the energy transfer rate ${\epsilon }_{{\ell }}$ has non-homogeneous scaling properties, such as the fractal dimension (which is directly related to the cascade efficiency) in different regions of space. The turbulent fields can be thus interpreted as a superposition of subsets, each characterized by a given fractal dimension, and with a typical energy transfer rate. Each region can then be reasonably assumed to have the same distribution P₀ of the field fluctuations, with variable width σ (depending on the cascade efficiency, and related to the local fractal dimension) and weight $L(\sigma )$ (depending on the fraction of space characterized by the same statistics). The Castaing model PDF consists thus of the continuous superposition of such distributions, each contributing to the statistics with its appropriate weight. The latter is introduced through the distribution function of the widths σ (Carbone et al. 2004). This leads, for each timescale ℓ, to the convolution

$$\begin{eqnarray}&&P({\rm{\Delta }}{\psi }_{{\ell }})={\displaystyle \int }_{0}^{\infty }{L}_{\lambda }({\sigma }_{{\ell }}){P}_{0}({\rm{\Delta }}{\psi }_{{\ell }},{\sigma }_{{\ell }})\ d\ \mathrm{ln}{\sigma }_{{\ell }}.\end{eqnarray} \tag{ 1 }$$

Based on empirical large-scale PDF shape, a Gaussian parent distribution is normally used. It is known from turbulence studies that PDFs of fluctuations have to be skewed. Indeed, symmetric PDFs would result in vanishing odd-order moments of the fluctuations, in contrast with experimental results (Frisch 1995). Asymmetric PDFs are also necessary to satisfy the linear scaling of the (non-vanishing) third-order moment of the fluctuations, as required by theoretical results (Frisch 1995; Politano & Pouquet 1998). Thus, in order to account for this, a skewness parameter a_s must also be included, so that

$$\begin{eqnarray}{P}_{0}({\rm{\Delta }}{\psi }_{{\ell }},{\sigma }_{{\ell }},{a}_{{\rm{s}}}) & = & \displaystyle \frac{1}{\sqrt{2\pi \sigma }}\mathrm{exp}\left[-\displaystyle \frac{{\rm{\Delta }}{\psi }_{{\ell }}^{2}}{2{\sigma }^{2}}\right.\\ & & \left.\left(1+{a}_{{\rm{s}}}\displaystyle \frac{{\rm{\Delta }}{\psi }_{{\ell }}/\sigma }{{(1+{\rm{\Delta }}{\psi }_{{\ell }}^{2}/{\sigma }^{2})}^{1/2}}\right)\right].\end{eqnarray} \tag{ 2 }$$

The PDF of variances ${L}_{\lambda }({\sigma }_{{\ell }})$ needs theoretical prescription. A lognormal ansatz has been often used, as conjectured in the framework of the multifractal cascade (Castaing et al. 1990)

$$\begin{eqnarray}&&{L}_{\lambda ({\sigma }_{{\ell }})}=\displaystyle \frac{1}{{\lambda }_{{\ell }}\sqrt{2\pi }}\mathrm{exp}\left[\displaystyle \frac{{\mathrm{ln}}^{2}{\sigma }_{{\ell }}/{\sigma }_{0,{\boldsymbol{r}}}}{2{\lambda }_{{\ell }}^{2}}\right].\end{eqnarray} \tag{ 3 }$$

Such a choice has been justified by assuming that the nonlinear energy transfer is the result of a multifractal fragmentation process, giving rise to a multiplicative hierarchy of energetic structures. By assuming random distribution of the multipliers (namely, of the local efficiency of the cascade), the central limit theorem suggests a lognormal distribution of the local energy transfer rate. Then, from dimensional considerations, the fluctuation variance can be expected to share the same statistical properties as the energy transfer rate, therefore giving the lognormal distribution (3) (Castaing et al. 1990). In Equation (3), for ${\lambda }_{{\ell }}^{2}=0$, the lognormal PDF is a δ-function, so that the convolution (1) gives one Gaussian of width ${\sigma }_{0,{\ell }}$ the most probable value of ${\sigma }_{{\ell }}$. As ${\lambda }_{{\ell }}^{2}$ increases, the convolution includes more and more values of ${\sigma }_{{\ell }}$, and the PDF tails are enhanced. Therefore, the scaling of the parameter ${\lambda }_{{\ell }}^{2}$ controls the shape of the PDF tails and describes the deviation from the parent distribution, characterizing the intermittency in the inertial range (Castaing et al. 1990; Sorriso-Valvo et al. 1999). In fully developed turbulence, a power-law scaling is usually observed for ${\lambda }_{{\ell }}^{2}$ (Castaing et al. 1990; Sorriso-Valvo et al. 1999). Finally, a relationship can be established between the scaling exponent of ${\lambda }_{{\ell }}^{2}$ and the multifractal properties of the flow (Castaing et al. 1990).

As briefly described above, the Castaing model is based on hypotheses on the physical processes governing the turbulent cascade. The main assumption is the choice of the weight distribution, which needs appropriate theoretical modeling. In this paper, after verifying that the multifractal cascade framework applies to solar wind turbulence, we show that it is possible to describe solar wind intermittency without any hypothesis on the shape of such distribution, but rather using empirical weights (Naert et al. 1998). In Section 2 we briefly introduce the data used for the analysis. In Section 3 we estimate the empirical distribution function of the local energy dissipation rate. Section 4 shows the conditioned analysis performed on the data in order to extract the empirical weights for the Castaing model; the self-consistent Castaing PDFs are built and compared with the experimental PDFs.

This work is based on the analysis of three different samples of in situ measurements of velocity ${\boldsymbol{v}}$, mass density ρ, estimated using proton and α particle, and magnetic field ${\boldsymbol{B}}$. The Elsasser variables ${{\boldsymbol{z}}}^{\pm }={\boldsymbol{v}}\pm {\boldsymbol{B}}/\sqrt{4\pi \rho }$ have also been evaluated from the time series. Two samples were taken in the ecliptic wind by the Helios 2 spacecraft during the first 4 months of 1976, when the spacecraft orbited from 1 AU on day 17 to 0.29 AU on day 108. Data resolution is 81 s, and 11 fast or slow wind streams, each about 49 hr long, have been extracted to avoid stream-stream interfaces and to ensure a better stationarity (Sorriso-Valvo et al. 1999). As pointed out in the literature, fast and slow wind turbulence should be studied separately, because of the different plasma conditions (Bruno & Carbone 2013). Therefore, for our statistical analysis we built two distinct samples by putting together six streams for the fast wind (which we name the fast sample, totalizing ${N}_{\mathrm{tot}}^{{\rm{F}}}=12288$ data points) and five streams for the slow wind (hereafter the slow sample, consisting of ${N}_{\mathrm{tot}}^{{\rm{S}}}=10240$ data points). The third sample was recorded in the solar wind out of the ecliptic, by instruments on board the Ulysses spacecraft, during the first 8 months of 1996. The spacecraft spanned distances in the range from 3.1 to 4.7 AU and latitudes from about 53° to 19°. Sampling resolution is 8 minutes. We refer to this data set as the polar sample. All samples were taken at solar minima, when the solar wind is more steady and free from disturbances of solar origin. We remind that, in order to study spacecraft time series, all scales ℓ are customarily transformed in the time lags $\tau ={\ell }/| \langle {\boldsymbol{v}}\rangle |$ through the bulk flow speed averaged over the entire data set. This is allowed by the Taylor hypothesis, which is generally valid for solar wind fluctuations in the inertial range (Perri & Balogh 2010; Bruno & Carbone 2013), and which we have tested in our data. In Figure 1, samples of the PDFs of standardized magnetic and velocity field fluctuations at different scales are shown for fast and polar samples (Sorriso-Valvo et al. 1999, 2001, 2004; Carbone et al. 2004). The typical tail enhancement toward small scales appears evident for all cases (Sorriso-Valvo et al. 1999). Similar results are observed for the velocity and for the Elsasser fields ${{\boldsymbol{z}}}^{\pm }$ (see, e.g., Sorriso-Valvo et al. 1999; Carbone et al. 2004). PDFs of magnetic field, velocity, and Elsasser field fluctuations have been successfully reproduced using the Castaing model given in Equation (1) (Sorriso-Valvo et al. 1999). In particular, the heavy tails are extremely well captured by the model, showing that the effect of intermittency has been well described. The fitting parameter ${\lambda }^{2}$ shows a power-law scaling extended over about one decade. As mentioned above, the scaling exponents are related to the fractal dimension of the most intermittent structures generated at the bottom of the cascade (Castaing et al. 1990), therefore providing important physical information on the turbulent flow properties.

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The first step for the self-consistent characterization of the PDFs is to map the properties of the plasma energy transfer rate using the experimental time series. In solar wind turbulence, understanding the mechanism responsible for the dissipation (and/or dispersion) of the energy at the bottom of the nonlinear cascade is still an open issue. Therefore, the actual expression of the dissipative (and/or dispersive) terms within MHD equations is not known. It should also be pointed out that, unlike ordinary turbulent flows, the solar wind plasma is only weakly collisional, so that molecular viscosity and resistivity cannot be defined in a simple way, nor estimated directly from the measurements. These considerations show that it is not possible, to date, to measure the local energy dissipation rate in solar wind turbulence. However, it is possible to define proxies of the energy dissipation rate, which can be reasonably used to represent the statistical properties of the field. In this paper, we use a definition based on the third-order moment scaling law for MHD (Politano & Pouquet 1998; Sorriso-Valvo et al. 2007), often referred to as the Politano–Pouquet law (PP). The PP law establishes, under given hypotheses (stationarity, homogeneity, isotropy, incompressibility), the linear scaling of the mixed third-order moment of the Elsasser fields,

$$\begin{eqnarray}&&{Y}^{\pm }(\tau )=\langle | {\rm{\Delta }}{{\boldsymbol{z}}}_{\tau }^{\pm }(t){| }^{2}\;{\rm{\Delta }}{z}_{\tau }^{\mp }(t)\rangle =-\displaystyle \frac{4}{3}\;\langle {\epsilon }^{\pm }\rangle \tau \langle v\rangle .\end{eqnarray} \tag{ 4 }$$

On the right-hand side of Equation (4), $\langle {\epsilon }^{\pm }\rangle$ is the mean energy transfer rate, estimated over the whole domain. By analogy, we define the "local" pseudo-energy transfer rate as

$$\begin{eqnarray}&&{\epsilon }_{\tau }^{\pm }(t)=\displaystyle \frac{| {\rm{\Delta }}{z}^{\pm }(t){| }^{2}{\rm{\Delta }}{z}_{R}^{\mp }(t)}{\tau \langle v\rangle },\end{eqnarray} \tag{ 5 }$$

so that the local energy transfer rate at the scale τ reads ${\epsilon }_{\tau }(t)=({\epsilon }_{\tau }^{+}(t)+{\epsilon }_{\tau }^{-}(t))/2$. At a given scale, each field increment can thus be associated with the local value of ${\epsilon }_{\tau }(t)$ (Marsch & Tu 1997). Since we are interested, in particular, in the small-scale intermittent effects, from now on we will only use the resolution scale values of $\epsilon (t)$ (namely, 81 s for Helios 2 data and 8 minutes for Ulysses data), which we will refer to as the energy dissipation rate. Figure 2 shows the variable $\epsilon (t)$, computed for fast and slow Helios 2 streams and for the Ulysses polar sample. Differences between the three samples are evident, in particular for the Ulysses data set, probably because of the lower data resolution. For all cases, the field is highly irregular and inhomogeneous, with spikes of large dissipation alternated with quiet periods. The PDFs of $\epsilon (t)$ are shown in Figure 3 for fast and slow wind, as obtained from Helios 2 data, and for polar Ulysses data. Because of the inhomogeneity, PDFs have been computed using bins of variable width, by imposing a fixed number of data points in each bin (N_p = 1500 for Helios data sets, N_p = 3000 for Ulysses data). The error bars are estimated as the counting (Poisson) error on N_p, and the three PDFs have been vertically shifted for clarity. Lognormal fits of the distributions are overplotted in light gray, showing good agreement with the framework of a multiplicative cascade (Castaing et al. 1990). Alternatively, PDFs are even better reproduced (with 2–10 times smaller ${\chi }^{2}$) by a stretched exponential fit $P(\epsilon )\sim \mathrm{exp}-{(\epsilon /{\epsilon }_{0})}^{c}$, where ${\epsilon }_{0}$ is a characteristic value of the energy dissipation rate and c is the parameter controlling the shape of the tails of the PDF. In particular, c = 2 indicates Gaussian, c = 1 indicates exponential, and $c\lt 1$ indicates heavy tailed, almost power-law distribution. In the present case, the shape parameters are indicated in Figure 3. Both fast and slow winds measured by Helios 2 show a strong deviation from Gaussian ($c\simeq 0.2$), with the presence of energy dissipation bursts that result in the heavy tails of the PDFs. On the contrary, Ulysses shows "thinner" tails, i.e., populated with less extreme events. This is to be expected, because of the different time resolution of the data. Indeed, while Ulysses sampling time sits in the middle of the inertial range, for Helios 2 it is closer to its bottom. It should be recalled that stretched exponential PDFs were predicted for a variable generated as the result of a multiplicative process controlled by a few extreme events, in the framework of the extreme deviation theory (Frisch & Sornette 1997). This supports once more the presence of an intermittent, multiplicative energy cascade in solar wind turbulence.

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In order to verify that the Castaing model can properly be applied to solar wind turbulence, a conditioned analysis of the data was performed. The range of values of the energy dissipation rate was divided into N_bin bins of variable width ${A}_{\epsilon }$ (with ${N}_{\mathrm{bin}}=8,7,10$ for fast, slow, and polar samples, respectively), in order to separate different regions of the time series. The PDFs of field fluctuations, at the resolution time lag, were thus estimated separately for each bin, i.e., conditioned to the values of the energy dissipation rate. The top left panel of Figure 4 shows, superimposed, all 10 conditioned PDFs $P({\rm{\Delta }}\psi | \epsilon )$ of the resolution scale radial velocity field fluctuations for the polar wind data set. Conditioned PDFs no longer have the heavy tails observed at small time lags (see Figure 1) and present a roughly Gaussian shape. The top right panel of the same figure shows examples of the Gaussian fits of the conditioned distributions. For clarity, only 4 out of 10 conditioning values of are plotted. This observation confirms the multifractal scenario, in which regions with the same energy dissipation rate are characterized by self-similar (Gaussian in this case) PDFs of the field fluctuations, even at small scale. Thus, the conditioned data lose intermittency, which indeed arises from the local fluctuations of the energy dissipation rate. The top panels of Figure 4 also qualitatively highlight the result of superimposing several Gaussian distributions of variable width and amplitude, obtaining the heavy tails observed in the usual PDF. The use of the Castaing distribution is therefore appropriate to describe solar wind intermittent turbulence. For each conditioned PDF, corresponding to a given set of constant dissipation rate, it is possible to obtain the probability density function of the standard deviations $L(\sigma )={N}_{{\rm{p}}}/{N}_{\mathrm{tot}}{A}_{\epsilon }$, where N_p is the (fixed) number of points in each dissipation bin. This is done for each data set by exploiting the correspondence between a given bin of energy dissipation rate and the standard deviation of the corresponding field fluctuations. In other words, for each field ψ and for each value of , a value of $\sigma ({\rm{\Delta }}\psi | \epsilon )$ is obtained through the Gaussian fit of the conditioned PDFs $P({\rm{\Delta }}\psi | \epsilon )$. Then, the probability $L(\sigma )$ is the fraction of data associated with that interval of values. This allows us to establish an explicit relationship between the statistical properties of the energy dissipation rate and of the standard deviation σ, postulated in the Castaing model. Central panels of Figure 4 show the PDFs of the standard deviation of the fields $L(\sigma )$ associated with the distributions of , along with lognormal and stretched exponential fits. In this case, the lognormal fit is not always able to capture the form of the distribution, while the stretched exponential fits are very good. For the latter, the smaller exponents c observed for the velocity fluctuations in all data sets, suggesting a lower intermittency, agree with the higher level of intermittency of the magnetic field (Sorriso-Valvo et al. 1999). It is also interesting to check directly from the data the validity of the assumption originally used in the Castaing model, namely, the phenomenological relationship between the energy dissipation rate and the variance of the fluctuations. This can be simply done by plotting versus $\sigma ({\rm{\Delta }}\psi | \epsilon )$ and verifying the power-law relation $\epsilon \sim {\sigma }^{\alpha }$ expected from a simple dimensional argument on the third-order scaling relation given by Equation (4), resulting in $\alpha =3$ (Castaing et al. 1990; Naert et al. 1998). As can be seen in the bottom panels of Figure 4, the power-law relationship is very well verified, with exponents compatible with the expected value $\alpha =3$. The error bars are estimated as half the size of the energy dissipation rate bin. Similar results apply to the other components of the fields and to the Elsasser fields (not shown). Note that a similar analysis performed on ordinary fluid turbulence provided instead $\alpha =5$ (Naert et al. 1998).

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With all this information at hand, the self-consistent Castaing distribution (1) can be finally discretized as follows:

$$\begin{eqnarray}&&P({\rm{\Delta }}{\psi }_{\tau })={A}_{\mathrm{norm}}\underset{i=1}{\overset{{N}_{\mathrm{bin}}}{\displaystyle \sum }}L(\sigma ){P}_{0}({\rm{\Delta }}{\psi }_{\tau },\sigma ,{a}_{{\rm{s}}})\ {A}_{\epsilon }.\end{eqnarray} \tag{ 6 }$$

Equation (6) only includes two free parameters, the normalization factor A_norm and the skewness factor a_s, while the information on the PDF shape is self-consistently included through the distribution $L(\sigma )$. The PDFs of the field fluctuations can now be compared with the self-consistent Castaing model, by fitting the data with Equation (6) in order to adjust only for the two free parameters. These are, however, irrelevant for the tails' curvature, i.e., for intermittency, so that there is no need to discuss them here. Figure 5 shows the PDFs of fluctuations, as in Figure 1 (markers), together with the fit with Equation (6) (solid line). The self-consistent model reproduces the shape of the PDF tails very satisfactorily in all cases. In order to test the goodness of the fit, a synthetic ${L}_{R}(\sigma )$ has been built by averaging over 10⁴ realizations of random distributions $L(\sigma )$, generated for σ in the same interval as each solar wind data set, $[{\sigma }_{\mathrm{min}},...,{\sigma }_{\mathrm{max}}]$. The fit of the data with the self-consistent Castaing model, using the synthetic ${L}_{R}(\sigma )$, is shown in Figure 5 as a dotted line. It is evident that a superposition of Gaussians with a random choice of the weigth distribution does not allow the correct reproduction of the data. Quantitatively, the random Gaussian superposition fits have ${\chi }^{2}$ values larger by a factor of 10 with respect to the self-consistent Castaing fit.

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In the attempt to understand the mechanism responsible for the generation of the highly intermittent turbulent fluctuations of solar wind fields, we have explored here the existence of a non-homogeneous energy cascade, a base hypothesis of several models. The intermittent solar wind field fluctuations, deeply investigated in the past decade, have often been described through the Castaing PDF model. In this work, we have evaluated empirically the weight distribution $L(\sigma )$ of the Castaing model (Equation (1)), rather than assuming an analytical prescription (usually a lognormal distribution). For the sake of generality, we have selected three different data sets, consisting of one fast and one slow ecliptic wind sample (Helios 2 data), and one fast polar wind sample (Ulysses data). Since it is not possible, to date, to measure the energy dissipation rate from the data, we have used a proxy derived from the exact statistical scaling law for the mixed third-order moment of the MHD fluctuations. Although such a law is expected to apply only for homogeneous, isotropic, incompressible MHD turbulence, its validity in the solar wind time series has recently been established by several authors (e.g., Sorriso-Valvo et al. 1999). Therefore, the use of the proxy introduced here is appropriate. After estimating the local value of the energy dissipation rate, we have evaluated its statistical properties. The PDFs are consistent with a lognormal function, predicted by the typical theoretical models used for the description of the turbulent fields as the result of a non-homogeneous, random multiplicative cascade. Alternatively, a stretched exponential fit was also used for all data sets. Stretched exponential distributions match the theoretical framework of the extreme deviations theory (Frisch & Sornette 1997), where the statistical properties of the multiplicative cascade process are controlled by the most extreme intermittent events (the heavy tails of the PDFs). The stretching parameter, obtained from the fit, can be used to describe quantitatively the degree of inhomogeneity, being related to the presence of correlations or clustering in the dissipation field. The results obtained here (see Figure 3) confirm the stronger intermittency of slow wind with respect to fast and polar wind (Sorriso-Valvo et al. 1999; Bruno & Carbone 2013).

Starting from this observation, we have then studied the conditioned PDFs of the field fluctuation, the conditioning parameter being the local energy dissipation rate. As expected within the multifractal picture (but never observed in the solar wind so far), conditioning results in loss of intermittency, so that self-similarity is restored when the inhomogeneity of the energy dissipation rate is removed. This result shows that the framework of the multifractal energy cascade applies to solar wind turbulence.

Upon observation of their Gaussian shape, the conditioned PDFs have finally been used to build the Castaing distribution self-consistently, through the empirical distribution of their standard deviations, $L(\sigma )$. The model PDF drawn following this procedure fits very well the shape of the experimental PDFs, and in particular can capture the curvature of the tails. The self-consistent Castaing model can therefore reproduce the intermittency of the fields. We remark that, with the selection process for the conditioned PDFs being based on the dissipation rate, the relevant physical information necessary to describe the intermittent PDFs is the non-homogeneous distribution of the local energy dissipation rate. The role of the shape parameter ${\lambda }^{2}$, used to describe quantitatively the degree of intermittency, is now played by the stretching parameter c of the distributions of the energy dissipation rate, not directly involved in the self-consistent Castaing model.

In conclusion, the stretched exponential distribution of the local energy dissipation rate, the Gaussian shape of the conditioned PDFs, and the successful application of the self-consistent Castaing model all strongly support the picture of the multifractal energy cascade being at the origin of the intermittency of solar wind turbulence. The validity of such a scenario holds for fast, slow, and polar samples, confirming that the turbulent cascade is always active in the solar wind.

The research leading to these results has received funding from the European Communitys Seventh Framework Programme (FP7/2007–2013) under grant agreement no. 269297/"TURBOPLASMAS."