The phenomenology of turbulence developed by Kolmogorov (1941) deals with some statistical hypotheses for fluctuations. The famous footnote remark by Landau (Landau and Lifshitz, 1971) pointed out a defect in the Kolmogorov theory, namely the fact that the theory does not take proper account of spatial fluctuations of local dissipation rate (Frisch, 1995). This led different authors to investigate the features related to scaling laws of fluctuations and, in particular, to investigate the departure from the Kolmogorov’s linear scaling of the structure functions (cf. Section 2.8). An up-to-date comprehensive review of these theoretical efforts can be found in the book by Frisch (1995).
Here we are interested in understanding what we can learn from solar wind turbulence about the basic features of scaling laws for fluctuations. We use velocity and magnetic fields time series, and we investigate the scaling behavior of the high-order moments of stochastic variables defined as variations of fields separated by a time8 interval . First of all, it is worthwhile to remark that scaling laws and, in particular, the exact relation (40) which defines the inertial range in fluid flows, is valid for longitudinal (streamwise) fluctuations. In common fluid flows the Kolmogorov linear scaling law is compared with the moments of longitudinal velocity differences. In the same way for the solar wind turbulence we investigate the scaling behavior of , where represents the component of the velocity field along the radial direction. As far as the magnetic differences are concerned , we are free for different choices and, in some sense, this is more interesting from an experimental point of view. We can use the reference system where represents the magnetic field projected along the radial direction, or the system where represents the magnetic field along the local background magnetic field, or represents the field along the minimum variance direction. As a different case we can simply investigate the scaling behavior of the fluctuations of the magnetic field intensity.
Let us consider the -th moment of both absolute values9 of velocity fluctuations and magnetic fluctuations , also called -th order structure function in literature (brackets being time average). Here we use magnetic fluctuations across structures at intervals calculated by using the magnetic field intensity. Typical structure functions of magnetic field fluctuations, for two different values of , for both a slow wind and a fast wind at 0.9 AU, are shown in Figures 80. The magnetic field we used is that measured by Helios 2 spacecraft. Structure functions calculated for the velocity fields have roughly the same shape. Looking at these Figures the typical scaling features of turbulence can be observed. Starting from low values at small scales, the structure functions increase towards a region where at the largest scales. This means that at these scales the field fluctuations are uncorrelated. A kind of “inertial range”, that is a region of intermediate scales where a power law can be recognized for both
Since as we have seen, Yaglom’s law is observed only in some few samples, the inertial range in the whole solar wind is not well defined. A look at Figure 80 clearly shows that we are in a situation similar to a low-Reynolds number fluid flow. In order to compare scaling exponents of the solar wind turbulent fluctuations with other experiments, it is perhaps better to try to recover exponents using the Extended Self-Similarity (ESS), introduced some time ago by Benzi et al. (1993), and used here as a tool to determine relative scaling exponents. In the fluid-like case, the third-order structure function can be regarded as a generalized scaling using the inverse of Equation (41) or of Equation (40) (Politano et al., 1998). Then, we can plot the -th order structure function vs. the third-order one to recover at least relative scaling exponents and (60). Quite surprisingly (see Figure 81), we find that the range where a power law can be recovered extends well beyond the inertial range, covering almost all the experimental range. In the fluid case the scaling exponents which can be obtained through ESS at low or moderate Reynolds numbers, coincide with the scaling exponents obtained for high Reynolds, where the inertial range is very well defined Benzi et al. (1993). This is due to the fact that, since by definition in the inertial range (Frisch, 1995), whatever its extension might be. In our case scaling exponents obtained through ESS can be used as a surrogate, since we cannot be sure that an inertial range exists.
It is worthwhile to remark (as shown in Figure 81) that we can introduce a general scaling relation between the -th order velocity structure function and the -th order structure function, with a relative scaling exponent . It has been found that this relation becomes an exact relation
when the velocity structure functions are normalized to the average velocity within each period used to calculate the structure function (Carbone et al., 1996a). This is very interesting because it implies (Carbone et al., 1996a) that the above relationship is satisfied by the following probability distribution function, if we assume that odd moments are much smaller than the even ones:
Relative scaling exponents, calculated by using data coming from Helios 2 at 0.9 AU, are reported in Table 1. As it can be seen, two main features can be noted:
- There is a significant departure from the Kolmogorov linear scaling, that is, real scaling exponents are anomalous and seem to be non-linear functions of , say for , while for . The same behavior can be observed for . In Table 1 we report also the scaling exponents obtained in usual fluid flows for velocity and temperature, the latter being a passive scalar. Scaling exponents for velocity field are similar to scaling exponents obtained in turbulent flows on Earth, showing a kind of universality in the anomaly. This effect is commonly attributed to the phenomenon of intermittency in fully developed turbulence (Frisch, 1995). Turbulence in the solar wind is intermittent, just like its fluid counterpart on Earth.
- The degree of intermittency is measured through the distance between the curve and the linear scaling . It can be seen that the magnetic field is more intermittent than the velocity field. The same difference is observed between the velocity field and a passive scalar (in our case the temperature) in ordinary fluid flows (Ruíz-Chavarría et al., 1995). That is the magnetic field, as long as intermittency properties are concerned, has the same scaling laws of a passive field. Of course this does not mean that the magnetic field plays the same role as a passive field. Statistical properties are in general different from dynamical properties.
|1||0.37 ± 0.06||0.56 ± 0.06||0.37||0.61|
|2||0.70 ± 0.05||0.83 ± 0.05||0.70||0.85|
|4||1.28 ± 0.02||1.14 ± 0.02||1.28||1.12|
|5||1.54 ± 0.03||1.25 ± 0.03||1.54||1.21|
|6||1.79 ± 0.05||1.35 ± 0.05||1.78||1.38|
In Table 1 we show scaling exponents up to the sixth order. Actually, a question concerns the validation of high-order moments estimates, say the maximum value of the order which can be determined with a finite number of points of our dataset. As the value of increases, we need an increasing number of points for an optimal determination of the structure function (Tennekes and Wyngaard, 1972). Anomalous scaling laws are generated by rare and intense events due to singularities in the gradients: the higher their intensity the more rare these events are. Of course, when the data set has a finite extent, the probability to get singularities stronger than a certain value approaches zero. In that case, scaling exponents of order higher than a certain value become linear functions of . Actually, the structure function depends on the probability distribution function through
Only few large structures are enough to generate the anomalous scaling laws. In fact, as shown by Salem et al. (2009), by suppressing through wavelets analysis just a few percentage of large structures on all scales, the scaling exponents become linear functions of , respectively and for the kinetic and magnetic fields.
|1||0.39 ± 0.01||0.38 ± 0.01||0.37 ± 0.01||0.36 ± 0.01|
|2||0.74 ± 0.01||0.73 ± 0.02||0.71 ± 0.01||0.70 ± 0.01|
|4||1.20 ± 0.02||1.24 ± 0.02||1.27 ± 0.01||1.28 ± 0.01|
|5||1.32 ± 0.03||1.41 ± 0.03||1.51 ± 0.03||1.55 ± 0.03|
|6||1.38 ± 0.04||1.50 ± 0.04||1.71 ± 0.03||1.78 ± 0.04|
As far as a comparison between different plasmas is concerned, the scaling exponents of magnetic structure functions, obtained from laboratory plasma experiments of a Reversed-Field Pinch at different distances from the external wall (Carbone et al., 2000) are shown in Table 2. In laboratory plasmas it is difficult to measure all the components of the vector field at the same time, thus, here we show only the scaling exponents obtained using magnetic field differences calculated from the radial component in a toroidal device where the -axis is directed along the axis of the torus. As it can be seen, intermittency in magnetic turbulence is not so strong as it appears to be in the solar wind, actually the degree of intermittency increases when going toward the external wall. This last feature appears to be similar to what is currently observed in channel flows, where intermittency also increases when going towards the external wall (Pope, 2000).
Scaling exponents of structure functions for Alfvén variables, velocity, and magnetic variables have been calculated also for high resolution 2D incompressible MHD numerical simulations (Politano et al., 1998). In this case, we are freed from the constraint of the Taylor hypothesis when calculating the fluctuations at a given scale. From 2D simulations we recover the fields and at some fixed times. We calculate the longitudinal fluctuations directly in space at a fixed time, namely (the same are made for different fields, namely the magnetic field or the Elsässer fields). Finally, averaging both in space and time, we calculate the scaling exponents through the structure functions. These scaling exponents are reported in Table 3. Note that, even in numerical simulations, intermittency for magnetic variables is stronger than for the velocity field.
|1||0.36 ± 0.06||0.56 ± 0.06||0.37 ± 0.01||0.46 ± 0.02|
|2||0.70 ± 0.05||0.83 ± 0.05||0.70 ± 0.01||0.78 ± 0.01|
|4||1.28 ± 0.02||1.14 ± 0.02||1.28 ± 0.02||1.18 ± 0.02|
|5||1.53 ± 0.03||1.25 ± 0.03||1.54 ± 0.03||1.31 ± 0.03|
|6||1.79 ± 0.05||1.35 ± 0.05||1.78 ± 0.05||1.40 ± 0.03|
The presence of scaling laws for fluctuations is a signature of the presence of self-similarity in the phenomenon. A given observable , which depends on a scaling variable , is invariant with respect to the scaling relation , when there exists a parameter such that . The solution of this last relation is a power law , where the scaling exponent is .
Since, as we have just seen, turbulence is characterized by scaling laws, this must be a signature of self-similarity for fluctuations. Let us see what this means. Let us consider fluctuations at two different scales, namely and . Their ratio depends only on the value of , and this should imply that fluctuations are self-similar. This means that PDFs are related through
Let us consider the normalized variables
When is unique or in a pure self-similar situation, PDFs are related through , say by changing scale PDFs coincide.
The PDFs relative to the normalized magnetic fluctuations , at three different scales , are shown in Figure 82. It appears evident that the global self-similarity in real turbulence is broken. PDFs do not coincide at different scales, rather their shape seems to depend on the scale . In particular, at large scales PDFs seem to be almost Gaussian, but they become more and more stretched as decreases. At the smallest scale PDFs are stretched exponentials. This scaling dependence of PDFs is a different way to say that scaling exponents of fluctuations are anomalous, or can be taken as a different definition of intermittency. Note that the wings of PDFs are higher than those of a Gaussian function. This implies that intense fluctuations have a probability of occurrence higher than that they should have if they were Gaussianly distributed. Said differently, intense stochastic fluctuations are less rare than we should expect from the point of view of a Gaussian approach to the statistics. These fluctuations play a key role in the statistics of turbulence. The same statistical behavior can be found in different experiments related to the study of the atmosphere (see Figure 83) and the laboratory plasma (see Figure 84).
Time dependence of and for three different scales is shown in Figures 85 and 86, respectively. These plots show that, as becomes small, intense fluctuations become more and more important, and they dominate the statistics. Fluctuations at large scales appear to be smooth while, as the scale becomes smaller, intense fluctuations becomes visible. These dominating fluctuations represent relatively rare events. Actually, at the smallest scales, the time behavior of both and is dominated by regions where fluctuations are low, in between regions where fluctuations are intense and turbulent activity is very high. Of course, this behavior cannot be described by a global self-similar behavior. Allowing the scaling laws to vary with the region of turbulence we are investigating would be more convincing.
The behavior we have just described is at the heart of the multifractal approach to turbulence (Frisch, 1995). In that description of turbulence, even if the small scales of fluid flow cannot be globally self-similar, self-similarity can be reintroduced as a local property. In the multifractal description it is conjectured that turbulent flows can be made by an infinite set of points , each set being characterized by a scaling law , that is, the scaling exponent can depend on the position . The usual dimension of that set is then not constant, but depends on the local value of , and is quoted as in literature. Then, the probability of occurrence of a given fluctuation can be calculated through the weight the fluctuation assumes within the whole flow, i.e.,
and the -th order structure function is immediately written through the integral over all (continuous) values of weighted by a smooth function , i.e.,
A moment of reflection allows us to realize that in the limit the integral is dominated by the minimum value (over ) of the exponent and, as shown by Frisch (1995), the integral can be formally solved using the usual saddle-point method. The scaling exponents of the structure function can then be written as
In this way, the departure of from the linear Kolmogorov scaling and thus intermittency, can be characterized by the continuous changing of as varies. That is, as varies we are probing regions of fluid where even more rare and intense events exist. These regions are characterized by small values of , that is, by stronger singularities of the gradient of the field.
Owing to the famous Landau footnote on the fact that fluctuations of the energy transfer rate must be taken into account in determining the statistics of turbulence, people tried to interpret the non-linear energy cascade typical of turbulence theory, within a geometrical framework. The old Richardson’s picture of the turbulent behavior as the result of a hierarchy of eddies at different scales has been modified and, as realized by Kraichnan (1974), once we leave the idea of a constant energy cascade rate we open a “Pandora’s box” of possibilities for modeling the energy cascade. By looking at scaling laws for and introducing the scaling exponents for the energy transfer rate , it can be found that (being when the Kolmogorov-like phenomenology is taken into account, or when the Iroshnikov-Kraichnan phenomenology holds). In this way the intermittency correction are determined by a cascade model for the energy transfer rate. When is a non-linear function of , the energy transfer rate can be described within the multifractal geometry (see, e.g., Meneveau, 1991, and references therein) characterized by the generalized dimensions (Hentschel and Procaccia, 1983). The scaling exponents of the structure functions are then related to by
The correction to the linear scaling is positive for , negative for , and zero for . A fractal behavior where gives a linear correction with a slope different from .
Cascade models view turbulence as a collection of fragments at a given scale , which results from the fragmentation of structures at the scale , down to the dissipative scale (Novikov, 1969). Sophisticated statistics are applied to obtain scaling exponents for the -th order structure function.
The starting point of fragmentation models is the old -model, a “pedagogical” fractal model introduced by Frisch et al. (1978) to account for the modification of the cascade in a simple way. In this model, the cascade is realized through the conjecture that active eddies and non-active eddies are present at each scale, the space-filling factor for the fragments being fixed for each scale. Since it is a fractal model, the -model gives a linear modification to . This can account for a fit on the data, as far as small values of are concerned. However, the whole curve is clearly nonlinear, and a multifractal approach is needed.
The random- model (Benzi et al., 1984), a multifractal modification of the -model, can be derived by invoking that the space-filling factor for the fragments at a given scale in the energy cascade is not fixed, but is given by a random variable . The probability of occurrence of a given is assumed to be a bimodal distribution where the eddies fragmentation process generates either space-filling eddies with probability or planar sheets with probability (for conservation ). It can be found that
The -model (Meneveau, 1991; Carbone, 1993) consists in an eddies fragmentation process described by a two-scale Cantor set with equal partition intervals. An eddy at the scale , with an energy derived from the transfer rate , breaks down into two eddies at the scale , with energies and . The parameter is not defined by the model, but is fixed from the experimental data. The model gives
In the model by She and Leveque (see, e.g., She and Leveque, 1994; Politano and Pouquet, 1998) one assumes an infinite hierarchy for the moments of the energy transfer rates, leading to , and a divergent scaling law for the infinite-order moment , which describes the most singular structures within the flow. The model readsPolitano and Pouquet, 1995) , so that , that is, the most singular dissipative structures are planar sheets. On the contrary, in fluid flows and the most dissipative structures are filaments. The large behavior of the -model is given by , so that Equations (63, 64) give the same results providing . As shown by Carbone et al. (1996b) all models are able to capture intermittency of fluctuations in the solar wind. The agreement between the curves and normalized scaling exponents is excellent, and this means that we realistically cannot discriminate between the models we reported above. The main problem is that all models are based on a conjecture which gives a curve as a function of a single free parameter, and that curve is able to fit the smooth observed behavior of . Statistics cannot prove, just disprove. We can distinguish between the fractal model and multifractal models, but we cannot realistically distinguish among the various multifractal models.
Besides the idea of self-similarity underlying the process of energy cascade in turbulence, a different point of view can be introduced. The idea is to characterize the behavior of the PDFs through the scaling laws of the parameters, which describe how the shape of the PDFs changes when going towards small scales. The model, originally introduced by Castaing et al. (2001), is based on a multiplicative process describing the cascade. In its simplest form the model can be introduced by saying that PDFs of increments , at a given scale, are made as a sum of Gaussian distributions with different widths . The distribution of widths is given by , namely
Shell models have remarkable properties which closely resemble those typical of MHD phenomena (Gloaguen et al., 1985; Biskamp, 1994; Giuliani and Carbone, 1998; Plunian et al., 2012). However, the presence of a constant forcing term always induces a dynamical alignment, unless the model is forced appropriately, which invariably brings the system towards a state in which velocity and magnetic fields are strongly correlated, that is, where and . When we want to compare statistical properties of turbulence described by MHD shell models with solar wind observations, this term should be avoided. It is possible to replace the constant forcing term by an exponentially time-correlated Gaussian random forcing which is able to destabilize the Alfvénic fixed point of the model (Giuliani and Carbone, 1998), thus assuring the energy cascade. The forcing is obtained by solving the following Langevin equation:
A statistically stationary state is reached by the system Gloaguen et al. (1985); Biskamp (1994); Giuliani and Carbone (1998); Plunian et al. (2012), with a well defined inertial range, say a region where Equation (48) is verified. Spectra for both the velocity and magnetic variables, as a function of , obtained in the stationary state using the GOY MHD shell model, are shown in Figures 87 and 88. Fluctuations are averaged over time. The Kolmogorov spectrum is also reported as a solid line. It is worthwhile to remark that, by adding a random term like to a little modified version of the MHD shell models ( is a random function with some statistical characteristics), a Kraichnan spectrum, say , where is the total energy, can be recovered (Biskamp, 1994; Hattori and Ishizawa, 2001). The term added to the model could represent the effect of the occurrence of a large-scale magnetic field.
Intermittency in the shell model is due to the time behavior of shell variables. It has been shown (Okkels, 1997) that the evolution of GOY model consists of short bursts traveling through the shells and long period of oscillations before the next burst arises. In Figures 89 and 90 we report the time evolution of the real part of both velocity variables and magnetic variables at three different shells. It can be seen that, while at smaller variables seems to be Gaussian, at larger variables present very sharp fluctuations in between very low fluctuations.
The time behavior of variables at different shells changes the statistics of fluctuations. In Figure 91 we report the probability distribution functions and , for different shells , of normalized variables
The same phenomenon gives rise to the departure of scaling laws of structure functions from a Kolmogorov scaling. Within the framework of the shell model the analogous of structure functions are defined as
Scaling exponents have been determined from a least square fit in the inertial range . The values of these exponents are reported in Table 4. It is interesting to notice that, while scaling exponents for velocity are the same as those found in the solar wind, scaling exponents for the magnetic field found in the solar wind reveal a more intermittent character. Moreover, we notice that velocity, magnetic and Elsässer variables are more intermittent than the mixed correlators and we think that this could be due to the cancelation effects among the different terms defining the mixed correlators.
|1||0.36 ± 0.01||0.35 ± 0.01||0.35 ± 0.01||0.36 ± 0.01||0.326||0.318|
|2||0.71 ± 0.02||0.69 ± 0.03||0.70 ± 0.02||0.70 ± 0.03||0.671||0.666|
|3||1.03 ± 0.03||1.01 ± 0.04||1.02 ± 0.04||1.02 ± 0.04||1.000||1.000|
|4||1.31 ± 0.05||1.31 ± 0.06||1.30 ± 0.05||1.32 ± 0.06||1.317||1.323|
|5||1.57 ± 0.07||1.58 ± 0.08||1.54 ± 0.07||1.60 ± 0.08||1.621||1.635|
|6||1.80 ± 0.08||1.8 ± 0.10||1.79 ± 0.09||1.87 ± 0.10||1.91||1.94|
Time intermittency in the shell model generates rare and intense events. These events are the result of the chaotic dynamics in the phase-space typical of the shell model (Okkels, 1997). That dynamics is characterized by a certain amount of memory, as can be seen through the statistics of waiting times between these events. The distributions of waiting times is reported in the bottom panels of Figures 91, at a given shell . The same statistical law is observed for the bursts of total dissipation (Boffetta et al., 1999).