7.1 Scaling exponents of structure functions

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 $r$ . First of all, it is worthwhile to remark that scaling laws and, in particular, the exact relation (26 ) 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 $dur = u(t + r)- u(t)$ , where $u(t)$ represents the component of the velocity field along the radial direction. As far as the magnetic differences are concerned $dbr = B(t + r) - B(t)$ , 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 $B(t)$ represents the magnetic field projected along the radial direction, or the system where $B(t)$ represents the magnetic field along the local background magnetic field, or $B(t)$ 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 $p$ -th moment of both absolute values of velocity fluctuations $Rp(r) = <|dur|p>$ and magnetic fluctuations $Sp(r) = <|dbr|p>$ , also called $p$ -th order structure function in literature (brackets being time average). Here we use magnetic fluctuations across structures at intervals $r$ calculated by using the magnetic field intensity. Typical structure functions of magnetic field fluctuations, for two different values of $p$ , for both a slow wind and a fast wind at $0.9 AU$ , are shown in Figures 72 . 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 three typical ranges of turbulence can be observed. Starting from low values at small scales, the structure functions increase towards a region where $Sp-- > const.$ 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 $r$ where a power law can be recognized for both $Sp(r) ~ rzp$ and $Rp(r) ~ rqp$ , is more or less visible only for the slow wind. In this range correlations exists, and we can obtain the scaling exponents $z p$ and $q p$ through a simple linear fit.

Figure 72:

Structure functions for the magnetic field intensity $Sn(r)$ for two different orders, $n = 3$ and $n = 5$ , for both slow wind and fast wind, as a function of the time scale $r$ . Data come from Helios 2 spacecraft at $0.9 AU$ .

Since the inertial range is not well defined⁹ , 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 $3$ -th order structure function can be regarded as a generalized scaling using the inverse of Equation (26

) or of Equation (27

) (Politano et al., 1998b

). Then, we can plot the $p$ -th order structure function vs. the $3$ -th order to recover at least relative scaling exponents $zp/z3$ and $qp/q3$ . Quite surprisingly (see Figure 73

), 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 the $4/5$ -law, by definition $z3 = 1$ 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.

Figure 73:

Structure functions $Sn(r)$ for two different orders, $n = 3$ and $n = 5$ , for both slow wind and high wind, as a function of the fourth-order structure function $S4(r)$ . Data come from Helios 2 spacecraft at $0.9 AU$ .

It is worthwhile to remark (as shown in Figure 73

) that we can introduce a general scaling relation between the $q$ -th order velocity structure function and the $p$ -th order structure function, with a relative scaling exponent $ap(q)$ . It has been found that this relation becomes an exact relation

a (q) Sq(r) = [Sp(r)]p ,

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:

That is, for each scale $r$ the knowledge of the relative scaling exponents $ap(q)$ completely determines the probability distribution of velocity differences as a function of a single parameter $S (r) p$ .

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 $p$ , say $zp/z3 > p/3$ for $p < 3$ , while $zp/z3 < p/3$ for $p > 3$ . The same behavior can be observed for $qp/q3$ . 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 $zp/z3$ and the linear scaling $p/3$ . It can be seen that magnetic field is more intermittent than 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, behaves like a passive field.

Table 1:

Scaling exponents for velocity $zp$ and magnetic $qp$ variables calculated through ESS. Errors represent the standard deviations of the linear fitting. The data used comes from a turbulent sample of slow wind at $0.9 AU$ from Helios 2 spacecraft. As a comparison we show the normalized scaling exponents of structure functions calculated in a wind tunnel on Earth (Ruíz-Chavarría et al., 1995 ) for velocity and temperature. The temperature is a passive scalar in this experiment.







$p$	$zp$	$qp$	$u(t)$ (fluid)	$T(t)$ (fluid)









$1$	$0.37 ± 0.06$	$0.56 ± 0.06$	$0.37$	$0.61$
$2$	$0.70 ± 0.05$	$0.83 ± 0.05$	$0.70$	$0.85$
$3$	$1.00$	$1.00$	$1.00$	$1.00$
$4$	$1.28 ± 0.02$	$1.14 ± 0.02$	$1.28$	$1.12$
$5$	$1.54 ± 0.03$	$1.25 ± 0.03$	$1.54$	$1.21$
$6$	$1.79 ± 0.05$	$1.35 ± 0.05$	$1.78$	$1.40$

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 $p$ which can be determined with a finite number of points of our dataset. As the value of $p$ 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 dataset has a finite extent, the probability to get singularities stronger than a certain value approaches zero. In that case, scaling exponents $zp$ of order higher than a certain value look to be linear function of $p$ . Actually, the structure function $S (r) p$ depends on the probability distribution function $P DF (du ) r$ through

integral p Sp(r) = durP DF (dur)ddur,

and the function $Sp$ is determined only when the integral converges. As $p$ increases, the function $Fp(dur) = dupP DF (dur) r$ becomes more and more disturbed, with some spikes, so that the integral becomes more and more undefined, as can be seen for example in Figure 1 of the paper by Dudok de Wit (2004 ). A simple calculation (Dudok de Wit, 2004 ) for the maximum value of the order $pm$ which can reliably be estimated with a given number $N$ of points in the dataset, gives the empirical criterion $pm -~ log N$ . Structure functions of order $p > pm$ cannot be determined accurately.

Table 2:

Normalized scaling exponents $qp/q3$ for radial magnetic fluctuations in a laboratory plasma, as measured at different distances $r/R$ ( $R -~ 0.45 cm$ being the minor radius of the torus in the experiment) from the external wall. Errors represent the standard deviations of the linear fitting. Scaling exponents have been obtained using the ESS.







$p$	$r/R = 0.96$	$r/R = 0.93$	$r/R = 0.90$	$r/R = 0.86$









$1$	$0.39 ± 0.01$	$0.38 ± 0.01$	$0.37 ± 0.01$	$0.36 ± 0.01$
$2$	$0.74 ± 0.01$	$0.73 ± 0.02$	$0.71 ± 0.01$	$0.70 ± 0.01$
$3$	$1.00$	$1.00$	$1.00$	$1.00$
$4$	$1.20 ± 0.02$	$1.24 ± 0.02$	$1.27 ± 0.01$	$1.28 ± 0.01$
$5$	$1.32 ± 0.03$	$1.41 ± 0.03$	$1.51 ± 0.03$	$1.55 ± 0.03$
$6$	$1.38 ± 0.04$	$1.50 ± 0.04$	$1.71 ± 0.03$	$1.78 ± 0.04$

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 $Br(t + t )- Br(t)$ calculated from the radial component in a toroidal device where the $z$ -axis is directed along the axis of the torus. As it can be seen, intermittency in magnetic turbulence is not so strong as appears in the solar wind, actually the degree of intermittency increases going toward the external wall. This last feature appears to be similar to what is currently observed in channel flows, where intermittency increases 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. (1998b ). In this case, we are free from the Taylor hypothesis in calculating the fluctuations at a given scale. From 2D simulations we recover the fields $u(r, t)$ and $b(r,t)$ at some fixed times. We calculate the longitudinal fluctuations directly in space at a fixed time, namely $dul = [u(r + l,t)- u(r,t)] .l/l$ (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.

Table 3:

Normalized scaling exponents $qp/q3$ for Alfvénic, velocity, and magnetic fluctuations obtained from data of high resolution 2D MHD numerical simulations. Scaling exponents have been calculated from spatial fluctuations; different times, in the statistically stationary state, have been used to improve statistics. The scaling exponents have been calculated by ESS using Equation (27

) as characteristic scale rather than the third-order structure function (cf. Politano et al.

, 1998b for details).







$p$	$Z+$	$Z -$	$v$	$B$









$1$	$0.36 ± 0.06$	$0.56 ± 0.06$	$0.37 ± 0.01$	$0.46 ± 0.02$
$2$	$0.70 ± 0.05$	$0.83 ± 0.05$	$0.70 ± 0.01$	$0.78 ± 0.01$
$3$	$1.00$	$1.00$	$1.00$	$1.00$
$4$	$1.28 ± 0.02$	$1.14 ± 0.02$	$1.28 ± 0.02$	$1.18 ± 0.02$
$5$	$1.53 ± 0.03$	$1.25 ± 0.03$	$1.54 ± 0.03$	$1.31 ± 0.03$
$6$	$1.79 ± 0.05$	$1.35 ± 0.05$	$1.78 ± 0.05$	$1.40 ± 0.03$