8.1 Structure functions

Apart from the earliest investigations on the fractal structure of magnetic field as observed in interplanetary space (Burlaga and Klein, 1986 ), the starting point for the investigation of intermittency in the solar wind dates back to 1991, when Burlaga (1991a

) started to look at the scaling of the bulk velocity fluctuations at $8.5 AU$ using Voyager 2 data. This author found that anomalous scaling laws for structure functions could be recovered in the range $0.85 < r < 13.6 h$ . This range of scales has been arbitrarily identified as a kind of “inertial range”, say a region were a linear scaling exists between $log S(p) r$ and $log r$ , and the scaling exponents have been calculated as the slope of these curves. However, structure functions of order $p < 20$ were determined on the basis of only about $4500$ data points. Nevertheless the scaling was found to be quite in agreement with that found in ordinary fluid flows. Although the data might be in agreement with the random- $b$ model, from a theoretical point of view Carbone (1993

, 1994b ) showed that normalized scaling exponents $zp/z4$ calculated by Burlaga (1991a

) would be better fitted by using a $p$ -model derived from the Kraichnan phenomenology (Kraichnan, 1965

; Carbone, 1993

), and considering the parameter $m -~ 0.77$ . The same author (Burlaga, 1991b ) investigated the multifractal structure of the interplanetary magnetic field near $25 AU$ and analyzed positive defined fields as magnetic field strength, temperature, and density using the multifractal machinery of dissipation fields (Paladin and Vulpiani, 1987 ; Meneveau, 1991 ). Burlaga (1991c ) showed that intermittent events observed in corotating streams at $1 AU$ should be described by a multifractal geometry. Even in this case the number of points used was very low to assure the reliability of high-order moments.

Marsch and Liu (1993 ) investigated the structure of intermittency of the turbulence observed in the inner heliosphere by using Helios 2 data. They analyzed both bulk velocity and Alfvén speed to calculate structure functions in the whole range $40.5 s$ (the instrument resolution) up to $24 h$ to estimate the $p$ -th order scaling exponents. Note that also in this analysis the number of data points used was too small to assure a reliability for order $p = 20$ structure functions as reported by Marsch and Liu (1993 ). From the analysis analogous to Burlaga (1991a ), authors found that anomalous scaling laws are present. A comparison between fast and slow streams at two heliocentric distances, namely $0.3 AU$ and $1 AU$ , allows authors to conjecture a scenario for high speed streams were Alfvénic turbulence, originally self-similar (or poorly intermittent) near the Sun, “ $...$ loses its self-similarity and becomes more multifractal in nature” (Marsch and Liu, 1993 ), which means that intermittent corrections increase from $0.3 AU$ to $1 AU$ . No such behavior seems to occur in the slow solar wind. From a phenomenological point of view, Marsch and Liu (1993 ) found that data can be fitted with a piecewise linear function for the scaling exponents $zp$ , namely a $b$ -model $zp = 3 - D + p(D - 2)/3$ , where $D -~ 3$ for $p < 6$ and $D -~ 2.6$ for $p > 6$ . Authors say that “We believe that we see similar indications in the data by Burlaga, who still prefers to fit his whole $zp$ dataset with a single fit according to the non-linear random $b$ -model.”. We like to comment that the impression by Marsch and Liu (1993 ) is due to the fact that the number of data points used was very small. As a matter of fact, only structure functions of order $p < 4$ are reliably described by the number of points used by Burlaga (1991a ).

However, the data analyses quoted above, which in some sense present some contradictory results, are based on high order statistics which is not supported by an adequate number of data points and the range of scales, where scaling laws have been recovered, is not easily identifiable. To overcome these difficulties Carbone et al. (1996a ) investigated the behavior of the normalized ratios $zp/z3$ through the ESS procedure described above, using data coming from low-speed streams measurements of Helios 2 spacecraft. Using ESS the whole range covered by measurements is linear, and scaling exponent ratios can be reliably calculated. Moreover, to have a dataset with a high number of points, authors mixed in the same statistics data coming from different heliocentric distances (from $0.3 AU$ up to $1 AU$ ). This is not correct as far as fast wind fluctuations are taken into account, because, as found by Marsch and Liu (1993 ) and Bruno et al. (2003b ), there is a radial evolution of intermittency. Results showed that intermittency is a real characteristic of turbulence in the solar wind, and that the curve $zp/z3$ is a non-linear function of $p$ as soon as values of $p < 6$ are considered.

Marsch et al. (1996 ) for the first time investigated the geometrical and scaling properties of the energy flux along the turbulent cascade and dissipation rate of kinetic energy. They showed the multifractal nature of the dissipation field and estimated, for the first time in solar wind MHD turbulence, the associated singularity spectrum which resulted to be very similar to those obtained for ordinary fluid turbulence (Meneveau and Sreenivasan, 1987b ). They also estimated the energy dissipation rate for time scales of $102 s$ to be around $5.4 × 10-16 erg cm -3 s- 1$ . This value was similar to the theoretical heating rate required in the model by Tu (1988 ) with Alfvén waves to explain the radial temperature dependence observed in fast solar wind.

Looking at the literature, it can be realized that often scaling exponents $zp$ , as observed mainly in the high-speed streams of the inner solar wind, cannot be explained properly by any cascade model for turbulence. This feature has been attributed to the fact that this kind of turbulence is not in a fully-developed state with a well defined spectral index. Models developed by Tu et al. (1984 ) and Tu (1988 ) were successful in describing the evolution of the observed power spectra. Using the same idea Tu et al. (1996 ) and Marsch and Tu (1997 ) investigated the behavior of an extended cascade model developed on the base of the $p$ -model (Meneveau and Sreenivasan, 1987a ; Carbone, 1993 ). Authors conjectured that: i) the scaling laws for fluctuations are still valid in the form $dZ ± ~ lh l$ , even when turbulence is not fully developed; ii) the energy cascade rate is not constant, its moments rather depend not only on the generalized dimensions $Dp$ but also on the spectral index $a$ of the power spectrum, say $<epr> ~ ep(l,a)l(p-1)Dp$ , where the averaged energy transfer rate is assumed to be

e(l,a) ~ l-(m/2+1)Pla/2,

being $P ~ la l$ the usual energy spectrum ( $l ~ 1/k$ ). The model gives

where the generalized dimensions are recovered from the usual $p$ -model

log2[mp-+-(1---m)p] Dp = (1- p) .

In the limit of “fully developed turbulence”, say when the spectral slope is $a = 2/m + 1$ the usual Equation (40 ) is recovered. The Helios 2 data are consistent with this model as far as the parameters are $m -~ 0.77$ and $a -~ 1.45$ , and the fit is relatively good (Tu et al., 1996 ). Recently, Horbury et al. (1997 ) and Horbury and Balogh (1997 ) studied the magnetic field fluctuations of the polar high-speed turbulence from Ulysses measurements at $3.1 AU$ and at $63o$ heliolatitude. These authors showed that the observed magnetic field fluctuations were in agreement with the intermittent turbulence p-model of Meneveau and Sreenivasan (1987a ). They also showed that the scaling exponents of structure functions of order $p < 6$ , in the scaling range $20 < r < 300 s$ followed the Kolmogorov scaling instead of Kraichnan scaling as expected. In addition, the same authors (Horbury et al., 1997 ) estimated the applicability of the model by Tu et al. (1996 ) and Marsch and Tu (1997 ) to the spectral transition range where the spectral index changes during the spectral evolution and concluded that this model was able to fit the observations much better than the $p$ -model when values of the parameters $p$ change continuously with the scale.

Analysis of scaling exponents of $p$ -th order structure functions has been performed using different spacecraft datasets of Ulysses spacecraft. Horbury et al. (1995a ) and Horbury et al. (1995c ) investigated the structure functions of magnetic field as obtained from observations recorded between $1.7$ and $4 AU$ , and covering a heliographic latitude between $o 40$ and $o 80$ south. By investigating the spectral index of the second order structure function, they found a decrease with heliocentric distance attributed to the radial evolution of fluctuations. Further investigations (see, e.g., Ruzmaikin et al., 1995 ) were obtained using structure functions to study the Ulysses magnetic field data in the range of scales $1 < r < 32 min$ . Ruzmaikin et al. (1995 ) showed that intermittency is at work and developed a bi-fractal model to describe Alfvénic turbulence. They found that intermittency may change the spectral index of the second order structure function and this modifies the calculation of the spectral index (Carbone, 1994a ). Ruzmaikin et al. (1995 ) found that polar Alfvénic turbulence should be described by a Kraichnan phenomenology (Kraichnan, 1965 ). However, the same data can be fitted also with a fluid-like scaling law (Tu et al., 1996 ) and, due to the relatively small amount of data, it is difficult to decide, on the basis of the second order structure function, which scaling relation describes appropriately intermittency in the solar wind.

In a further paper Carbone et al. (1995b ) provided evidence for differences in the ESS scaling laws between ordinary fluid flows and solar wind turbulence. Through the analysis of different datasets collected in the solar wind and in ordinary fluid flows, it was shown that normalized scaling exponents $zp/z3$ are the same as far as $p < 8$ are considered. This indicates a kind of universality in the scaling exponents for the velocity structure functions. Differences between scaling exponents calculated in ordinary fluid flows and solar wind turbulence are confined to high-order moments. Nevertheless, the differences found in the datasets were related to different kind of singular structures in the model described by Equation (41 ). Solar wind data can be fitted by that model as soon as the most intermittent structures are assumed to be planar sheets $C = 1$ and $m = 4$ , that is a Kraichnan scaling is used. On the contrary, ordinary fluid flows can be fitted only when $C = 2$ and $m = 3$ , that is, structures are filaments and the Kolmogorov scaling have been used. However it is worthwhile to remark that differences have been found for high-order structure functions, just where measurements are unreliable.