"The Solar Wind as a Turbulence Laboratory"
Roberto Bruno and Vincenzo Carbone 
1 Introduction
1.1 What does turbulence stand for?
1.2 Dynamics vs. statistics
2 Equations and Phenomenology
2.1 The Navier–Stokes equation and the Reynolds number
2.2 The coupling between a charged fluid and the magnetic field
2.3 Scaling features of the equations
2.4 The non-linear energy cascade
2.5 The inhomogeneous case
2.6 Dynamical system approach to turbulence
2.7 Shell models for turbulence cascade
2.8 The phenomenology of fully developed turbulence: Fluid-like case
2.9 The phenomenology of fully developed turbulence: Magnetically-dominated case
2.10 Some exact relationships
2.11 Yaglom’s law for MHD turbulence
2.12 Density-mediated Elsässer variables and Yaglom’s law
2.13 Yaglom’s law in the shell model for MHD turbulence
3 Early Observations of MHD Turbulence in the Ecliptic
3.1 Turbulence in the ecliptic
3.2 Turbulence studied via Elsässer variables
4 Observations of MHD Turbulence in the Polar Wind
4.1 Evolving turbulence in the polar wind
4.2 Polar turbulence studied via Elsässer variables
5 Numerical Simulations
5.1 Local production of Alfvénic turbulence in the ecliptic
5.2 Local production of Alfvénic turbulence at high latitude
6 Compressive Turbulence
6.1 On the nature of compressive turbulence
6.2 Compressive turbulence in the polar wind
6.3 The effect of compressive phenomena on Alfvénic correlations
7 A Natural Wind Tunnel
7.1 Scaling exponents of structure functions
7.2 Probability distribution functions and self-similarity of fluctuations
7.3 What is intermittent in the solar wind turbulence? The multifractal approach
7.4 Fragmentation models for the energy transfer rate
7.5 A model for the departure from self-similarity
7.6 Intermittency properties recovered via a shell model
8 Observations of Yaglom’s Law in Solar Wind Turbulence
9 Intermittency Properties in the 3D Heliosphere: Taking a Look at the Data
9.1 Structure functions
9.2 Probability distribution functions
10 Turbulent Structures
10.1 On the statistics of magnetic field directional fluctuations
10.2 Radial evolution of intermittency in the ecliptic
10.3 Radial evolution of intermittency at high latitude
11 Solar Wind Heating by the Turbulent Energy Cascade
11.1 Dissipative/dispersive range in the solar wind turbulence
12 The Origin of the High-Frequency Region
12.1 A dissipation range
12.2 A dispersive range
13 Two Further Questions About Small-Scale Turbulence
13.1 Whistler modes scenario
13.2 Kinetic Alfvén waves scenario
13.3 Where does the fluid-like behavior break down in solar wind turbulence?
13.4 What physical processes replace “dissipation” in a collisionless plasma?
14 Conclusions and Remarks
A Some Characteristic Solar Wind Parameters
B Tools to Analyze MHD Turbulence in Space Plasmas
B.1 Statistical description of MHD turbulence
B.2 Spectra of the invariants in homogeneous turbulence
B.3 Introducing the Elsässer variables
C Wavelets as a Tool to Study Intermittency
D Reference Systems
D.1 Minimum variance reference system
D.2 The mean field reference system
E On-board Plasma and Magnetic Field Instrumentation
E.1 Plasma instrument: The top-hat
E.2 Measuring the velocity distribution function
E.3 Computing the moments of the velocity distribution function
E.4 Field instrument: The flux-gate magnetometer
F Spacecraft and Datasets

9 Intermittency Properties in the 3D Heliosphere: Taking a Look at the Data

In this section, we present a reasoned look at the main aspect of what has been reported in literature about the problem of intermittency in the solar wind turbulence. In particular, we present results from data analysis.

9.1 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 (1991aJump To The Next Citation Point) 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 (p) log Sr 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-β model, from a theoretical point of view Carbone (1993Jump To The Next Citation Point, 1994b) showed that normalized scaling exponents ζp∕ζ4 calculated by Burlaga (1991aJump To The Next Citation Point) would be better fitted by using a p-model derived from the Kraichnan phenomenology (Kraichnan, 1965Jump To The Next Citation Point; Carbone, 1993Jump To The Next Citation Point), and considering the parameter μ ≃ 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 co-rotating 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 (1993Jump To The Next Citation Point) 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 (1993Jump To The Next Citation Point). From the analysis analogous to Burlaga (1991aJump To The Next Citation Point), 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, 1993Jump To The Next Citation Point), 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 (1993Jump To The Next Citation Point) found that data can be fitted with a piecewise linear function for the scaling exponents ζp, namely a β-model ζp = 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 ζp dataset with a single fit according to the non-linear random β-model.”. We like to comment that the impression by Marsch and Liu (1993Jump To The Next Citation Point) 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 ζp∕ζ3 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 (1993Jump To The Next Citation Point) and Bruno et al. (2003bJump To The Next Citation Point), 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 ζp∕ ζ3 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, 1987Jump To The Next Citation Point). 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 (1988Jump To The Next Citation Point) 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 ζp, 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. (1996Jump To The Next Citation Point) and Marsch and Tu (1997Jump To The Next Citation Point) investigated the behavior of an extended cascade model developed on the base of the p-model (Meneveau and Sreenivasan, 1987Jump To The Next Citation Point; Carbone, 1993). Authors conjectured that: i) the scaling laws for fluctuations are still valid in the form δZ ± ∼ 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 α of the power spectrum, say ⟨𝜖pr⟩ ∼ 𝜖p(ℓ,α)ℓ(p−1)Dp, where the averaged energy transfer rate is assumed to be

𝜖(ℓ,α) ∼ ℓ−(m ∕2+1)P α∕2, ℓ

being P ∼ ℓα ℓ the usual energy spectrum (ℓ ∼ 1∕k). The model gives

( ) [ ( )] p- m- m- p- ζp = 1 + m − 1 Dp ∕m + α 2 − 1 + 2 m , (67 )
where the generalized dimensions are recovered from the usual p-model
log2[μp-+-(1 −-μ)p] Dp = (1 − p) .

In the limit of “fully developed turbulence”, say when the spectral slope is α = 2∕m + 1 the usual Equation (63View Equation) is recovered. The Helios 2 data are consistent with this model as far as the parameters are μ ≃ 0.77 and α ≃ 1.45, and the fit is relatively good (Tu et al., 1996Jump To The Next Citation Point). Recently, Horbury et al. (1997Jump To The Next Citation Point) 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 63∘ heliolatitude. These authors showed that the observed magnetic field fluctuations were in agreement with the intermittent turbulence p-model of Meneveau and Sreenivasan (1987). 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. (1996Jump To The Next Citation Point) 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 ∘ 40 and ∘ 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., 1995Jump To The Next Citation Point) were obtained using structure functions to study the Ulysses magnetic field data in the range of scales 1 ≤ r ≤ 32 min. Ruzmaikin et al. (1995Jump To The Next Citation Point) 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., 1996Jump To The Next Citation Point) 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. (1995bJump To The Next Citation Point) 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 ζp∕ζ3 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 (64View Equation). 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.

9.2 Probability distribution functions

As said in Section 7.2 the statistics of turbulent flows can be characterized by the PDF of field differences over varying scales. At large scales PDFs are Gaussian, while tails become higher than Gaussian (actually, PDFs decay as exp[− δZ ±] ℓ) at smaller scales.

Marsch and Tu (1994Jump To The Next Citation Point) started to investigate the behavior of PDFs of fluctuations against scales and they found that PDFs are rather spiky at small scales and quite Gaussian at large scales. The same behavior have been obtained by Sorriso-Valvo et al. (1999Jump To The Next Citation Point, 2001) who investigated Helios 2 data for both velocity and magnetic field.

In order to make a quantitative analysis of the energy cascade leading to the scaling dependence of PDFs just described, the distributions obtained in the solar wind have been fitted (Sorriso-Valvo et al., 1999Jump To The Next Citation Point) by using the log-normal ansatz

( ) 1 ln2σ ∕σ0 G λ(σ) = √----- exp − --2λ2--- . (68 ) 2π λ
The width of the log-normal distribution of σ is given by ∘ ------- λ2(ℓ) = ⟨(δσ)2⟩, while σ0 is the most probable value of σ.

Table 5: The values of the parameters σ0, μ, and γ, in the fit of λ2(τ) (see Equation (68View Equation) as a kernel for the scaling behavior of PDFs. FW and SW refer to fast and slow wind, respectively, as obtained from the Helios 2 spacecraft, by collecting in a single dataset all periods.
parameter B field (SW) V field (SW) B field (FW) V field (FW)
σ0 0.90 ± 0.05 0.95 ± 0.05 0.85 ± 0.05 0.90 ± 0.05
μ 0.75 ± 0.03 0.38 ± 0.02 0.90 ± 0.03 0.54 ± 0.03
γ 0.18 ± 0.03 0.20 ± 0.04 0.19 ± 0.02 0.44 ± 0.05

The Equation (65View Equation) has been fitted to the experimental PDFs of both velocity and magnetic intensity, and the corresponding values of the parameter λ have been recovered. In Figure 94View Image the solid lines show the curves relative to the fit. It can be seen that the scaling behavior of PDFs, in all cases, is very well described by Equation (65View Equation). At every scale r, we get a single value for the width 2 λ (r), which can be approximated by a power law 2 −γ λ (r) = μr for r < 1 h, as it can be seen in Figure 95View Image. The values of parameters μ and γ obtained in the fit, along with the values of σ0, are reported in Table 5. The fits have been obtained in the range of scales τ ≤ 0.72 h for the magnetic field, and τ ≤ 1.44 h for the velocity field. The analysis of PDFs shows once more that magnetic field is more intermittent than the velocity field.

View Image

Figure 94: Left: normalized PDFs of fluctuations of the longitudinal velocity field at four different scales τ. Right: normalized PDFs of fluctuations of the magnetic field magnitude at four different scales τ. Solid lines represent the fit made by using the log-normal model. Image reproduced by permission from Sorriso-Valvo et al. (1999Jump To The Next Citation Point), copyright by AGU.
View Image

Figure 95: Scaling laws of the parameter λ2 (τ) as a function of the scales τ, obtained by the fits of the PDFs of both velocity and magnetic variables (see Figure 94View Image). Solid lines represent fits made by power laws. Image reproduced by permission from Sorriso-Valvo et al. (1999), copyright by AGU.

The same analysis has been repeated by Forman and Burlaga (2003Jump To The Next Citation Point). These authors used 64 s averages of radial solar wind speed reported by the SWEPAM instrument on the ACE spacecraft, increments have been calculated over a range of lag times from 64 s to several days. From the PDF obtained through the Equation (68View Equation) authors calculated the structure functions and compared the free parameters of the model with the scaling exponents of the structure functions. Then a fit on the scaling exponents allows to calculate the values of 2 λ and σ0. Once these parameters have been calculated, the whole PDF is evaluated. The same authors found that the PDFs do not precisely fit the data, at least for large values of the moment order. Interesting enough, Forman and Burlaga (2003) investigated the behavior of PDFs when different kernels G (σ) λ, derived from different cascade models, are taken into account in Equation (65View Equation). They discussed the physical content of each model, concluding that a cascade model derived from lognormal or log-Lévy theories,10 modified by self-organized criticality proposed by Schertzer et al. (1997), seems to avoid all problems present in other cascade models.

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