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"The Solar Wind as a Turbulence Laboratory"
Roberto Bruno and Vincenzo Carbone 
Abstract
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
Acknowledgments
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
References
Footnotes
Updates
Figures
Tables

8 Observations of Yaglom’s Law in Solar Wind Turbulence

To avoid the risk of misunderstanding, let us start by recalling that Yaglom’s law (39View Equation) has been derived from a set of equations (MHD) and under assumptions which are far from representing an exact mathematical model for the solar wind plasma. Yaglom’s law is valid in MHD under the hypotheses of incompressibility, stationarity, homogeneity, and isotropy. Also, the form used for the dissipative terms of MHD equations is only valid for collisional plasmas, characterized by quasi-Maxwellian distribution functions, and in case of equal kinematic viscosity and magnetic diffusivity coefficients (Biskamp, 2003). In solar wind plasmas the above hypotheses are only rough approximations, and MHD dissipative coefficients are not even defined (Tu and Marsch, 1995aJump To The Next Citation Point). At frequencies higher than the ion cyclotron frequency, kinetic processes are indeed present, and a number of possible dissipation mechanisms can be discussed. When looking for the Yaglom’s law in the SW, the strong conjecture that the law remains valid for any form of the dissipative term is needed.

Despite the above considerations, Yaglom’s law results surprisingly verified in some solar wind samples. Results of the occurrence of Yaglom’s law in the ecliptic plane, has been reported by MacBride et al. (2008Jump To The Next Citation Point, 2010Jump To The Next Citation Point) and Smith et al. (2009Jump To The Next Citation Point) and, independently, in the polar wind by Sorriso-Valvo et al. (2007Jump To The Next Citation Point). It is worthwhile to note that, the occurrence of Yaglom’s law in polar wind, where fluctuations are Alfvénic, represents a double surprising feature because, according to the usual phenomenology of MHD turbulence, a nonlinear energy cascade should be absent for Alfénic turbulence.

In a first attempt to evaluate phenomenologically the value of the energy dissipation rate, MacBride et al. (2008Jump To The Next Citation Point) analyzed the data from ACE to evaluate the occurrence of both the Kolmogorov’s 4/5-law and their MHD analog (39View Equation). Although some words of caution related to spikes in wind speed, magnetic field strength caused by shocks and other imposed heliospheric structures that constitute inhomogeneities in the data, authors found that both relations are more or less verified in solar wind turbulence. They found a distribution for the energy dissipation rate, defined in the above paper as šœ– = (šœ–+ + šœ–− )āˆ•2 ii ii, with an average of about 4 šœ– ā‰ƒ 1.22 × 10 Jāˆ•Kg s.

In order to avoid variations of the solar activity and ecliptic disturbances (like slow wind sources, coronal mass ejections, ecliptic current sheet, and so on), and mainly mixing between fast and slow wind, Sorriso-Valvo et al. (2007Jump To The Next Citation Point) used high speed polar wind data measured by the Ulysses spacecraft. In particular, authors analyze the first seven months of 1996, when the heliocentric distance slowly increased from 3 AU to 4 AU, while the heliolatitude decreased from about 55āˆ˜ to 30āˆ˜. The third-order mixed structure functions have been obtained using 10-days moving averages, during which the fields can be considered as stationary. A linear scaling law, like the one shown in Figure 92View Image, has been observed in a significant fraction of samples in the examined period, with a linear range spanning more than two decades. The linear law generally extends from few minutes up to 1 day or more, and is present in about 20 periods of a few days in the 7 months considered. This probably reflects different regimes of driving of the turbulence by the Sun itself, and it is certainly an indication of the nonstationarity of the energy injection process. According to the formal definition of inertial range in the usual fluid flows, authors attribute to the range where Yaglom’s law appear the role of inertial range in the solar wind turbulence (Sorriso-Valvo et al., 2007Jump To The Next Citation Point). This range extends on scales larger than the usual range of scales where a Kolmogorov relation has been observed, say up to about few hours (cf. Figure 25View Image).

View Image

Figure 92: An example of the linear scaling for the third-order mixed structure functions Y ±, obtained in the polar wind using Ulysses measurements. A linear scaling law represents a range of scales where Yaglom’s law is satisfied. Image reproduced by permission from Sorriso-Valvo et al. (2007Jump To The Next Citation Point), copyright by APS.

Several other periods are found where the linear scaling range is reduced and, in particular, the sign of Y ± ā„“ is observed to be either positive or negative. In some other periods the linear scaling law is observed either for Y + ā„“ or Y − ā„“ rather than for both quantities. It is worth noting that in a large fraction of cases the sign switches from negative to positive (or viceversa) at scales of about 1 day, roughly indicating the scale where the small scale Alfvénic correlations between velocity and magnetic fields are lost. This should indicate that the nature of fluctuations changes across the break. The values of the pseudo-energies dissipation rates šœ–± has been found to be of the order of magnitude about few hundreds of J/Kg s, higher than that found in usual fluid flows which result of the order of 1 ÷ 50 Jāˆ•Kg s.

The occurrence of Yaglom’s law in solar wind turbulence has been evidenced by a systematic study by MacBride et al. (2010Jump To The Next Citation Point), which, using ACE data, found a reasonable linear scaling for the mixed third-order structure functions, from about 64 s. to several hours at 1 AU in the ecliptic plane. Assuming that the third-order mixed structure function is perpendicular to the mean field, or assuming that this function varies only with the component of the scale ā„“α that is perpendicular to the mean field, and is cylindrically symmetric, the Yaglom’s law would reduce to a 2D state. On the other hand, if the third-order function is parallel to the mean field or varies only with the component of the scale that is parallel to the mean field, the Yaglom’slaw would reduce to a 1D-like case. In both cases the result will depend on the angle between the average magnetic field and the flow direction. In both cases the energy cascade rate varies in the range 103 ÷ 104 Jāˆ•Kg s (see MacBride et al., 2010Jump To The Next Citation Point, for further details).

Quite interestingly, Smith et al. (2009) found that the pseudo-energy cascade rates derived from Yaglom’s scaling law reveal a strong dependence on the amount of cross-helicity. In particular, they showed that when the correlation between velocity and magnetic fluctuations are higher than about 0.75, the third-order moment of the outward-propagating component, as well as of the total energy and cross-helicity are negative. As already made by Sorriso-Valvo et al. (2007Jump To The Next Citation Point), they attribute this phenomenon to a kind of inverse cascade, namely a back-transfer of energy from small to large scales within the inertial range of the dominant component. We should point out that experimental values of energy transfer rate in the incompressive case, estimated with different techniques from different data sets (Vasquez et al., 2007Jump To The Next Citation Point; MacBride et al., 2010Jump To The Next Citation Point), are only partially in agreement with that obtained by Sorriso-Valvo et al. (2007). However, the different nature of wind (ecliptic vs. polar, fast vs. slow, at different radial distances from the Sun) makes such a comparison only indicative.

As far as the scaling law (46View Equation) is concerned, Carbone et al. (2009aJump To The Next Citation Point) found that a linear scaling for ± W ā„“ as defined in (46View Equation), appears almost in all Ulysses dataset. In particular, the linear scaling for ± W ā„“ is verified even when there is no scaling at all for ± Yā„“ (39View Equation). In particular, it has been observed (Carbone et al., 2009aJump To The Next Citation Point) that a linear scaling for W +ā„“ appears in about half the whole signal, while W −ā„“ displays scaling on about a quarter of the sample. The linear scaling law generally extends on about two decades, from a few minutes up to one day or more, as shown in Figure 93View Image. At variance to the incompressible case, the two fluxes ± W ā„“ coexist in a large number of cases. The pseudo-energies dissipation rates so obtained are considerably larger than the relative values obtained in the incompressible case. In fact it has been found that on average + 3 šœ– ā‰ƒ 3 × 10 J āˆ•Kg s. This result shows that the nonlinear energy cascade in solar wind turbulence is considerably enhanced by density fluctuations, despite their small amplitude within the Alfvénic polar turbulence. Note that the new variables Δw ±i are built by coupling the Elsässer fields with the density, before computing the scale-dependent increments. Moreover, the third-order moments are very sensitive to intense field fluctuations, that could arise when density fluctuations are correlated with velocity and magnetic field. Similar results, but with a considerably smaller effect, were found in numerical simulations of compressive MHD (Mac Low and Klessen, 2004).

View Image

Figure 93: The linear scaling relation is reported for both the usual third-order structure function Y + ā„“ and the same quantity build up with the density-mediated variables W + ā„“. A linear relation full line is clearly observed. Data refer to the Ulysses spacecraft. Image reproduced by permission from Carbone et al. (2009a), copyright by APS.

Finally, it is worth reporting that the presence of Yaglom’s law in solar wind turbulence is an interesting theoretical topic, because this is the first real experimental evidence that the solar wind turbulence, at least at large-scales, can be described within the magnetohydrodynamic model. In fact, Yaglom’s law is an exact law derived from MHD equations and, let us say once more, their occurrence in a medium like the solar wind is a welcomed surprise. By the way, the presence of the law in the polar wind solves the paradox of the presence of Alfvénic turbulence as first pointed out by Dobrowolny et al. (1980a). Of course, the presence of Yaglom’s law generates some controversial questions about data selection, reliability and a brief discussion on the extension of the inertial range. The interested reader can find some questions and relative answers in Physical Review Letters (Forman et al., 2010; Sorriso-Valvo et al., 2010a).


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