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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

List of Tables

Table 1:
Scaling exponents for velocity ζp and magnetic ξp 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.
Table 2:
Normalized scaling exponents ξpāˆ•ξ3 for radial magnetic fluctuations in a laboratory plasma, as measured at different distances a āˆ•R (R ā‰ƒ 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.
Table 3:
Normalized scaling exponents ξpāˆ•ξ3 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 (40View Equation) as characteristic scale rather than the third-order structure function (cf. Politano et al., 1998, for details).
Table 4:
Scaling exponents for velocity and magnetic variables, Elsässer variables, and fluxes. Errors on β ± p are about one order of magnitude smaller than the errors shown.
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.
Table 6:
Typical values of several solar wind parameters as measured by Helios 2 at 1 AU.
Table 7:
Typical values of different speeds obtained at 1 AU. The Alfvén speed has been measured, while all the others have been obtained from the parameters reported in Table 6.
Table 8:
Typical values of different frequencies at 1 AU. These values have been obtained from the parameters reported in Table 6.
Table 9:
Typical values of different lengths at 1 AU plus the distance traveled by a proton before colliding with another proton. These values have been obtained from the parameters reported in Table 6.
Table 10:
Expected values for Alfvèn ratio rA, normalized cross-helicity σc, and normalized residual energy σ r for a pure Alfvèn wave outward or inward oriented.