"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

A Some Characteristic Solar Wind Parameters

Although solar wind is a highly variable medium, it is possible to identify some characteristic values for its most common parameters. Since the wind is an expanding medium, we ought to choose one heliocentric distance to refer to and, usually, this distance is 1 AU. In the following, we will provide different tables referring to several solar wind parameters, velocities, characteristic times, and lengths.

As it can be seen, the solar wind is a super-Alfvénic, collisionless plasma, and MHD turbulence can be investigated for frequencies smaller than −1 ∼ 10 Hz.

Table 6: Typical values of several solar wind parameters as measured by Helios 2 at 1 AU.
Wind Parameter Slow wind Fast wind
number density ∼ 15 cm–3 ∼ 4 cm–3
bulk velocity ∼ 350 km s–1 ∼ 600 km s–1
proton temperature ∼ 5 × 104 K ∼ 2 × 105 K
electron temperature ∼ 2 × 105 K ∼ 1 × 105 K
α-particles temperature ∼ 2 × 105 K ∼ 8 × 105 K
magnetic field ∼ 6 nT ∼ 6 nT

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.
Speed Slow wind Fast wind
Alfvén ∼ 30 km s–1 ∼ 60 km s–1
ion sound ∼ 60 km s–1 ∼ 60 km s–1
proton thermal ∼ 35 km s–1 ∼ 70 km s–1
electron thermal ∼ 3000 km s–1 ∼ 2000 km s–1

Table 8: Typical values of different frequencies at 1 AU. These values have been obtained from the parameters reported in Table 6.
Frequency Slow wind Fast wind
proton cyclotron ∼ 0.1 Hz ∼ 0.1 Hz
electron cyclotron ∼ 2 × 102 Hz ∼× 102 Hz
plasma ∼× 105 Hz ∼ 1 × 105 Hz
proton-proton collision ∼ 2 × 10–6 Hz ∼ 1 × 10–7 Hz

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.
Length Slow wind Fast wind
Debye ∼ 4 m ∼ 15 m
proton gyroradius ∼ 130 km ∼ 260 km
electron gyroradius ∼ 2 km ∼ 1.3 km
distance between 2 proton collisions ∼ 1.2 AU ∼ 40 AU

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