"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

F Spacecraft and Datasets

Measurements performed by spacecrafts represent a unique chance to investigate a wide range of scales of low-frequency turbulence in a magnetized medium. The interested readers are strongly encouraged to visit the web pages of each specific space mission or, more simply, the Heliophysics Data Portal (formerly VSPO) (External Link as a wide source of information. This portal represents an easy way to access all the available datasets, related to magnetospheric and heliospheric missions, allowing the user to quickly find data files and interfaces to data from a large number of space missions and ground-based observations.

Two of the s/c which have contributed most to the study of MHD turbulence are the old Helios and Voyager spacecraft, which explored the inner and outer heliosphere, respectively, providing us with an almost complete map of the gross features of low-frequency plasma turbulence. The Helios project was a German-American mission consisting in two interplanetary probes: Helios 1, which was launched in December 1974, and Helios 2, launched one year later. These s/c had a highly elliptic orbit, lying in the ecliptic, which brought the s/c from 1 AU to 0.3 AU in only 6 months. Helios dataset is, with no doubt, the most important and unique one to study MHD turbulence in the inner heliosphere. Most of the knowledge we have today about this topic is based on Helios data mainly because this s/c is the only one that has gone so close to the Sun. As a matter of fact, the orbit of this s/c allowed to observe the radial evolution of turbulence within regions of space (< 0.7 AU) where dynamical processes between fast and slow streams have not yet reprocessed the plasma.

The two Voyagers were launched in 1977. One of them, Voyager 1 will soon reach the termination shock and enter the interstellar medium. As a consequence, for the first time, we will be able to measure interstellar particles and fields not affected by the solar wind. Within the study of MHD turbulence, the importance of the two Voyagers in the outer heliosphere is equivalent to that of the two Helios in the inner regions of the heliosphere. However, all these s/c have been limited to orbit in the, or close to the ecliptic plane.

Finally, in October 1990, Ulysses was launched and, after a fly-by with Jupiter it reached its final orbit tilted at 80.2∘ with respect to the solar equator. For the first time, we were able to sample the solar wind coming from polar coronal holes, the pure fast wind not “polluted” by the dynamical interaction with slow equatorial wind. As a matter of fact, the Ulysses scientific mission has been dedicated to investigate the heliospheric environment out of the ecliptic plane. This mission is still providing exciting results.

Another spacecraft called WIND was launched in November 1994 and is part of the ISTP Project. WIND, was initially located at the Earth-Sun Lagrangian point L1 to sample continuously the solar wind. Afterwards, it was moved to a much more complicated orbit which allows the spacecraft to repeatedly visit different regions of space around Earth, while continuing to sample the solar wind. The high resolution of magnetic field and plasma measurements of WIND makes this spacecraft very useful to investigate small scales phenomena, where kinetic effects start to play a key role.

The Advanced Composition Explorer (ACE) represents another solar wind monitor located at L1. This spacecraft was launched by NASA in 1997 and its solar wind instruments are characterized by a high rate sampling. Finally, we like to call the attention of the reader on the possibility to easily view and retrieve from the web real time solar wind data from both WIND and ACE.

A few years ago, Voyager 1 and Voyager 2 reached the termination shock, extending our exploration to almost the whole heliosphere. However, the exploration will not be complete until we will reach the base of the solar corona. All the fundamental physical processes concurring during the birth of the solar wind take place in this part of the heliosphere. Moreover, this is a key region also for the study of turbulence, since here non-linear interactions between inward and outward modes start to be active and produce the turbulence spectrum that we observe in the heliosphere.

This region is so important for our understanding of the solar wind that both ESA and NASA are planning space mission dedicated to explore it. In particular, the European Space Agency is planning to launch the Solar Orbiter mission in January 2017 (External Link

Solar Orbiter is proposed as a space mission dedicated to study the solar surface, the corona, and the solar wind by means of remote sensing and in-situ measurements, respectively. Consequently, the s/c will carry a heliospheric package primarily designed to measure ions and electrons of the solar wind, energetic particles, radio waves, and magnetic fields and a remote sensing package for for EUV/X-ray spectroscopy and imaging of the disk and the extended corona. In particular, the high resolution imaging of the Sun will give close-up observations of the solar atmosphere, with the capability of resolving fine structures (of the order of 100 km) in the transition region and corona. This will certainly represent a major step forward in investigating the occurrence of intermittent structures of turbulence at very small scales.

The observations provided by Helios 25 years ago and, more recently, by Ulysses suggest that the local production of Alfvén waves is much stronger in the region just inside 0.3 AU, and Solar Orbiter, repeatedly reaching 0.28 AU, will provide excellent observations to study problems related to local generation and non-linear coupling between outward and inward waves. Moreover, the high data sampling will provide extremely useful and totally new insight about wave dissipation via wave-particle coupling mechanism and the role that the damping of slow, fast, and Alfvén waves can have in the heating of the solar wind ions. Finally, the opportunity given by Solar Orbiter to correlate in-situ plasma measures with the simultaneous imaging of the same flow element of the solar wind during the co-rotation phase, will provide the possibility to separate temporal effects from spatial effects for the first time in the solar wind. This will be of primary importance for finally understand the physical mechanisms at the basis of the solar wind generation.

A similar mission, Solar Probe Plus (External Link, is under development by NASA, on a schedule to launch no later than 2018. Solar Probe Plus will orbit the Sun 24 times, gradually approaching the Sun with each pass. On the final three orbits, Solar Probe Plus will fly to within 8.5 solar radii of the Sun’s surface. This mission, although very risky, will allow us to tremendously advance our knowledge about the physical processes that heat and accelerate the solar wind.

Thus, future key missions for investigating turbulence properties in the solar wind plasma are not just behind the corner and, for the time being, we have to use observations from already flown or still flying spacecraft. This does not mean that exciting results are over, while we wait for these new missions. The main difference with the past is that now we are in a different phase of our research. This phase aims to refine many of the concepts we learned in the past, especially those concerning the radial evolution and the local production of turbulence. As a consequence more refined data analysis and computer simulations are now discovering new and very interesting aspects of MHD turbulence which, we hope, we contributed to illustrate in this review.

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