"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

13 Two Further Questions About Small-Scale Turbulence

The most “conservative” way to describe the presence of a dissipative/dispersive region in the solar wind turbulence, as we reported before, is for example through the Hall-MHD model. While when dealing with large scale we can successfully approach the problem of turbulence by saying that some form of dissipation must exist at small scales, the dissipationless character of solar wind cannot be avoided when we deal with small scales. The full understanding of the physical mechanisms that allow the dissipation of energy in the absence of collisional viscosity would be a step of crucial importance in the problem of high frequency turbulence in space plasmas. Another fundamental question concerns the dispersive properties of small-scale turbulence beyond the spectral break. This last question has been reformulated by saying: what are the principal constituent modes of small-scale turbulence? This approach explicitly assumes that small-scale fluctuations in solar wind can be described through a weak turbulence framework. In other words, a dispersion relation, namely a precise relationship between the frequency ω and the wave-vector k, is assumed.

As it is well known from basic plasma physics, linear theory for homogeneous, collisionless plasma yields three kind of modes at and below the proton cyclotron frequency Ωp. At wave-vectors transverse to the background magnetic field and Ωp > ωr (being ωr the real part of the frequency of fluctuation), two modes are present, namely a left-hand polarized Alfvén cyclotron mode and a right-hand polarized magnetosonic mode. A third ion-acoustic (slow) mode exists but is damped, except when T ≫ T e p, which is not common in solar wind turbulence. At quasi-perpendicular propagation the Alfvénic branch evolves into Kinetic Alfvén Waves (KAW), while magnetosonic modes may propagate at Ωp ≪ ωr as whistler modes. As the wave-vector becomes oblique to the background magnetic field both modes develop a nonzero magnetic compressibility where parallel fluctuations becomes important. There are two distinct scenarios for the subsequent energy cascade of KAW and whistlers (Gary and Smith, 2009Jump To The Next Citation Point).

13.1 Whistler modes scenario

This scenario involves a two-mode cascade process, both Alfvénic and magnetosonic modes which are only weakly damped as the plasma β ≤ 1, transfer energy to quasi-perpendicular propagating wave-vectors. The KAW are damped by Landau damping which is proportional to k2⊥, so that they cannot contribute to the formation of dispersive region (unless for fluctuations propagating along the perpendicular direction). Even left-hand polarized Alfvén modes at quasi-parallel propagation suffer for proton cyclotron damping at scales k∥ ∼ ωp ∕c and do not contribute. Quasi-parallel magnetosonic modes are not damped at the above scale, so that a weak cascade of right-hand polarized fluctuations can generate a dispersive region of whistler modes (Stawicki et al., 2001; Gary and Borovsky, 2004, 2008; Goldstein et al., 1994). The cascade of weakly damped whistler modes has been reproduced through electron MHD numerical simulations (Biskamp et al., 1996, 1999; Wareing and Hollerbach, 2009; Cho and Lazarian, 2004) and Particle-in-Cell (PIC) codes (Gary et al., 2008; Saito et al., 2008).

13.2 Kinetic Alfvén waves scenario

In this scenario (Howes, 2008; Schekochihin et al., 2009) long-wavelength Alfvénic turbulence transfer energy to quasi-perpendicular propagation for the primary turbulent cascade up to the thermal proton gyroradius where fluctuations are subject to the proton Landau damping. The remaining fluctuation energy continues the cascade to small scales as KAW at quasi-perpendicular propagation and at frequencies ωr < Ωp Bale et al. (2005Jump To The Next Citation Point); Sahraoui et al. (2009Jump To The Next Citation Point). Fluctuations are completely damped via electron Landau resonance at wavelength of the order of the electron gyroradius. This scenario has been observed through gyrokinetic numerical simulations Howes et al. (2008bJump To The Next Citation Point), where the spectral breakpoint k ⊥ ∼ Ωp∕vth (being vth the proton thermal speed) has been observed.

13.3 Where does the fluid-like behavior break down in solar wind turbulence?

Up to now spacecraft observations do not allow us to unambiguously distinguish between both previous scenarios. As stated by Gary and Smith (2009) at our present level of understanding of linear theory, the best we can say is that quasi-parallel whistlers, quasi-perpendicular whistlers, and KAW all probably could contribute to dispersion range turbulence in solar wind. Thus, the critical question is not which mode is present (if any exists in a nonlinear, collisionless medium as solar wind), but rather, what are the conditions which favor one mode over the others. On the other hand, starting from observations, we cannot rule out the possibility that strong turbulence rather than “modes” are at work to account for the high-frequency part of the magnetic energy spectrum. One of the most striking observations of small-scale turbulence is the fact that the electric field is strongly enhanced after the spectral break (Bale et al., 2005Jump To The Next Citation Point). This means that turbulence at small scales is essentially electrostatic in nature, even if weak magnetic fluctuations are present. The enhancement of the electrostatic part has been viewed as a strong indication for the presence of KAW, because gyrokinetic simulations show the same phenomenon Howes et al. (2008b). However, as pointed out by Matthaeus et al. (2008Jump To The Next Citation Point) (see also the Reply by Howes et al., 2008a to the comment by Matthaeus et al., 2008), the enhancement of electrostatic fluctuations can be well reproduced by Hall-MHD turbulence, without the presence of KAW modes. Actually, the enhancement of the electric field turns out to be a statistical property of the inviscid Hall MHD (Servidio et al., 2008), that is in the absence of viscous and dissipative terms the statistical equilibrium ensemble of Hall-MHD equations in the wave-vectors space is build up with an enhancement of the electric field at large wave-vectors. This represents a thermodynamic equilibrium property of equations, and has little to do with a non-equilibrium turbulent cascade13. This would means that the enhancement of the electrostatic part of fluctuations cannot be seen as a proof firmly establishing that KAW are at work in the dispersive region.

One of the most peculiar possibility from the Cluster spacecraft was the possibility to separate the time domain from the space domain, using the tetrahedral formation of the four spacecrafts which form the Cluster mission (Escoubet et al., 2001). This allows us to obtain a 3D wavevector spectrum and the possibility to identify the actual dispersion relation of solar wind turbulence, if any exists, at small scales. This can be made by using the k-filtering technique which is based on the strong assumption of plane-wave propagation (Glassmeier et al., 2001). Of course, due to the relatively small distances between spacecrafts, this cannot be applied to large-scale turbulence.

Apart for the spectral break identified by Leamon et al. (1998), a new break has been identified in the solar wind turbulence using high-frequency Cluster data, at about few tens of Hz. In fact, Cluster data at the burst mode can reach the characteristic electron inertial scale λe and the electron Larmor radius ρe. Using FluxGate Magnetometer and Spatiotemporal Analysis of Field Fluctuations experiment/search coil, Sahraoui et al. (2009) showed that the turbulent spectrum changes shape at wavevectors of about kρe ∼ kλe ≃ 1. This result, which perhaps identify the occurrence of a dissipative range in solar wind turbulence, has been obtained in the upstream solar wind magnetically connected to the bow shock. However, in these studies the plasma β was of the order of β ≃ 1, thus not allowing the separation between both scales. Alexandrova et al. (2009), using three instruments onboard Cluster spacecrafts operating in different frequency ranges, resolved the spectrum up to 300 Hz. They confirmed the presence of the high-frequency spectral break at about kρe ∼ [0.1,1] and, what is mainly interesting, they fitted this part of the spectrum through an exponential decay √ ---- ∼ exp[− kρe], thus indicating the onset of dissipation.

The 3D spectral shape reveals poor surprise, that is the energy distribution exhibits anisotropic features characterized by a prominently extended structure perpendicular to the mean magnetic field preferring the ecliptic north direction and also by a moderately extended structure parallel to the mean field (Narita et al., 2010). Results of the 3D energy distribution suggest the dominance of quasi 2D turbulence toward smaller spatial scales, overall symmetry to changing the sign of the wave vector (reflectional symmetry) and absence of spherical and axial symmetry. This last was one of the main hypothesis for the Maltese Cross (Matthaeus et al., 1990), even if bias due to satellite fly through can generate artificial deviations from axisymmetry (Turner et al., 2011).

More interestingly, (Sahraoui et al., 2010bJump To The Next Citation Point) investigate the occurrence of a dispersion relation. They claim that the energy cascade should be carried by highly oblique KAW with doppler-shifted plasma frequency ωplas ≤ 0.1 ωci down to k⊥ρi ∼ 2. Each wavevector spectrum in the direction perpendicular to an “average” magnetic field B0 shows two scaling ranges separated by a breakpoint in the interval [0.1,1]k ⊥ρi, say a Kolmogorov scaling followed by a steeper scaling. The authors conjecture that the turbulence undergoes a transition-range, where part of energy is dissipated into proton heating via Landau damping, and the remaining energy cascades down to electron scales where Electron Landau damping may dominate. The dispersion relation, compared with linear solutions of the Maxwell–Vlasov equations (Sahraoui et al., 2010bJump To The Next Citation Point, cf. Figure 5 of), seems to identify KAW as responsible for the cascade at small scales. The conjecture by Sahraoui et al. (2010b) does not take into account the fact that Landau damping is rapidly saturating under solar wind conditions (Marsch, 2006Jump To The Next Citation Point; Valentini et al., 2008Jump To The Next Citation Point).

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Figure 112: Observed dispersion relations (dots), with estimated error bars, compared to linear solutions of the Maxwell–Vlasov equations for three observed angles between the k vector and the local magnetic field direction (damping rates are represented by the dashed lines). Proton and electron Landau resonances are represented by the black curves Lp,e. Proton cyclotron resonance are shown by the curves Cp. (the electron cyclotron resonance lies out of the plotted frequency range). Image reproduced by permission from Sahraoui et al. (2010a), copyright by APS.

The question of the existence of a dispersion relation was investigated by Narita et al. (2011aJump To The Next Citation Point), which investigated three selected time intervals of magnetic field data of CLUSTER FGM in the solar wind. They used a refined version of the k-filtering technique, called MSR technique, to obtain high-resolution energy spectra in the wavevector domain. Like the wave telescope, the MSR technique performs fitting of the measured data with a propagating plane wave as a function of frequency and wave vector. The main result is the strong spread in the frequency-wavevector domain, namely none of the three intervals exhibits a clear organization of dispersion relation (see Figure 113View Image). Frequencies and wave vectors appear to be strongly scattered, thus not allowing for the identification of wave-like behavior.

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Figure 113: Top: Angles between the wave vectors and the mean magnetic field as a function of the wave number. Bottom: Frequency-wave number diagram of the identified waves in the plasma rest frame. Magnetosonic (MS), whistler (WHL), and kinetic Alfvén waves (KAW)dispersion relations are represented by dashed, straight, and dotted lines, respectively. Image reproduced by permission from Narita et al. (2011a), copyright by AGU.

The above discussed papers shed some “darkness” on the scenario of small scales solar wind turbulence as made by “modes”, or at least they indicate that solar wind turbulence, at least at small scales, is far from universality. As a further stroke of the grey brush, Perri et al. (2011Jump To The Next Citation Point) simply calculated the frequency of the spectral break as a function of radial distances from the Sun. In fact, since plasma parameters, and in particular the magnetic field intensity, changes when going towards large radial distances, the frequency break should change accordingly. They used Messenger data, as far as the inner heliosphere is concerned, and Ulysses data for outer heliosphere. Data from 0.5 AU, up to 5 AU, are summarized in Figure 2 of Perri et al. (2011). While the characteristic frequencies of plasma lower going to higher radial distances, the position of the spectral break remains constant over all the interval of distances investigated. That is the observed high-frequency spectral break seems to be independent of the distance from the Sun, and then of both the ion-cyclotron frequency and the proton gyroradius. So, where does the fluid-like behavior break down in solar wind turbulence?

13.4 What physical processes replace “dissipation” in a collisionless plasma?

As we said before, the understanding of the small-scale termination of the turbulent energy cascade in collisionless plasmas is nowadays one of the outstanding unsolved problem in space plasma physics. In the absence of collisional viscosity and resistivity the dynamics of small scales is kinetic in nature and must be described by the kinetic theory of plasma. The identification of the physical mechanism that “replaces” dissipation in the collisionless solar wind plasma and establishes a link between the macroscopic and the microscopic scales would open new scenarios in the study of the turbulent heating in space plasmas. This problem is yet in its infancy. Kinetic theory is known since long time from plasma physics, the interested reader can read the excellent review by Marsch (2006Jump To The Next Citation Point). However, it is restricted mainly to linear theoretical arguments. The fast technological development of supercomputers gives nowadays the possibility of using kinetic Eulerian Vlasov codes that solve the Vlasov–Maxwell equations in multi-dimensional phase space. The only limitation to the “dream” of solving 3D-3V problems (3D in real space and 3D in velocity space) resides in the technological development of fast enough solvers. The use of almost noise-less codes is crucial and allows for the first time the possibility of analyzing kinetic nonlinear effects as the nonlinear evolution of particles distribution function, nonlinear saturation of Landau damping, etc. Of course, faster numerical way to solve the dissipation issue in collisionless plasmas might consist in using intermediate gyrokinetic descriptions (Brizard and Hahm, 2007) based on a gyrotropy and strong anisotropy assumptions k∥ ≪ k⊥.

As we said before, observations of small-scale turbulence showed the presence of a significant level of electrostatic fluctuations (Gurnett and Anderson, 1977Jump To The Next Citation Point; Gurnett and Frank, 1978Jump To The Next Citation Point; Gurnett et al., 1979Jump To The Next Citation Point; Bale et al., 2005). Old observations of plasma wave measurements on the Helios 1 and 2 spacecrafts (Gurnett and Anderson, 1977; Gurnett and Frank, 1978Jump To The Next Citation Point; Gurnett et al., 1979Jump To The Next Citation Point) have revealed the occurrence of electric field wave-like turbulence in the solar wind at frequencies between the electron and ion plasma frequencies. Wavelength measurements using the IMP 6 spacecraft provided strong evidence for the presence of electric fluctuations which were identified as ion acoustic waves which are Doppler-shifted upward in frequency by the motion of the solar wind (Gurnett and Frank, 1978Jump To The Next Citation Point). Comparison of the Helios results showed that the ion acoustic wave-like turbulence detected in interplanetary space has characteristics essentially identical to those of bursts of electrostatic turbulence generated by protons streaming into the solar wind from the earth’s bow shock (Gurnett and Frank, 1978Jump To The Next Citation Point; Gurnett et al., 1979). Gurnett and Frank (1978) observed that in a few cases of Helios data, ion acoustic wave intensities are enhanced in direct association with abrupt increases in the anisotropy of the solar wind electron distribution. This relationship strongly suggests that the ion acoustic wave-like structures detected by Helios far from the earth are produced by an electron heat flux instability or by protons streaming into the solar wind from the earth’s bow shock. Further evidences (Marsch, 2006Jump To The Next Citation Point) revealed the strong association between the electrostatic peak and nonthermal features of the velocity distribution function of particles like temperature anisotropy and generation of accelerated beams.

Araneda et al. (2008Jump To The Next Citation Point) using Vlasov kinetic theory and one-dimensional Particle-in-Cell hybrid simulations provided a novel explanation of the bursts of ion-acoustic activity occurring in the solar wind. These authors studied the effect on the proton velocity distributions in a low-β plasma of compressible fluctuations driven by the parametric instability of Alfvén-cyclotron waves. Simulations showed that field-aligned proton beams are generated during the saturation phase of the wave-particle interaction, with a drift speed which is slightly greater than the Alfvén speed. As a consequence, the main part of the distribution function becomes anisotropic due to phase mixing. This observation is relevant, because the same anisotropy is typically observed in the velocity distributions measured in the fast solar wind (Marsch, 2006).

In recent papers, Valentini et al. (2008Jump To The Next Citation Point) and Valentini and Veltri (2009Jump To The Next Citation Point) used hybrid Vlasov–Maxwell model where ions are considered as kinetic particles, while electrons are treated as a fluid. Numerical simulations have been obtained in 1D-3V phase space (1D in the physical space and 3D in the velocity space) where a turbulent cascade is triggered by the nonlinear coupling of circularly left-hand polarized Alfvén waves, in the perpendicular plane and in parallel propagation, at plasma-β of the order of unity. Numerical results show that energy is transferred to short scales in longitudinal electrostatic fluctuations of the acoustic form. The numerical dispersion relation in the k − ω plane displays the presence of two branches of electrostatic waves. The upper branch, at higher frequencies, consists of ion-acoustic waves while the new lower frequency branch consists of waves propagating with a phase speed of the order of the ion thermal speed. This new branch is characterized by the presence of a plateau around the thermal speed in the ion distribution function, which is a typical signature of the nonlinear saturation of wave-particle interaction process.

Numerical simulations show that energy should be “dissipated” at small scales through the generation of an ion-beam in the velocity distribution function as a consequence of the trapping process and the nonlinear saturation of Landau damping, which results in bursts of electrostatic activity. Whether or not this picture, which seems to be confirmed by recent numerical simulations (Araneda et al., 2008; Valentini et al., 2008; Valentini and Veltri, 2009), represents the final fate of the real turbulent energy cascade observed at macroscopic scales, requires further investigations. Available measurements in the interplanetary space, even using Cluster spacecrafts, do not allow analysis at typical kinetic scales.

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