"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

5 Numerical Simulations

Numerical simulations currently represent one of the main source of information about non-linear evolution of fluid flows. The actual super-computers are now powerful enough to simulate equations (NS or MHD) that describe turbulent flows with Reynolds numbers of the order of 104 in two-dimensional configurations, or 103 in three-dimensional one. Of course, we are far from achieving realistic values, but now we are able to investigate turbulence with an inertial range extended for more than one decade. Rather the main source of difficulties to get results from numerical simulations is the fact that they are made under some obvious constraints (say boundary conditions, equations to be simulated, etc.), mainly dictated by the limited physical description that we are able to use when numerical simulations are made, compared with the extreme richness of the phenomena involved: numerical simulations, even in standard conditions, are used tout court as models for the solar wind behavior. Perhaps the only exception, to our knowledge, is the attempt to describe the effects of the solar wind expansion on turbulence evolution like, for example, in the papers by Velli et al. (1989, 1990); Hellinger and Trávníček (2008). Even with this far too pessimistic point of view, used here solely as a few words of caution, simulations in some cases were able to reproduce some phenomena observed in the solar wind.

Nevertheless, numerical simulations have been playing a key role, and will continue to do so in our seeking an understanding of turbulent flows. Numerical simulations allows us to get information that cannot be obtained in laboratory. For example, high resolution numerical simulations provide information at every point on a grid and, for some times, about basic vector quantities and their derivatives. The number of degree of freedom required to resolve the smaller scales is proportional to a power of the Reynolds number, say to Re9 ∕4, although the dynamically relevant number of modes may be much less. Then one of the main challenge remaining is how to handle and analyze the huge data files produced by large simulations (of the order of Terabytes). Actually a lot of papers appeared in literature on computer simulations related to MHD turbulence. The interested reader can look at the book by Biskamp (1993Jump To The Next Citation Point) and the reviews by Pouquet (1993, 1996).

5.1 Local production of Alfvénic turbulence in the ecliptic

The discovery of the strong correlation between velocity and magnetic field fluctuations has represented the motivation for some MHD numerical simulations, aimed to confirm the conjecture by Dobrowolny et al. (1980bJump To The Next Citation Point). The high level of correlation seems to be due to a kind of self-organization (dynamical alignment) of MHD turbulence, generated by the natural evolution of MHD towards the strongest attractive fixed point of equations (Ting et al., 1986Jump To The Next Citation Point; Carbone and Veltri, 1987, 1992Jump To The Next Citation Point). Numerical simulations (Carbone and Veltri, 1992; Ting et al., 1986) confirmed this conjecture, say MHD turbulence spontaneously can tends towards a state were correlation increases, that is, the quantity σc = 2Hc ∕E, where Hc is the cross-helicity and E the total energy of the flow (see Appendix B.1), tends to be maximal.

The picture of the evolution of incompressible MHD turbulence, which comes out is rather nice but solar wind turbulence displays a more complicated behavior. In particular, as we have reported above, observations seems to point out that solar wind evolves in the opposite way. The correlation is high near the Sun, at larger radial distances, from 1 to 10 AU the correlation is progressively lower, while the level in fluctuations of mass density and magnetic field intensity increases. What is more difficult to understand is why correlation is progressively destroyed in the solar wind, while the natural evolution of MHD is towards a state of maximal normalized cross-helicity. A possible solution can be found in the fact that solar wind is neither incompressible nor statistically homogeneous, and some efforts to tentatively take into account more sophisticated effects have been made.

A mechanism, responsible for the radial evolution of turbulence, was suggested by Roberts and Goldstein (1988); Goldstein et al. (1989); and Roberts et al. (1991Jump To The Next Citation Point, 1992Jump To The Next Citation Point) and was based on velocity shear generation. The suggestion to adopt such a mechanism came from a detailed analysis made by Roberts et al. (1987aJump To The Next Citation Point,bJump To The Next Citation Point) of Helios and Voyager interplanetary observations of the radial evolution of the normalized cross-helicity σ c at different time scales. Moreover, Voyager’s observations showed that plasma regions, which had not experienced dynamical interactions with neighboring plasma, kept the Alfvénic character of the fluctuations at distances as far as 8 AU (Roberts et al., 1987bJump To The Next Citation Point). In particular, the vicinity of Helios trajectory to the interplanetary current sheet, characterized by low velocity flow, suggested Roberts et al. (1991Jump To The Next Citation Point) to include in his simulations a narrow low speed flow surrounded by two high speed flows. The idea was to mimic the slow, equatorial solar wind between north and south fast polar wind. Magnetic field profile and velocity shear were reconstructed using the six lowest Z ± Fourier modes as shown in Figure 67View Image. An initial population of purely outward propagating Alfvénic fluctuations (z+) was added at large k and was characterized by a spectral slope of k −1. No inward modes were present in the same range. Results of Figure 67View Image show that the time evolution of + z spectrum is quite rapid at the beginning, towards a steeper spectrum, and slows down successively. At the same time, − z modes are created by the generation mechanism at higher and higher k but, along a Kolmogorov-type slope k− 5∕3.

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Figure 67: Time evolution of the power density spectra of z+ and z− showing the turbulent evolution of the spectra due to velocity shear generation (from Roberts et al., 1991Jump To The Next Citation Point).

These results, although obtained from simulations performed using 2D incompressible spectral and pseudo-spectral codes, with fairly small Reynolds number of Re ≃ 200, were similar to the spectral evolution observed in the solar wind (Marsch and Tu, 1990a). Moreover, spatial averages across the simulation box revealed a strong cross-helicity depletion right across the slow wind, representing the heliospheric current sheet. However, magnetic field inversions and even relatively small velocity shears would largely affect an initially high Alfvénic flow (Roberts et al., 1992Jump To The Next Citation Point). However, Bavassano and Bruno (1992) studied an interaction region, repeatedly observed between 0.3 and 0.9 AU, characterized by a large velocity shear and previously thought to be a good candidate for shear generation (Bavassano and Bruno, 1989Jump To The Next Citation Point). They concluded that, even in the hypothesis of a very fast growth of the instability, inward modes would not have had enough time to fill up the whole region as observed by Helios 2.

The above simulations by Roberts et al. (1991Jump To The Next Citation Point) were successively implemented with a compressive pseudo-spectral code (Ghosh and Matthaeus, 1990) which provided evidence that, during this turbulence evolution, clear correlations between magnetic field magnitude and density fluctuations, and between z− and density fluctuations should arise. However, such a clear correlation, by-product of the non-linear evolution, was not found in solar wind data (Marsch and Tu, 1993bJump To The Next Citation Point; Bruno et al., 1996Jump To The Next Citation Point). Moreover, their results did not show the flattening of e spectrum at higher frequency, as observed by Helios (Tu et al., 1989b). As a consequence, velocity shear alone cannot explain the whole phenomenon, other mechanisms must also play a relevant role in the evolution of interplanetary turbulence.

Compressible numerical simulations have been performed by Veltri et al. (1992) and Malara et al. (1996Jump To The Next Citation Point, 2000Jump To The Next Citation Point) which invoked the interactions between small scale waves and large scale magnetic field gradients and the parametric instability, as characteristic effects to reduce correlations. In a compressible, statistically inhomogeneous medium such as the heliosphere, there are many processes which tend to destroy the natural evolution toward a maximal correlation, typical of standard MHD. In such a medium an Alfvén wave is subject to parametric decay instability (Viñas and Goldstein, 1991; Del Zanna et al., 2001Jump To The Next Citation Point; Del Zanna, 2001), which means that the mother wave decays in two modes: i) a compressive mode that dissipates energy because of the steepening effect, and ii) a backscattered Alfvénic mode with lower amplitude and frequency. Malara et al. (1996) showed that in a compressible medium, the correlation between the velocity and the magnetic field fluctuations is reduced because of the generation of the backward propagating Alfvénic fluctuations, and of a compressive component of turbulence, characterized by density fluctuations δρ ⁄= 0 and magnetic intensity fluctuations δ|B | ⁄= 0.

From a technical point of view it is worthwhile to remark that, when a large scale field which varies on a narrow region is introduced (typically a tanh-like field), periodic boundaries conditions should be used with some care. Roberts et al. (1991Jump To The Next Citation Point, 1992Jump To The Next Citation Point) used a double shear layer, while Malara et al. (1992) introduced an interesting numerical technique based on both the glue between two simulation boxes and a Chebyshev expansion, to maintain a single shear layer, say non periodic boundary conditions, and an increased resolution where the shear layer exists.

Grappin et al. (1992) observed that the solar wind expansion increases the lengths normal to the radial direction, thus producing an effect similar to a kind of inverse energy cascade. This effect perhaps should be able to compete with the turbulent cascade which transfers energy to small scales, thus stopping the non-linear interactions. In absence of non-linear interactions, the natural tendency towards an increase of σc is stopped. These inferences have been corroborated by further studies like those by Grappin and Velli (1996) and Goldstein and Roberts (1999). A numerical model treating the evolution of e+ and e, including parametric decay of e+, was presented by Marsch and Tu (1993aJump To The Next Citation Point). The parametric decay source term was added in order to reproduce the decreasing cross-helicity observed during the wind expansion. As a matter of fact, the cascade process, when spectral equations for both e+ and e are included and solved self-consistently, can only steepen the spectra at high frequency. Results from this model, shown in Figure 68View Image, partially reproduce the observed evolution of the normalized cross-helicity. While the radial evolution of e+ is correctly reproduced, the behavior of e shows an over-production of inward modes between 0.6 and 0.8 AU probably due to an overestimation of the strength of the pump-wave. However, the model is applied to the situation observed by Helios at 0.3 AU where a rather flat e spectrum already exists.

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Figure 68: Radial evolution of e+ and e spectra obtained from the Marsch and Tu (1993aJump To The Next Citation Point) model, in which a parametric decay source term was added to the Tu’s model (Tu et al., 1984Jump To The Next Citation Point) that was, in turn, extended by including both spectrum equations for e+ and e and solved them self-consistently. Image reproduced by permission from Marsch and Tu (1993aJump To The Next Citation Point), copyright by AGU.

5.2 Local production of Alfvénic turbulence at high latitude

An interesting solution to the radial behavior of the minority modes might be represented by local generation mechanisms, like parametric decay (Malara et al., 2001a; Del Zanna et al., 2001Jump To The Next Citation Point), which might saturate and be inhibited beyond 2.5 AU.

Parametric instability has been studied in a variety of situations depending on the value of the plasma β (among others Sagdeev and Galeev, 1969; Goldstein, 1978; Hoshino and Goldstein, 1989; Malara and Velli, 1996). Malara et al. (2000) and Del Zanna et al. (2001) recently studied the non-linear growth of parametric decay of a broadband Alfvén wave, and showed that the final state strongly depends on the value of the plasma β (thermal to magnetic pressure ratio). For β < 1 the instability completely destroys the initial Alfvénic correlation. For β ∼ 1 (a value close to solar wind conditions) the instability is not able to go beyond some limit in the disruption of the initial correlation between velocity and magnetic field fluctuations, and the final state is σc ∼ 0.5 as observed in the solar wind (see Section 4.2).

These authors solved numerically the fully compressible, non-linear MHD equations in a one-dimensional configuration using a pseudo-spectral numerical code. The simulation starts with a non-monochromatic, large amplitude Alfvén wave polarized on the yz plane, propagating in a uniform background magnetic field. Successively, the instability was triggered by adding some noise of the order 10–6 to the initial density level.

During the first part of the evolution of the instability the amplitude of unstable modes is small and, consequently, non-linear couplings are negligible. A subsequent exponential growth, predicted by the linear theory, increases the level of both e and density compressive fluctuations. During the second part of the development of the instability, non-linear couplings are not longer disregardable and their effect is firstly to slow down the exponential growth of unstable modes and then to saturate the instability to a level that depends on the value of the plasma β.

Spectra of ± e are shown in Figure 69View Image for different times during the development of the instability. At the beginning the spectrum of the mother-wave is peaked at k = 10, and before the instability saturation (t ≤ 35) the back-scattered e and the density fluctuations eρ are peaked at k = 1 and k = 11, respectively. After saturation, as the run goes on, the spectrum of e approaches that of e+ towards a common final state characterized by a Kolmogorov-like spectrum and e+ slightly larger than e.

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Figure 69: Spectra of e+ (thick line), e (dashed line), and ρ e (thin line) are shown for 6 different times during the development of the instability. For t ≥ 50 a typical Kolmogorov slope appears. These results refer to β = 1. Image reproduced by permission from Malara et al. (2001bJump To The Next Citation Point), copyright by EGU.

The behavior of outward and inward modes, density and magnetic magnitude variances and the normalized cross-helicity σc is summarized in the left column of Figure 70View Image. The evolution of σc, when the instability reaches saturation, can be qualitatively compared with Ulysses observations (courtesy of B. Bavassano) in the right panel of the same figure, which shows a similar trend.

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Figure 70: Top left panel: time evolution of e+ (solid line) and e (dashed line). Middle left panel: density (solid line) and magnetic magnitude (dashed line) variances. Bottom left panel: normalized cross helicity σc. Right panel: Ulysses observations of σc radial evolution within the polar wind (left column is from Malara et al., 2001bJump To The Next Citation Point, right panel is a courtesy of B. Bavassano).

Obviously, making this comparison, one has to take into account that this model has strong limitations like the presence of a peak in e+ not observed in real polar turbulence. Another limitation, partly due to dissipation that has to be included in the model, is that the spectra obtained at the end of the instability growth are steeper than those observed in the solar wind. Finally, a further limitation is represented by the fact that this code is 1D. However, although for an incompressible 1-D simulation we do not expect to have turbulence development, in this case, since parametric decay is based on compressive phenomena, an energy transfer along the spectrum might be at work.

In addition, Umeki and Terasawa (1992) studying the non-linear evolution of a large-amplitude incoherent Alfvén wave via 1D magnetohydrodynamic simulations, reported that while in a low beta plasma (β ≈ 0.2) the growth of backscattered Alfvén waves, which are opposite in helicity and propagation direction from the original Alfvén waves, could be clearly detected, in a high beta plasma (β ≈ 2) there was no production of backscattered Alfvén waves. Consequently, although numerical results obtained by Malara et al. (2001b) are very encouraging, the high beta plasma (β ≈ 2), characteristic of fast polar wind at solar minimum, plays against a relevant role of parametric instability in developing solar wind turbulence as observed by Ulysses. However, these simulations do remain an important step forward towards the understanding of turbulent evolution in the polar wind until other mechanisms will be found to be active enough to justify the observations shown in Figure 63View Image.

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