"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

6 Compressive Turbulence

Interplanetary medium is slightly compressive, magnetic field intensity and proton number density experience fluctuations over all scales and the compression depends on both the scale and the nature of the wind. As a matter of fact, slow wind is generally more compressive than fast wind, as shown in Figure 71View Image where, following Bavassano et al. (1982aJump To The Next Citation Point) and Bruno and Bavassano (1991Jump To The Next Citation Point), we report the ratio between the power density associated with magnetic field intensity fluctuations and that associated with the fluctuations of the three components. In addition, as already shown by Bavassano et al. (1982aJump To The Next Citation Point), this parameter increases with heliocentric distance for both fast and slow wind as shown in the bottom panel, where the ratio between the compression at 0.9 AU and that at 0.3 AU is generally greater than 1. It is also interesting to notice that within the Alfvénic fast wind, the lowest compression is observed in the middle frequency range, roughly between −4 − 3 10 –10 Hz. On the other hand, this frequency range has already been recognized as the most Alfvénic one, within the inner heliosphere (Bruno et al., 1996).
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Figure 71: The first two rows show magnetic field compression (see text for definition) for fast (left column) and slow (right column) wind at 0.3 AU (upper row) and 0.9 AU (middle row). The bottom panels show the ratio between compression at 0.9 AU and compression at 0.3 AU. This ratio is generally greater than 1 for both fast and slow wind.

As a matter of fact, it seems that high Alfvénicity is correlated with low compressibility of the medium (Bruno and Bavassano, 1991Jump To The Next Citation Point; Klein et al., 1993Jump To The Next Citation Point; Bruno and Bavassano, 1993Jump To The Next Citation Point) although compressibility is not the only cause for a low Alfvénicity (Roberts et al., 1991, 1992; Roberts, 1992).

The radial dependence of the normalized number density fluctuations δn ∕n for the inner and outer heliosphere were studied by Grappin et al. (1990Jump To The Next Citation Point) and Roberts et al. (1987bJump To The Next Citation Point) for the hourly frequency range, but no clear radial trend emerged from these studies. However, interesting enough, Grappin et al. (1990) found that values of e were closely associated with enhancements of δn ∕n on scales longer than 1 h.

On the other hand, a spectral analysis of proton number density, magnetic field intensity, and proton temperature performed by Marsch and Tu (1990bJump To The Next Citation Point) and Tu et al. (1991Jump To The Next Citation Point) in the inner heliosphere, separately for fast and slow wind (see Figure 72View Image), showed that normalized spectra of the above parameters within slow wind were only marginally dependent on the radial distance. On the contrary, within fast wind, magnetic field and proton density normalized spectra showed not only a clear radial dependence but also similar level of power for −4 −1 k < 4 × 10 km s. For larger k these spectra show a flattening that becomes steeper for increasing distance, as was already found by Bavassano et al. (1982b) for magnetic field intensity. Normalized temperature spectra does not suffer any radial dependence neither in slow wind nor in fast wind.

Spectral index is around − 5∕3 for all the spectra in slow wind while, fast wind spectral index is around − 5∕3 for −4 −1 k < 4 × 10 km and slightly less steep for larger wave numbers.

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Figure 72: From left to right: normalized spectra of number density, magnetic field intensity fluctuations (adapted from Marsch and Tu, 1990b), and proton temperature (adapted from Tu et al., 1991). Different lines refer to different heliocentric distances for both slow and fast wind.

6.1 On the nature of compressive turbulence

Considerable efforts, both theoretical and observational, have been made in order to disclose the nature of compressive fluctuations. It has been proposed (Montgomery et al., 1987Jump To The Next Citation Point; Matthaeus and Brown, 1988Jump To The Next Citation Point; Zank et al., 1990; Zank and Matthaeus, 1990; Matthaeus et al., 1991Jump To The Next Citation Point; Zank and Matthaeus, 1992Jump To The Next Citation Point) that most of compressive fluctuations observed in the solar wind could be accounted for by the Nearly Incompressible (NI) model. Within the framework of this model, (Montgomery et al., 1987) showed that a spectrum of small scale density fluctuations follows a k−5∕3 when the spectrum of magnetic field fluctuations follows the same scaling. Moreover, it was showed (Matthaeus and Brown, 1988; Zank and Matthaeus, 1992) that if compressible MHD equations are expanded in terms of small turbulent sonic Mach numbers, pressure balanced structures, Alfvénic and magnetosonic fluctuations naturally arise as solutions and, in particular, the RMS of small density fluctuations would scale like M 2, being M = δv∕Cs the turbulent sonic Mach number, δv the RMS of velocity fluctuations and C s the sound speed. In addition, if heat conduction is allowed in the approximation, temperature fluctuations dominate over magnetic and density fluctuations, temperature and density are anticorrelated and would scale like M. However, in spite of some examples supporting this theory (Matthaeus et al., 1991Jump To The Next Citation Point reported 13% of cases satisfied the requirements of NI-theory), wider statistical studies, conducted by Tu and Marsch (1994Jump To The Next Citation Point), Bavassano et al. (1995Jump To The Next Citation Point) and Bavassano and Bruno (1995Jump To The Next Citation Point), showed that NI theory is not applicable sic et simpliciter to the solar wind. The reason might be in the fact that interplanetary medium is highly inhomogeneous because of the presence of an underlying structure convected by the wind. As a matter of fact, Thieme et al. (1989Jump To The Next Citation Point) showed evidence for the presence of time intervals characterized by clear anti-correlation between kinetic pressure and magnetic pressure while the total pressure remained fairly constant. These pressure balance structures were for the first time observed by Burlaga and Ogilvie (1970) for a time scale of roughly one to two hours. Later on, Vellante and Lazarus (1987) reported strong evidence for anti-correlation between field intensity and proton density, and between plasma and field pressure on time scales up to 10 h. The anti-correlation between kinetic and magnetic pressure is usually interpreted as indicative of the presence of a pressure balance structure since slow magnetosonic modes are readily damped (Barnes, 1979).

These features, observed also in their dataset, were taken by Thieme et al. (1989Jump To The Next Citation Point) as evidence of stationary spatial structures which were supposed to be remnants of coronal structures convected by the wind. Different values assumed by plasma and field parameters within each structure were interpreted as a signature characterizing that particular structure and not destroyed during the expansion. These intervals, identifiable in Figure 73View Image by vertical dashed lines, were characterized by pressure balance and a clear anti-correlation between magnetic field intensity and temperature.

These structures were finally related to the fine ray-like structures or plumes associated with the underlying chromospheric network and interpreted as the signature of interplanetary flow-tubes. The estimated dimension of these structures, back projected onto the Sun, suggested that they over-expand in the solar wind. In addition, Grappin et al. (2000) simulated the evolution of Alfvén waves propagating within such pressure equilibrium ray structures in the framework of global Eulerian solar wind approach and found that the compressive modes in these simulations are very much reduced within the ray structures, which indeed correspond to the observational findings (Buttighoffer et al., 1995, 1999).

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Figure 73: From top to bottom: field intensity |B |; proton and alpha particle velocity vp and vα; corrected proton velocity vpc = vp − δvA, where vA is the Alfvén speed; proton and alpha number density np and nα; proton and alpha temperature Tp and T α; kinetic and magnetic pressure Pk and Pm, which the authors call Pgas and Pmag; total pressure Ptot and β = Pgas∕Pmag (from Tu and Marsch, 1995aJump To The Next Citation Point).

The idea of filamentary structures in the solar wind dates back to Parker (1964Jump To The Next Citation Point), followed by other authors like McCracken and Ness (1966), Siscoe et al. (1968), and more recently has been considered again in the literature with new results (see Section 10). These interplanetary flow tubes would be of different sizes, ranging from minutes to several hours and would be separated from each other by tangential discontinuities and characterized by different values of plasma parameters and a different magnetic field orientation and intensity. This kind of scenario, because of some similarity to a bunch of tangled, smoking “spaghetti” lifted by a fork, was then named “spaghetti-model”.

A spectral analysis performed by Marsch and Tu (1993aJump To The Next Citation Point) in the frequency range 6 × 10−3– 6 × 10−6 showed that the nature and intensity of compressive fluctuations systematically vary with the stream structure. They concluded that compressive fluctuations are a complex superposition of magnetoacoustic fluctuations and pressure balance structures whose origin might be local, due to stream dynamical interaction, or of coronal origin related to the flow tube structure. These results are shown in Figure 74View Image where the correlation coefficient between number density n and total pressure P tot (indicated with the symbols pT in the figure), and between kinetic pressure Pk and magnetic pressure Pm (indicated with the symbols pk and pb, respectively) is plotted for both Helios s/c relatively to fast wind. Positive values of correlation coefficients C(n, pT) and C (pk,pb) identify magnetosonic waves, while positive values of C (n, pT) and negative values of C(pk,pb) identify pressure balance structures. The purest examples of each category are located at the upper left and right corners.

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Figure 74: Correlation coefficient between number density n and total pressure pT plotted vs. the correlation coefficient between kinetic pressure and magnetic pressure for both Helios relatively to fast wind. Image reproduced by permission from Marsch and Tu (1993b).

Following these observations, Tu and Marsch (1994Jump To The Next Citation Point) proposed a model in which fluctuations in temperature, density, and field directly derive from an ensemble of small amplitude pressure balanced structures and small amplitude fast perpendicular magnetosonic waves. These last ones should be generated by the dynamical interaction between adjacent flow tubes due to the expansion and, eventually, they would experience also a non-linear cascade process to smaller scales. This model was able to reproduce most of the correlations described by Marsch and Tu (1993a) for fast wind.

Later on, Bavassano et al. (1996aJump To The Next Citation Point) tried to characterize compressive fluctuations in terms of their polytropic index, which resulted to be a useful tool to study small scale variations in the solar wind. These authors followed the definition of polytropic fluid given by Chandrasekhar (1967): “a polytropic change is a quasi-static change of state carried out in such a way that the specific heat remains constant (at some prescribed value) during the entire process”. For such a variation of state the adiabatic laws are still valid provided that the adiabatic index γ is replaced by a new adiabatic index ′ γ = (cP − c)∕(cV − c) where c is the specific heat of the polytropic variation, and cP and cV are the specific heat at constant pressure and constant volume, respectively. This similarity is lost if we adopt the definition given by Courant and Friedrichs (1976), for whom a fluid is polytropic if its internal energy is proportional to the temperature. Since no restriction applies to the specific heats, relations between temperature, density, and pressure do not have a simple form as in Chandrasekhar approach (Zank and Matthaeus, 1991Jump To The Next Citation Point). Bavassano et al. (1996aJump To The Next Citation Point) recovered the polytropic index from the relation between density n and temperature T changes for the selected scale ′ Tn1 −γ = const. and used it to determine whether changes in density and temperature were isobaric (γ′ = 0), isothermal (γ′ = 1), adiabatic (γ′ = γ), or isochoric (γ′ = ∞). Although the role of the magnetic field was neglected, reliable conclusions could be obtained whenever the above relations between temperature and density were strikingly clear. These authors found intervals characterized by variations at constant thermal pressure P. They interpreted these intervals as a subset of total-pressure balanced structures where the equilibrium was assured by the thermal component only, perhaps tiny flow tubes like those described by Thieme et al. (1989Jump To The Next Citation Point) and Tu and Marsch (1994Jump To The Next Citation Point). Adiabatic changes were probably related to magnetosonic waves excited by contiguous flow tubes (Tu and Marsch, 1994Jump To The Next Citation Point). Proton temperature changes at almost constant density were preferentially found in fast wind, close to the Sun. These regions were characterized by values of B and N remarkable stable and by strong Alfvénic fluctuations (Bruno et al., 1985). Thus, they suggested that these temperature changes could be remnants of thermal features already established at the base of the corona.

Thus, the polytropic index offers a very simple way to identify basic properties of solar wind fluctuations, provided that the magnetic field does not play a major role.

6.2 Compressive turbulence in the polar wind

Compressive fluctuations in high latitude solar wind have been extensively studied by Bavassano et al. (2004Jump To The Next Citation Point) looking at the relationship between different parameters of the solar wind and comparing these results with predictions by existing models.

These authors indicated with N, Pm, Pk, and Pt the proton number density n, magnetic pressure, kinetic pressure and total pressure (Ptot = Pm + Pk ), respectively, and computed correlation coefficients ρ between these parameters. Figure 75View Image clearly shows that a pronounced positive correlation for N − Pt and a negative pronounced correlation for Pm − Pk is a constant feature of the observed compressive fluctuations. In particular, the correlation for N − Pt is especially strong within polar regions at small heliocentric distance. In mid-latitude regions the correlation weakens, while almost disappears at low latitudes. In the case of P − P m k, the anticorrelation remains strong throughout the whole latitudinal excursion. For polar wind the anticorrelation appears to be less strong at small distances, just where the N − Pt correlation is highest.

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Figure 75: Histograms of ρ(N − Pt) and ρ(Pm − Pk ) per solar rotation. The color bar on the left side indicates polar (red), mid-latitude (blue), and low latitude (green) phases. Moreover, universal time UT, heliocentric distance, and heliographic latitude are also indicated on the left side of the plot. Occurrence frequency is indicated by the color bar shown on the right hand side of the figure. Image reproduced by permission from Bavassano et al. (2004Jump To The Next Citation Point), copyright EGU.

The role played by density and temperature in the anticorrelation between magnetic and thermal pressures is investigated in Figure 76View Image, where the magnetic field magnitude is directly compared with proton density and temperature. As regards the polar regions, a strong B-T anticorrelation is clearly apparent at all distances (right panel). For B-N an anticorrelation tends to emerge when solar distance increases. This means that the magnetic-thermal pressure anticorrelation is mostly due to an anticorrelation of the magnetic field fluctuations with respect to temperature fluctuations, rather than density (see, e.g., Bavassano et al., 1996a,b). Outside polar regions the situation appears in part reversed, with a stronger role for the B-N anticorrelation.

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Figure 76: Solar rotation histograms of B-N and B-T in the same format of Figure 75View Image. Image reproduced by permission from Bavassano et al. (2004Jump To The Next Citation Point), copyright EGU.

In Figure 77View Image scatter plots of total pressure vs. density fluctuations are used to test a model by Tu and Marsch (1994Jump To The Next Citation Point), based on the hypothesis that the compressive fluctuations observed in solar wind are mainly due to a mixture of pressure-balanced structures (PBS) and fast magnetosonic waves (W). Waves can only contribute to total pressure fluctuations while both waves and pressure-balanced structures may contribute to density fluctuations. A tunable parameter in the model is the relative PBS/W contribution to density fluctuations α. Straight lines in Figure 77View Image indicate the model predictions for different values of α. It is easily seen that for all polar wind samples the great majority of experimental data fall in the α > 1 region. Thus, pressure-balanced structures appear to play a major role with respect to magnetosonic waves. This is a feature already observed by Helios in the ecliptic wind (Tu and Marsch, 1994Jump To The Next Citation Point), although in a less pronounced way. Different panels of Figure 77View Image refer to different heliocentric distances within the polar wind. Namely, going from P1 to P4 is equivalent to move from 1.4 to 4 AU. A comparison between these panels indicates that the observed distribution tends to shift towards higher values of α (i.e., pressure-balanced structures become increasingly important), which probably is a radial distance effect.

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Figure 77: Scatter plots of the relative amplitudes of total pressure vs. density fluctuations for polar wind samples P1 to P4. Straight lines indicate the Tu and Marsch (1994) model predictions for different values of α, the relative PBS/W contribution to density fluctuations. Image reproduced by permission from Bavassano et al. (2004Jump To The Next Citation Point), copyright EGU.

Finally, the relative density fluctuations dependence on the turbulent Mach number M (the ratio between velocity fluctuation amplitude and sound speed) is shown in Figure 78View Image. The aim is to look for the presence, in the observed fluctuations, of nearly incompressible MHD behaviors. In the framework of the NI theory (Zank and Matthaeus, 1991, 1993) two different scalings for the relative density fluctuations are possible, as M or as 2 M, depending on the role that thermal conduction effects may play in the plasma under study (namely a heat-fluctuation-dominated or a heat-fluctuation-modified behavior, respectively). These scalings are shown in Figure 78View Image as solid (for M) and dashed (for M 2) lines.

It is clearly seen that for all the polar wind samples no clear trend emerges in the data. Thus, NI-MHD effects do not seem to play a relevant role in driving the polar wind fluctuations. This confirms previous results in the ecliptic by Helios in the inner heliosphere (Bavassano et al., 1995; Bavassano and Bruno, 1995) and by Voyagers in the outer heliosphere (Matthaeus et al., 1991). It is worthy of note that, apart from the lack of NI trends, the experimental data from Ulysses, Voyagers, and Helios missions in all cases exhibit quite similar distributions. In other words, for different heliospheric regions, solar wind regimes, and solar activity conditions, the behavior of the compressive fluctuations in terms of relative density fluctuations and turbulent Mach numbers seems almost to be an invariant feature.

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Figure 78: Relative amplitude of density fluctuations vs. turbulent Mach number for polar wind. Solid and dashed lines indicate the M and 2 M scalings, respectively. Image reproduced by permission from Bavassano et al. (2004), copyright EGU.

The above observations fully support the view that compressive fluctuations in high latitude solar wind are a mixture of MHD modes and pressure balanced structures. It has to be reminded that previous studies (McComas et al., 1995, 1996; Reisenfeld et al., 1999) indicated a relevant presence of pressure balanced structures at hourly scales. Moreover, nearly-incompressible (see Section 6.1) effects do not seem to play any relevant role. Thus, polar observations do not show major differences when compared with ecliptic observations in fast wind, the only possible difference being a major role of pressure balanced structures.

6.3 The effect of compressive phenomena on Alfvénic correlations

A lack of δV − δB correlation does not strictly indicate a lack of Alfvénic fluctuations since a superposition of both outward and inward oriented fluctuations of the same amplitude would produce a very low correlation as well. In addition, the rather complicated scenario at the base of the corona, where both kinetic and magnetic phenomena contribute to the birth of the wind, suggest that the imprints of such a structured corona is carried away by the wind during its expansion. At this point, we would expect that solar wind fluctuations would not solely be due to the ubiquitous Alfvénic and other MHD propagating modes but also to an underlying structure convected by the wind, not necessarily characterized by Alfvén-like correlations. Moreover, dynamical interactions between fast and slow wind, built up during the expansion, contribute to increase the compressibility of the medium.

It has been suggested that disturbances of the mean magnetic field intensity and plasma density act destructively on δV − δB correlation. Bruno and Bavassano (1993Jump To The Next Citation Point) analyzed the loss of the Alfvénic character of interplanetary fluctuations in the inner heliosphere within the low frequency part of the Alfvénic range, i.e., between 2 and 10 h. Figure 79View Image, from their work, shows the wind speed profile, σc, the correlation coefficients, phase and coherence for the three components (see Appendix B.2.1), the angle between magnetic field and velocity minimum variance directions, and the heliocentric distance. Magnetic field sectors were rectified (see Appendix B.3) and magnetic field and velocity components were rotated into the magnetic field minimum variance reference system (see Appendix D). Although the three components behave in a similar way, the most Alfvénic ones are the two components Y and Z transverse to the minimum variance component X. As a matter of fact, for an Alfvén mode we would expect a high δV − δB correlation, a phase close to zero for outward waves and a high coherence. Moreover, it is rather clear that the most Alfvénic intervals are located within the trailing edges of high velocity streams. However, as the radial distance increases, the Alfvénic character of the fluctuations decreases and the angle Θbv increases. The same authors found that high values of Θbv are associated with low values of σc and correspond to the most compressive intervals. They concluded that the depletion of the Alfvénic character of the fluctuations, within the hourly frequency range, might be driven by the interaction with static structures or magnetosonic perturbations able to modify the homogeneity of the background medium on spatial scales comparable to the wavelength of the Alfvénic fluctuations. A subsequent paper by Klein et al. (1993) showed that the δV − δB decoupling increases with the plasma β, suggesting that in regions where the local magnetic field is less relevant, compressive events play a major role in this phenomenon.

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Figure 79: Wind speed profile V and |σc|V are shown in the top panel. The lower three panels refer to correlation coefficient, phase angle and coherence for the three components of δV and δB fluctuations, respectively. The successive panel indicates the value of the angle between magnetic field and velocity fluctuations minimum variance directions. The bottom panel refers to the heliocentric distance (from Bruno and Bavassano, 1993).

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