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"The Solar Wind as a Turbulence Laboratory"
Roberto Bruno and Vincenzo Carbone 
Abstract
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
Acknowledgments
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
References
Footnotes
Updates
Figures
Tables

3 Early Observations of MHD Turbulence in the Ecliptic

Here we briefly present the history, since the first Mariner missions during the 1960s, of the main steps towards the completion of an observational picture of turbulence in interplanetary space. This retrospective look at all the advances made in this field shows that space flights allowed us to discover a very large laboratory in space. As a matter of fact, in a wind tunnel we deal with characteristic dimensions of the order of L ≤ 10 m and probes of the size of about d ≃ 1 cm. In space, L ≃ 108 m, while “probes” (say spacecrafts) are about d ≃ 5 m. Thus, space provides a much larger laboratory. Most measurements are single point measurements, the ESA-Cluster project providing for multiple measurements only recently.

3.1 Turbulence in the ecliptic

When dealing with laboratory turbulence it is important to know all the aspects of the experimental device where turbulent processes take place in order to estimate related possible effects driven or influenced by the environment. In the solar wind, the situation is, in some aspects, similar although the plasma does not experience any confinement due to the “experimental device”, which would be represented by free interplanetary space. However, it is a matter of fact that the turbulent state of the wind fluctuations and the subsequent radial evolution during the wind expansion greatly differ from fast to slow wind, and it is now well accepted that the macrostructure convected by the wind itself plays some role (see reviews by Tu and Marsch, 1995aJump To The Next Citation Point; Goldstein et al., 1995b).

Fast solar wind originates from the polar regions of the Sun, within the open magnetic field line regions identified by coronal holes. Beautiful observations by SOHO spacecraft (see animation of Figure 14Watch/download Movie) have localized the birthplace of the solar wind within the intergranular lane, generally where three or more granules get together. Clear outflow velocities of up to 10 km s–1 have been recorded by SOHO/SUMER instrument (Hassler et al., 1999Jump To The Next Citation Point).


Figure 14: mpg-Movie (2363 KB) An animation built on SOHO/EIT and SOHO/SUMER observations of the solar-wind source regions and magnetic structure of the chromospheric network. Outflow velocities, at the network cell boundaries and lane junctions below the polar coronal hole, reach up to 10 km s–1 are represented by the blue colored areas (original figures from Hassler et al., 1999).

Slow wind, on the contrary, originates from the equatorial zone of the Sun. The slow wind plasma leaks from coronal features called “helmets”, which can be easily seen protruding into the Sun’s atmosphere during a solar eclipse (see Figure 15View Image). Moreover, plasma emissions due to violent and abrupt phenomena also contribute to the solar wind in these regions of the Sun. An alternative view is that both high- and low-speed winds come from coronal holes (defined as open field regions) and that the wind speed at 1 AU is determined by the rate of flux-tube expansion near the Sun as firstly suggested by Levine et al. (1977) (Wang and Sheeley Jr, 1990; Bravo and Stewart, 1997; Arge and Pizzo, 2000; Poduval and Zhao, 2004; Whang et al., 2005, see also:) and/or by the location and strength of the coronal heating (Leer and Holzer, 1980; Hammer, 1982; Hollweg, 1986; Withbroe, 1988; Wang, 1993, 1994; Sandbaek et al., 1994; Hansteen and Leer, 1995; Cranmer et al., 2007).

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Figure 15: Helmet streamer during a solar eclipse. Slow wind leaks into the interplanetary space along the flanks of this coronal structure. Image reproduced from MSFC.

However, this situation greatly changes during different phases of the solar activity cycle. Polar coronal holes, which during the maximum of activity are limited to small and not well defined regions around the poles, considerably widen up during solar minimum, reaching the equatorial regions (Forsyth et al., 1997; Forsyth and Breen, 2002; Balogh et al., 1999). This new configuration produces an alternation of fast and slow wind streams in the ecliptic plane, the plane where most of the spacecraft operate and record data. During the expansion, a dynamical interaction between fast and slow wind develops, generating the so called “stream interface”, a thin region ahead of the fast stream characterized by strong compressive phenomena.

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Figure 16: High velocity streams and slow wind as seen in the ecliptic during solar minimum as function of time [yyddd ]. Streams identified by labels are the same co-rotating stream observed by Helios 2, during its primary mission to the Sun in 1976, at different heliocentric distances. These streams, named “The Bavassano–Villante streams” after Tu and Marsch (1995aJump To The Next Citation Point), have been of fundamental importance in understanding the radial evolution of MHD turbulence in the solar wind.

Figure 16View Image shows a typical situation in the ecliptic where fast streams and slow wind were observed by Helios 2 s/c during its primary mission to the Sun. At that time, the spacecraft moved from 1 AU (around day 17) to its closest approach to the Sun at 0.29 AU (around day 108). During this radial excursion, Helios 2 had a chance to observe the same co-rotating stream, that is plasma coming from the same solar source, at different heliocentric distances. This fortuitous circumstance, gave us the unique opportunity to study the radial evolution of turbulence under the reasonable hypothesis of time-stationarity of the source regions. Obviously, similar hypotheses decay during higher activity phase of the solar cycle since, as shown in Figure 17View Image, the nice and regular alternation of fast co-rotating streams and slow wind is replaced by a much more irregular and spiky profile also characterized by a lower average speed.

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Figure 17: High velocity streams and slow wind as seen in the ecliptic during solar maximum. Data refer to Helios 2 observations in 1979.
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Figure 18: High velocity streams and slow wind as seen in the ecliptic during solar minimum.

Figure 18View Image focuses on a region centered on day 75, recognizable in Figure 16View Image, when the s/c was at approximately 0.7 AU from the Sun. Slow wind on the left-hand side of the plot, fast wind on the right hand side, and the stream interface in between, can be clearly seen. This is a sort of canonical situation often encountered in the ecliptic, within the inner heliosphere, during solar activity minimum. Typical solar wind parameters, like proton number density ρp, proton temperature Tp, magnetic field intensity |B |, azimuthal angle Φ, and elevation angle Θ are shown in the panels below the wind speed profile. A quick look at the data reveals that fast wind is less dense but hotter than slow wind. Moreover, both proton number density and magnetic field intensity are more steady and, in addition, the bottom two panels show that magnetic field vector fluctuates in direction much more than in slow wind. This last aspect unravels the presence of strong Alfvénic fluctuations which act mainly on magnetic field and velocity vector direction, and are typically found within fast wind (Belcher and Davis Jr, 1971Jump To The Next Citation Point; Belcher and Solodyna, 1975Jump To The Next Citation Point). The region just ahead of the fast wind, namely the stream interface, where dynamical interaction between fast and slow wind develops, is characterized by compressive effects which enhance proton density, temperature and field intensity. Within slow wind, a further compressive region precedes the stream interface but it is not due to dynamical effects but identifies the heliospheric current sheet, the surface dividing the two opposite polarities of the interplanetary magnetic field. As a matter of fact, the change of polarity can be noted within the first half of day 73 when the azimuthal angle Φ rotates by about 180 ∘. Detailed studies (Bavassano et al., 1997Jump To The Next Citation Point) based on interplanetary scintillations (IPS) and in-situ measurements have been able to find a clear correspondence between the profile of path-integrated density obtained from IPS measurements and in-situ measurements by Helios 2 when the s/c was around 0.3 AU from the Sun.

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Figure 19: Left panel: a simple sketch showing the configuration of a helmet streamer and the density profile across this structure. Right panel: Helios 2 observations of magnetic field and plasma parameters across the heliospheric current sheet. From top to bottom: wind speed, magnetic field azimuthal angle, proton number density, density fluctuations and normalized density fluctuations, proton temperature, magnetic field magnitude, total pressure, and plasma beta, respectively. Image reproduced by permission from Bavassano et al. (1997), copyright by AGU.

Figure 19View Image shows measurements of several plasma and magnetic field parameters. The third panel from the top is the proton number density and it shows an enhancement within the slow wind just preceding the fast stream, as can be seen at the top panel. In this case the increase in density is not due to the dynamical interaction between slow and fast wind but it represents the profile of the heliospheric current sheet as sketched on the left panel of Figure 19View Image. As a matter of fact, at these short distances from the Sun, dynamical interactions are still rather weak and this kind of compressive effects can be neglected with respect to the larger density values proper of the current sheet.

3.1.1 Spectral properties

First evidences of the presence of turbulent fluctuations were showed by Coleman (1968Jump To The Next Citation Point), who, using Mariner 2 magnetic and plasma observations, investigated the statistics of interplanetary fluctuations during the period August 27 – October 31, 1962, when the spacecraft orbited from 1.0 to 0.87 AU. At variance with Coleman (1968Jump To The Next Citation Point), Barnes and Hollweg (1974) analyzed the properties of the observed low-frequency fluctuations in terms of simple waves, disregarding the presence of an energy spectrum. Here we review the gross features of turbulence as observed in space by Mariner and Helios spacecraft. By analyzing spectral densities, Coleman (1968Jump To The Next Citation Point) concluded that the solar wind flow is often turbulent, energy being distributed over an extraordinarily wide frequency range, from one cycle per solar rotation to 0.1 Hz. The frequency spectrum, in a range of intermediate frequencies [2 × 10− 5– 2.3 × 10− 3], was found to behave roughly as f−1.2, the difference with the expected Kraichnan f −1.5 spectral slope was tentatively attributed to the presence of high-frequency transverse fluctuations resulting from plasma garden-hose instability (Scarf et al., 1967). Waves generated by this instability contribute to the spectrum only in the range of frequencies near the proton cyclotron frequency and would weaken the frequency dependence relatively to the Kraichnan scaling. The magnetic spectrum obtained by Coleman (1968Jump To The Next Citation Point) is shown in Figure 20View Image.

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Figure 20: The magnetic energy spectrum as obtained by Coleman (1968Jump To The Next Citation Point).

Spectral properties of the interplanetary medium have been summarized by Russell (1972Jump To The Next Citation Point), who published a composite spectrum of the radial component of magnetic fluctuations as observed by Mariner 2, Mariner 4, and OGO 5 (see Figure 21View Image). The frequency spectrum so obtained was divided into three main ranges: i) up to about 10 −4 Hz the spectral slope is about 1∕f; ii) at intermediate frequencies −4 −1 10 ≤ f ≤ 10 Hz a spectrum which roughly behaves as −3∕2 f has been found; iii) the high-frequency part of the spectrum, up to 1 Hz, behaves as 2 1∕f. The intermediate range7 of frequencies shows the same spectral properties as that introduced by Kraichnan (1965Jump To The Next Citation Point) in the framework of MHD turbulence. It is worth reporting that scatter plots of the values of the spectral index of the intermediate region do not allow us to distinguish between a Kolmogorov spectrum f−5∕3 and a Kraichnan spectrum f− 3∕2 (Veltri, 1980).

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Figure 21: A composite figure of the magnetic spectrum obtained by Russell (1972).

Only lately, Podesta et al. (2007Jump To The Next Citation Point) addressed again the problem of the spectral exponents of kinetic and magnetic energy spectra in the solar wind. Their results, instead of clarifying once forever the ambiguity between f −5∕3 and f− 3∕2 scaling, placed new questions about this unsolved problem.

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Figure 22: Magnetic energy spectra, velocity spectra and kinetic energy spectra obtained by Podesta et al. (2007Jump To The Next Citation Point). Image reproduced by permission, copyright by AAS.

As a matter of fact, Podesta et al. (2007) chose different time intervals between 1995 and 2003 lasting 2 or 3 solar rotations during which WIND spacecraft recorded solar wind velocity and magnetic field conditions. Figure 22View Image shows the results obtained for the time interval that lasted about 3 solar rotations between November 2000 and February 2001, and is representative also of the other analyzed time intervals. Quite unexpectedly, these authors found that the power law exponents of velocity and magnetic field fluctuations often have values near 3∕2 and 5∕3, respectively. In addition, the kinetic energy spectrum is characterized by a power law exponent slightly greater than or equal to 3∕2 due to the effects of density fluctuations.

It is worth mentioning that this difference was first observed by Salem (2000) years before, but, at that time, the accuracy of the data was questioned Salem et al. (2009Jump To The Next Citation Point). Thus, to corroborate previous results, Salem et al. (2009Jump To The Next Citation Point) investigated anomalous scaling and intermittency effects of both magnetic field and solar wind velocity fluctuations in the inertial range using WIND data. These authors used a wavelet technique for a systematic elimination of intermittency effects on spectra and structure functions in order to recover the actual scaling properties in the inertial range. They found that magnetic field and velocity fluctuations exhibit a well-defined, although different, monofractal behavior, following a Kolmogorov − 5∕3 scaling and a Iroshnikov–Kraichnan − 3∕2 scaling, respectively. These results are clearly opposite to the expected scaling for kinetic and magnetic fluctuations which should follow Kolmogorov and Kraichnan scaling, respectively (see Section 2.8). However, as remarked by Roberts (2007Jump To The Next Citation Point), Voyager observations of the velocity spectrum have demonstrated a likely asymptotic state in which the spectrum steepens towards a spectral index of − 5∕3, finally matching the magnetic spectrum and the theoretical expectation of Kolmogorov turbulence. Moreover, the same authors examined Ulysses spectra to determine if the Voyager result, based on a very few sufficiently complete intervals, were correct. Preliminary results confirmed the − 5 ∕3 slope for velocity fluctuations at ∼ 5 AU from the Sun in the ecliptic.

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Figure 23: Velocity spectral index vs. heliocentric distance (Roberts, 2007Jump To The Next Citation Point).

Figure 23View Image, taken from Roberts (2007Jump To The Next Citation Point), shows the evolution of the spectral index during the radial excursion of Ulysses. These authors examined many intervals in order to develop a more general picture of the spectral evolution in various conditions, and how magnetic and velocity spectra differ in these cases. The general trend shown in Figure 23View Image is towards − 5∕3 as the distance increases. Lower values are due to the highly Alfvénic fast polar wind while higher values, around 2, are mainly due to the jumps at the stream fronts as previously shown by Roberts (2007). Thus, the discrepancy between magnetic and velocity spectral slope is only temporary and belongs to the evolutionary phase of the spectra towards a well developed Kolmogorov like turbulence spectrum.

Horbury et al. (2008Jump To The Next Citation Point) performed a study on the anisotropy of the energy spectrum of magnetohydrodynamic (MHD) turbulence with respect to the magnetic field orientation to test the validity of the critical balance theory (Goldreich and Sridhar, 1995) in space plasma environment. This theory predicts that the power spectrum P (k ) would scale as f− 5∕3 when the angle 𝜃B between the mean field direction and the flow direction is 90∘. On the other hand, in case 𝜃 = 0∘ B the scaling would follow −2 f. Moreover, the latter spectrum would also have a smaller energy content.

Horbury et al. (2008Jump To The Next Citation Point) used 30 days of Ulysses magnetic field observations (1995, days 100 – 130) with a resolution of 1 second. At that time, Ulysses was immersed in the steady high speed solar wind coming from the Sun’s Northern polar coronal hole at 1.4 AU from the Sun. These authors studied the anisotropies of the turbulence by measuring how the spacecraft frame spectrum of magnetic fluctuations varies with 𝜃B. They adopted a method based on wavelet analysis which was sensitive to the frequent changes of the local magnetic field direction.

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Figure 24: Top panel: Trace of power in the magnetic field as a function of the angle between the local magnetic field and the sampling direction at a spacecraft frequency of 61 mHz. The larger scatter for 𝜃 > 90 B is the result of fewer data points at these angles. Bottom panel: spectral index of the trace, fitted over spacecraft frequencies from 15.98 mHz. Image reproduced by permission from Horbury et al. (2008), copyright by APS.

The lower panel of Figure 24View Image clearly shows that for angles larger than about ∘ 45 the spectral index smoothly fluctuates around − 5 ∕3 while, for smaller angles, it tends to a value of − 2, as predicted by the critical balance type of cascade. However, although the same authors recognize that a spectral index of − 2 has not been routinely observed in the fast solar wind and that the range of 𝜃B over which the spectral index deviates from − 5∕3 is wider than expected, they consider these findings to be a robust evidence of the validity of critical balance theory in space plasma environment.

3.1.2 Experimental evaluation of Reynolds number in the solar wind

Properties of solar wind fluctuations have been widely studied in the past, relying on the “frozen-in approximation” (Taylor, 1938). The hypothesis at the basis of Taylor’s approximation is that, since large integral scales in turbulence contain most of the energy, the advection due to the smallest turbulent scales fluctuations can be disregarded and, consequently, the advection of a turbulent field past an observer in a fixed location is considered solely due to the larger scales. In experimental physics, this hypothesis allows time series measured at a single point in space to be interpreted as spatial variations in the mean flow being swept past the observer. However, the canonical way to establish the presence of spatial structures relies in the computation of two-point single time measurements. Only recently, the simultaneous presence of several spacecraft sampling solar wind parameters allowed to correlate simultaneous in-situ observations in two different observing locations in space. Matthaeus et al. (2005Jump To The Next Citation Point) and Weygand et al. (2007) firstly evaluated the two-point correlation function using simultaneous measurements of interplanetary magnetic field from the Wind, ACE, and Cluster spacecraft. Their technique allowed to compute for the first time fundamental turbulence parameters previously determined from single spacecraft measurements. In particular, these authors evaluated the correlation scale λC and the Taylor microscale λT which allow to determine empirically the effective magnetic Reynolds number.

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Figure 25: Typical interplanetary magnetic field power spectrum at 1 AU. The low frequency range refers to Helios 2 observations (adapted from Bruno et al., 2009Jump To The Next Citation Point) while the high frequency refers to WIND observations (adapted from Leamon et al., 1998Jump To The Next Citation Point). Vertical dashed lines indicate the correlative, Taylor and Kolmogorov length scales.

As a matter of fact, there are three standard turbulence length scales which can be identified in a typical turbulence power spectrum as shown in Figure 25View Image: the correlation length λC, the Taylor scale λT and the Kolmogorov scale λK. The Correlation or integral length scale represents the largest separation distance over which eddies are still correlated, i.e., the largest turbulent eddy size. The Taylor scale is the scale size at which viscous dissipation begins to affect the eddies, it is several times larger than Kolmogorov scale and marks the transition from the inertial range to the dissipation range. The Kolmogorov scale is the one that characterizes the smallest dissipation-scale eddies.

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Figure 26: Typical two-point correlation function. The Taylor scale λT and the correlation length λC are the radius of curvature of the Correlation function at the origin (see inset graph) and the scale at which turbulent fluctuation are no longer correlated, respectively.

The Taylor scale λT and the correlation length λC, as indicated in Figure 26View Image, can be obtained from the two-point correlation function being the former the radius of curvature of the Correlation function at the origin and the latter the scale at which turbulent fluctuation are no longer correlated. Thus, λT can be obtained from from Taylor expansion of the two point correlation function for r → 0 (Tennekes and Lumely, 1972):

2 R (r) ≈ 1 − -r--+ ... (49 ) 2λ2T
where r is the spacecraft separation and R (r) = ⟨b(x ) ⋅ b (x + r )⟩ is the auto-correlation function computed along the x direction for the fluctuating field b (x ). On the other hand, the correlation length λC can be obtained integrating the normalized correlation function along a chosen direction of integration ξ:
∫ ∞ R (ξ) λC = R-(0) d ξ. (50 ) 0

At this point, following Batchelor (1970Jump To The Next Citation Point) it is possible to obtain the effective magnetic Reynolds number:

( λ )2 Remff = -C- . (51 ) λT

Figure 27View Image shows estimates of the correlation function from ACE-Wind for separation distances 20 –350 RE and two sets of Cluster data for separations 0.02– 0.04RE and 0.4–1.2 RE, respectively.

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Figure 27: Estimates of the correlation function from ACE-Wind for separation distances 20 –350 RE and two sets of Cluster data for separations 0.02 –0.04RE and 0.4– 1.2 RE, respectively. Image adapted from Matthaeus et al. (2005Jump To The Next Citation Point).

Following the definitions of λC and λT given above, Matthaeus et al. (2005Jump To The Next Citation Point) were able to fit the first data set of Cluster, i.e., the one with shorter separations, with a parabolic fit while they used an exponential fit for ACE-Wind and the second Cluster data set. These fits provided estimates for λC and λT from which these authors obtained the first empirical determination of Re ff m which resulted to be of the order of 5 2.3 × 10, as illustrated in Figure 28View Image.

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Figure 28: Left panel: parabolic fit at small scales in order to estimate λ T. Right panel: exponential fit at intermediate and large scales in order to estimate λC. The square of the ratio of these two length scales gives an estimate of the effective magnetic Reynolds number. Image adapted from Matthaeus et al. (2005).

3.1.3 Evidence for non-linear interactions

As we said previously, Helios 2 s/c gave us the unique opportunity to study the radial evolution of turbulent fluctuations in the solar wind within the inner heliosphere. Most of the theoretical studies which aim to understand the physical mechanism at the base of this evolution originate from these observations (Bavassano et al., 1982bJump To The Next Citation Point; Denskat and Neubauer, 1983Jump To The Next Citation Point).

In Figure 29View Image we consider again similar observations taken by Helios 2 during its primary mission to the Sun together with observations taken by Ulysses in the ecliptic at 1.4 and 4.8 AU in order to extend the total radial excursion.

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Figure 29: Left panel: power density spectra of magnetic field fluctuations observed by Helios 2 between 0.3 and 1 AU within the trailing edge of the same corotating stream shown in Figure 16View Image, during the first mission to the Sun in 1976 and by Ulysses between 1.4 and 4.8 AU during the ecliptic phase. Ulysses observations at 4.8 AU refer to the end of 1991 while observations taken at 1.4 AU refer to the end of August of 2007. While the spectral index of slow wind does not show any radial dependence, the spectral break, clearly present in fast wind and marked by a blue dot, moves to lower and lower frequency as the heliocentric distance increases. Image adapted from Bruno et al. (2009).

Helios 2 power density spectra were obtained from the trace of the spectral matrix of magnetic field fluctuations, and belong to the same co-rotating stream observed on day 49, at a heliocentric distance of 0.9 AU, on day 75 at 0.7 AU and, finally, on day 104 at 0.3 AU. Ulysses spectra, constructed in the same way as those of Helios 2, were taken at 1.4 and 4.8 AU during the ecliptic phase of the orbit. Observations at 4.8 AU refer to the end of 1991 (fast wind period started on day 320, slow wind period started on day 338) while observations taken at 1.4 AU refer to fast wind observed at the end of August of 2007, starting on day 241:12.

While the spectral index of slow wind does not show any radial dependence, being characterized by a single Kolmogorov type spectral index, fast wind is characterized by two distinct spectral slopes: about − 1 within low frequencies and about a Kolmogorov like spectrum at higher frequencies. These two regimes are clearly separated by a knee in the spectrum often referred to as “frequency break”. As the wind expands, the frequency break moves to lower and lower frequencies so that larger and larger scales become part of the Kolmogorov-like turbulence spectrum, i.e., of what we will indicate as “inertial range” (see discussion at the end of the previous section). Thus, the power spectrum of solar wind fluctuations is not solely function of frequency f, i.e., P (f), but it also depends on heliocentric distance r, i.e., P (f) → P(f,r ).

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Figure 30: Radial dependence of the frequency break observed in the ecliptic within fast wind as shown in the previous Figure 29View Image. The radial dependence seems to be governed by a power-law of the order of − 1.5 R.

Figure 30View Image shows the frequency location of the spectral breaks observed in the left-hand-side panel of Figure 29View Image as a function of heliocentric distance. The radial distribution of these 5 points suggests that the frequency break moves at lower and lower frequencies during the wind expansion following a power-law of the order of R −1.5. Previous results, obtained for long data sets spanning hundreds of days and inevitably mixing fast and slow wind, were obtained by Matthaeus and Goldstein (1986) who found the breakpoint around 10 h at 1 AU, and Klein et al. (1992) who found that the breakpoint was near 16 h at 4 AU. Obviously, the frequency location of the breakpoint provided by these early determinations is strongly affected by the fact that mixing fast and slow wind would shift the frequency break to lower frequencies with respect to solely fast wind. In any case, this frequency break is strictly related to the correlation length (Klein, 1987) and the shift to lower frequency, during the wind expansion, is consistent with the growth of the correlation length observed in the inner (Bruno and Dobrowolny, 1986Jump To The Next Citation Point) and outer heliosphere (Matthaeus and Goldstein, 1982aJump To The Next Citation Point). Analogous behavior for the low frequency shift of the spectral break, similar to the one observed in the ecliptic, has been reported by Horbury et al. (1996aJump To The Next Citation Point) studying the rate of turbulent evolution over the Sun’s poles. These authors used Ulysses magnetic field observations between 1.5 and 4.5 AU selecting mostly undisturbed, high speed polar flows. They found a radial gradient of the order of −1.1 R, clearly slower than the one reported in Figure 30View Image or that can be inferred from results by Bavassano et al. (1982bJump To The Next Citation Point) confirming that the turbulence evolution in the polar wind is slower than the one in the ecliptic, as qualitatively predicted by Bruno (1992Jump To The Next Citation Point), because of the lack of large scale stream shears. However, these results will be discussed more extensively in in Section 4.1.

However, the phenomenology described above only apparently resembles hydrodynamic turbulence where the large eddies, below the frequency break, govern the whole process of energy cascade along the spectrum (Tu and Marsch, 1995b). As a matter of fact, when the relaxation time increases, the largest eddies provide the energy to be transferred along the spectrum and dissipated, with a decay rate approximately equal to the transfer rate and, finally, to the dissipation rate at the smallest wavelengths where viscosity dominates. Thus, we expect that the energy containing scales would loose energy during this process but would not become part of the turbulent cascade, say of the inertial range. Scales on both sides of the frequency break would remain separated. Accurate analysis performed in the solar wind (Bavassano et al., 1982bJump To The Next Citation Point; Marsch and Tu, 1990bJump To The Next Citation Point; Roberts, 1992Jump To The Next Citation Point) have shown that the low frequency range of the solar wind magnetic field spectrum radially evolves following the WKB model, or geometrical optics, which predicts a radial evolution of the power associated with the fluctuations ∼ r−3. Moreover, a steepening of the spectrum towards a Kolmogorov like spectral index can be observed. On the contrary, the same in-situ observations established that the radial decay for the higher frequencies was faster than −3 ∼ r and the overall spectral slope remained unchanged. This means that the energy contained in the largest eddies does not decay as it would happen in hydrodynamic turbulence and, as a consequence, the largest eddies cannot be considered equivalent to the energy containing eddies identified in hydrodynamic turbulence. So, this low frequency range is not separated from the inertial range but becomes part of it as the turbulence ages. These observations cast some doubts on the applicability of hydrodynamic turbulence paradigm to interplanetary MHD turbulence. A theoretical help came from adopting a local energy transfer function (Tu et al., 1984Jump To The Next Citation Point; Tu, 1987a,b, 1988Jump To The Next Citation Point), which would take into account the non-linear effects between eddies of slightly differing wave numbers, together with a WKB description which would mainly work for the large scale fluctuations. This model was able to reproduce the displacement of the frequency break with distance by combining the linear WKB law and a model of nonlinear coupling besides most of the features observed in the magnetic power spectra P(f,r ) observed by Bavassano et al. (1982bJump To The Next Citation Point). In particular, the concept of the “frequency break”, just mentioned, was pointed out for the first time by Tu et al. (1984Jump To The Next Citation Point) who, developing the analytic solution for the radially evolving power spectrum P(f, r) of fluctuations, obtained a critical frequency “f c” such that for frequencies f ≪ f ,P (f,r) ∝ f−1 c and for − 1.5 f ≫ fc,P (f,r) ∝ f.

3.1.4 Fluctuations anisotropy

Interplanetary magnetic field (IMF) and velocity fluctuations are rather anisotropic as for the first time observed by Belcher and Davis Jr (1971Jump To The Next Citation Point); Belcher and Solodyna (1975Jump To The Next Citation Point); Chang and Nishida (1973Jump To The Next Citation Point); Burlaga and Turner (1976); Solodyna and Belcher (1976Jump To The Next Citation Point); Parker (1980); Bavassano et al. (1982aJump To The Next Citation Point); Tu et al. (1989a); and Marsch and Tu (1990aJump To The Next Citation Point). This feature can be better observed if fluctuations are rotated into the minimum variance reference system (see Appendix D).

Sonnerup and Cahill (1967Jump To The Next Citation Point) introduced the minimum variance analysis which consists in determining the eigenvectors of the matrix

S = ⟨B B ⟩ − ⟨B ⟩⟨B ⟩, ij i j i j

where i and j denote the components of magnetic field along the axes of a given reference system. The statistical properties of eigenvalues approximately satisfy the following statements:

  • One of the eigenvalues of the variance matrix is always much smaller than the others, say λ1 ≪ (λ2, λ3), and the corresponding eigenvector &tidle; V1 is the minimum-variance direction (see Appendix D.1 for more details). This indicates that, at least locally, the magnetic fluctuations are confined in a plane perpendicular to the minimum-variance direction.
  • In the plane perpendicular to &tidle; V1, fluctuations appear to be anisotropically distributed, say λ3 > λ2. Typical values for eigenvalues are λ3 : λ2 : λ1 = 10 : 3.5 : 1.2 (Chang and Nishida, 1973; Bavassano et al., 1982aJump To The Next Citation Point).
  • The direction &tidle; V1 is nearly parallel to the average magnetic field B0, that is, the distribution of the angles between V&tidle;1 and B0 is narrow with width of about 10∘ and centered around zero.

As shown in Figure 31View Image, in this new reference system it is readily seen that the maximum and intermediate components have much more power compared with the minimum variance component. Generally, this kind of anisotropy characterizes Alfvénic intervals and, as such, it is more commonly found within high velocity streams (Marsch and Tu, 1990aJump To The Next Citation Point).

View Image

Figure 31: Power density spectra of the three components of IMF after rotation into the minimum variance reference system. The black curve corresponds to the minimum variance component, the blue curve to the maximum variance, and the red one to the intermediate component. This case refers to fast wind observed at 0.3 AU and the minimum variance direction forms an angle of ∘ ∼ 8 with respect to the ambient magnetic field direction. Thus, most of the power is associated with the two components quasi-transverse to the ambient field.

A systematic analysis for both magnetic and velocity fluctuations was performed by Klein et al. (1991Jump To The Next Citation Point, 1993Jump To The Next Citation Point) between 0.3 and 10 AU. These studies showed that magnetic field and velocity minimum variance directions are close to each other within fast wind and mainly clustered around the local magnetic field direction. The effects of expansion are such as to separate field and velocity minimum variance directions. While magnetic field fluctuations keep their minimum variance direction loosely aligned with the mean field direction, velocity fluctuations tend to have their minimum variance direction oriented along the radial direction. The depleted alignment to the background magnetic field would suggest a smaller anisotropy of the fluctuations. As a matter of fact, Klein et al. (1991Jump To The Next Citation Point) found that the degree of anisotropy, which can be defined as the ratio between the power perpendicular to and that along the minimum variance direction, decreases with heliocentric distance in the outer heliosphere.

At odds with these conclusions were the results by Bavassano et al. (1982aJump To The Next Citation Point) who showed that the ratio λ ∕ λ 1 3, calculated in the inner heliosphere within a co-rotating high velocity stream, clearly decreased with distance, indicating that the degree of magnetic anisotropy increased with distance. Moreover, this radial evolution was more remarkable for fluctuations of the order of a few hours than for those around a few minutes. Results by Klein et al. (1991Jump To The Next Citation Point) in the outer heliosphere and by Bavassano et al. (1982aJump To The Next Citation Point) in the inner heliosphere remained rather controversial until recent studies (see Section 10.2), performed by Bruno et al. (1999bJump To The Next Citation Point), found a reason for this discrepancy.

A different approach to anisotropic fluctuations in solar wind turbulence have been made by Bigazzi et al. (2006) and Sorriso-Valvo et al. (2006Jump To The Next Citation Point, 2010bJump To The Next Citation Point). In these studies the full tensor of the mixed second-order structure functions has been used to quantitatively measure the degree of anisotropy and its effect on small-scale turbulence through a fit of the various elements of the tensor on a typical function (Sorriso-Valvo et al., 2006). Moreover three different regions of the near-Earth space have been studied, namely the solar wind, the Earth’s foreshock and magnetosheath showing that, while in the undisturbed solar wind the observed strong anisotropy is mainly due to the large-scale magnetic field, near the magnetosphere other sources of anisotropy influence the magnetic field fluctuations (Sorriso-Valvo et al., 2010b).

3.1.5 Simulations of anisotropic MHD

In the presence of a DC background magnetic field B0 which, differently from the bulk velocity field, cannot be eliminated by a Galilean transformation, MHD incompressible turbulence becomes anisotropic (Shebalin et al., 1983Jump To The Next Citation Point; Montgomery, 1982Jump To The Next Citation Point; Zank and Matthaeus, 1992Jump To The Next Citation Point; Carbone and Veltri, 1990Jump To The Next Citation Point; Oughton, 1993). The main effect produced by the presence of the background field is to generate an anisotropic distribution of wave vectors as a consequence of the dependence of the characteristic time for the non-linear coupling on the angle between the wave vector and the background field. This effect can be easily understood if one considers the MHD equation. Due to the presence of a term (B0 ⋅ ∇ )z±, which describes the convection of perturbations in the average magnetic field, the non-linear interactions between Alfvénic fluctuations are weakened, since convection decorrelates the interacting eddies on a time of the order −1 (k ⋅ B0 ). Clearly fluctuations with wave vectors almost perpendicular to B0 are interested by such an effect much less than fluctuations with k ∥ B0. As a consequence, the former are transferred along the spectrum much faster than the latter (Shebalin et al., 1983Jump To The Next Citation Point; Grappin, 1986; Carbone and Veltri, 1990Jump To The Next Citation Point).

To quantify anisotropy in the distribution of wave vectors k for a given dynamical variable Q (k,t) (namely the energy, cross-helicity, etc.), it is useful to introduce the parameter

∘ --2---- Ω = tan −1 ⟨k⊥-⟩Q- (52 ) Q 2⟨k2∥⟩Q
(Shebalin et al., 1983Jump To The Next Citation Point; Carbone and Veltri, 1990Jump To The Next Citation Point), where the average of a given quantity g(k) is defined as
∫ 3 ⟨g(k )⟩ = --d∫-k-g(k)Q-(k,t). Q d3k Q (k,t)

For a spectrum with wave vectors perpendicular to B0 we have a spectral anisotropy Ω = 90 ∘, while for an isotropic spectrum Ω = 45∘. Numerical simulations in 2D configuration by Shebalin et al. (1983Jump To The Next Citation Point) confirmed the occurrence of anisotropy, and found that anisotropy increases with the Reynolds number. Unfortunately, in these old simulations, the Reynolds numbers used are too small to achieve a well defined spectral anisotropy. Carbone and Veltri (1990Jump To The Next Citation Point) started from the spectral equations obtained through the Direct Interaction Approximation closure by Veltri et al. (1982Jump To The Next Citation Point), and derived a shell model analogous for the anisotropic MHD turbulence. Of course the anisotropy is over–simplified in the model, in particular the Alfvén time is assumed isotropic. However, the model was useful to investigate spectral anisotropy at very high Reynolds numbers. The phenomenological anisotropic spectrum obtained from the model, for both pseudo-energies obtained through polarizations a = 1,2 defined through Equation (17View Equation), can be written as

± ± [ 2 2 2 2]−μ± E a (k, t) ∼ Ca ℓ∥k ∥ + ℓ⊥k ⊥ . (53 )

The spectral anisotropy is different within the injection, inertial, and dissipative ranges of turbulence (Carbone and Veltri, 1990). Wave vectors perpendicular to B0 are present in the spectrum, but when the process of energy transfer generates a strong anisotropy (at small times), a competing process takes place which redistributes the energy over all wave vectors. The dynamical balance between these tendencies fixes the value of the spectral anisotropy Ω ≃ 55∘ in the inertial range. On the contrary, since the redistribution of energy cannot take place, in the dissipation domain the spectrum remains strongly anisotropic, with ∘ Ω ≃ 80. When the Reynolds number increases, the contribution of the inertial range extends, and the increases of the total anisotropy tends to saturate at about Ω ≃ 60∘ at Reynolds number of 105. This value corresponds to a rather low value for the ratio between parallel and perpendicular correlation lengths ℓ ∕ℓ ≃ 2 ∥ ⊥, too small with respect to the observed value ℓ∥∕ℓ⊥ ≥ 10. This suggests that the non-linear dynamical evolution of an initially isotropic spectrum of turbulence is perhaps not sufficient to explain the observed anisotropy. These results have been confirmed numerically (Oughton et al., 1994).

3.1.6 Spectral anisotropy in the solar wind

The correlation time, as defined in Appendix A, estimates how much an element of our time series x (t) at time t1 depends on the value assumed by x (t) at time t0, being t1 = t0 + δt. This concept can be transferred from the time domain to the space domain if we adopt the Taylor hypothesis and, consequently, we can talk about spatial scales.

Correlation lengths in the solar wind generally increase with heliocentric distance (Matthaeus and Goldstein, 1982bJump To The Next Citation Point; Bruno and Dobrowolny, 1986Jump To The Next Citation Point), suggesting that large scale correlations are built up during the wind expansion. This kind of evolution is common to both fast and slow wind as shown in Figure 32View Image, where we can observe the behavior of the Bz correlation function for fast and slow wind at 0.3 and 0.9 AU.

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Figure 32: Correlation function just for the Z component of interplanetary magnetic field as observed by Helios 2 during its primary mission to the Sun. The blue color refers to data recorded at 0.9 AU while the red color refers to 0.3 AU. Solid lines refer to fast wind, dashed lines refer to slow wind.

Moreover, the fast wind correlation functions decrease much faster than those related to slow wind. This behavior reflects also the fact that the stochastic character of Alfvénic fluctuations in the fast wind is very efficient in decorrelating the fluctuations of each of the magnetic field components.

More detailed studies performed by Matthaeus et al. (1990Jump To The Next Citation Point) provided for the first time the two-dimensional correlation function of solar wind fluctuations at 1 AU. The original dataset comprised approximately 16 months of almost continuous magnetic field 5-min averages. These results, based on ISEE 3 magnetic field data, are shown in Figure 33View Image, also called the “The Maltese Cross”.

View Image

Figure 33: Contour plot of the 2D correlation function of interplanetary magnetic field fluctuations as a function of parallel and perpendicular distance with respect to the mean magnetic field. The separation in r∥ and r⊥ is in units of 1010 cm. Image reproduced by permission from Matthaeus et al. (1990Jump To The Next Citation Point), copyright by AGU.

This figure has been obtained under the hypothesis of cylindrical symmetry. Real determination of the correlation function could be obtained only in the positive quadrant, and the whole plot was then made by mirroring these results on the remaining three quadrants. The iso-contour lines show contours mainly elongated along the ambient field direction or perpendicular to it. Alfvénic fluctuations with k ∥ B0 contribute to contours elongated parallel to r⊥. Fluctuations in the two-dimensional turbulence limit (Montgomery, 1982Jump To The Next Citation Point) contribute to contours elongated parallel to r∥. This two-dimensional turbulence is characterized for having both the wave vector k and the perturbing field δb perpendicular to the ambient field B0. Given the fact that the analysis did not select fast and slow wind, separately, it is likely that most of the slab correlations came from the fast wind while the 2D correlations came from the slow wind. As a matter of fact, Dasso et al. (2005), using 5 years of spacecraft observations at roughly 1 AU, showed that fast streams are dominated by fluctuations with wavevectors quasi-parallel to the local magnetic field, while slow streams are dominated by quasi-perpendicular fluctuation wavevectors. Anisotropic turbulence has been observed in laboratory plasmas and reverse pinch devices (Zweben et al., 1979).

Bieber et al. (1996) formulated an observational test to distinguish the slab (Alfvénic) from the 2D component within interplanetary turbulence. These authors assumed a mixture of transverse fluctuations, some of which have wave vectors perpendicular k ⊥ B0 and polarization of fluctuations δB (k⊥) perpendicular to both vectors (2D geometry with k ≃ 0 ∥), and some parallel to the mean magnetic field k ∥ B0, the polarization of fluctuations δB (k∥) being perpendicular to the direction of B0 (slab geometry with k⊥ ≃ 0). The magnetic field is then rotated into the same mean field coordinate system used by Belcher and Davis Jr (1971Jump To The Next Citation Point) and Belcher and Solodyna (1975Jump To The Next Citation Point), where the y-coordinate is perpendicular to both B0 and the radial direction, while the x-coordinate is perpendicular to B0 but with a component also in the radial direction. Using that geometry, and defining the power spectrum matrix as

1 ∫ Pij(k) = ----3- d3r⟨Bi(x)Bj (x + r)⟩e −ik⋅r, (2π )

it can be found that, assuming axisymmetry, a two-component model can be written in the frequency domain

( )1 −q ( )1 −q fPyy(f) = rCs --2πf---- + (1 − r)Cs--2q--- --2-πf-- , (54 ) Uw cos ψ (1 + q) Uw sin ψ
( )1 −q ( )1−q --2πf---- ---2--- --2-πf-- f Pxx(f) = rCs Uw cos ψ + (1 − r)Cs(1 + q) Uw sin ψ , (55 )
where the anisotropic energy spectrum is the sum of both components:
( ) ( ) 2πf 1−q 2πf 1−q fT (f) = 2rCs --------- + 2(1 − r)Cs -------- . (56 ) Uw cos ψ Uw sin ψ
Here f is the frequency, Cs is a constant defining the overall spectrum amplitude in wave vector space, Uw is the bulk solar wind speed and ψ is the angle between B0 and the wind direction. Finally, r is the fraction of slab components and (1 − r) is the fraction of 2D components.

The ratio test adopted by these authors was based on the ratio between the reduced perpendicular spectrum (fluctuations ⊥ to the mean field and solar wind flow direction) and the reduced quasi-parallel spectrum (fluctuations ⊥ to the mean field and in the plane defined by the mean field and the flow direction). This ratio, expected to be 1 for slab turbulence, resulted to be ∼ 1.4 for fluctuations within the inertial range, consistent with 74% of 2D turbulence and 26% of slab. A further test, the anisotropy test, evaluated how the spectrum should vary with the angle between the mean magnetic field and the flow direction of the wind. The measured slab spectrum should decrease with the field angle while the 2D spectrum should increase, depending on how these spectra project on the flow direction. The results from this test were consistent with with 95% of 2D turbulence and 5% of slab. In other words, the slab turbulence due to Alfvénic fluctuations would be a minor component of interplanetary MHD turbulence. A third test derived from Mach number scaling associated with the nearly incompressible theory (Zank and Matthaeus, 1992Jump To The Next Citation Point), assigned the same fraction ∼ 80% to the 2D component. However, the data base for this analysis was derived from Helios magnetic measurements, and all data were recorded near times of solar energetic particle events. Moreover, the quasi totality of the data belonged to slow solar wind (Wanner and Wibberenz, 1993) and, as such, this analysis cannot be representative of the whole phenomenon of turbulence in solar wind. As a matter of fact, using Ulysses observations, Smith (2003) found that in the polar wind the percentage of slab and 2D components is about the same, say the high latitude slab component results unusually higher as compared with ecliptic observations.

Successive theoretical works by Ghosh et al. (1998a,b) in which they used compressible models in large variety of cases were able to obtain, in some cases, parallel and perpendicular correlations similar to those obtained in the solar wind. However, they concluded that the “Maltese” cross does not come naturally from the turbulent evolution of the fluctuations but it strongly depends on the initial conditions adopted when the simulation starts. It seems that the existence of these correlations in the initial data represents an unavoidable constraint. Moreover, they also stressed the importance of time-averaging since the interaction between slab waves and transverse pressure-balanced magnetic structures causes the slab turbulence to evolve towards a state in which a two-component correlation function emerges during the process of time averaging.

The presence of two populations, i.e., a slab-like and a quasi-2D like, was also inferred by Dasso et al. (2003). These authors computed the reduced spectra of the normalized cross-helicity and the Alfvén ratio from ACE dataset. These parameters, calculated for different intervals of the angle 𝜃 between the flow direction and the orientation of the mean field B0, showed a remarkable dependence on 𝜃.

The geometry used in these analyses assumes that the energy spectrum in the rest frame of the plasma is axisymmetric and invariant for rotations about the direction of B 0. Even if these assumption are good when we want to translate results coming from 2D numerical simulations to 3D geometry, these assumptions are quite in contrast with the observational fact that the eigenvalues of the variance matrix are different, namely λ3 ⁄= λ2.

Going back from the correlation tensor to the power spectrum is a complicated technical problem. However, Carbone et al. (1995aJump To The Next Citation Point) derived a description of the observed anisotropy in terms of a model for the three-dimensional energy spectra of magnetic fluctuations. The divergence-less of the magnetic field allows to decompose the Fourier amplitudes of magnetic fluctuations in two independent polarizations: The first one I[1](k) corresponds, in the weak turbulence theory, to the Alfvénic mode, while the second polarization [2] I (k ) corresponds to the magnetosonic mode. By using only the hypothesis that the medium is statistically homogeneous and some algebra, authors found that the energy spectra of both polarizations can be related to the two-points correlation tensor and to the variance matrix. Through numerical simulations of the shell model (see later in the review) it has been shown that the anisotropic energy spectrum can be described in the inertial range by a phenomenological expression

[ ]− 1− μ∕2 I [s](k ) = Cs (ℓ[s]kx)2 + (ℓ[s]ky)2 + (ℓ[s]kz)2 s , (57 ) x y z
where ki are the Cartesian components of the wave vector k, and Cs, ℓ[s] i, and μs (s = 1,2 indicates both polarizations; i = x, y,z) are free parameters. In particular, C s gives information on the energy content of both polarizations, [s] ℓi represent the spectral extensions along the direction of a given system of coordinates, and μs are two spectral indices.

A fit to the eigenvalues of the variance matrix allowed Carbone et al. (1995aJump To The Next Citation Point) to fix the free parameters of the spectrum for both polarizations. They used data from Bavassano et al. (1982aJump To The Next Citation Point) who reported the values of λi at five wave vectors calculated at three heliocentric distances, selecting periods of high correlation (Alfvénic periods) using magnetic field measured by the Helios 2 spacecraft. They found that the spectral indices of both polarizations, in the range 1.1 ≤ μ1 ≤ 1.3 and 1.46 ≤ μ2 ≤ 1.8 increase systematically with increasing distance from the Sun, the polarization [2] spectra are always steeper than the corresponding polarization [1] spectra, while polarization [1] is always more energetic than polarization [2]. As far as the characteristic lengths are concerned, it can be found that ℓ[1x]> ℓ[1y]≫ ℓz[1 ], indicating that wave vectors k ∥ B0 largely dominate. Concerning polarization [2], it can be found that ℓ [2] ≫ ℓ[2]≃ ℓ[2] x y z, indicating that the spectrum [2] I (k) is strongly flat on the plane defined by the directions of B0 and the radial direction. Within this plane, the energy distribution does not present any relevant anisotropy.

Let us compare these results with those by Matthaeus et al. (1990Jump To The Next Citation Point), the comparison being significant as far as the plane yz is taken into account. The decomposition of Carbone et al. (1995aJump To The Next Citation Point) in two independent polarizations is similar to that of Matthaeus et al. (1990Jump To The Next Citation Point), a contour plot of the trace of the correlation tensor Fourier transform T (k) = I[1](k) + I[2](k) on the plane (ky;kz) shows two populations of fluctuations, with wave vectors nearly parallel and nearly perpendicular to B0, respectively. The first population is formed by all the polarization [1] fluctuations and by the fluctuations with k ∥ B0 belonging to polarization [2]. The latter fluctuations are physically indistinguishable from the former, in that when k is nearly parallel to B0, both polarization vectors are quasi-perpendicular to B 0. On the contrary, the second population is almost entirely formed by fluctuations belonging to polarization [2]. While it is clear that fluctuations with k nearly parallel to B0 are mainly polarized in the plane perpendicular to B0 (a consequence of ∇ ⋅ B = 0), fluctuations with k nearly perpendicular to B0 are polarized nearly parallel to B0.

Although both models yield to the occurrence of two populations, Matthaeus et al. (1990Jump To The Next Citation Point) give an interpretation of their results which is in contrast with that of Carbone et al. (1995aJump To The Next Citation Point). Namely Matthaeus et al. (1990Jump To The Next Citation Point) suggest that a nearly 2D incompressible turbulence characterized by wave vectors and magnetic fluctuations, both perpendicular to B0, is present in the solar wind. However, this interpretation does not arise from data analysis, rather from the 2D numerical simulations by Shebalin et al. (1983Jump To The Next Citation Point) and from analytical studies (Montgomery, 1982Jump To The Next Citation Point). Let us note, however, that in the former approach, which is strictly 2D, when k ⊥ B0 magnetic fluctuations are necessarily parallel to B0. In the latter one, along with incompressibility, it is assumed that the energy in the fluctuations is much less than in the DC magnetic field; both hypotheses do not apply to the solar wind case. On the contrary, results by Carbone et al. (1995aJump To The Next Citation Point) can be directly related to the observational data. In any case, it is worth reporting that a model like that discussed here, that is a superposition of fluctuations with both slab and 2D components, has been used to describe turbulence also in the Jovian magnetosphere (Saur et al., 2002, 2003). In addition, several theoretical and observational works indicate that there is a competition between the radial axis and the mean field axis in shaping the polarization and spectral anisotropies in the solar wind.

In this respect, Grappin and Velli (1996Jump To The Next Citation Point) used numerical simulations of MHD equations which included expansion effects (Expanding Box Model) to study the formation of anisotropy in the wind and the interaction of Alfvén waves within a transverse magnetic structures. These authors found that a large-scale isotropic Alfvénic eddy stretched by expansion naturally mixes with smaller scale transverse Alfvén waves with a different anisotropy.

Saur and Bieber (1999), on the other hand, employed three different tests on about three decades of solar wind observations at 1 AU in order to better understand the anisotropic nature of solar wind fluctuations. Their data analysis strongly supported the composite model of a turbulence made of slab and 2-D fluctuations.

Narita et al. (2011b), using the four Cluster spacecraft, determined the three-dimensional wave-vector spectra of fluctuating magnetic fields in the solar wind within the inertial range. These authors found that the spectra are anisotropic throughout the analyzed frequency range and the power is extended primarily in the directions perpendicular to the mean magnetic field, as might be expected of 2-D turbulence, however, the analyzed fluctuations cannot be considered axisymmetric.

Finally, Turner et al. (2011Jump To The Next Citation Point) suggested that the non-axisymmetry anisotropy of the frequency spectrum observed using in-situ observations may simply arise from a sampling effect related to the fact that the s/c samples three dimensional fluctuations as a one-dimensional series and that the energy density is not equally distributed among the different scales (i.e., spectral index > 1).

3.1.7 Magnetic helicity

Magnetic helicity Hm, as defined in Appendix B.1, measures the “knottedness” of magnetic field lines (Moffatt, 1978Jump To The Next Citation Point). Moreover, Hm is a pseudo scalar and changes sign for coordinate inversion. The plus or minus sign, for circularly polarized magnetic fluctuations in a slab geometry, indicates right or left-hand polarization. Statistical information about the magnetic helicity is derived from the Fourier transform of the magnetic field auto-correlation matrix Rij(r) = ⟨Bi(x) ⋅ Bj(x + r )⟩ as shown by Matthaeus and Goldstein (1982bJump To The Next Citation Point). While the trace of the symmetric part of the spectral matrix accounts for the magnetic energy, the imaginary part of the spectral matrix accounts for the magnetic helicity (Batchelor, 1970Jump To The Next Citation Point; Montgomery, 1982; Matthaeus and Goldstein, 1982bJump To The Next Citation Point). However, what is really available from in-situ measurements in space experiments are data from a single spacecraft, and we can obtain values of R only for collinear sequences of r along the x direction which corresponds to the radial direction from the Sun. In these conditions the Fourier transform of R allows us to obtain only a reduced spectral tensor along the radial direction so that Hm (k) will depend only on the wave-number k in this direction. Although the reduced spectral tensor does not carry the complete spectral information of the fluctuations, for slab and isotropic symmetries it contains all the information of the full tensor. The expression used by Matthaeus and Goldstein (1982bJump To The Next Citation Point) to compute the reduced Hm is given in Appendix B.2. In the following, we will drop the suffix r for sake of simplicity.

The general features of the reduced magnetic helicity spectrum in the solar wind were described for the first time by Matthaeus and Goldstein (1982bJump To The Next Citation Point) in the outer heliosphere, and by Bruno and Dobrowolny (1986Jump To The Next Citation Point) in the inner heliosphere. A useful dimensionless way to represent both the degree of and the sense of polarization is the normalized magnetic helicity σm (see Appendix B.2). This quantity can randomly vary between +1 and − 1, as shown in Figure 34View Image from the work by Matthaeus and Goldstein (1982bJump To The Next Citation Point) and relative to Voyager’s data taken at 1 AU. However, net values of ±1 are reached only for pure circularly polarized waves.

View Image

Figure 34: σm vs. frequency and wave number relative to an interplanetary data sample recorded by Voyager 1 at approximately 1 AU. Image reproduced by permission from Matthaeus and Goldstein (1982bJump To The Next Citation Point), copyright by AGU.

Based on these results, Goldstein et al. (1991) were able to reproduce the distribution of the percentage of occurrence of values of σm (f) adopting a model where the magnitude of the magnetic field was allowed to vary in a random way and the tip of the vector moved near a sphere. By this way they showed that the interplanetary magnetic field helicity measurements were inconsistent with the previous idea that fluctuations were randomly circularly polarized at all scales and were also magnitude preserving.

However, evidence for circular polarized MHD waves in the high frequency range was provided by Polygiannakis et al. (1994), who studied interplanetary magnetic field fluctuations from various datasets at various distances ranging from 1 to 20 AU. They also concluded that the difference between left- and right-hand polarizations is significant and continuously varying.

As already noticed by Smith et al. (1983Jump To The Next Citation Point, 1984), knowing the sign of σm and the sign of the normalized cross-helicity σ c it is possible to infer the sense of polarization of the fluctuations. As a matter of fact, a positive cross-helicity indicates an Alfvén mode propagating outward, while a negative cross-helicity indicates a mode propagating inward. On the other hand, we know that a positive magnetic-helicity indicates a right-hand polarized mode, while a negative magnetic-helicity indicates a left-hand polarized mode. Thus, since the sense of polarization depends on the propagating direction with respect to the observer, σ (f )σ (f) < 0 m c will indicate right circular polarization while σ (f)σ (f) > 0 m c will indicate left circular polarization. Thus, each time magnetic helicity and cross-helicity are available from measurements in a super-Alfvénic flow, it is possible to infer the rest frame polarization of the fluctuations from a single point measurements, assuming the validity of the slab geometry.

The high variability of σm, observable in Voyager’s data (see Figure 34View Image), was equally observed in Helios 2 data in the inner heliosphere (Bruno and Dobrowolny, 1986). The authors of this last work computed the difference (M H > 0) − |M H < 0| of magnetic helicity for different frequency bands and noticed that most of the resulting magnetic helicity was contained in the lowest frequency band. This result supported the theoretical prediction of an inverse cascade of magnetic helicity from the smallest to the largest scales during turbulence development (Pouquet et al., 1976Jump To The Next Citation Point).

Numerical simulations of the incompressible MHD equations by Mininni et al. (2003aJump To The Next Citation Point), discussed in Section 3.1.9, clearly confirm the tendency of magnetic helicity to follow an inverse cascade. The generation of magnetic field in turbulent plasmas and the successive inverse cascade has strong implications in the emergence of large scale magnetic fields in stars, interplanetary medium and planets (Brandenburg, 2001Jump To The Next Citation Point).

This phenomenon was firstly demonstrated in numerical simulations based on the eddy damped quasi normal Markovian (EDQNM) closure model of three-dimensional MHD turbulence by Pouquet et al. (1976). Successively, other investigators confirmed such a tendency for the magnetic helicity to develop an inverse cascade (Meneguzzi et al., 1981; Cattaneo and Hughes, 1996; Brandenburg, 2001).

Mininni et al. (2003aJump To The Next Citation Point) performed the first direct numerical simulations of turbulent Hall dynamo. They showed that the Hall current can have strong effects on turbulent dynamo action, enhancing or even suppressing the generation of the large-scale magnetic energy. These authors injected a weak magnetic field at small scales in a system kept in a stationary regime of hydrodynamic turbulence and followed the exponential growth of magnetic energy due to the dynamo action. This evolution can be seen in Figure 35Watch/download Movie in the same format described for Figure 40Watch/download Movie, shown in Section 3.1.9. Now, the forcing is applied at wave number kforce = 10 in order to give enough room for the inverse cascade to develop. The fluid is initially in a strongly turbulent regime as a result of the action of the external force at wave number kforce = 10. An initial magnetic fluctuation is introduced at t = 0 at kseed = 35. The magnetic energy starts growing exponentially fast and, when the saturation is reached, the magnetic energy is larger than the kinetic energy. Notably, it is much larger at the largest scales of the system (i.e., k = 1). At these large scales, the system is very close to a magnetostatic equilibrium characterized by a force-free configuration.


Figure 35: avi-Movie (1752 KB) A numerical simulation of the incompressible MHD equations in three dimensions, assuming periodic boundary conditions (see details in Mininni et al., 2003aJump To The Next Citation Point). The left panel shows the power spectra for kinetic energy (green), magnetic energy (red), and total energy (blue) vs. time. The right panel shows the spatially integrated kinetic, magnetic, and total energies vs. time. The vertical (orange) line indicates the current time. These results correspond to a 1283 simulation with an external force applied at wave number kforce = 10 (movie kindly provided by D. Gómez).

3.1.8 Alfvénic correlations as incompressive turbulence

In a famous paper, Belcher and Davis Jr (1971Jump To The Next Citation Point) showed that a strong correlation exists between velocity and magnetic field fluctuations, in the form

δB δv ≃ ± √-----, (58 ) 4π ρ
where the sign of the correlation is given by the sign [− k ⋅ B0 ], being k the wave vector and B0 the background magnetic field vector. These authors showed that in about 25 d of data from Mariner 5, out of the 160 d of the whole mission, fluctuations were described by Equation (58View Equation), and the sign of the correlation was such to indicate always an outward sense of propagation with respect to the Sun. Authors also noted that these periods mainly occur within the trailing edges of high-speed streams. Moreover, in the regions where Equation (58View Equation) is verified to a high degree, the magnetic field magnitude is almost constant (B2 ∼ const.).
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Figure 36: Alfvénic correlation in fast solar wind. Left panel: large scale Alfvénic fluctuations found by Bruno et al. (1985Jump To The Next Citation Point). Right panel: small scale Alfvénic fluctuations for the first time found by Belcher and Solodyna (1975Jump To The Next Citation Point). Image reproduced by permission, copyright by AGU.

Today we know that Alfvénic correlations are ubiquitous in the solar wind and that these correlations are much stronger and are found at lower and lower frequencies, as we look at shorter and shorter heliocentric distances. In the right panel of Figure 36View Image we show results from Belcher and Solodyna (1975Jump To The Next Citation Point) obtained on the basis of 5 min averages of velocity and magnetic field recorded by Mariner 5 in 1967, during its mission to Venus. On the left panel of Figure 36View Image we show results from a similar analysis performed by Bruno et al. (1985Jump To The Next Citation Point) obtained on the basis of 1 h averages of velocity and magnetic field recorded by Helios 2 in 1976, when the s/c was at 0.29 AU from the Sun. These last authors found that, in their case, Alfvénic correlations extended to time periods as low as 15 h in the s/c frame at 0.29 AU, and to periods a factor of two smaller near the Earth’s orbit. Now, if we think that this long period of the fluctuations at 0.29 AU was larger than the transit time from the Sun to the s/c, this results might be the first evidence for a possible solar origin for these fluctuations, probably caused by the shuffling of the foot-points of the solar surface magnetic field.

Alfvén modes are not the only low frequency plasma fluctuations allowed by the MHD equations but they certainly are the most frequent fluctuations observed in the solar wind. The reason why other possible propagating modes like the slow sonic mode and the fast magnetosonic mode cannot easily be found, besides the fact that the eigenvectors associated with these modes are not directly identifiable because they necessitate prior identification of wavevectors, contrary to the simple Alfvén eigenvectors, depends also on the fact that these compressive modes are strongly damped in the solar wind shortly after they are generated (see Section 6). On the contrary, Alfvénic fluctuations, which are difficult to be damped because of their incompressive nature, survive much longer and dominate solar wind turbulence. Nevertheless, there are regions where Alfvénic correlations are much stronger like the trailing edge of fast streams, and regions where these correlations are weak like intervals of slow wind (Belcher and Davis Jr, 1971Jump To The Next Citation Point; Belcher and Solodyna, 1975Jump To The Next Citation Point). However, the degree of Alfvénic correlations unavoidably fades away with increasing heliocentric distance, although it must be reported that there are cases when the absence of strong velocity shears and compressive phenomena favor a high Alfvénic correlation up to very large distances from the Sun (Roberts et al., 1987aJump To The Next Citation Point; see Section 5.1).

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Figure 37: Alfvénic correlation in fast and slow wind. Notice the different degree of correlation between these two types of wind.

Just to give a qualitative quick example about Alfvénic correlations in fast and slow wind, we show in Figure 37View Image the speed profile for about 100 d of 1976 as observed by Helios 2, and the traces of velocity and magnetic field Z components (see Appendix D for the orientation of the reference system) V Z and B Z (this last one expressed in Alfvén units, see Appendix B.1) for two different time intervals, which have been enlarged in the two inserted small panels. The high velocity interval shows a remarkable anti-correlation which, since the mean magnetic field B0 is oriented away from the Sun, suggests a clear presence of outward oriented Alfvénic fluctuations given that the sign of the correlation is the sign [− k ⋅ B0 ]. At odds with the previous interval, the slow wind shows that the two traces are rather uncorrelated. For sake of brevity, we omit to show the very similar behavior for the other two components, within both fast and slow wind.

The discovery of Alfvénic correlations in the solar wind stimulated fundamental remarks by Kraichnan (1974Jump To The Next Citation Point) who, following previous theoretical works by Kraichnan (1965Jump To The Next Citation Point) and Iroshnikov (1963), showed that the presence of a strong correlation between velocity and magnetic fluctuations renders non-linear transfer to small scales less efficient than for the Navier–Stokes equations, leading to a turbulent behavior which is different from that described by Kolmogorov (1941Jump To The Next Citation Point). In particular, when Equation (58View Equation) is exactly satisfied, non-linear interactions in MHD turbulent flows cannot exist. This fact introduces a problem in understanding the evolution of MHD turbulence as observed in the interplanetary space. Both a strong correlation between velocity and magnetic fluctuations and a well defined turbulence spectrum (Figures 29View Image, 37View Image) are observed, and the existence of the correlations is in contrast with the existence of a spectrum which in turbulence is due to a non-linear energy cascade. Dobrowolny et al. (1980bJump To The Next Citation Point) started to solve the puzzle on the existence of Alfvénic turbulence, say the presence of predominately outward propagation and the fact that MHD turbulence with the presence of both Alfvénic modes present will evolve towards a state where one of the mode disappears. However, a lengthy debate based on whether the highly Alfvénic nature of fluctuations is what remains of the turbulence produced at the base of the corona or the solar wind itself is an evolving turbulent magnetofluid, has been stimulating the scientific community for quite a long time.

3.1.9 Radial evolution of Alfvénic turbulence

The degree of correlation not only depends on the type of wind we look at, i.e., fast or slow, but also on the radial distance from the Sun and on the time scale of the fluctuations.

Figure 38View Image shows the radial evolution of σc (see Appendix B.1) as observed by Helios and Voyager s/c (Roberts et al., 1987bJump To The Next Citation Point). It is clear enough that σc not only tends to values around 0 as the heliocentric distance increases, but larger and larger time scales are less and less Alfvénic. Values of σc ∼ 0 suggest a comparable amount of “outward” and “inward” correlations.

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Figure 38: Histograms of normalized cross-helicity σc showing its evolution between 0.3 (circles), 2 (triangles), and 20 (squares) AU for different time scales: 3 h (top panel), 9 h (middle panel), and 81 h (bottom panel). Image reproduced by permission from Roberts et al. (1987bJump To The Next Citation Point), copyright by AGU.

The radial evolution affects also the Alfvén ratio rA (see Appendix B.3.1) as it was found by Bruno et al. (1985Jump To The Next Citation Point). However, early analyses (Belcher and Davis Jr, 1971Jump To The Next Citation Point; Solodyna and Belcher, 1976; Matthaeus and Goldstein, 1982bJump To The Next Citation Point) had already shown that this parameter is usually less than unit. Spectral studies by Marsch and Tu (1990aJump To The Next Citation Point), reported in Figure 39View Image, showed that within slow wind it is the lowest frequency range the one that experiences the strongest decrease with distance, while the highest frequency range remains almost unaffected. Moreover, the same study showed that, within fast wind, the whole frequency range experiences a general depletion. The evolution is such that close to 1 AU the value of rA in fast wind approaches that in slow wind.

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Figure 39: Values of the Alfvén ratio rA as a function of frequency and heliocentric distance, within slow (left column) and fast (right column) wind. Image reproduced by permission from Marsch and Tu (1990aJump To The Next Citation Point), copyright by AGU.

Moreover, comparing these results with those by Matthaeus and Goldstein (1982bJump To The Next Citation Point) obtained from Voyager at 2.8 AU, it seems that the evolution recorded within fast wind tends to a sort of limit value around 0.4 – 0.5.

Also Roberts et al. (1990Jump To The Next Citation Point), analyzing fluctuations between 9 h and 3 d found a similar radial trend. These authors showed that rA dramatically decreases from values around unit at the Earth’s orbit towards 0.4 – 0.5 at approximately 8 AU. For larger heliocentric distances, rA seems to stabilize around this last value.

The reason why rA tends to a value less than unit is still an open question although MHD computer simulations (Matthaeus, 1986) showed that magnetic reconnection and high plasma viscosity can produce values of rA < 1 within the inertial range. Moreover, the magnetic energy excess can be explained as a competing action between the equipartition trend due to linear propagation (or Alfvén effect, Kraichnan (1965Jump To The Next Citation Point)), and a local dynamo effect due to non-linear terms (Grappin et al., 1991Jump To The Next Citation Point), see closure calculations by Grappin et al. (1983); DNS by Müller and Grappin (2005).

However, this argument forecasts an Alfvén ratio rA ⁄= 1 but, it does not say whether it would be larger or smaller than “1”, i.e., we could also have a final excess of kinetic energy.

Similar unbalance between magnetic and kinetic energy has recently been found in numerical simulations by Mininni et al. (2003aJump To The Next Citation Point), already cited in Section 3.1.7. These authors studied the effect of a weak magnetic field at small scales in a system kept in a stationary regime of hydrodynamic turbulence. In these conditions, the dynamo action causes the initial magnetic energy to grow exponentially towards a state of quasi equipartition between kinetic and magnetic energy. This simulation was aiming to provide more insights on a microscopic theory of the alpha-effect, which is responsible to convert part of the toroidal magnetic field on the Sun back to poloidal to sustain the cycle. However, when the simulation saturates, the unbalance between kinetic and magnetic energy reminds the conditions in which the Alfvén ratio is found in interplanetary space. Results from the above study can be viewed in the animation of Figure 40Watch/download Movie. At very early time the fluid is in a strongly turbulent regime as a result of the action of the external force at wave number kforce = 3. An initial magnetic fluctuation is introduced at t = 0 at kseed = 35. The magnetic energy starts growing exponentially fast and, when the simulation reaches the saturation stage, the magnetic power spectrum exceeds the kinetic power spectrum at large wave numbers (i.e., k > kforce), as also observed in Alfvénic fluctuations of the solar wind (Bruno et al., 1985Jump To The Next Citation Point; Tu and Marsch, 1990aJump To The Next Citation Point) as an asymptotic state (Roberts et al., 1987aJump To The Next Citation Point,bJump To The Next Citation Point; Bavassano et al., 2000bJump To The Next Citation Point) of turbulence.


Figure 40: avi-Movie (1781 KB) A 1283 numerical simulation, as in Figure 35Watch/download Movie, but with an external force applied at wave number kforce = 3 (movie kindly provided by D. Gómez).

However, when the two-fluid effect, such as the Hall current and the electron pressure (Mininni et al., 2003b), is included in the simulation, the dynamo can work more efficiently and the final stage of the simulation is towards equipartition between kinetic and magnetic energy.

On the other hand, Marsch and Tu (1993aJump To The Next Citation Point) analyzed several intervals of interplanetary observations to look for a linear relationship between the mean electromotive force 𝜀 = ⟨δV δB ⟩, generated by the turbulent motions, and the mean magnetic field B0, as predicted by simple dynamo theory (Krause and Rädler, 1980). Although sizable electromotive force was found in interplanetary fluctuations, these authors could not establish any simple linear relationship between B0 and 𝜀.

Lately, Bavassano and Bruno (2000Jump To The Next Citation Point) performed a three-fluid analysis of solar wind Alfvénic fluctuations in the inner heliosphere, in order to evaluate the effect of disregarding the multi-fluid nature of the wind on the factor relating velocity and magnetic field fluctuations. It is well known that converting magnetic field fluctuations into Alfvén units we divide by the factor Fp = (4πMpNp )1∕2. However, fluctuations in velocity tend to be smaller than fluctuations in Alfvén units. In Figure 41View Image we show scatter plots between the z-component of the Alfvén velocity and the proton velocity fluctuations. The z-direction has been chosen as the same of Vp × B, where Vp is the proton bulk flow velocity and B is the mean field direction. The reason for such a choice is due to the fact that this direction is the least affected by compressive phenomena deriving from the wind dynamics. These results show that although the correlation coefficient in both cases is around − 0.95, the slope of the best fit straight line passes from 1 at 0.29 AU to a slope considerably different from 1 at 0.88 AU.

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Figure 41: Scatter plot between the z-component of the Alfvén velocity and the proton velocity fluctuations at about 2 mHz. Data refer to Helios 2 observations at 0.29 AU (left panel) and 0.88 AU (right panel). Image adapted from Bavassano and Bruno (2000Jump To The Next Citation Point).

Belcher and Davis Jr (1971Jump To The Next Citation Point) suggested that this phenomenon had to be ascribed to the presence of α particles and to an anisotropy in the thermal pressure. Moreover, taking into account the multi-fluid nature of the solar wind, the dividing factor should become F = FpFiFa, where Fi would take into account the presence of other species besides protons, and F a would take into account the presence of pressure anisotropy P∥ ⁄= P ⊥, where ∥ and ⊥ refer to the background field direction. In particular, following Bavassano and Bruno (2000Jump To The Next Citation Point), the complete expressions for Fi and Fa are

[ ]1∕2 ∑ Fi = 1 + (MsNs )∕(MpNp ) s

and

[ ∑ ] −1∕2 Fa = 1 − 4-π (P∥s − P⊥s + MsNsU2s ) , B20 s

where the letter “s” stands for the s-th species, being Us = Vs − V its velocity in the center of mass frame of reference. Vs is the velocity of the species “s” in the s/c frame and ∑ ∑ V = ( sMsNsVs )∕( s MsNs ) is the velocity of the center of mass.

Bavassano and Bruno (2000) analyzed several time intervals within the same co-rotating high velocity stream observed at 0.3 and 0.9 AU and performed the analysis using the new factor “F” to express magnetic field fluctuations in Alfvén units, taking into account the presence of α particles and electrons, besides the protons. However, the correction resulted to be insufficient to bring back to “1” the slope of the δVPz − δVAz relationship shown in the right panel of Figure 41View Image. In conclusion, the radial variation of the Alfvén ratio rA towards values less than 1 is not completely due to a missed inclusion of multi-fluid effects in the conversion from magnetic field to Alfvén units. Thus, we are left with the possibility that the observed depletion of rA is due to a natural evolution of turbulence towards a state in which magnetic energy becomes dominant (Grappin et al., 1991Jump To The Next Citation Point; Roberts et al., 1992Jump To The Next Citation Point; Roberts, 1992Jump To The Next Citation Point), as observed in the animation of Figure 40Watch/download Movie taken from numerical simulations by Mininni et al. (2003a) or, it is due to the increased presence of magnetic structures like MFDT (Tu and Marsch, 1993Jump To The Next Citation Point).

3.2 Turbulence studied via Elsässer variables

The Alfvénic character of solar wind fluctuations,especially within co-rotating high velocity streams, suggests to use the Elsässer variables (Appendix B.3) to separate the “outward” from the “inward” contribution to turbulence. These variables, used in theoretical studies by Dobrowolny et al. (1980aJump To The Next Citation Point,bJump To The Next Citation Point); Veltri et al. (1982); Marsch and Mangeney (1987); and Zhou and Matthaeus (1989Jump To The Next Citation Point), were for the first time used in interplanetary data analysis by Grappin et al. (1990Jump To The Next Citation Point) and Tu et al. (1989bJump To The Next Citation Point). In the following, we will describe and discuss several differences between “outward” and “inward” modes, but the most important one is about their origin. As a matter of fact, the existence of the Alfvénic critical point implies that only “outward” propagating waves of solar origin will be able to escape from the Sun. “Inward” waves, being faster than the wind bulk speed, will precipitate back to the Sun if they are generated before this point. The most important implication due to this scenario is that “inward” modes observed beyond the Alfvénic point cannot have a solar origin but they must have been created locally by some physical process. Obviously, for the other Alfvénic component, both solar and local origins are still possible.

3.2.1 Ecliptic scenario

Early studies by Belcher and Davis Jr (1971), performed on magnetic field and velocity fluctuations recorded by Mariner 5 during its trip to Venus in 1967, already suggested that the majority of the Alfvénic fluctuations are characterized by an “outward” sense of propagation, and that the best regions where to observe these fluctuations are the trailing edge of high velocity streams. Moreover, Helios spacecraft, repeatedly orbiting around the Sun between 0.3 to 1 AU, gave the first and unique opportunity to study the radial evolution of turbulence (Bavassano et al., 1982bJump To The Next Citation Point; Denskat and Neubauer, 1983). Successively, when Elsässer variables were introduced in the analysis (Grappin et al., 1989Jump To The Next Citation Point), it was finally possible not only to evaluate the “inward” and “outward” Alfvénic contribution to turbulence but also to study the behavior of these modes as a function of the wind speed and radial distance from the Sun.

Figure 42View Image (Tu et al., 1990Jump To The Next Citation Point) clearly shows the behavior of e± (see Appendix B.3) across a high speed stream observed at 0.3 AU. Within fast wind e+ is much higher than e− and its spectral slope shows a break. Lower frequencies have a flatter slope while the slope of higher frequencies is closer to a Kolmogorov-like. − e has a similar break but the slope of lower frequencies follows the Kolmogorov slope, while higher frequencies form a sort of plateau.

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Figure 42: Power density spectra ± e computed from ± δz fluctuations for different time intervals indicated by the arrows. Image reproduced by permission from Tu et al. (1990), copyright by AGU.

This configuration vanishes when we pass to the slow wind where both spectra have almost equivalent power density and follow the Kolmogorov slope. This behavior, for the first time reported by Grappin et al. (1990Jump To The Next Citation Point), is commonly found within co-rotating high velocity streams, although much more clearly expressed at shorter heliocentric distances, as shown below.

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Figure 43: Power density spectra − e and + e computed from − δz and + δz fluctuations. Spectra have been computed within fast (H) and slow (L) streams around 0.4 and 0.9 AU as indicated by different line styles. The thick line represents the average power spectrum obtained from all the about 50 e spectra, regardless of distances and wind speed. The shaded area is the 1σ width related to the average. Image reproduced by permission from Tu and Marsch (1990bJump To The Next Citation Point), copyright by AGU.

Spectral power associated with outward (right panel) and inward (left panel) Alfvénic fluctuations, based on Helios 2 observations in the inner heliosphere, are concisely reported in Figure 43View Image. The e spectrum, if we exclude the high frequency range of the spectrum relative to fast wind at 0.4 AU, shows an average power law profile with a slope of − 1.64, consistent with Kolmogorov’s scaling. The lack of radial evolution of e spectrum brought Tu and Marsch (1990aJump To The Next Citation Point) to name it “the background spectrum” of solar wind turbulence.

Quite different is the behavior of e+ spectrum. Close to the Sun and within fast wind, this spectrum appears to be flatter at low frequency and steeper at high frequency. The overall evolution is towards the “background spectrum” by the time the wind reaches 0.8 AU.

In particular, Figure 43View Image tells us that the radial evolution of the normalized cross-helicity has to be ascribed mainly to the radial evolution of e+ rather than to both Alfvénic fluctuations (Tu and Marsch, 1990aJump To The Next Citation Point). In addition, Figure 44View Image, relative to the Elsässer ratio rE, shows that the hourly frequency range, up to ∼ 2 × 10−3 Hz, is the most affected by this radial evolution.

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Figure 44: Ratio of e over e+ within fast wind at 0.3 and 0.9 AU in the left and right panels, respectively. Image reproduced by permission from Marsch and Tu (1990aJump To The Next Citation Point), copyright by AGU.

As a matter of fact, this radial evolution can be inferred from Figure 45View Image where values of e and e+ together with solar wind speed, magnetic field intensity, and magnetic field and particle density compression are shown between 0.3 and 1 AU during the primary mission of Helios 2. It clearly appears that enhancements of e and depletion of e+ are connected to compressive events, particularly within slow wind. Within fast wind the average level of e is rather constant during the radial excursion while the level of e+ dramatically decreases with a consequent increase of the Elsässer ratio (see Appendix B.3.1).

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Figure 45: Upper panel: solar wind speed and solar wind speed multiplied by σc. In the lower panels the authors reported: σc, rE, e, e+, magnetic compression, and number density compression, respectively. Image reproduced by permission from Bruno and Bavassano (1991Jump To The Next Citation Point), copyright by AGU.

Further ecliptic observations (see Figure 46View Image) do not indicate any clear radial trend for the Elsässer ratio between 1 and 5 AU, and its value seems to fluctuate between 0.2 and 0.4.

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Figure 46: Ratio of e over e+ within fast wind between 1 and 5 AU as observed by Ulysses in the ecliptic. Image reproduced by permission from Bavassano et al. (2001Jump To The Next Citation Point), copyright by AGU.

However, low values of the normalized cross-helicity can also be associated with a particular type of incompressive events, which Tu and Marsch (1991Jump To The Next Citation Point) called Magnetic Field Directional Turnings or MFDT. These events, found within slow wind, were characterized by very low values of σc close to zero and low values of the Alfvén ratio, around 0.2. Moreover, the spectral slope of e+, e and the power associated with the magnetic field fluctuations was close to the Kolmogorov slope. These intervals were only scarcely compressive, and short period fluctuations, from a few minutes to about 40 min, were nearly pressure balanced. Thus, differently from what had previously been observed by Bruno et al. (1989), who found low values of cross-helicity often accompanied by compressive events, these MFDTs were mainly incompressive. In these structures most of the fluctuating energy resides in the magnetic field rather than velocity as shown in Figure 47View Image taken from Tu and Marsch (1991Jump To The Next Citation Point). It follows that the amplitudes of the fluctuating Alfvénic fields ± δz result to be comparable and, consequently, the derived parameter σc → 0. Moreover, the presence of these structures would also be able to explain the fact that rA < 1. Tu and Marsch (1991Jump To The Next Citation Point) suggested that these fluctuations might derive from a special kind of magnetic structures, which obey the MHD equations, for which (B ⋅ ∇ )B = 0, field magnitude, proton density, and temperature are all constant. The same authors suggested the possibility of an interplanetary turbulence mainly made of outwardly propagating Alfvén waves and convected structures represented by MFDTs. In other words, this model assumed that the spectrum of e would be caused by MFDTs. The different radial evolution of the power associated with these two kind of components would determine the radial evolution observed in both σc and rA. Although the results were not quantitatively satisfactory, they did show a qualitative agreement with the observations.

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Figure 47: Left column: e+ and e spectra (top) and σc (bottom) during a slow wind interval at 0.9 AU. Right column: kinetic e v and magnetic e B energy spectra (top) computed from the trace of the relative spectral tensor, and spectrum of the Alfvén ratio rA (bottom) Image reproduced by permission from Tu and Marsch (1991Jump To The Next Citation Point).

These convected structures are an important ingredient of the turbulent evolution of the fluctuations and can be identified as the 2D incompressible turbulence suggested by Matthaeus et al. (1990Jump To The Next Citation Point) and Tu and Marsch (1991Jump To The Next Citation Point).

As a matter of fact, a statistical analysis by Bruno et al. (2007Jump To The Next Citation Point) showed that magnetically dominated structures represent an important component of the interplanetary fluctuations within the MHD range of scales. As a matter of fact, these magnetic structures and Alfvénic fluctuations dominate at scales typical of MHD turbulence. For instance, this analysis suggested that more than 20% of all analyzed intervals of 1 hr scale are magnetically dominated and only weakly Alfvénic. Observations in the ecliptic performed by Helios and WIND s/c and out of the ecliptic, performed by Ulysses, showed that these advected, mostly incompressive structures are ubiquitous in the heliosphere and can be found in both fast and slow wind.

It proves interesting enough to look at the radial evolution of interplanetary fluctuations in terms of normalized cross-helicity σc and normalized residual energy σr (see Appendix B.3).

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Figure 48: Left, from top to bottom: frequency histograms of σr vs. σc (here σC and σR) for fast wind observed by Helios 2 at 0.29, 0.65 and 0.88 AU, respectively. The color code, for each panel, is normalized to the maximum of the distribution. The yellow circle represents the limiting value given by σ2 + σ2 = 1 c r while, the yellow dashed line represents the relation σr = σc − 1, see text for details. Right, from top to bottom: frequency histograms of σr vs. σc (here σC and σR) for slow wind observed by Helios 2 at 0.32, 0.69 and 0.90 AU, respectively. The color code, for each panel, is normalized to the maximum of the distribution. Image reproduced by permission from Bruno et al. (2007), copyright EGU.

These results, shown in the left panels of Figure 48View Image, highlight the presence of a radial evolution of the fluctuations towards a double-peaked distribution during the expansion of the solar wind. The relative analysis has been performed on a co-rotating fast stream observed by Helios 2 at three different heliocentric distances over consecutive solar rotations (see Figure 16View Image and related text). Closer to the Sun, at 0.3 AU, the distribution is well centered around σr = 0 and σc = 1, suggesting that Alfvénic fluctuations, outwardly propagating, dominate the scenario. By the time the wind reaches 0.7 AU, the appearance of a tail towards negative values of σr and lower values of σc indicates a partial loss of the Alfvénic character in favor of fluctuations characterized by a stronger magnetic energy content. This clear tendency ends up with the appearance of a secondary peak by the time the wind reaches 0.88 AU. This new family of fluctuations forms around σr = − 1 and σc = 0. The values of σr and σc which characterize this new population are typical of MFDT structures described by Tu and Marsch (1991Jump To The Next Citation Point). Together with the appearance of these fluctuations, the main peak characterized by Alfvén like fluctuations looses much of its original character shown at 0.3 AU. The yellow straight line that can be seen in the left panels of Figure 48View Image would be the linear relation between σr and σc in case fluctuations were made solely by Alfvén waves outwardly propagating and advected MFDTs (Tu and Marsch, 1991Jump To The Next Citation Point) and it would replace the canonical, quadratic relation 2 2 σ r + σ c ≤ 1 represented by the yellow circle drawn in each panel. However, the yellow dashed line shown in the left panels of Figure 48View Image does not seem to fit satisfactorily the observed distributions.

Quite different is the situation within slow wind, as shown in the right panels of Figure 48View Image. As a matter of fact, these histograms do not show any striking radial evolution like in the case of fast wind. High values of σc are statistically much less relevant than in fast wind and a well defined population characterized by σr = − 1 and σc = 0, already present at 0.3 AU, becomes one of the dominant peaks of the histogram as the wind expands. This last feature is really at odds with what happens in fast wind and highlights the different nature of the fluctuations which, in this case, are magnetically dominated. The same authors obtained very similar results for fast and slow wind also from the same type of analysis performed on WIND and Ulysses data which, in addition, confirmed the incompressive character of the Alfvénic fluctuations and highlighted a low compressive character also for the populations characterized by σ ∼ − 1 r and σc ∼ 0.

About the origin of these structures, these authors suggest that they might be not only created locally during the non linear evolution of the fluctuations but they might also have a solar origin. The reason why they are not seen close to the Sun, within fast wind, might be due to the fact that these fluctuations, mainly non-compressive, change the direction of the magnetic field similarly to Alfvénic fluctuations but produce a much smaller effect since the associated δb is smaller than the one corresponding to Alfvénic fluctuations. As the wind expands, the Alfvénic component undergoes non-linear interactions which produce a transfer of energy to smaller and smaller scales while, these structures, being advected, have a much longer lifetime. As the expansion goes on, the relative weight of these fluctuations grows and they start to be detected.

3.2.2 On the nature of Alfvénic fluctuations

The Alfvénic nature of outward modes has been widely recognized through several frequency decades up to periods of the order of several hours in the s/c rest frame (Bruno et al., 1985Jump To The Next Citation Point). Conversely, the nature of those fluctuations identified by δz−, called “inward Alfvén modes”, is still not completely clear. There are many clues which would suggest that these fluctuations, especially in the hourly frequencies range, have a non-Alfvénic nature. Several studies on this topic in the low frequency range have suggested that structures convected by the wind could well mimic non-existent inward propagating modes (see the review by Tu and Marsch, 1995aJump To The Next Citation Point). However, other studies (Tu et al., 1989bJump To The Next Citation Point) have also found, in the high frequency range and within fast streams, a certain anisotropy in the components which resembles the same anisotropy found for outward modes. So, these observations would suggest a close link between inward modes at high frequency and outward modes, possibly the same nature.

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Figure 49: Power density spectra for e+ and e during a high velocity stream observed at 0.3 AU. Best fit lines for different frequency intervals and related spectral indices are also shown. Vertical lines fix the limits of five different frequency intervals analyzed by Bruno et al. (1996Jump To The Next Citation Point). Image reproduced by permission, copyright by AIP.

Figure 49View Image shows power density spectra for e+ and e during a high velocity stream observed at 0.3 AU (similar spectra can be also found in the paper by Grappin et al., 1990Jump To The Next Citation Point and Tu et al., 1989bJump To The Next Citation Point). The observed spectral indices, reported on the plot, are typically found within high velocity streams encountered at short heliocentric distances. Bruno et al. (1996Jump To The Next Citation Point) analyzed the power relative to e+ and e modes, within five frequency bands, ranging from roughly 12 h to 3 min, delimited by the vertical solid lines equally spaced in log-scale. The integrated power associated with e+ and e within the selected frequency bands is shown in Figure 50View Image. Passing from slow to fast wind e+ grows much more within the highest frequency bands. Moreover, there is a good correlation between the profiles of e and e+ within the first two highest frequency bands, as already noticed by Grappin et al. (1990Jump To The Next Citation Point) who looked at the correlation between daily averages of e and e+ in several frequency bands, even widely separated in frequency. The above results stimulated these authors to conclude that it was reminiscent of the non-local coupling in k-space between opposite modes found by Grappin et al. (1982) in homogeneous MHD. Expansion effects were also taken into account by Velli et al. (1990Jump To The Next Citation Point) who modeled inward modes as that fraction of outward modes back-scattered by the inhomogeneities of the medium due to expansion effects (Velli et al., 1989Jump To The Next Citation Point). However, following this model we would often expect the two populations to be somehow related to each other but, in situ observations do not favor this kind of forecast (Bavassano and Bruno, 1992Jump To The Next Citation Point).

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Figure 50: Left panel: wind speed profile is shown in the top panel. Power density associated with e+ (thick line) and e (thin line), within the five frequency bands chosen, is shown in the lower panels. Right panel: wind speed profile is shown in the top panel. Values of the angle 𝜃 ± between the minimum variance direction of + δz (thick line) and − δz (thin line) and the direction of the ambient magnetic field are shown in the lower panels, relatively to each frequency band. Image reproduced by permission from Bruno et al. (1996Jump To The Next Citation Point), copyright by AIP.

An alternative generation mechanism was proposed by Tu et al. (1989bJump To The Next Citation Point) based on the parametric decay of e+ in high frequency range (Galeev and Oraevskii, 1963). This mechanism is such that large amplitude Alfvén waves, unstable to perturbations of random field intensity and density fluctuations, would decay into two secondary Alfvén modes propagating in opposite directions and a sound-like wave propagating in the same direction of the pump wave. Most of the energy of the mother wave would go into the sound-like fluctuation and the backward propagating Alfvén mode. On the other hand, the production of e modes by parametric instability is not particularly fast if the plasma β ∼ 1, like in the case of solar wind (Goldstein, 1978Jump To The Next Citation Point; Derby, 1978), since this condition slows down the growth rate of the instability. It is also true that numerical simulations by Malara et al. (2000Jump To The Next Citation Point, 2001aJump To The Next Citation Point, 2002Jump To The Next Citation Point), and Primavera et al. (2003Jump To The Next Citation Point) have shown that parametric decay can still be thought as a possible mechanism of local production of turbulence within the polar wind (see Section 4). However, the strong correlation between e+ and e profiles found only within the highest frequency bands would support this mechanism and would suggest that e modes within these frequency bands would have an Alfvénic nature. Another feature shown in Figure 50View Image that favors these conclusions is the fact that both δz+ and δz− keep the direction of their minimum variance axis aligned with the background magnetic field only within the fast wind, and exclusively within the highest frequency bands. This would not contradict the view suggested by Barnes (1981). Following this model, the majority of Alfvénic fluctuations propagating in one direction have the tip of the magnetic field vector randomly wandering on the surface of half a sphere of constant radius, and centered along the ambient field B∘. In this situation the minimum variance would be oriented along B ∘, although this would not represent the propagation direction of each wave vector which could propagate even at large angles from this direction. This situation can be seen in the right hand panel of Figure 98View Image of Section 10, which refers to a typical Alfvénic interval within fast wind. Moreover, δz+ fluctuations show a persistent anisotropy throughout the fast stream since the minimum variance axis remains quite aligned to the background field direction. This situation downgrades only at the very low frequencies where + 𝜃, the angle between the minimum variance direction of + δz and the direction of the ambient magnetic field, starts wandering between 0∘ and 90∘. On the contrary, in slow wind, since Alfvénic modes have a smaller amplitude, compressive structures due to the dynamic interaction between slow and fast wind or, of solar origin, push the minimum variance direction to larger angles with respect to B ∘, not depending on the frequency range.

In a way, we can say that within the stream, both 𝜃+ and 𝜃−, the angle between the minimum variance direction of − δz and the direction of the ambient magnetic field, show a similar behavior as we look at lower and lower frequencies. The only difference is that − 𝜃 reaches higher values at higher frequencies than 𝜃+. This was interpreted (Bruno et al., 1996Jump To The Next Citation Point) as due to the fact that transverse fluctuations of δz− carry much less power than those of δz+ and, consequently, they are more easily influenced by perturbations represented by the background, convected structure of the wind (e.g., TD’s and PBS’s). As a consequence, at low frequency − δz fluctuations may represent a signature of the compressive component of the turbulence while, at high frequency, they might reflect the presence of inward propagating Alfvén modes. Thus, while for periods of several hours δz+ fluctuations can still be considered as the product of Alfvén modes propagating outward (Bruno et al., 1985Jump To The Next Citation Point), δz− fluctuations are rather due to the underlying convected structure of the wind. In other words, high frequency turbulence can be looked at mainly as a mixture of inward and outward Alfvénic fluctuations plus, presumably, sound-like perturbations (Marsch and Tu, 1993aJump To The Next Citation Point). On the other hand, low frequency turbulence would be made of outward Alfvénic fluctuations and static convected structures representing the inhomogeneities of the background medium.


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