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, 1995a

; 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 15 ) 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 $-1 10 km s$ have been recorded by SOHO/SUMER instrument (Hassler et al., 1999 ).

Figure 15:

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 16

). Moreover, plasma emissions due to violent and abrupt phenomena also contribute to the solar wind in these regions of the Sun.

Figure 16:

Helmet streamer during a solar eclipse. Slow wind leaks into the interplanetary space along the flanks of this coronal structure. (Figure taken from High Altitude Observatory, 1991 ).

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.

Figure 17:

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 corotating 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 (1995a

), have been of fundamental importance in understanding the radial evolution of MHD turbulence in the solar wind.

Figure 17

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 corotating 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 18

, the nice and regular alternation of fast corotating streams and slow wind is replaced by a much more irregular and spikey profile also characterized by a lower average speed.

Figure 18:

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

Figure 19:

High velocity streams and slow wind as seen in the ecliptic during solar minimum

Figure 19

focuses on a region centered on day 75, recognizable in Figure 17

, 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 $rp$ , proton temperature $Tp$ , magnetic field intensity $|B|$ , azimuthal angle $P$ , and elevation angle $Q$ 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, 1971

; Belcher and Solodyna, 1975

). 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 $P$ rotates by about $180o$ . Detailed studies (Bavassano et al., 1997

) 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.

Figure 20:

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 (adopted from Bavassano et al., 1997 , © 1997 American Geophysical Union, reproduced by permission of American Geophysical Union).

Figure 20

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 20

. 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 (1968 ) 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 (1968 ), 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 spacecrafts. By analyzing spectral densities, Coleman (1968 ) 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, was found to behave roughly as $- 1.2 f$ , 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 is shown in Figure 21 .

Figure 21:

The magnetic energy spectrum as obtained by Coleman (1968 ).

Spectral properties of the interplanetary medium have been summarized by Russell (1972

), who published a composite spectrum of the radial component of magnetic fluctuations as observed by Mariner 2, Mariner 4 and OGO 5 (see Figure 22

). The frequency spectrum so obtained was divided into three main ranges: up to about $10 -4 Hz$ the spectral slope was about $f- 1$ ; at intermediate frequencies $10- 4 < f < 10- 1 Hz$ a spectral slope of about $f-3/2$ was found; finally, the high-frequency part of the spectrum, up to $1 Hz$ , was characterized by a $-2 f$ dependence. The intermediate range⁷ of frequencies recalls spectral properties similar to those introduced by Kraichnan (1965

) in the framework of MHD turbulence. It is worth reporting that scatter plots in 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 ). Then, as far as the solar wind turbulence concerns we do not think we should long discuss here whether or not solar wind developed turbulence be represented by $- 5/3 f$ or $- 3/2 f$ , since observations showed that the slope is usually around $f- 1.6$ (Bavassano et al., 1982b

; Tu and Marsch, 1995a

) which, irony of fate, is just between the two cited values. Although we prefer to postpone to a future version of the present paper a detailed discussion on this topic and the related inertial range of solar wind fluctuations, it is worth citing that Tu et al. (1989c

) already discussed this problem on the basis of Tu’s model (Tu, 1988

), using a variable ratio of the inward to outward Alfvénic energy as determined by observations of normalized cross-helicity. These values were then used to find the cascade constant that determines the level of the energy spectrum. The value they found for this constant resulted to be very close to the value observed in ordinary fluid turbulence, assuming that the correspondence between fluid and magnetofluid theories is reached by imposing zero cross-helicity for the MHD turbulence.

As a final comment, the situation of spectral indices determination in MHD turbulence is not changed since the ’70s (cf. Carbone and Pouquet, 2005 ), numerical simulations deal with MHD flows of moderate Reynolds numbers and an inertial range is scarcely observed. The debate, after thirty years, is always open and contributions are welcome.

Figure 22:

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

3.1.2 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., 1982b ; Denskat and Neubauer, 1983 ). In Figure 23 we re-propose similar observations taken by Helios 2 during its primary mission to the Sun.

Figure 23:

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 17

, during the first mission to the Sun in 1976. The spectral break (blue dot) shown by each spectrum, moves to lower and lower frequency as the heliocentric distance increases.

These power density spectra were obtained from the trace of the spectral matrix of magnetic field fluctuations, and belong to the same corotating stream observed by Helios 2 on day 49, at a heliocentric distance of $0.9 AU$ , on day 75 at $0.7 AU$ and, finally, on day 104 at $0.3 AU$ . All the spectra are 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., $P (f) --> P(f,r)$ .

Matthaeus and Goldstein (1986 ) found the breakpoint around $10 h$ at $1 AU$ , and Klein et al. (1992 ) found that the breakpoint was near $16 h$ at $4 AU$ . 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, 1986 ) and outer heliosphere (Matthaeus and Goldstein, 1982a ). This phenomenology 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., 1982b ; Marsch and Tu, 1990b ; Roberts, 1992 ) 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 $-3 ~ r$ . 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 $~ r-3$ 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., 1984 ; Tu, 1987a ,b , 1988 ), 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 most of the features observed in the magnetic power spectra $P (f,r)$ observed by Bavassano et al. (1982b ). In particular, the concept of the “frequency break”, just mentioned, was pointed out for the first time by Tu et al. (1984 ) who, developing the analytic solution for the radially evolving power spectrum $P (f,r)$ of fluctuations, obtained a critical frequency “ $fc$ ” such that for frequencies $f « fc,P (f,r) oc f- 1$ and for $f » fc,P (f,r) oc f- 1.5$ . In addition, their model was the first model able to explain the decreasing of the “break frequency” with increasing heliocentric distance.

3.1.3 Fluctuations anisotropy

Interplanetary magnetic field (IMF) and velocity fluctuations are rather anisotropic as for the first time observed by Belcher and Davis Jr (1971 ), Belcher and Solodyna (1975 ), Chang and Nishida (1973 ), Burlaga and Turner (1976 ), Solodyna and Belcher (1976 ), Parker (1980 ), Bavassano et al. (1982a ), Tu et al. (1989a ), and Marsch and Tu (1990a ). Moreover, this feature can be better observed if fluctuations are rotated into the minimum variance reference system (see Appendix 15).

Sonnerup and Cahill (1967 ) introduced the minimum variance analysis which consists of determining the eigenvectors of the matrix

Sij = <BiBj >- <Bi><Bj>,

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 $c1 « (c2, c3)$ , and the corresponding eigenvector $~V1$ is the minimum-variance direction (see Appendix 15.1 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 $~ V1$ , fluctuations appear to be anisotropically distributed, say $c3 > c2$ . Typical values for eigenvalues are $c3 : c2 : c1 = 10 : 3.5 : 1.2$ (Chang and Nishida, 1973 ; Bavassano et al., 1982a ).
The direction $V~1$ is nearly parallel to the average magnetic field $B0$ , that is, the distribution of the angles between $V~1$ and $B0$ is narrow with width of about $10o$ and centered around zero.

As shown in Figure 24 , 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, 1990a ).

Figure 24:

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 $~ 8o$ 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. (1991

, 1993

) 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 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. (1991

) 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. (1982a ) who showed that the ratio $c1/c3$ , calculated in the inner heliosphere within a corotating 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. (1991 ) in the outer heliosphere and by Bavassano et al. (1982a ) in the inner heliosphere remained rather controversial until recent studies (see Section 9.1), performed by Bruno et al. (1999b ), found a reason for this discrepancy.

3.1.4 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., 1983 ; Carbone and Veltri, 1990 ). 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 $(k .B0) -1$ . Clearly fluctuations with wave vectors almost perpendicular to $B0$ are interested by such an effect much less than fluctuations with $k || B 0$ . As a consequence, the former are transferred along the spectrum much faster than the latter (Shebalin et al., 1983 ; Grappin, 1986 ; Carbone and Veltri, 1990 ).

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

(Shebalin et al., 1983

; Carbone and Veltri, 1990

), where the average of a given quantity $g(k)$ is defined as

integral d3k g(k)Q(k, t) <g(k) >Q = -- integral -d3k-Q(k,-t)-.

For a spectrum with wave vectors perpendicular to $B0$ we have a spectral anisotropy $o _O_ = 90$ , while for an isotropic spectrum $o _O_ = 45$ . Numerical simulations in 2D configuration by Shebalin et al. (1983 ) 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 (1990 ) started from the spectral equations obtained through the Direct Interaction Approximation closure by Veltri et al. (1982 ), and derived a shell model analogous for the anisotropic MHD turbulence. The phenomenological anisotropic spectrum obtained from the model, for both pseudo-energies obtained through polarizations $a = 1,2$ defined through Equation (14 ), can be written as

Authors showed that spectral anisotropy is different within the three ranges of turbulence. Wave vectors perpendicular to $B 0$ 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 $_O_ -~ 55o$ 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 $_O_ -~ 80o$ . When the Reynolds number increases, the contribution of the inertial range extends, and the increases of the total anisotropy tends to saturate at about $o _O_ -~ 60$ at Reynolds number of $5 10$ . This value corresponds to a rather low value for the ratio between parallel and perpendicular correlation lengths $l||/l _L -~ 2$ , too small with respect to the observed value $l||/l _L > 10$ . This suggests that the non-linear dynamical evolution of an initially isotropic spectrum of turbulence is perhaps not sufficient to explain the observed anisotropy. Recent numerical simulations confirmed these results (Oughton et al., 1994 ).

3.1.5 Fluctuations correlation length and the Maltese Cross

The correlation time, as defined in Appendix 12, estimates how much an element of our time series $x(t)$ at time $t 1$ depends on the value assumed by $x(t)$ at time $t 0$ , being $t = t + dt 1 0$ . 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, 1982b ; Bruno and Dobrowolny, 1986 ), 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 25 , where we can observe the behavior of the $Bz$ correlation function for fast and slow wind at $0.3$ and $0.9 AU$ .

Figure 25:

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 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. (1990 ) 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 26 , also called the “The Maltese Cross”.

Figure 26:

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 $r _L$ is in units of $1010 cm$ (adopted from Matthaeus et al., 1990

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 $k ||B 0$ contribute to contours elongated parallel to $r _L$ . Fluctuations in the two-dimensional turbulence limit (Montgomery, 1982

) contribute to contours elongated parallel to $r||$ . This two-dimensional turbulence is characterized for having both the wave vector $k$ and the perturbing field $db$ 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.

Anisotropic turbulence has been observed in laboratory plasmas and reverse pinch devices (Zweben et al., 1979 ), and has been studied both theoretically (Montgomery, 1982 ; Zank and Matthaeus, 1992 ) and through numerical simulations (Shebalin et al., 1983 ; Oughton, 1993 ). In particular, these simulations focused on non-linear spectral transfer within MHD turbulence in presence of a relevant mean magnetic field. They observed that a strong anisotropy is created during the turbulent process and much of the power is transferred to fluctuations with higher $k_ L$ and less to fluctuations with higher $k ||$ . 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 $k _L B0$ and polarization of fluctuations $dB(k_ L )$ perpendicular to both vectors (2D geometry with $k|| -~ 0$ ), and some parallel to the mean magnetic field $k || B0$ , the polarization of fluctuations $dB(k ||)$ being perpendicular to the direction of $B0$ (slab geometry with $k_ L -~ 0$ ). The magnetic field is then rotated into the same mean field coordinate system used by Belcher and Davis Jr (1971 ) and Belcher and Solodyna (1975 ), 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

integral --1--- 3 - ik.r Pij(k) = (2p)3 d r <Bi(x)Bj(x + r)>e ,

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

where the anisotropic energy spectrum is the sum of both components:

Here $f$ is the frequency, $Cs$ is a constant defining the overall spectrum amplitude in wave vector space, $U w$ is the bulk solar wind speed and $y$ is the angle between $B 0$ 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 $_L$ to the mean field and solar wind flow direction) and the reduced quasi-parallel spectrum (fluctuations $_L$ 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, 1992 ), 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 was 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 individual 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 $h$ between the flow direction and the orientation of the mean field $B0$ , showed a remarkable dependence on $h$ .

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 $B0$ . 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 $c3 /= c2$ .

Going back from the correlation tensor to the power spectrum is a complicated technical problem. However, Carbone et al. (1995a ) 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 $[1] I (k)$ corresponds, in the weak turbulence theory, to the Alfvénic mode, while the second polarization $I[2](k)$ 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 (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

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

A fit to the eigenvalues of the variance matrix allowed Carbone et al. (1995a ) to fix the free parameters of the spectrum for both polarizations. They used data from Bavassano et al. (1982a ) who reported the values of $c i$ 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 $1.1 < m1 < 1.3$ and $1.46 < m2 < 1.8$ 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 $[1] [1] lx > ly » lz[1]$ , indicating that wave vectors $k || B0$ largely dominate. Concerning polarization $[2]$ , it can be found that $lx[2] » l[2y] -~ l[z2]$ , indicating that the spectrum $I[2](k)$ 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. (1990 ), the comparison being significant as far as the plane $yz$ is taken into account. The decomposition of Carbone et al. (1995a ) in two independent polarizations is similar to that of Matthaeus et al. (1990 ), a contour plot of the trace of the correlation tensor Fourier transform $[1] [2] T (k) = I (k) + I (k)$ 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 $B0$ . 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. (1990 ) give an interpretation of their results, which is in contrast with that of Carbone et al. (1995a ). Namely Matthaeus et al. (1990 ) 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. (1983 ) and of analytical studies (Montgomery, 1982 ). Let us note, however, that in the former approach, which is strictly 2D, when $k _L B 0$ magnetic fluctuations are necessarily parallel to $B 0$ . 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. (1995a ) can be directly related to the observational data. To conclude, 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 in the Jovian magnetosphere (Saur et al., 2002 , 2003 ).

3.1.6 Magnetic helicity

Magnetic helicity $Hm$ , as defined in Appendix 13.1, measures the “knottedness” of magnetic field lines (Moffat, 1978 ). 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. The general features of the magnetic helicity spectrum in the solar wind were for the first time described by Matthaeus and Goldstein (1982b ) in the outer heliosphere, and by Bruno and Dobrowolny (1986 ) in the inner heliosphere. A useful dimensionless way to represent both the degree of and the sense of polarization is the normalized magnetic helicity $sm$ (see Appendix 13.1). This quantity can randomly vary between $+1$ and $- 1$ , as shown in Figure 27 from the work by Matthaeus and Goldstein (1982b ) and relative to Voyager’s data taken at $1 AU$ . However, net values of $± 1$ are reached only for pure circularly polarized waves.

Figure 27:

$sm$ vs. frequency and wave number relative to an interplanetary data sample recorded by Voyager 1 at approximately $1 AU$ (adopted from Matthaeus and Goldstein, 1982b

Based on these results, Goldstein et al. (1991 ) were able to reproduce the distribution of the percentage of occurrence of values of $sm(f )$ 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. (1983 , 1984 ), knowing the sign of $sm$ and the sign of the normalized cross-helicity $sc$ 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, $s (f )s (f) < 0 m c$ will indicate right circular polarization while $s (f)s (f) > 0 m c$ will indicate left circular polarization. Thus, any time magnetic helicity and cross-helicity are available from measurements in a super-Alfv&eac