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 have been recorded by SOHO/SUMER instrument (Hassler et al., 1999).




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 to . At variance with Coleman (1968), Barnes and Hollweg (1974) analyzed the properties of the observed lowfrequency 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 !. The frequency spectrum, in a range of intermediate frequencies, was found to behave roughly as , the difference with the expected Kraichnan spectral slope was tentatively attributed to the presence of highfrequency transverse fluctuations resulting from plasma gardenhose 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.

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.

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 repropose similar observations taken by Helios 2 during its primary mission to the Sun.

Matthaeus and Goldstein (1986) found the breakpoint around at , and Klein et al. (1992) found that the breakpoint was near at . 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 . Moreover, a steepening of the spectrum towards a Kolmogorov like spectral index can be observed. On the contrary, the same insitu observations established that the radial decay for the higher frequencies was faster than 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 nonlinear 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 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 of fluctuations, obtained a critical frequency “” such that for frequencies and for . In addition, their model was the first model able to explain the decreasing of the “break frequency” with increasing heliocentric distance.
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
where and denote the components of magnetic field along the axes of a given reference system. The statistical properties of eigenvalues approximately satisfy the following statements:
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).
At odds with these conclusions were the results by Bavassano et al. (1982a) who showed that the ratio , 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.
In the presence of a DC background magnetic field 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 nonlinear 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 , which describes the convection of perturbations in the average magnetic field, the nonlinear interactions between Alfvénic fluctuations are weakened, since convection decorrelates the interacting eddies on a time of the order . Clearly fluctuations with wave vectors almost perpendicular to are interested by such an effect much less than fluctuations with . 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 for a given dynamical variable (namely the energy, crosshelicity, etc.), it is useful to introduce the parameter
(Shebalin et al., 1983; Carbone and Veltri, 1990), where the average of a given quantity is defined asFor a spectrum with wave vectors perpendicular to we have a spectral anisotropy , while for an isotropic spectrum . 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 pseudoenergies obtained through polarizations 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 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 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 . When the Reynolds number increases, the contribution of the inertial range extends, and the increases of the total anisotropy tends to saturate at about at Reynolds number of . This value corresponds to a rather low value for the ratio between parallel and perpendicular correlation lengths , too small with respect to the observed value . This suggests that the nonlinear 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).
The correlation time, as defined in Appendix 12, estimates how much an element of our time series at time depends on the value assumed by at time , being . 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 correlation function for fast and slow wind at and .
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 twodimensional correlation function of solar wind fluctuations at . The original dataset comprised approximately 16 months of almost continuous magnetic field averages. These results, based on ISEE 3 magnetic field data, are shown in Figure 26, also called the “The Maltese Cross”.

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 nonlinear 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 and less to fluctuations with higher . 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 and polarization of fluctuations perpendicular to both vectors (2D geometry with ), and some parallel to the mean magnetic field , the polarization of fluctuations being perpendicular to the direction of (slab geometry with ). 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 ycoordinate is perpendicular to both and the radial direction, while the xcoordinate is perpendicular to but with a component also in the radial direction. Using that geometry, and defining the power spectrum matrix as
it can be found that, assuming axisymmetry, a twocomponent model can be written in the frequency domain
where the anisotropic energy spectrum is the sum of both components: Here is the frequency, is a constant defining the overall spectrum amplitude in wave vector space, is the bulk solar wind speed and is the angle between and the wind direction. Finally, is the fraction of slab components and 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 quasiparallel spectrum (fluctuations to the mean field and in the plane defined by the mean field and the flow direction). This ratio, expected to be for slab turbulence, resulted to be for fluctuations within the inertial range, consistent with of 2D turbulence and 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 of 2D turbulence and 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 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 timeaveraging since the interaction between slab waves and transverse pressurebalanced magnetic structures causes the slab turbulence to evolve towards a state in which a twocomponent correlation function emerges during the process of time averaging.
The presence of two populations, i.e., a slablike and a quasi2D like, was also inferred by Dasso et al. (2003). These authors computed the reduced spectra of the normalized crosshelicity and the Alfvén ratio from ACE dataset. These parameters, calculated for different intervals of the angle between the flow direction and the orientation of the mean field , showed a remarkable dependence on .
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 . 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 .
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 threedimensional energy spectra of magnetic fluctuations. The divergenceless of the magnetic field allows to decompose the Fourier amplitudes of magnetic fluctuations in two independent polarizations: The first one corresponds, in the weak turbulence theory, to the Alfvénic mode, while the second polarization 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 twopoints 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 are the Cartesian components of the wave vector , and , , and ( indicates both polarizations; ) are free parameters. In particular, gives information on the energy content of both polarizations, represent the spectral extensions along the direction of a given system of coordinates, and 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 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 and increase systematically with increasing distance from the Sun, the polarization spectra are always steeper than the corresponding polarization spectra, while polarization is always more energetic than polarization . As far as the characteristic lengths are concerned, it can be found that , indicating that wave vectors largely dominate. Concerning polarization , it can be found that , indicating that the spectrum is strongly flat on the plane defined by the directions of 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 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 on the plane shows two populations of fluctuations, with wave vectors nearly parallel and nearly perpendicular to , respectively. The first population is formed by all the polarization [1] fluctuations and by the fluctuations with belonging to polarization [2]. The latter fluctuations are physically indistinguishable from the former, in that when is nearly parallel to , both polarization vectors are quasiperpendicular to . On the contrary, the second population is almost entirely formed by fluctuations belonging to polarization [2]. While it is clear that fluctuations with nearly parallel to are mainly polarized in the plane perpendicular to (a consequence of ), fluctuations with nearly perpendicular to are polarized nearly parallel to .
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 , 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 magnetic fluctuations are necessarily parallel to . 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).
Magnetic helicity , as defined in Appendix 13.1, measures the “knottedness” of magnetic field lines (Moffat, 1978). Moreover, 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 (see Appendix 13.1). This quantity can randomly vary between and , as shown in Figure 27 from the work by Matthaeus and Goldstein (1982b) and relative to Voyager’s data taken at . However, net values of are reached only for pure circularly polarized waves.

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 to . 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 and the sign of the normalized crosshelicity it is possible to infer the sense of polarization of the fluctuations. As a matter of fact, a positive crosshelicity indicates an Alfvén mode propagating outward, while a negative crosshelicity indicates a mode propagating inward. On the other hand, we know that a positive magnetichelicity indicates a right hand polarized mode, while a negative magnetichelicity indicates a left hand polarized mode. Thus, since the sense of polarization depends on the propagating direction with respect to the observer, will indicate right circular polarization while will indicate left circular polarization. Thus, any time magnetic helicity and crosshelicity are available from measurements in a superAlfvénic 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 , observable in Voyager’s data (see Figure 27), was equally observed in Helios 2 data in the inner heliosphere (Bruno and Dobrowolny, 1986). The authors of this last work computed the difference 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., 1976).
Numerical simulations of the incompressible MHD equations by Mininni et al. (2003a), discussed in Section 3.1.8, clearly show the tendency of magnetic helicity to follow an inverse cascade. 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 28 in the same format described for Figure 33, shown in Section 3.1.8. Now, the forcing is applied at wave number 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 . An initial magnetic fluctuation is introduced at at . 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., ). At these large scales, the system is very close to a magnetostatic equilibrium characterized by a forcefree configuration.

In a famous paper, Belcher and Davis Jr (1971) showed that a strong correlation exists between velocity and magnetic field fluctuations, in the form
where the sign of the correlation is given by the , being the wave vector and the background magnetic field vector. These authors showed that in about of data from Mariner 5, out of the of the whole mission, fluctuations were described by Equation (36), 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 highspeed streams. Moreover, in the regions where Equation (36) is verified to a high degree, the magnetic field magnitude is almost constant .

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, depends 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 uncompressive 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, 1971; Belcher and Solodyna, 1975). 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., 1987a; see Section 5.1).
Just to give a qualitative quick example about Alfvénic correlations in fast and slow wind, we show in Figure 30 the speed profile for about of 1976 as observed by Helios 2, and the traces of velocity and magnetic field components (see Appendix 15 for the orientation of the reference system) and (this last one expressed in Alfvén units, see Appendix 13.1) for two different time intervals, which have been enlarged in the two inserted small panels. The high velocity interval shows a remarkable anticorrelation which, since the mean magnetic field is oriented away from the Sun, suggests a clear presence of outward oriented Alfvénic fluctuations given that the sign of the correlation is the . 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 (1974) who, following previous theoretical works by Kraichnan (1965) and Iroshnikov (1963), showed that the presence of a strong correlation between velocity and magnetic fluctuations renders nonlinear transfer to small scales less efficient than for the NavierStokes equations, leading to a turbulent behavior which is different from that described by Kolmogorov (1941). In particular, when Equation (36) is exactly satisfied, nonlinear 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 23, 30) are observed, and the existence of the correlations is in contrast with the existence of a spectrum which in turbulence is due to a nonlinear energy cascade. Dobrowolny et al. (1980b) 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.
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 31 shows the radial evolution of (see Appendix 13.1) as observed by Helios and Voyager s/c (Roberts et al., 1987b). It is clear enough that not only tends to values around as the heliocentric distance increases, but larger and larger time scales are less and less Alfvénic. Values of suggest a comparable amount of “outward” and “inward” correlations.


Also Roberts et al. (1990), analysing fluctuations between and found a similar radial trend. These authors showed that dramatically decreases from values around unit at the Earth’s orbit towards at approximately . For larger heliocentric distances, seems to stabilize around this last value.
The reason why 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 within the inertial range. Moreover, as pointed out by Grappin et al. (1991), the magnetic energy excess can be explained as a competing action between the “Alfvén effect” (Kraichnan, 1965), which would work towards equipartition, and the nonlinear terms (Grappin et al., 1983). However, this argument forecasts an Alfvén ratio but, it does not say whether it would be larger or smaller than ””, 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. (2003a), already cited in Section 3.1.6. 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 alphaeffect, 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 33. 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 . An initial magnetic fluctuation is introduced at t=0 at . 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., ), as also observed in Alfvénic fluctuations of the solar wind (Bruno et al., 1985; Tu and Marsch, 1990a) as an asymptotic state (Roberts et al., 1987a,b; Bavassano et al., 2000b) of turbulence.

On the other hand, Marsch and Tu (1993a) analyzed several intervals of interplanetary observations to look for a linear relationship between the mean electromotive force , generated by the turbulent motions, and the mean magnetic field , 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 and .
Lately, Bavassano and Bruno (2000) performed a threefluid analysis of solar wind Alfvénic fluctuations in the inner heliosphere, in order to evaluate the effect of disregarding the multifluid 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 . However, fluctuations in velocity tend to be smaller than fluctuations in Alfvén units. In Figure 34 we show scatter plots between the component of the Alfvén velocity and the proton velocity fluctuations. The direction has been chosen as the same of , where is the proton bulk flow velocity and 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 , the slope of the best fit straight line passes from at to a slope considerably different from at .

and
where the letter “s” stands for the th species, being its velocity in the center of mass frame of reference. is the velocity of the species “s” in the s/c frame and is the velocity of the center of mass.
Bavassano and Bruno (2000) analyzed several time intervals within the same corotating high velocity stream observed at and and performed the analysis using the new factor “F” to express magnetic field fluctuations in Alfvén units, taking into account the presence of particles and electrons, besides the protons. However, the correction resulted to be insufficient to bring back to “” the slope of the relationship shown in the right panel of Figure 34. In conclusion, the radial variation of the Alfvén ratio towards values less than is not completely due to a missed inclusion of multifluid effects in the conversion from magnetic field to Alfvén units. Thus, we are left with the possibility that the observed depletion of is due to a natural evolution of turbulence towards a state in which magnetic energy becomes dominant (Grappin et al., 1991; Roberts et al., 1992; Roberts, 1992), as observed in the animation of Figure 33 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, 1993).
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