The Solar Wind as a Turbulence Laboratory
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Abstract
In this review we will focus on a topic of fundamental importance for both astrophysics and plasma physics, namely the occurrence of largeamplitude lowfrequency fluctuations of the fields that describe the plasma state. This subject will be treated within the context of the expanding solar wind and the most meaningful advances in this research field will be reported emphasizing the results obtained in the past decade or so. As a matter of fact, Helios inner heliosphere and Ulysses’ high latitude observations, recent multispacecrafts measurements in the solar wind (Cluster four satellites) and new numerical approaches to the problem, based on the dynamics of complex systems, brought new important insights which helped to better understand how turbulent fluctuations behave in the solar wind. In particular, numerical simulations within the realm of magnetohydrodynamic (MHD) turbulence theory unraveled what kind of physical mechanisms are at the basis of turbulence generation and energy transfer across the spectral domain of the fluctuations. In other words, the advances reached in these past years in the investigation of solar wind turbulence now offer a rather complete picture of the phenomenological aspect of the problem to be tentatively presented in a rather organic way.
1 Introduction
The whole heliosphere is permeated by the solar wind, a supersonic and superAlfvén plasma flow of solar origin which continuously expands into the heliosphere. This medium offers the best opportunity to study directly collisionless plasma phenomena, mainly at low frequencies where highamplitude fluctuations have been observed. During its expansion, the solar wind develops a strong turbulent character, which evolves towards a state that resembles the well known hydrodynamic turbulence described by Kolmogorov (1941, (1991). Because of the presence of a strong magnetic field carried by the wind, lowfrequency fluctuations in the solar wind are usually described within a magnetohydrodynamic (MHD, hereafter) benchmark (Kraichnan, (1965; Biskamp, (1993; Tu and Marsch, (1995a; Biskamp, (2003; Petrosyan et al., (2010). However, due to some peculiar characteristics, the solar wind turbulence contains some features hardly classified within a general theoretical framework.
Turbulence in the solar heliosphere plays a relevant role in several aspects of plasma behavior in space, such as solar wind generation, highenergy particles acceleration, plasma heating, and cosmic rays propagation. In the 1970s and 80s, impressive advances have been made in the knowledge of turbulent phenomena in the solar wind. However, at that time, spacecraft observations were limited by a small latitudinal excursion around the solar equator and, in practice, only a thin slice above and below the equatorial plane was accessible, i.e., a sort of 2D heliosphere. A rather exhaustive survey of the most important results based on insitu observations in the ecliptic plane has been provided in an excellent review by Tu and Marsch (1995a) and we invite the reader to refer to that paper. This one, to our knowledge, has been the last large review we find in literature related to turbulence observations in the ecliptic.
In the 1990s, with the launch of the Ulysses spacecraft, investigations have been extended to the highlatitude regions of the heliosphere, allowing us to characterize and study how turbulence evolves in the polar regions. An overview of Ulysses results about polar turbulence can also be found in Horbury and Tsurutani (2001). With this new laboratory, relevant advances have been made. One of the main goals of the present work will be that of reviewing observations and theoretical efforts made to understand the nearequatorial and polar turbulence in order to provide the reader with a rather complete view of the lowfrequency turbulence phenomenon in the 3D heliosphere.
New interesting insights in the theory of turbulence derive from the point of view which considers a turbulent flow as a complex system, a sort of benchmark for the theory of dynamical systems. The theory of chaos received the fundamental impulse just through the theory of turbulence developed by Ruelle and Takens (1971) who, criticizing the old theory of Landau and Lifshitz (1971), were able to put the numerical investigation by Lorenz (1963) in a mathematical framework. Gollub and Swinney (1975) set up accurate experiments on rotating fluids confirming the point of view of Ruelle and Takens (1971) who showed that a strange attractor in the phase space of the system is the best model for the birth of turbulence This gave a strong impulse to the investigation of the phenomenology of turbulence from the point of view of dynamical systems (Bohr et al., (1998). For example, the criticism by Landau leading to the investigation of intermittency in fully developed turbulence was worked out through some phenomenological models for the energy cascade (cf. Frisch, (1995). Recently, turbulence in the solar wind has been used as a big wind tunnel to investigate scaling laws of turbulent fluctuations, multifractals models, etc. The review by Tu and Marsch (1995a) contains a brief introduction to this important argument, which was being developed at that time relatively to the solar wind (Burlaga, (1993; Carbone, (1993; Biskamp, (1993, (2003; Burlaga, (1995). The reader can convince himself that, because of the wide range of scales excited, space plasma can be seen as a very big laboratory where fully developed turbulence can be investigated not only per se, rather as far as basic theoretical aspects are concerned.
Turbulence is perhaps the most beautiful unsolved problem of classical physics, the approaches used so far in understanding, describing, and modeling turbulence are very interesting even from a historic point of view, as it clearly appears when reading, for example, the book by Frisch (1995). History of turbulence in interplanetary space is, perhaps, even more interesting since its knowledge proceeds together with the human conquest of space Thus, whenever appropriate, we will also introduce some historical references to show the way particular problems related to turbulence have been faced in time, both theoretically and technologically. Finally, since turbulence is a phenomenon visible everywhere in nature, it will be interesting to compare some experimental and theoretical aspects among different turbulent media in order to assess specific features which might be universal, not limited only to turbulence in space plasmas. In particular, we will compare results obtained in interplanetary space with results obtained from ordinary fluid flows on Earth, and from experiments on magnetic turbulence in laboratory plasmas designed for thermonuclear fusion.
1.1 What does turbulence stand for?
Turbulent features can be recognized even in natural turbulent systems like, for example, the atmosphere of Jupiter (see Figure 4). A different example of turbulence in plasmas is reported in Figure 5 where we show the result of a typical high resolution numerical simulations of 2D MHD turbulence In this case the turbulent field shown is the current density. These basic features of mixing between order and chaos make the investigation of properties of turbulence terribly complicated, although extraordinarily fascinating.
Turbulence increases the properties of transport in a flow. For example, the urban pollution, without atmospheric turbulence, would not be spread (or eliminated) in a relatively short time. Results from numerical simulations of the concentration of a passive scalar transported by a turbulent flow is shown in Figure 6. On the other hand, in laboratory plasmas inside devices designed to achieve thermonuclear controlled fusion, anomalous transport driven by turbulent fluctuations is the main cause for the destruction of magnetic confinement. Actually, we are far from the achievement of controlled thermonuclear fusion. Turbulence, then, acquires the strange feature of something to be avoided in some cases, or to be invoked in some other cases.
1.2 Dynamics vs. statistics
In Figure 8 we report a typical sample of turbulence as observed in a fluid flow in the Earth’s atmosphere. Time evolution of both the longitudinal velocity component and the temperature is shown. Measurements in the solar wind show the same typical behavior. A typical sample of turbulence as measured by Helios 2 spacecraft is shown in Figure 9. A further sample of turbulence, namely the radial component of the magnetic field measured at the external wall of an experiment in a plasma device realized for thermonuclear fusion, is shown in Figure 10.
A different property of turbulence is that all dynamically interesting scales are excited, that is, energy is spread over all scales. This can be seen in Figure 11 where we show the magnetic field intensity within a typical solar wind stream (see top panel). In the middle and bottom panels we show fluctuations at two different detailed scales. A kind of selfsimilarity (say a similarity at all scales) is observed.
It should be emphasized that there are no firm mathematical arguments for the existence of an asymptotic statistical state. As we have just seen, reproducible statistical results are obtained from observations, that is, it is suggested experimentally and from physical plausibility. Apart from physical plausibility, it is embarrassing that such an important feature of fully developed turbulence, as the existence of a statistical stability, should remain unsolved. However, such is the complex nature of turbulence
2 Equations and Phenomenology
In this section, we present the basic equations that are used to describe charged fluid flows, and the basic phenomenology of lowfrequency turbulence Readers interested in examining closely this subject can refer to the very wide literature on the subject of turbulence in fluid flows, as for example the recent books by, e.g., Pope (2000); McComb (1990); Frisch (1995) or many others, and the less known literature on MHD flows (Biskamp, (1993; Boyd and Sanderson, (2003; Biskamp, (2003). In order to describe a plasma as a continuous medium it will be assumed collisional and, as a consequence, all quantities will be functions of space r and time t. Apart for the required quasineutrality, the basic assumption of MHD is that fields fluctuate on the same time and length scale as the plasma variables, say ωτ_{H} ≃ 1 and kL_{H} ≃ 1 (k and ω are, respectively, the wave number and the frequency of the fields, while τ_{H} and L_{H} are the hydrodynamic time and length scale, respectively). Since the plasma is treated as a single fluid, we have to take the slow rates of ions. A simple analysis shows also that the electrostatic force and the displacement current can be neglected in the nonrelativistic approximation. Then, MHD equations can be derived as shown in the following sections.
2.1 The NavierStokes equation and the Reynolds number
The Reynolds number Re = UL/ν is evidently the only parameter of the fluid flow. This defines a Reynolds number similarity for fluid flows, namely fluids with the same value of the Reynolds number behaves in the same way. Looking at Equation (5) it can be realized that the Reynolds number represents a measure of the relative strength between the nonlinear convective term and the viscous term in Equation (4). The higher Re, the more important the nonlinear term is in the dynamics of the flow. Turbulence is a genuine result of the nonlinear dynamics of fluid flows.
2.2 The coupling between a charged fluid and the magnetic field
It is worthwhile to remark that in the classical hydrodynamics, dissipative processes are defined through three coefficients, namely two viscosities and one thermoconduction coefficient. In the hydromagnetic case the number of coefficients increases considerably. Apart from few additional electrical coefficients, we have a largescale (background) magnetic field B_{0}. This makes the MHD equations intrinsically anisotropic. Furthermore, the stress tensor (8) is deeply modified by the presence of a magnetic field B_{0}, in that kinetic viscous coefficients must depend on the magnitude and direction of the magnetic field (Braginskii, (1965). This has a strong influence on the determination of the Reynolds number.
2.3 Scaling features of the equations
The scaled Euler equations are the same as Equations (4 and 5), but without the term proportional to R^{−1}. The scaled variables obtained from the Euler equations are, then, the same. Thus, scaled variables exhibit scaling similarity, and the Euler equations are said to be invariant with respect to scale transformations. Said differently, this means that NS Equations (4) show scaling properties (Frisch, (1995), that is, there exists a class of solutions which are invariant under scaling transformations. Introducing a length scale ℓ, it is straightforward to verify that the scaling transformations ℓ ↑ λ ℓ’ and u → λ^{ h }u’ (λ is a scaling factor and h is a scaling index) leave invariant the inviscid NS equation for any scaling exponent h, providing P → λ^{2h}P’. When the dissipative term is taken into account, a characteristic length scale exists, say the dissipative scale ℓ_{D}. From a phenomenological point of view, this is the length scale where dissipative effects start to be experienced by the flow. Of course, since ℓ_{D} is in general very low, we expect that ℓ_{D} is very small. Actually, there exists a simple relationship for the scaling of .D with the Reynolds number, namely ℓ_{D} ~ LRe^{−3/4}. The larger the Reynolds number, the smaller the dissipative length scale.
As it is easily verified, ideal MHD equations display similar scaling features. Say the following scaling transformations u → λ^{ h }u’ and B → λ^{ β }B’ (β here is a new scaling index different from h), leave the inviscid MHD equations unchanged, providing P → λ^{2β}P’, T → λ^{2h}T’, and ρ → λ^{2(β−h)}ρ’. This means that velocity and magnetic variables have different scalings, say h ≠ β, only when the scaling for the density is taken into account. In the incompressible case, we cannot distinguish between scaling laws for velocity and magnetic variables.
2.4 The nonlinear energy cascade
Looking at Equation (20), we see that the role played by the nonlinear term is that of a redistribution of energy among the various wave vectors. This is the physical meaning of the nonlinear energy cascade of turbulence
2.5 The inhomogeneous case
Equations (20) refer to the standard homogeneous and incompressible MHD. Of course, the solar wind is inhomogeneous and compressible and the energy transfer equations can be as complicated as we want by modeling all possible physical effects like, for example, the wind expansion or the inhomogeneous largescale magnetic field. Of course, simulations of all turbulent scales requires a computational effort which is beyond the actual possibilities. A way to overcome this limitation is to introduce some turbulence modeling of the various physical effects. For example, a set of equations for the crosscorrelation functions of both Elsäasser fluctuations have been developed independently by Marsch and Tu (1989), Zhou and Matthaeus (1990), Oughton and Matthaeus (1992), and Tu and Marsch (1990a), following Marsch and Mangeney (1987) (see review by Tu and Marsch, (1996), and are based on some rather strong assumptions: i) a twoscale separation, and ii) smallscale fluctuations are represented as a kind of stochastic process (Tu and Marsch, (1996). These equations look quite complicated, and just a comparison based on orderofmagnitude estimates can be made between them and solar wind observations (Tu and Marsch, (1996).
A different approach, introduced by Grappin et al. (1993), is based on the socalled “expandingbox model” (Grappin and Velli, (1996; Liewer et al., (2001; Hellinger et al., (2005). The model uses transformation of variables to the moving solar wind frame that expands together with the size of the parcel of plasma as it propagates outward from the Sun. Despite the model requires several simplifying assumptions, like for example lateral expansion only for the wavepackets and constant solar wind speed, as well as a secondorder approximation for coordinate transformation Liewer et al. (2001) to remain tractable, it provides qualitatively good description of the solar wind expansions, thus connecting the disparate scales of the plasma in the various parts of the heliosphere.
2.6 Dynamical system approach to turbulence
In the limit of fully developed turbulence, when dissipation goes to zero, an infinite range of scales are excited, that is, energy lies over all available wave vectors. Dissipation takes place at a typical dissipation length scale which depends on the Reynolds number Re through ℓ_{D} ~ LRe^{−3/4} (for a Kolmogorov spectrum E(k) ~ k^{−5/3}). In 3D numerical simulations the minimum number of grid points necessary to obtain information on the fields at these scales is given by N ~ (L/ℓ_{D})^{3} ~ Re^{9/4}. This rough estimate shows that a considerable amount of memory is required when we want to perform numerical simulations with high Re. At present, typical values of Reynolds numbers reached in 2D and 3D numerical simulations are of the order of 10^{4} and 10^{3}, respectively. At these values the inertial range spans approximately one decade or a little more.
Given the situation described above, the question of the best description of dynamics which results from original equations, using only a small amount of degree of freedom, becomes a very important issu. This can be achieved by introducing turbulence models which are investigated using tools of dynamical system theory (Bohr et al., (1998). Dynamical systems, then, are solutions of minimal sets of ordinary differential equations that can mimic the gross features of energy cascade turbulence These studies are motivated by the famous Lorenz’s model (Lorenz, (1963) which, containing only three degrees of freedom, simulates the complex chaotic behavior of turbulent atmospheric flows, becoming a paradigm for the study of chaotic systems.
The Lorenz’s model has been used as a paradigm as far as the transition to turbulence is concerned. Actually, since the solar wind is in a state of fully developed turbulence, the topic of the transition to turbulence is not so close to the main goal of this review. However, since their importance in the theory of dynamical systems, we spend few sentences abut this central topic. Up to the Lorenz’s chaotic model, studies on the birth of turbulence dealt with linear and, very rarely, with weak nonlinear evolution of external disturbances. The first physical model of laminarturbulent transition is due to Landau and it is reported in the fourth volume of the course on Theoretical Physics (Landau and Lifshitz, (1971). According to this model, as the Reynolds number is increased, the transition is due to a infinite series of Hopf bifurcations at fixed values of the Reynolds number. Each subsequent bifurcation adds a new incommensurate frequency to the flow whose dynamics become rapidly quasiperiodic. Due to the infinite number of degree of freedom involved, the quasiperiodic dynamics resembles that of a turbulent flow.
In this context turbulence can evolve via two distinct routes: Pomeau.Manneville intermittency (Pomeau and Manneville, (1980) and crisisinduced intermittency (Ott and Sommerer, (1994). Both types of chaotic transitions follow episodic switching between different temporal behaviors. In one case (Pomeau.Manneville) the behavior of the magnetic fluctuations evolve from nearly periodic to chaotic while, in the other case the behavior intermittently assumes weakly chaotic or strongly chaotic features.
2.7 Shell models for turbulence cascade
Since numerical simulations, in some cases, cannot be used, simple dynamical systems can be introduced to investigate, for example, statistical properties of turbulent flows which can be compared with observations. These models, which try to mimic the gross features of the time evolution of spectral NavierStokes or MHD equations, are often called “shell models” or “discrete cascade models”. Starting from the old papers by Siggia (1977) different shell models have been introduced in literature for 3D fluid turbulence (Biferale, (2003). MHD shell models have been introduced to describe the MHD turbulent cascade (Plunian et al., (2012), starting from the paper by Gloaguen et al. (1985).
The most used shell model is usually quoted in literature as the GOY model, and has been introduced some time ago by Gledzer (1973) and by Ohkitani and Yamada (1989). Apart from the first MHD shell model (Gloaguen et al., (1985), further models, like those by Frick and Sokoloff (1998) and Giuliani and Carbone (1998) have been introduced and investigated in detail. In particular, the latter ones represent the counterpart of the hydrodynamic GOY model, that is they coincide with the usual GOY model when the magnetic variables are set to zero.
 a)
Introduce discrete wave vectors:
As a first step we divide the wave vector space in a discrete number of shells whose radii grow according to a power k_{ n } = k_{0}λ^{ n }, where λ > 1 is the intershell ratio, k_{0} is the fundamental wave vector related to the largest available length scale L, and n = 1, 2, ..., N.
 b)
Assign to each shell discrete scalar variables:
Each shell is assigned two or more complex scalar variables u_{ n }(t) and b_{ n }(t), or Elsäasser variables Z _{ n } ^{±} (t) = u_{ n } ± b_{ n }(t). These variables describe the chaotic dynamics of modes in the shell of wave vectors between k_{ n } and k_{n+1}. It is worth noting that the discrete variable, mimicking the average behavior of Fourier modes within each shell, represents characteristic fluctuations across eddies at the scale ℓ_{ n } ~ k _{ n } ^{−1} . That is, the fields have the same scalings as field differences, for example Z _{ n } ^{±} ~ Z^{±}(x + ℓ_{ n }) − Z^{±}(x) ~ ℓ _{ n } ^{ h } in fully developed turbulence In this way, the possibility to describe spatial behavior within the model is ruled out. We can only get, from a dynamical shell model, time series for shell variables at a given k_{ n }, and we loose the fact that turbulence is a typical temporal and spatial complex phenomenon.
 c)
Introduce a dynamical model which describes nonlinear evolution:
Looking at Equation (19) a model must have quadratic nonlinearities among opposite variables Z _{ n } ^{±} (t) and Z _{ n } ^{∓} (t), and must couple different shells with free coupling coefficients.
 d)
Fix as much as possible the coupling coefficients:
This last step is not standard. A numerical investigation of the model might require the scanning of the properties of the system when all coefficients are varied. Coupling coefficients can be fixed by imposing the conservation laws of the original equations, namely the total pseudoenergiesthat means the conservation of both the total energy and the crosshelicity:$$E^ \pm (t) = \frac{1} {2}\sum\limits_n {\left {Z_n^ \pm } \right^2 },$$(22a)where Re indicates the real part of the product u_{ n }b_{ n }*. As we said before, shell models cannot describe spatial geometry of nonlinear interactions in turbulence, so that we loose the possibility of distinguishing between twodimensional and threedimensional turbulent behavior. The distinction is, however, of primary importance, for example as far as the dynamo effect is concerned in MHD. However, there is a third invariant which we can impose, namely$$\begin{array}{*{20}c} {E(t) = \frac{1} {2}\sum\limits_n {\left {u_n } \right^2 + \left {b_n } \right^2 ;} } & {H_c (t) = \sum\limits_n {2\Re e(u_n b_n^* )} } \\ \end{array},$$(22b)which can be dimensionally identified as the magnetic helicity when α = 1, so that the shell model so obtained is able to mimic a kind of 3D MHD turbulence (Giuliani and Carbone (1998).$$H(t) = \sum\limits_n {\left {  1} \right^n \frac{{\left {b_n } \right^2 }} {{k_n^\alpha }}},$$(23)
2.8 The phenomenology of fully developed turbulence: Fluidlike case
In the real space turbulence properties can be described using either the probability distribution functions (PDFs hereafter) of increments, or the longitudinal structure functions, which represents nothing but the higher order moments of the field. Disregarding the magnetic field, in a purely fully developed fluid turbulence, this is defined as S _{ℓ} ^{(p)} = 〈Δu _{ℓ} ^{ p } 〉. These quantities, in the inertial range, behave as a power law S _{ℓ} ^{(p)} ~ ℓ^{ ξp }, so that it is interesting to compute the set of scaling exponent ξ_{ p }. Using, from a phenomenological point of view, the scaling for field increments (see Equation (26)), it is straightforward to compute the scaling laws S _{ℓ} ^{(p)} ~ ℓ^{p/3}. Then ξ_{ p } = p/3 results to be a linear function of the order p.
2.9 The phenomenology of fully developed turbulence: Magneticallydominated case
2.10 Some exact relationships
This important relation can be obtained in a more general framework from MHD equations. A Yaglom’s relation for MHD can be obtained using the analogy of MHD equations with a transport equation, so that we can obtain a relation similar to the Yaglom’s equation for the transport of a passive quantity (Monin and Yaglom, (1975). Using the above analogy, the Yaglom’s relation has been extended some time ago to MHD turbulence by Chandrasekhar (1967), and recently it has been revised by Politano et al. (1998) and Politano and Pouquet (1998) in the framework of solar wind turbulence In the following section we report an alternative and more general derivation of the Yaglom’s law using structure functions (SorrisoValvo et al., (2007; Carbone et al., (2009c).
2.11 Yaglom’s law for MHD turbulence
The relations we obtained can be used, or better, in a certain sense they might be used, as a formal definition of inertial range. Since they are exact relationships derived from NavierStokes and MHD equations under usual hypotheses, they represent a kind of “zerothorder” conditions on experimental and theoretical analysis of the inertial range properties of turbulence It is worthwhile to remark the two main properties of the Yaglom’s laws. The first one is the fact that, as it clearly appears from the Kolmogorov’s relation (Kolmogorov, (1941), the thirdorder moment of the velocity fluctuations is different from zero. This means that some nonGaussian features must be at work, or, which is the same, some hidden phase correlations. Turbulence is something more complicated than random fluctuations with a certain slope for the spectral density. The second feature is the minus sign which appears in the various relations. This is essential when the sign of the energy cascade must be inferred from the Yaglom relations, the negative asymmetry being a signature of a direct cascade towards smaller scales. Note that, Equation (40) has been obtained in the limit of zero viscosity assuming that the pseudoenergy dissipation rates ∈ _{ ii } ^{±} remain finite in this limit. In usual fluid flows the analogous hypothesis, namely ν remains finite in the limit ν → 0, is an experimental evidence, confirmed by experiments in different conditions (Frisch, (1995). In MHD turbulent flows this remains a conjecture, confirmed only by high resolution numerical simulations (Mininni and Pouquet, (2009).
2.12 Densitymediated Elsäasser variables and Yaglom’s law
Relation (40), which is of general validity within MHD turbulence, requires local characteristics of the turbulent fluid flow which can be not always satisfied in the solar wind flow, namely, largescale homogeneity, isotropy, and incompressibility. Density fluctuations in solar wind have a low amplitude, so that nearly incompressible MHD framework is usually considered (Montgomery et al., (1987; Matthaeus and Brown, (1988; Zank and Matthaeus, (1993; Matthaeus et al., (1991; Bavassano and Bruno, (1995). However, compressible fluctuations are observed, typically convected structures characterized by anticorrelation between kinetic pressure and magnetic pressure (Tu and Marsch, (1994). Properties and interaction of the basic MHD modes in the compressive case have also been considered (Goldreich and Sridhar, (1995; Cho and Lazarian, (2002).
2.13 Yaglom’s law in the shell model for MHD turbulence
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 ≃ 10^{8} m, while “probes” (say spacecrafts) are about d ≃ 5 m. Thus, space provides a much larger laboratory. Most measurements are single point measurements, the ESACluster 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, (1995a; Goldstein et al., (1995b).
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 16 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 timestationarity of the source regions. Obviously, similar hypotheses decay during higher activity phase of the solar cycle since, as shown in Figure 17, the nice and regular alternation of fast corotating streams and slow wind is replaced by a much more irregular and spiky profile also characterized by a lower average speed.
Figure 19 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 19. 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
Only lately, Podesta et al. (2007) 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.
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 22 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.
Horbury et al. (2008) 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 θ_{ B } between the mean field direction and the flow direction is 90°. On the other hand, in case θ_{ B } = 0° the scaling would follow θ^{−2}. Moreover, the latter spectrum would also have a smaller energy content.
Horbury et al. (2008) 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 θ_{ B }. They adopted a method based on wavelet analysis which was sensitive to the frequent changes of the local magnetic field direction.
The lower panel of Figure 24 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 θ_{ B } 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
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 25: the correlation length λ_{ C }, the Taylor scale λ_{ T } and the Kolmogorov scale λ_{ K }. 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 dissipationscale eddies.
Following the definitions of λ_{ C } and λ_{ T } given above, Matthaeus et al. (2005) 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 ACEWind and the second Cluster data set. These fits provided estimates for λ_{ C } and λ_{ T } from which these authors obtained the first empirical determination of R _{ m } ^{eff} which resulted to be of the order of 2.3 × 10^{5}, as illustrated in Figure 28.
3.1.3 Evidence for nonlinear 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 29 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.
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 Kolmogorovlike 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).
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 (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). This feature can be better observed if fluctuations are rotated into the minimum variance reference system (see Appendix D).

One of the eigenvalues of the variance matrix is always much smaller than the others, say λ_{1} ≪ (λ_{2}, λ_{3}), and the corresponding eigenvector Ṽ_{1} is the minimumvariance 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 minimumvariance direction.

In the plane perpendicular to Ṽ_{1}, fluctuations appear to be anisotropically distributed, say λ_{3} > λ_{2}. Typical values for eigenvalues are λ_{3} : λ_{2} : λ_{1} = 10 : 3.5 : 1.2 (Chang and Nishida, (1973; Bavassano et al., (1982a).

The direction Ṽ_{1} is nearly parallel to the average magnetic field B_{0}, that is, the distribution of the angles between Ṽ_{1} and B_{0} is narrow with width of about 10° and centered around zero.
As shown in Figure 31, 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).
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 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. (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 λ_{1}/λ_{3}, 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 10.2), performed by Bruno et al. (1999b), found a reason for this discrepancy.
3.1.5 Simulations of anisotropic MHD
In the presence of a DC background magnetic field B_{0} which, differently from the bulk velocity field, cannot be eliminated by a Galilean transformation, MHD incompressible turbulence becomes anisotropic (Shebalin et al., (1983; Montgomery, (1982; Zank and Matthaeus, (1992; Carbone and Veltri, (1990; 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 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 (B_{0} · ∇)z^{±}, which describes the convection of perturbations in theaverage magnetic field, the nonlinear interactions between Alfvénic fluctuations are weakened, since convection decorrelates the interacting eddies on a time of the order (k · B_{0})^{−1}. Clearly fluctuations with wave vectors almost perpendicular to B_{0} 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).
The spectral anisotropy is different within the injection, inertial, and dissipative ranges of turbulence (Carbone and Veltri, (1990). 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 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 nonlinear 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 t_{1} depends on the value assumed by x(t) at time t_{0}, being t_{1} = t_{0} + 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.
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. (1990) provided for the first time the twodimensional correlation function of solar wind fluctuations at 1 AU. The original dataset comprised approximately 16 months of almost continuous magnetic field 5min averages. These results, based on ISEE 3 magnetic field data, are shown in Figure 33, also called the “The Maltese Cross”.
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 isocontour 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_{⊥}. Fluctuations in the twodimensional turbulence limit (Montgomery, (1982) contribute to contours elongated parallel to r_{⊥}. This twodimensional turbulence is characterized for having both the wave vector k and the perturbing field δb perpendicular to the ambient field B_{0}. 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 quasiparallel to the local magnetic field, while slow streams are dominated by quasiperpendicular fluctuation wavevectors. Anisotropic turbulence has been observed in laboratory plasmas and reverse pinch devices (Zweben et al., (1979).
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 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 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 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 θ between the flow direction and the orientation of the mean field B_{0}, 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 λ_{3} ≠ λ_{2}.
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 λ_{ 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 ≤ μ_{1} ≤ 1.3 and 1.46 ≤ μ_{2} ≤ 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 _{ x } ^{[1]} > ℓ _{ y } ^{[1]} ≫ ℓ_{ z }[1], indicating that wave vectors k ∥ B_{0} largely dominate. Concerning polarization [2], it can be found that ℓ_{ x }[2] ≫ ℓ _{ y } ^{[2]} ≃ ∓ _{ z } ^{[2]} , indicating that the spectrum I^{[2]}(k) is strongly flat on the plane defined by the directions of B_{0} 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 T(k) = I^{[1]}(k) + I^{[2]}(k) on the plane (k_{ y }; k_{ z }) shows two populations of fluctuations, with wave vectors nearly parallel and nearly perpendicular to B_{0}, respectively. The first population is formed by all the polarization [1] fluctuations and by the fluctuations with k ∥ B_{0} belonging to polarization [2]. The latter fluctuations are physically indistinguishable from the former, in that when k is nearly parallel to B_{0}, both polarization vectors are quasiperpendicular 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 B_{0} are mainly polarized in the plane perpendicular to B_{0} (a consequence of ∇ · B = 0), fluctuations with k nearly perpendicular to B_{0} are polarized nearly parallel to B_{0}.
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 B_{0}, 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 from analytical studies (Montgomery, (1982). Let us note, however, that in the former approach, which is strictly 2D, when k ⊥ 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. 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 (1996) 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 largescale 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 2D fluctuations.
Narita et al. (2011b), using the four Cluster spacecraft, determined the threedimensional wavevector 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 2D turbulence, however, the analyzed fluctuations cannot be considered axisymmetric.
Finally, Turner et al. (2011) suggested that the nonaxisymmetry anisotropy of the frequency spectrum observed using insitu observations may simply arise from a sampling effect related to the fact that the s/c samples three dimensional fluctuations as a onedimensional 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 H_{m}, as defined in Appendix B.1, measures the “knottedness” of magnetic field lines (Moffatt, (1978). Moreover, H_{m} 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 lefthand polarization. Statistical information about the magnetic helicity is derived from the Fourier transform of the magnetic field autocorrelation matrix R_{ ij }(r) = 〈B_{ i }(x) · B_{ j }(x+r)〉 as shown by Matthaeus and Goldstein (1982b). 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, (1970; Montgomery, (1982; Matthaeus and Goldstein, (1982b). However, what is really available from insitu 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 H_{m}(k) will depend only on the wavenumber 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 (1982b) to compute the reduced H_{m} 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 (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 σ_{m} (see Appendix B.2). This quantity can randomly vary between +1 and −1, as shown in Figure 34 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.
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 righthand polarizations is significant and continuously varying.
As already noticed by Smith et al. (1983, (1984), knowing the sign of σ_{m} and the sign of the normalized crosshelicity σ_{c} 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 righthand polarized mode, while a negative magnetichelicity indicates a lefthand polarized mode. Thus, since the sense of polarization depends on the propagating direction with respect to the observer, σ_{m}(f)σ_{c}(f) < 0 will indicate right circular polarization while σ_{m}(f)σ_{c}(f) > 0 will indicate left circular polarization. Thus, each 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 σ_{m}, observable in Voyager’s data (see Figure 34), was equally observed in Helios 2 data in the inner heliosphere (Bruno and Dobrowolny, (1986). The authors of this last work computed the difference (MH > 0) − MH < 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., (1976).
Numerical simulations of the incompressible MHD equations by Mininni et al. (2003a), 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, (2001).
This phenomenon was firstly demonstrated in numerical simulations based on the eddy damped quasi normal Markovian (EDQNM) closure model of threedimensional 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).
3.1.8 Alfvén correlations as incompressive turbulence
Today we know that Alfvén 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 36 we show results from Belcher and Solodyna (1975) 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 36 we show results from a similar analysis performed by Bruno et al. (1985) 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, Alfvén 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 footpoints of the solar surface magnetic field.
Just to give a qualitative quick example about Alfvénic correlations in fast and slow wind, we show in Figure 37 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 anticorrelation which, since the mean magnetic field B_{0} 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 · B_{0}]. 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 Alfvén 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 (59) 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 29, 37) 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 Alfvén turbulence, say the presence of predominately outward propagation and the fact that MHD turbulence with the presence of both Alfvén modes present will evolve towards a state where one of the mode disappears. However, a lengthy debate based on whether the highly Alfvén 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 38 shows the radial evolution of σ_{c} (see Appendix B.1) as observed by Helios and Voyager s/c (Roberts et al., (1987b). It is clear enough that σ_{c} 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 σ_{c} ~ 0 suggest a comparable amount of “outward” and “inward” correlations.
The radial evolution affects also the Alfvén ratio r_{ A } (see Appendix B.3.1) as it was found by Bruno et al. (1985). However, early analyses (Belcher and Davis Jr, (1971; Solodyna and Belcher, (1976; Matthaeus and Goldstein, (1982b) had already shown that this parameter is usually less than unit. Spectral studies by Marsch and Tu (1990a), reported in Figure 39, 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 r_{A} in fast wind approaches that in slow wind.
Moreover, comparing these results with those by Matthaeus and Goldstein (1982b) 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. (1990), analyzing fluctuations between 9 h and 3 d found a similar radial trend. These authors showed that r_{ A } dramatically decreases from values around unit at the Earth’s orbit towards 0.4 . 0.5 at approximately 8 AU. For larger heliocentric distances, r_{A} seems to stabilize around this last value.
The reason why r_{ A } 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 r_{ A } < 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 (1965)), and a local dynamo effect due to nonlinear terms (Grappin et al., (1991), see closure calculations by Grappin et al. (1983); DNS by Müller and Grappin (2005).
However, when the twofluid 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 (1993a) analyzed several intervals of interplanetary observations to look for a linear relationship between the mean electromotive force ε = δVδB, generated by the turbulent motions, and the mean magnetic field B_{0}, 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 B_{0} and ε.
Bavassano and Bruno (2000) analyzed several time intervals within the same corotating 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 δV_{Pz} ∓ δV_{Az} relationship shown in the right panel of Figure 41. In conclusion, the radial variation of the Alfvén ratio r_{A} towards values less than 1 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 r_{A} 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 40 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).
3.2 Turbulence studied via Elsässer variables
The Alfvénic character of solar wind fluctuations,especially within corotating 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. (1980a,b); Veltri et al. (1982); Marsch and Mangeney (1987); and Zhou and Matthaeus (1989), were for the first time used in interplanetary data analysis by Grappin et al. (1990) and Tu et al. (1989b). 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., (1982b; Denskat and Neubauer, (1983). Successively, when Elsässer variables were introduced in the analysis (Grappin et al., (1989), 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 42 (Tu et al., (1990) 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 Kolmogorovlike. e^{−} has a similar break but the slope of lower frequencies follows the Kolmogorov slope, while higher frequencies form a sort of plateau.
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. (1990), is commonly found within corotating high velocity streams, although much more clearly expressed at shorter heliocentric distances, as shown below.
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.
As a matter of fact, this radial evolution can be inferred from Figure 45 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).
Further ecliptic observations (see Figure 46) 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.
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. (1990) and Tu and Marsch (1991).
As a matter of fact, a statistical analysis by Bruno et al. (2007) 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 crosshelicity σ_{c} and normalized residual energy σ_{r} (see Appendix B.3).
Quite different is the situation within slow wind, as shown in the right panels of Figure 48. As a matter of fact, these histograms do not show any striking radial evolution like in the case of fast wind. High values of σ_{c} are statistically much less relevant than in fast wind and a well defined population characterized by σ_{c} = −1 and σ_{c} = 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_{r} ~ −1 and σ_{c} ~ 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 noncompressive, 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 nonlinear 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
Figure 49 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., (1990 and Tu et al., (1989b). The observed spectral indices, reported on the plot, are typically found within high velocity streams encountered at short heliocentric distances. Bruno et al. (1996) 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 logscale. The integrated power associated with e^{+} and e^{−} within the selected frequency bands is shown in Figure 50. 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. (1990) 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 nonlocal coupling in kspace between opposite modes found by Grappin et al. (1982) in homogeneous MHD. Expansion effects were also taken into account by Velli et al. (1990) who modeled inward modes as that fraction of outward modes backscattered by the inhomogeneities of the medium due to expansion effects (Velli et al., (1989). 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, (1992).
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., (1996) 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., (1985), 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, soundlike perturbations (Marsch and Tu, (1993a). 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.
4 Observations of MHD Turbulence in the Polar Wind
In 1994 – 1995, Ulysses gave us the opportunity to look at the solar wind outoftheecliptic, providing us with new exciting observations. For the first time heliospheric instruments were sampling pure, fast solar wind, free of any dynamical interaction with slow wind. There is one figure that within our scientific community has become as popular as “La Gioconda” by Leonardo da Vinci within the world of art. This figure produced at LANL (McComas et al., (1998) is shown in the upper left panel of Figure 51, which has been taken from a successive paper by (McComas et al., (2003), and summarizes the most important aspects of the large scale structure of the polar solar wind during the minimum of the solar activity phase, as indicated by the low value of the Wolf’s number reported in the lower panel. It shows speed profile, proton number density profile and magnetic field polarity vs. heliographic latitude during the first complete Ulysses’ polar orbit. Fast wind fills up north and south hemispheres of the Sun almost completely, except a narrow latitudinal belt around the equator, where the slow wind dominates. Flow velocity, which rapidly increases from the equator towards higher latitudes, quickly reaches a plateau and the wind escapes the polar regions with a rather uniform speed. Moreover, polar wind is characterized by a lower number density and shows rather uniform magnetic polarity of opposite sign, depending on the hemisphere. Thus, the main difference between ecliptic and polar wind is that this last one completely lacks of dynamical interactions with slower plasma and freely flows into the interplanetary space The presence or not of this phenomenon, as we will see in the following pages, plays a major role in the development of MHD turbulence during the wind expansion.
4.1 Evolving turbulence in the polar wind
 i.
Alfvénic turbulence would have not relaxed towards standard turbulence because the large scale velocity shears would have been much less relevant (Grappin et al., (1991);
 ii.
since the magnetic field would be smaller far from the ecliptic, at large heliocentric distances, even small shears would lead to an isotropization of the fluctuations and produce a turbulent cascade faster than the one observed at low latitudes, and the subsequent evolution would take less time (Roberts et al., (1990);
 iii.
there would still be evolution due to interaction with convected plasma and field structures but it would be slower than in the ecliptic since the power associated with Alfvénic fluctuations would largely dominate over the inhomogeneities of the medium. Thus, Alfvénic correlations should last longer than in the ecliptic plane, with a consequent slower evolution of the normalized crosshelicity (Bruno, (1992).
A fourth possibility was added by Tu and Marsch (1995a), based on their model (Tu and Marsch, (1993). Following this model they assumed that polar fluctuations were composed by outward Alfvénic fluctuations and MFDT. The spectra of these components would decrease with radial distance because of a WKB evolution and convective effects of the diverging flow. As the distance increases, the field becomes more transverse with respect to the radial direction, the s/c would sample more convective structures and, as a consequence, would observe a decrease of both σ_{c} and r_{A}.
Today we know that polar Alfvénic turbulence evolves in the same way it does in the ecliptic plane, but much more slowly. Moreover, the absence of strong velocity shears and enhanced compressive phenomena suggests that also some other mechanism based on parametric decay instability might play some role in the local production of turbulence (Bavassano et al., (2000a; Malara et al., (2001a, (2002; Primavera et al., (2003).
The first results of Ulysses magnetic field and plasma measurements in the polar regions, i.e., above ±30. latitude (left panel of Figure 51), revealed the presence of Alfvénic correlations in a frequency range from less than 1 to more than 10 h (Balogh et al., (1995; Smith et al., (1995; Goldstein et al., (1995a) in very good agreement with ecliptic observations (Bruno et al., (1985). However, it is worth noticing that Helios observations referred to very short heliocentric distances around 0.3 AU while the above Ulysses observations were taken up to 4 AU. As a matter of fact, these long period Alfvén waves observed in the ecliptic, in the inner solar wind, become less prominent as the wind expands due to streamstream dynamical interaction effects (Bruno et al., (1985) and strong velocity shears (Roberts et al., (1987a). At high latitude, the relative absence of enhanced dynamical interaction between flows at different speed and, as a consequence, the absence of strong velocity shears favors the survival of these extremely low frequency Alfvénic fluctuations for larger heliocentric excursions.
Horbury et al. (1995c) and Forsyth et al. (1996) showed that the interplanetary magnetic field fluctuations observed by Ulysses continuously evolve within the fast polar wind, at least out to 4 AU. Since this evolution was observed within the polar wind, rather free of corotating and transient events like those characterizing low latitudes, they concluded that some other mechanism was at work and this evolution was an intrinsic property of turbulence.
Results in Figure 54 show the evolution of the spectral slope computed across three different time scale intervals. The smallest time scales show a clear evolution that keeps on going past the highest latitude on day 256, strongly suggesting that this evolution is radial rather than latitudinal effect. Horbury et al. (1996a) worked on determining the rate of turbulent evolution for the polar wind.
Forsyth et al. (1996) studied the radial dependence of the normalized hourly variances of the components B_{ R }, B_{ T } and B_{ N } and the magnitude B of the magnetic field (see Appendix D to learn about the B reference system). The variance along the radial direction was computed as σ_{ R }^{2} = 〈B_{ R }^{2} > − < B_{ R }^{2} and successively normalized to B^{2} to remove the field strength dependence Moreover, variances along the other two directions T and N were similarly defined. Fitting the radial dependence with a power law of the form r^{−α}, but limiting the fit to the radial excursion between 1.5 and 3 AU, these authors obtained α = 3.39 ± 0.07 for σ _{r} ^{2} , α = 3.45 ± 0.09 for σ _{ T } ^{2} , α = 3.37 ± 0.09 for σ _{ N } ^{2} , and α = 2.48 ± 0.14 for σ _{ B } ^{2} . Thus, for hourly variances, the power associated with the components showed a radial dependence stronger than the one predicted by the WKB approximation, which would provide α = 3. These authors also showed that including data between 3 and 4 AU, corresponding to intervals characterized by compressional features mainly due to high latitude CMEs, they would obtain less steep radial gradients, much closer to a WKB type. These results suggested that compressive effects can feed energy at the smallest scales, counteracting dissipative phenomena and mimicking a WKBlike behavior of the fluctuations. However, they concluded that for lower frequencies, below the frequency break point, fluctuations do follow the WKB radial evolution.
On the basis of variations of spectral index and radial and latitudinal dependencies, these authors were able to identify four wave number ranges as indicated by the circled numbers in the top panel of Figure 58. Range 1 was characterized by a radial power decrease weaker than WKB (−3), positive latitudinal trend for components (more power at higher latitude) and negative for magnitude (less compressive events at higher latitudes). Range 2 showed a more rapid radial decrease of power for both magnitude and components and a negative latitudinal power trend, which implies less power at higher latitudes. Moreover, the spectral index of the components (bottom panel) is around 0.5 and tends to 0 at larger scales. Within range 3 the power of the components follows a WKB radial trend and the spectral index is around −1 for both magnitude and components. This hourly range has been identified as the most Alfvénic at low latitudes and its radial evolution has been recognized to be consistent with WKB radial index (Roberts, (1989; Marsch and Tu, (1990a). Even within this range, and also within the next one, the latitude power trend is slightly negative for both components and magnitude. Finally, range 4 is clearly indicative of turbulent cascade with a radial power trend of the components much faster than WKB expectation and becoming even stronger at higher wave numbers. Moreover, the radial spectral index reveals that steepening is at work only for the previous wave number ranges as expected since the breakpoint moves to smaller wave number during spectrum evolution. The spectral index of the components tends to −5/3 with increasing wave number while that of the magnitude is constantly flatter. The same authors gave an estimate of the radial scaleshift of the breakpoint during the wind expansion around k ∝ r^{1.1}, in agreement with earlier estimates (Horbury et al., 1996a).
Horbury and Balogh (2001) and Horbury and Tsurutani (2001) estimated that the power observed at 80° is about 30% less than that observed at 30°. These authors proposed a possible effect due to the overexpansion of the polar coronal hole at higher latitudes. In addition, within the fourth range, field magnitude fluctuations radially decrease less rapidly than the fluctuations of the components, but do not show significant latitudinal variations. Finally, the smaller spectral index reveals that the high frequency range of the field magnitude spectrum shows a flattening.
As already shown by Horbury et al. (1995b), around the 5 min range, magnetic field fluctuations are transverse to the mean field direction the majority of the time. The minimum variance direction lies mainly within an angle of about 26° from the average background field direction and fluctuations are highly anisotropic, such that the ratio between perpendicular to parallel power is about 30. Since during the observations reported in Horbury and Balogh (2001) and Horbury and Tsurutani (2001) the mean field resulted to be radially oriented most of the time, the radial minimum variance direction at short time scales is an effect induced by larger scales behavior.
Anyhow, radial and latitudinal anisotropy trends tend to disappear for higher frequencies. In the mean time, interesting enough, there is a strong radial increase of magnetic field compression (top panel of Figure 59), defined as the ratio between the power density associated with magnetic field intensity fluctuations and that associated with the fluctuations of the three components (Bavassano et al., (1982a; Bruno and Bavassano, (1991). The attempt to attribute this phenomenon to parametric decay of large amplitude Alfvén waves or dynamical interactions between adjacent flux tubes or interstellar pickup ions was not satisfactory in all cases.
Comparing high latitude with low latitude results for high speed streams, Horbury and Balogh (2001) found remarkable good agreement between observations by Ulysses at 2.5 AU and by Helios at 0.7 AU. In particular, Figure 60 shows Ulysses and Helios 1 spectra projected to 1 AU for comparison.
4.2 Polar turbulence studied via Elsässer variables
Goldstein et al. (1995a) for the first time showed a spectral analysis of Ulysses observations based on Elsässer variables during two different time intervals, at 4 AU and close to −40°, and at 2 AU and around the maximum southern pass, as shown in Figure 61. Comparing the two Ulysses observations it clearly appears that the spectrum closer to the Sun is less evolved than the spectrum measured farther out, as will be confirmed by the next Figure 62, where these authors reported the normalized crosshelicity and the Alfvén ratio for the two intervals. Moreover, following these authors, the comparison between Helios spectra at 0.3 AU and Ulysses at 2 and 4 AU suggests that the radial scaling of e^{+} at the low frequency end of the spectrum follows the WKB prediction of 1/r decrease (Heinemann and Olbert, 1980). However, the selected time interval for Helios s/c was characterized by rather slow wind taken during the rising phase the solar cycle, two conditions which greatly differ from those referring to Ulysses data. As a consequence, comparing Helios results with Ulysses results obtained within the fast polar wind might be misleading. It would be better to choose Helios observations within high speed corotating streams which resemble much better solar wind conditions at high latitude.
Anyhow, results relative to the normalized crosshelicity σ_{c} (see Figure 62) clearly show high values of σ_{c}, around 0.8, which normally we observe in the ecliptic at much shorter heliocentric distances (Tu and Marsch, (1995a). A possible radial effect would be responsible for the depleted level of σ_{c} at 4 AU. Moreover, a strong anisotropy can also be seen for frequencies between 10^{−6} − 10^{−5} Hz with the transverse σ_{c} much larger than the radial one. This anisotropy is somewhat lost during the expansion to 4 AU.
This different radial behavior is readily seen in the radial plot of the Elsässer ratio r_{E} shown in the top panel of the right column of Figure 63. Before 2.5 AU this ratio continuously grows to about 0.5 near 2.5 AU. Beyond this region, since the radial gradient of the inward and outward components is approximately the same, r_{E} stabilizes around 0.5.
In a successive paper, Bavassano et al. (2002a) tested whether or not the radial dependence observed in e^{±} was to be completely ascribed to the radial expansion of the wind or possible latitudinal dependencies also contributed to the turbulence evolution in the polar wind.
5 Numerical Simulations
Numerical simulations currently represent one of the main source of information about nonlinear evolution of fluid flows. The actual supercomputers are now powerful enough to simulate equations (NS or MHD) that describe turbulent flows with Reynolds numbers of the order of 10^{4} in twodimensional configurations, or 10^{3} in threedimensional one. Of course, we are far from achieving realistic values, but now we are able to investigate turbulence with an inertial range extended for more than one decade. Rather the main source of difficulties to get results from numerical simulations is the fact that they are made under some obvious constraints (say boundary conditions, equations to be simulated, etc.), mainly dictated by the limited physical description that we are able to use when numerical simulations are made, compared with the extreme richness of the phenomena involved: numerical simulations, even in standard conditions, are used tout court as models for the solar wind behavior. Perhaps the only exception, to our knowledge, is the attempt to describe the effects of the solar wind expansion on turbulence evolution like, for example, in the papers by Velli et al. (1989, (1990); Hellinger and Trávníček (2008). Even with this far too pessimistic point of view, used here solely as a few words of caution, simulations in some cases were able to reproduce some phenomena observed in the solar wind.
Nevertheless, numerical simulations have been playing a key role, and will continue to do so in our seeking an understanding of turbulent flows. Numerical simulations allows us to get information that cannot be obtained in laboratory. For example, high resolution numerical simulations provide information at every point on a grid and, for some times, about basic vector quantities and their derivatives. The number of degree of freedom required to resolve the smaller scales is proportional to a power of the Reynolds number, say to Re^{9/4}, although the dynamically relevant number of modes may be much less. Then one of the main challenge remaining is how to handle and analyze the huge data files produced by large simulations (of the order of Terabytes). Actually a lot of papers appeared in literature on computer simulations related to MHD turbulence The interested reader can look at the book by Biskamp (1993) and the reviews by Pouquet (1993, (1996).
5.1 Local production of Alfvénic turbulence in the ecliptic
The discovery of the strong correlation between velocity and magnetic field fluctuations has represented the motivation for some MHD numerical simulations, aimed to confirm the conjecture by Dobrowolny et al. (1980b). The high level of correlation seems to be due to a kind of selforganization (dynamical alignment) of MHD turbulence, generated by the natural evolution of MHD towards the strongest attractive fixed point of equations (Ting et al., (1986; Carbone and Veltri, (1987, (1992). Numerical simulations (Carbone and Veltri, (1992; Ting et al., (1986) confirmed this conjecture, say MHD turbulence spontaneously can tends towards a state were correlation increases, that is, the quantity σ_{c} = 2H_{c}/E, where H_{c} is the crosshelicity and E the total energy of the flow (see Appendix B.1), tends to be maximal.
The picture of the evolution of incompressible MHD turbulence, which comes out is rather nice but solar wind turbulence displays a more complicated behavior. In particular, as we have reported above, observations seems to point out that solar wind evolves in the opposite way. The correlation is high near the Sun, at larger radial distances, from 1 to 10 AU the correlation is progressively lower, while the level in fluctuations of mass density and magnetic field intensity increases. What is more difficult to understand is why correlation is progressively destroyed in the solar wind, while the natural evolution of MHD is towards a state of maximal normalized crosshelicity. A possible solution can be found in the fact that solar wind is neither incompressible nor statistically homogeneous, and some efforts to tentatively take into account more sophisticated effects have been made.
A mechanism, responsible for the radial evolution of turbulence, was suggested by Roberts and Goldstein (1988); Goldstein et al. (1989); and Roberts et al. (1991, (1992) and was based on velocity shear generation. The suggestion to adopt such a mechanism came from a detailed analysis made by Roberts et al. (1987a,b) of Helios and Voyager interplanetary observations of the radial evolution of the normalized crosshelicity σ_{c} at different time scales. Moreover, Voyager’s observations showed that plasma regions, which had not experienced dynamical interactions with neighboring plasma, kept the Alfvénic character of the fluctuations at distances as far as 8 AU (Roberts et al., (1987b). In particular, the vicinity of Helios trajectory to the interplanetary current sheet, characterized by low velocity flow, suggested Roberts et al. (1991) to include in his simulations a narrow low speed flow surrounded by two high speed flows. The idea was to mimic the slow, equatorial solar wind between north and south fast polar wind. Magnetic field profile and velocity shear were reconstructed using the six lowest Z^{±} Fourier modes as shown in Figure 67. An initial population of purely outward propagating Alfvénic fluctuations (z^{+}) was added at large k and was characterized by a spectral slope of k^{−1}. No inward modes were present in the same range. Results of Figure 67 show that the time evolution of z^{+} spectrum is quite rapid at the beginning, towards a steeper spectrum, and slows down successively. At the same time, z^{−} modes are created by the generation mechanism at higher and higher k but, along a Kolmogorovtype slope k^{−5/3}.
These results, although obtained from simulations performed using 2D incompressible spectral and pseudospectral codes, with fairly small Reynolds number of Re ≃ 200, were similar to the spectral evolution observed in the solar wind (Marsch and Tu, (1990a). Moreover, spatial averages across the simulation box revealed a strong crosshelicity depletion right across the slow wind, representing the heliospheric current sheet. However, magnetic field inversions and even relatively small velocity shears would largely affect an initially high Alfvénic flow (Roberts et al., (1992). However, Bavassano and Bruno (1992) studied an interaction region, repeatedly observed between 0.3 and 0.9 AU, characterized by a large velocity shear and previously thought to be a good candidate for shear generation (Bavassano and Bruno, (1989). They concluded that, even in the hypothesis of a very fast growth of the instability, inward modes would not have had enough time to fill up the whole region as observed by Helios 2.
Compressible numerical simulations have been performed by Veltri et al. (1992) and Malara et al. (1996, (2000) which invoked the interactions between small scale waves and large scale magnetic field gradients and the parametric instability, as characteristic effects to reduce correlations. In a compressible, statistically inhomogeneous medium such as the heliosphere, there are many processes which tend to destroy the natural evolution toward a maximal correlation, typical of standard MHD. In such a medium an Alfvén wave is subject to parametric decay instability (Viñas and Goldstein, 1991; Del Zanna et al., 2001; Del Zanna, 2001), which means that the mother wave decays in two modes: i) a compressive mode that dissipates energy because of the steepening effect, and ii) a backscattered Alfvénic mode with lower amplitude and frequency. Malara et al. (1996) showed that in a compressible medium, the correlation between the velocity and the magnetic field fluctuations is reduced because of the generation of the backward propagating Alfvénic fluctuations, and of a compressive component of turbulence, characterized by density fluctuations δρ ≠ 0 and magnetic intensity fluctuations δB ≠ 0.
From a technical point of view it is worthwhile to remark that, when a large scale field which varies on a narrow region is introduced (typically a tanhlike field), periodic boundaries conditions should be used with some care. Roberts et al. (1991, 1992) used a double shear layer, while Malara et al. (1992) introduced an interesting numerical technique based on both the glue between two simulation boxes and a Chebyshev expansion, to maintain a single shear layer, say non periodic boundary conditions, and an increased resolution where the shear layer exists.
5.2 Local production of Alfvénic turbulence at high latitude
An interesting solution to the radial behavior of the minority modes might be represented by local generation mechanisms, like parametric decay (Malara et al., (2001a; Del Zanna et al., 2001), which might saturate and be inhibited beyond 2.5 AU.
Parametric instability has been studied in a variety of situations depending on the value of the plasma β (among others Sagdeev and Galeev, (1969; Goldstein, (1978; Hoshino and Goldstein, (1989; Malara and Velli, (1996). Malara et al. (2000) and Del Zanna et al. (2001) recently studied the nonlinear growth of parametric decay of a broadband Alfvén wave, and showed that the final state strongly depends on the value of the plasma β (thermal to magnetic pressure ratio). For β < 1 the instability completely destroys the initial Alfvénic correlation. For β ~ 1 (a value close to solar wind conditions) the instability is not able to go beyond some limit in the disruption of the initial correlation between velocity and magnetic field fluctuations, and the final state is σ_{c} ~ 0.5 as observed in the solar wind (see Section 4.2).
These authors solved numerically the fully compressible, nonlinear MHD equations in a onedimensional configuration using a pseudospectral numerical code. The simulation starts with a nonmonochromatic, large amplitude Alfvén wave polarized on the yz plane, propagating in a uniform background magnetic field. Successively, the instability was triggered by adding some noise of the order 10^{−6} to the initial density level.
During the first part of the evolution of the instability the amplitude of unstable modes is small and, consequently, nonlinear couplings are negligible. A subsequent exponential growth, predicted by the linear theory, increases the level of both e^{−} and density compressive fluctuations. During the second part of the development of the instability, nonlinear couplings are not longer disregardable and their effect is firstly to slow down the exponential growth of unstable modes and then to saturate the instability to a level that depends on the value of the plasma β.
Spectra of e^{±} are shown in Figure 69 for different times during the development of the instability. At the beginning the spectrum of the motherwave is peaked at k = 10, and before the instability saturation (t ≤ 35) the backscattered e^{−} and the density fluctuations e^{ ρ } are peaked at k = 1 and k = 11, respectively. After saturation, as the run goes on, the spectrum of e^{−} approaches that of e^{+} towards a common final state characterized by a Kolmogorovlike spectrum and e^{+} slightly larger than e^{−}.
The behavior of outward and inward modes, density and magnetic magnitude variances and the normalized crosshelicity σ_{c} is summarized in the left column of Figure 70. The evolution of σ_{c}, when the instability reaches saturation, can be qualitatively compared with Ulysses observations (courtesy of B. Bavassano) in the right panel of the same figure, which shows a similar trend.
Obviously, making this comparison, one has to take into account that this model has strong limitations like the presence of a peak in e^{+} not observed in real polar turbulence Another limitation, partly due to dissipation that has to be included in the model, is that the spectra obtained at the end of the instability growth are steeper than those observed in the solar wind. Finally, a further limitation is represented by the fact that this code is 1D. However, although for an incompressible 1D simulation we do not expect to have turbulence development, in this case, since parametric decay is based on compressive phenomena, an energy transfer along the spectrum might be at work.
6 Compressive Turbulence
Interplanetary medium is slightly compressive, magnetic field intensity and proton number density experience fluctuations over all scales and the compression depends on both the scale and the nature of the wind. As a matter of fact, slow wind is generally more compressive than fast wind, as shown in Figure 71 where, following Bavassano et al. (1982a) and Bruno and Bavassano (1991), we report the ratio between the power density associated with magnetic field intensity fluctuations and that associated with the fluctuations of the three components. In addition, as already shown by Bavassano et al. (1982a), this parameter increases with heliocentric distance for both fast and slow wind as shown in the bottom panel, where the ratio between the compression at 0.9 AU and that at 0.3 AU is generally greater than 1. It is also interesting to notice that within the Alfvénic fast wind, the lowest compression is observed in the middle frequency range, roughly between 10^{−4} − 10^{−3} Hz. On the other hand, this frequency range has already been recognized as the most Alfvénic one, within the inner heliosphere (Bruno et al., (1996).
As a matter of fact, it seems that high Alfvénicity is correlated with low compressibility of the medium (Bruno and Bavassano, (1991; Klein et al., (1993; Bruno and Bavassano, (1993) although compressibility is not the only cause for a low Alfvénicity (Roberts et al., (1991, (1992; Roberts, (1992).
The radial dependence of the normalized number density fluctuations δn/n for the inner and outer heliosphere were studied by Grappin et al. (1990) and Roberts et al. (1987b for the hourly frequency range, but no clear radial trend emerged from these studies. However, interesting enough, Grappin et al. (1990) found that values of e^{−} were closely associated with enhancements of δn/n on scales longer than 1 h.
On the other hand, a spectral analysis of proton number density, magnetic field intensity, and proton temperature performed by Marsch and Tu (1990b) and Tu et al. (1991) in the inner heliosphere, separately for fast and slow wind (see Figure 72), showed that normalized spectra of the above parameters within slow wind were only marginally dependent on the radial distance On the contrary, within fast wind, magnetic field and proton density normalized spectra showed not only a clear radial dependence but also similar level of power for k < 4×10^{−4} km s^{−1}. For larger k these spectra show a flattening that becomes steeper for increasing distance, as was already found by Bavassano et al. (1982b) for magnetic field intensity. Normalized temperature spectra does not suffer any radial dependence neither in slow wind nor in fast wind.
Spectral index is around .5/3 for all the spectra in slow wind while, fast wind spectral index is around −5/3 for k < 4 × 10^{−4} km.1 and slightly less steep for larger wave numbers.
6.1 On the nature of compressive turbulence
These features, observed also in their dataset, were taken by Thieme et al. (1989) as evidence of stationary spatial structures which were supposed to be remnants of coronal structures convected by the wind. Different values assumed by plasma and field parameters within each structure were interpreted as a signature characterizing that particular structure and not destroyed during the expansion. These intervals, identifiable in Figure 73 by vertical dashed lines, were characterized by pressure balance and a clear anticorrelation between magnetic field intensity and temperature.
The idea of filamentary structures in the solar wind dates back to Parker (1964), followed by other authors like McCracken and Ness (1966), Siscoe et al. (1968), and more recently has been considered again in the literature with new results (see Section 10). These interplanetary flow tubes would be of different sizes, ranging from minutes to several hours and would be separated from each other by tangential discontinuities and characterized by different values of plasma parameters and a different magnetic field orientation and intensity. This kind of scenario, because of some similarity to a bunch of tangled, smoking “spaghetti” lifted by a fork, was then named “spaghettimodel”.
Following these observations, Tu and Marsch (1994) proposed a model in which fluctuations in temperature, density, and field directly derive from an ensemble of small amplitude pressure balanced structures and small amplitude fast perpendicular magnetosonic waves. These last ones should be generated by the dynamical interaction between adjacent flow tubes due to the expansion and, eventually, they would experience also a nonlinear cascade process to smaller scales. This model was able to reproduce most of the correlations described by Marsch and Tu (1993a) for fast wind.
Later on, Bavassano et al. (1996a) tried to characterize compressive fluctuations in terms of their polytropic index, which resulted to be a useful tool to study small scale variations in the solar wind. These authors followed the definition of polytropic fluid given by Chandrasekhar (1967): “a polytropic change is a quasistatic change of state carried out in such a way that the specific heat remains constant (at some prescribed value) during the entire process”. For such a variation of state the adiabatic laws are still valid provided that the adiabatic index γ is replaced by a new adiabatic index γ’ = (cp − c)/(cv − c) where c is the specific heat of the polytropic variation, and cp and cv are the specific heat at constant pressure and constant volume, respectively. This similarity is lost if we adopt the definition given by Courant and Friedrichs (1976), for whom a fluid is polytropic if its internal energy is proportional to the temperature. Since no restriction applies to the specific heats, relations between temperature, density, and pressure do not have a simple form as in Chandrasekhar approach (Zank and Matthaeus, (1991). Bavassano et al. (1996a) recovered the polytropic index from the relation between density n and temperature T changes for the selected scale Tn^{1−γ’} = const. and used it to determine whether changes in density and temperature were isobaric (γ’ = 0), isothermal (γ’ = 1), adiabatic (γ’ = γ), or isochoric (γ’ = ∞). Although the role of the magnetic field was neglected, reliable conclusions could be obtained whenever the above relations between temperature and density were strikingly clear. These authors found intervals characterized by variations at constant thermal pressure P. They interpreted these intervals as a subset of totalpressure balanced structures where the equilibrium was assured by the thermal component only, perhaps tiny flow tubes like those described by Thieme et al. (1989) and Tu and Marsch (1994). Adiabatic changes were probably related to magnetosonic waves excited by contiguous flow tubes (Tu and Marsch, (1994). Proton temperature changes at almost constant density were preferentially found in fast wind, close to the Sun. These regions were characterized by values of B and N remarkable stable and by strong Alfvénic fluctuations (Bruno et al., (1985). Thus, they suggested that these temperature changes could be remnants of thermal features already established at the base of the corona.
Thus, the polytropic index offers a very simple way to identify basic properties of solar wind fluctuations, provided that the magnetic field does not play a major role.
6.2 Compressive turbulence in the polar wind
Compressive fluctuations in high latitude solar wind have been extensively studied by Bavassano et al. (2004) looking at the relationship between different parameters of the solar wind and comparing these results with predictions by existing models.
These authors indicated with N, P_{ m }, P_{ k }, and P_{ t } the proton number density n, magnetic pressure, kinetic pressure and total pressure (P_{tot} = P_{ m } + P_{ k }), respectively, and computed correlation coefficients ρ between these parameters. Figure 75 clearly shows that a pronounced positive correlation for N − P_{ t } and a negative pronounced correlation for P_{ m } − P_{ k } is a constant feature of the observed compressive fluctuations. In particular, the correlation for N − P_{ t } is especially strong within polar regions at small heliocentric distance In midlatitude regions the correlation weakens, while almost disappears at low latitudes. In the case of P_{ m } − P_{ k }, the anticorrelation remains strong throughout the whole latitudinal excursion. For polar wind the anticorrelation appears to be less strong at small distances, just where the N − P_{ t } correlation is highest.
The role played by density and temperature in the anticorrelation between magnetic and thermal pressures is investigated in Figure 76, where the magnetic field magnitude is directly compared with proton density and temperature. As regards the polar regions, a strong BT anticorrelation is clearly apparent at all distances (right panel). For BN an anticorrelation tends to emerge when solar distance increases. This means that the magneticthermal pressure anticorrelation is mostly due to an anticorrelation of the magnetic field fluctuations with respect to temperature fluctuations, rather than density (see, e.g., Bavassano et al., (1996a,b). Outside polar regions the situation appears in part reversed, with a stronger role for the BN anticorrelation.
Finally, the relative density fluctuations dependence on the turbulent Mach number M (the ratio between velocity fluctuation amplitude and sound speed) is shown in Figure 78. The aim is to look for the presence, in the observed fluctuations, of nearly incompressible MHD behaviors. In the framework of the NI theory (Zank and Matthaeus, (1991, (1993) two different scalings for the relative density fluctuations are possible, as M or as M^{2}, depending on the role that thermal conduction effects may play in the plasma under study (namely a heatfluctuationdominated or a heatfluctuationmodified behavior, respectively). These scalings are shown in Figure 78 as solid (for M) and dashed (for M^{2}) lines.
The above observations fully support the view that compressive fluctuations in high latitude solar wind are a mixture of MHD modes and pressure balanced structures. It has to be reminded that previous studies (McComas et al., (1995, (1996; Reisenfeld et al., (1999) indicated a relevant presence of pressure balanced structures at hourly scales. Moreover, nearlyincompressible (see Section 6.1) effects do not seem to play any relevant role. Thus, polar observations do not show major differences when compared with ecliptic observations in fast wind, the only possible difference being a major role of pressure balanced structures.
6.3 The effect of compressive phenomena on Alfvénic correlations
A lack of δV − δB correlation does not strictly indicate a lack of Alfvénic fluctuations since a superposition of both outward and inward oriented fluctuations of the same amplitude would produce a very low correlation as well. In addition, the rather complicated scenario at the base of the corona, where both kinetic and magnetic phenomena contribute to the birth of the wind, suggest that the imprints of such a structured corona is carried away by the wind during its expansion. At this point, we would expect that solar wind fluctuations would not solely be due to the ubiquitous Alfvénic and other MHD propagating modes but also to an underlying structure convected by the wind, not necessarily characterized by Alfvénlike correlations. Moreover, dynamical interactions between fast and slow wind, built up during the expansion, contribute to increase the compressibility of the medium.
7 A Natural Wind Tunnel
The solar wind has been used as a wind tunnel by Burlaga who, at the beginning of the 1990s, started to investigate anomalous fluctuations (Burlaga, (1991a,b,c, (1995) as observed by measurements in the outer heliosphere by the Voyager spacecraft. In 1991, Marsch, in a review on solar wind turbulence given at the Solar Wind Seven conference, underlined the importance of investigating scaling laws in the solar wind and we like to report his sentence: “The recent work by Burlaga (1991a,b) opens in my mind a very promising avenue to analyze and understand solar wind turbulence from a new theoretical vantage point. ...This approach may also be useful for MHD turbulence Possible connections between intermittent turbulence and deterministic chaos have recently been investigated ...We are still waiting for applications of these modern concepts of chaos theory to solar wind MHD fluctuations.” (cf. Marsch, (1992, p. 503). A few years later Carbone (1993) and, independently, Biskamp (1993) faced the question of anomalous scaling from a theoretical point of view. More than ten years later the investigation of statistical mechanics of MHD turbulence from one side, and of lowfrequency solar wind turbulence on the other side, has produced a lot of papers, and is now mature enough to be tentatively presented in a more organic way.
7.1 Scaling exponents of structure functions
The phenomenology of turbulence developed by Kolmogorov (1941) deals with some statistical hypotheses for fluctuations. The famous footnote remark by Landau (Landau and Lifshitz, (1971) pointed out a defect in the Kolmogorov theory, namely the fact that the theory does not take proper account of spatial fluctuations of local dissipation rate (Frisch, (1995). This led different authors to investigate the features related to scaling laws of fluctuations and, in particular, to investigate the departure from the Kolmogorov’s linear scaling of the structure functions (cf. Section 2.8). An uptodate comprehensive review of these theoretical efforts can be found in the book by Frisch (1995).
Here we are interested in understanding what we can learn from solar wind turbulence about the basic features of scaling laws for fluctuations. We use velocity and magnetic fields time series, and we investigate the scaling behavior of the highorder moments of stochastic variables defined as variations of fields separated by a time^{8} interval τ. First of all, it is worthwhile to remark that scaling laws and, in particular, the exact relation (41) which defines the inertial range in fluid flows, is valid for longitudinal (streamwise) fluctuations. In common fluid flows the Kolmogorov linear scaling law is compared with the moments of longitudinal velocity differences. In the same way for the solar wind turbulence we investigate the scaling behavior of Δu_{ τ } = u(t+τ)−u(t), where u(t) represents the component of the velocity field along the radial direction. As far as the magnetic differences are concerned Δb_{ τ } = B(t+τ) − B(t), we are free for different choices and, in some sense, this is more interesting from an experimental point of view. We can use the reference system where B(t) represents the magnetic field projected along the radial direction, or the system where B(t) represents the magnetic field along the local background magnetic field, or B(t) represents the field along the minimum variance direction. As a different case we can simply investigate the scaling behavior of the fluctuations of the magnetic field intensity.
Scaling exponents for velocity ζ_{ p } and magnetic ξ_{ p } variables calculated through ESS. Errors represent the standard deviations of the linear fitting. The data used comes from a turbulent sample of slow wind at 0.9 AU from Helios 2 spacecraft. As a comparison we show the normalized scaling exponents of structure functions calculated in a wind tunnel on Earth (RuízChavarría et al., (1995) for velocity and temperature. The temperature is a passive scalar in this experiment.
p  ζ_{ p }  ξ_{ p }  u(t)(fluid)  T(t) (fluid) 

1  0.37 ± 0.06  0.56 ± 0.06  0.37  0.61 
2  0.70 ± 0.05  0.83 ± 0.05  0.70  0.85 
3  1.00  1.00  1.00  1.00 
4  1.28 ± 0.02  1.14 ± 0.02  1.28  1.12 
5  1.54 ± 0.03  1.25 ± 0.03  1.54  1.21 
6  1.79 ± 0.05  1.35 ± 0.05  1.78  1.38 
 i.
There is a significant departure from the Kolmogorov linear scaling, that is, real scaling exponents are anomalous and seem to be nonlinear functions of p, say ζ_{ p }/ζ_{3} > p/3 for p < 3, while ζ_{ p }/ζ_{3} < p/3 for p > 3. The same behavior can be observed for ξ_{ p }/ξ_{3}. In Table 1 we report also the scaling exponents obtained in usual fluid flows for velocity and temperature, the latter being a passive scalar. Scaling exponents for velocity field are similar to scaling exponents obtained in turbulent flows on Earth, showing a kind of universality in the anomaly. This effect is commonly attributed to the phenomenon of intermittency in fully developed turbulence (Frisch, (1995). Turbulence in the solar wind is intermittent, just like its fluid counterpart on Earth.
 ii.
The degree of intermittency is measured through the distance between the curve ζ_{ p }/ζ_{3} and the linear scaling p/3. It can be seen that the magnetic field is more intermittent than the velocity field. The same difference is observed between the velocity field and a passive scalar (in our case the temperature) in ordinary fluid flows (RuízChavarría et al., (1995). That is the magnetic field, as long as intermittency properties are concerned, has the same scaling laws of a passive field. Of course this does not mean that the magnetic field plays the same role as a passive field. Statistical properties are in general different from dynamical properties.
Normalized scaling exponents ξ_{ p }/ξ_{3} for radial magnetic fluctuations in a laboratory plasma, as measured at different distances a/R (R ≃ 0.45 cm being the minor radius of the torus in the experiment) from the external wall. Errors represent the standard deviations of the linear fitting. Scaling exponents have been obtained using the ESS.
p  a/R = 0.96  a/R = 0.93  a/R = 0.90  a/R = 0.86 

1  0.39 ± 0.01  0.38 ± 0.01  0.37 ± 0.01  0.36 ± 0.01 
2  0.74 ± 0.01  0.73 ± 0.02  0.71 ± 0.01  0.70 ± 0.01 
3  1.00  1.00  1.00  1.00 
4  1.20 ± 0.02  1.24 ± 0.02  1.27 ± 0.01  1.28 ± 0.01 
5  1.32 ± 0.03  1.41 ± 0.03  1.51 ± 0.03  1.55 ± 0.03 
6  1.38 ± 0.04  1.50 ± 0.04  1.71 ± 0.03  1.78 ± 0.04 
Only few large structures are enough to generate the anomalous scaling laws. In fact, as shown by Salem et al. (2009), by suppressing through wavelets analysis just a few percentage of large structures on all scales, the scaling exponents become linear functions of p, respectively p/4 and p/3 for the kinetic and magnetic fields.
As far as a comparison between different plasmas is concerned, the scaling exponents of magnetic structure functions, obtained from laboratory plasma experiments of a ReversedField Pinch at different distances from the external wall (Carbone et al., (2000) are shown in Table 2. In laboratory plasmas it is difficult to measure all the components of the vector field at the same time, thus, here we show only the scaling exponents obtained using magnetic field differences B_{ r }(t+τ)−B_{ r }(t) calculated from the radial component in a toroidal device where the zaxis is directed along the axis of the torus. As it can be seen, intermittency in magnetic turbulence is not so strong as it appears to be in the solar wind, actually the degree of intermittency increases when going toward the external wall. This last feature appears to be similar to what is currently observed in channel flows, where intermittency also increases when going towards the external wall (Pope, (2000).
Scaling exponents of structure functions for Alfvén variables, velocity, and magnetic variables have been calculated also for high resolution 2D incompressible MHD numerical simulations (Politano et al., (1998). In this case, we are freed from the constraint of the Taylor hypothesis when calculating the fluctuations at a given scale. From 2D simulations we recover the fields u(r, t) and b(r, t) at some fixed times. We calculate the longitudinal fluctuations directly in space at a fixed time, namely Δu_{∓} = [u(r+ℓ, t)− u(r, t)] · ℓ/ℓ (the same are made for different fields, namely the magnetic field or the Elsässer fields). Finally, averaging both in space and time, we calculate the scaling exponents through the structure functions. These scaling exponents are reported in Table 3. Note that, even in numerical simulations, intermittency for magnetic variables is stronger than for the velocity field.
7.2 Probability distribution functions and selfsimilarity of fluctuations
The presence of scaling laws for fluctuations is a signature of the presence of selfsimilarity in the phenomenon. A given observable u(ℓ), which depends on a scaling variable ℓ, is invariant with respect to the scaling relation ℓ → λℓ, when there exists a parameter μ(λ) such that u(ℓ) = μ(λ)u(λℓ). The solution of this last relation is a power law u(ℓ) = Cℓ^{ h }, where the scaling exponent is h = −log_{λ}μ.
Normalized scaling exponents ξ_{ p }/ξ_{3} for Alfvénic, velocity, and magnetic fluctuations obtained from data of high resolution 2D MHD numerical simulations. Scaling exponents have been calculated from spatial fluctuations; different times, in the statistically stationary state, have been used to improve statistics. The scaling exponents have been calculated by ESS using Equation (41) as characteristic scale rather than the thirdorder structure function (cf. Politano et al., (1998, for details).
p  Z ^{+}  Z ^{−}  v  B 

1  0.36 ± 0.06  0.56 ± 0.06  0.37 ± 0.01  0.46 ± 0.02 
2  0.70 ± 0.05  0.83 ± 0.05  0.70 ± 0.01  0.78 ± 0.01 
3  1.00  1.00  1.00  1.00 
4  1.28 ± 0.02  1.14 ± 0.02  1.28 ± 0.02  1.18 ± 0.02 
5  1.53 ± 0.03  1.25 ± 0.03  1.54 ± 0.03  1.31 ± 0.03 
6  1.79 ± 0.05  1.35 ± 0.05  1.78 ± 0.05  1.40 ± 0.03 
7.3 What is intermittent in the solar wind turbulence? The multifractal approach
Time dependence of Δu_{ τ } and Δb_{ τ } for three different scales τ is shown in Figures 85 and 86, respectively. These plots show that, as τ becomes small, intense fluctuations become more and more important, and they dominate the statistics. Fluctuations at large scales appear to be smooth while, as the scale becomes smaller, intense fluctuations becomes visible. These dominating fluctuations represent relatively rare events. Actually, at the smallest scales, the time behavior of both Δu_{ τ } and Δb_{ τ } is dominated by regions where fluctuations are low, in between regions where fluctuations are intense and turbulent activity is very high. Of course, this behavior cannot be described by a global selfsimilar behavior. Allowing the scaling laws to vary with the region of turbulence we are investigating would be more convincing.
7.4 Fragmentation models for the energy transfer rate
Cascade models view turbulence as a collection of fragments at a given scale ℓ, which results from the fragmentation of structures at the scale ℓ’ > ℓ, down to the dissipative scale (Novikov, (1969). Sophisticated statistics are applied to obtain scaling exponents ζ_{ p } for the pth order structure function.
The starting point of fragmentation models is the old βmodel, a “pedagogical” fractal model introduced by Frisch et al. (1978) to account for the modification of the cascade in a simple way. In this model, the cascade is realized through the conjecture that active eddies and nonactive eddies are present at each scale, the spacefilling factor for the fragments being fixed for each scale. Since it is a fractal model, the βmodel gives a linear modification to ζ_{ p }. This can account for a fit on the data, as far as small values of p are concerned. However, the whole curve ζ_{ p } is clearly nonlinear, and a multifractal approach is needed.
7.5 A model for the departure from selfsimilarity
7.6 Intermittency properties recovered via a shell model
A statistically stationary state is reached by the system Gloaguen et al. (1985); Biskamp (1994); Giuliani and Carbone (1998); Plunian et al. (2012), with a well defined inertial range, say a region where Equation (49) is verified. Spectra for both the velocity u_{ n }(t)^{2} and magnetic b_{ n }(t)^{2} variables, as a function of k_{ n }, obtained in the stationary state using the GOY MHD shell model, are shown in Figures 87 and 88. Fluctuations are averaged over time. The Kolmogorov spectrum is also reported as a solid line. It is worthwhile to remark that, by adding a random term like ik_{ n }B_{0}(t)Z _{ n } ^{±} to a little modified version of the MHD shell models (B_{0} is a random function with some statistical characteristics), a Kraichnan spectrum, say E(k_{ n }) ~ k _{ n } ^{−3/2} , where E(k_{ n }) is the total energy, can be recovered (Biskamp, (1994; Hattori and Ishizawa, (2001). The term added to the model could represent the effect of the occurrence of a largescale magnetic field.
Scaling exponents for velocity and magnetic variables, Elsässer variables, and fluxes. Errors on β _{ p } ^{±} are about one order of magnitude smaller than the errors shown.
p  ζ_{ p }  η_{ p }  ξ _{ p } ^{+}  ξ _{ p } ^{−}  β _{ p } ^{+}  β _{ p } ^{−} 

1  0.36 ± 0.01  0.35 ± 0.01  0.35 ± 0.01  0.36 ± 0.01  0.326  0.318 
2  0.71 ± 0.02  0.69 ± 0.03  0.70 ± 0.02  0.70 ± 0.03  0.671  0.666 
3  1.03 ± 0.03  1.01 ± 0.04  1.02 ± 0.04  1.02 ± 0.04  1.000  1.000 
4  1.31 ± 0.05  1.31 ± 0.06  1.30 ± 0.05  1.32 ± 0.06  1.317  1.323 
5  1.57 ± 0.07  1.58 ± 0.08  1.54 ± 0.07  1.60 ± 0.08  1.621  1.635 
6  1.80 ± 0.08  1.8 ± 0.10  1.79 ± 0.09  1.87 ± 0.10  1.91  1.94 
8 Observations of Yaglom’s Law in Solar Wind Turbulence
To avoid the risk of misunderstanding, let us start by recalling that Yaglom’s law (40) has been derived from a set of equations (MHD) and under assumptions which are far from representing an exact mathematical model for the solar wind plasma. Yaglom’s law is valid in MHD under the hypotheses of incompressibility, stationarity, homogeneity, and isotropy. Also, the form used for the dissipative terms of MHD equations is only valid for collisional plasmas, characterized by quasiMaxwellian distribution functions, and in case of equal kinematic viscosity and magnetic diffusivity coefficients (Biskamp, (2003). In solar wind plasmas the above hypotheses are only rough approximations, and MHD dissipative coefficients are not even defined (Tu and Marsch, (1995a). At frequencies higher than the ion cyclotron frequency, kinetic processes are indeed present, and a number of possible dissipation mechanisms can be discussed. When looking for the Yaglom’s law in the SW, the strong conjecture that the law remains valid for any form of the dissipative term is needed.
Despite the above considerations, Yaglom’s law results surprisingly verified in some solar wind samples. Results of the occurrence of Yaglom’s law in the ecliptic plane, has been reported by MacBride et al. (2008, (2010) and Smith et al. (2009) and, independently, in the polar wind by SorrisoValvo et al. (2007). It is worthwhile to note that, the occurrence of Yaglom’s law in polar wind, where fluctuations are Alfvénic, represents a double surprising feature because, according to the usual phenomenology of MHD turbulence, a nonlinear energy cascade should be absent for Alfénic turbulence.
In a first attempt to evaluate phenomenologically the value of the energy dissipation rate, MacBride et al. (2008) analyzed the data from ACE to evaluate the occurrence of both the Kolmogorov’s 4/5law and their MHD analog (40). Although some words of caution related to spikes in wind speed, magnetic field strength caused by shocks and other imposed heliospheric structures that constitute inhomogeneities in the data, authors found that both relations are more or less verified in solar wind turbulence They found a distribution for the energy dissipation rate, defined in the above paper as ∈ = (∈ _{ii} ^{+} + ∈ _{ii} ^{−} )/2, with an average of about ∈ ≃ 1.22 × 10^{4} J/Kg s.
Several other periods are found where the linear scaling range is reduced and, in particular, the sign of Y _{ℓ} ^{±} is observed to be either positive or negative. In some other periods the linear scaling law is observed either for Y _{ℓ} ^{+} or Y _{ℓ} ^{−} rather than for both quantities. It is worth noting that in a large fraction of cases the sign switches from negative to positive (or viceversa) at scales of about 1 day, roughly indicating the scale where the small scale Alfvénic correlations between velocity and magnetic fields are lost. This should indicate that the nature of fluctuations changes across the break. The values of the pseudoenergies dissipation rates ∈^{±} has been found to be of the order of magnitude about few hundreds of J/Kg s, higher than that found in usual fluid flows which result of the order of 1 ÷ 50 J/Kg s.
The occurrence of Yaglom’s law in solar wind turbulence has been evidenced by a systematic study by MacBride et al. (2010), which, using ACE data, found a reasonable linear scaling for the mixed thirdorder structure functions, from about 64 s. to several hours at 1 AU in the ecliptic plane. Assuming that the thirdorder mixed structure function is perpendicular to the mean field, or assuming that this function varies only with the component of the scale ℓ_{ α } that is perpendicular to the mean field, and is cylindrically symmetric, the Yaglom’s law would reduce to a 2D state. On the other hand, if the thirdorder function is parallel to the mean field or varies only with the component of the scale that is parallel to the mean field, the Yaglom’slaw would reduce to a 1Dlike case. In both cases the result will depend on the angle between the average magnetic field and the flow direction. In both cases the energy cascade rate varies in the range 10^{3} ÷ 10^{4} J/Kg s (see MacBride et al., (2010, for further details).
Quite interestingly, Smith et al. (2009) found that the pseudoenergy cascade rates derived from Yaglom’s scaling law reveal a strong dependence on the amount of crosshelicity. In particular, they showed that when the correlation between velocity and magnetic fluctuations are higher than about 0.75, the thirdorder moment of the outwardpropagating component, as well as of the total energy and crosshelicity are negative. As already made by SorrisoValvo et al. (2007), they attribute this phenomenon to a kind of inverse cascade, namely a backtransfer of energy from small to large scales within the inertial range of the dominant component. We should point out that experimental values of energy transfer rate in the incompressive case, estimated with different techniques from different data sets (Vasquez et al., (2007; MacBride et al., (2010), are only partially in agreement with that obtained by SorrisoValvo et al. (2007). However, the different nature of wind (ecliptic vs. polar, fast vs. slow, at different radial distances from the Sun) makes such a comparison only indicative.
Finally, it is worth reporting that the presence of Yaglom’s law in solar wind turbulence is an interesting theoretical topic, because this is the first real experimental evidence that the solar wind turbulence, at least at largescales, can be described within the magnetohydrodynamic model. In fact, Yaglom’s law is an exact law derived from MHD equations and, let us say once more, their occurrence in a medium like the solar wind is a welcomed surprise. By the way, the presence of the law in the polar wind solves the paradox of the presence of Alfvénic turbulence as first pointed out by Dobrowolny et al. (1980a). Of course, the presence of Yaglom’s law generates some controversial questions about data selection, reliability and a brief discussion on the extension of the inertial range. The interested reader can find some questions and relative answers in Physical Review Letters (Forman et al., (2010; SorrisoValvo et al., (2010a).
9 Intermittency Properties in the 3D Heliosphere: Taking a Look at the Data
In this section, we present a reasoned look at the main aspect of what has been reported in literature about the problem of intermittency in the solar wind turbulence In particular, we present results from data analysis.
9.1 Structure functions
Apart from the earliest investigations on the fractal structure of magnetic field as observed in interplanetary space (Burlaga and Klein, (1986), the starting point for the investigation of intermittency in the solar wind dates back to 1991, when Burlaga (1991a) started to look at the scaling of the bulk velocity fluctuations at 8.5 AU using Voyager 2 data. This author found that anomalous scaling laws for structure functions could be recovered in the range 0.85 ≤ r ≤ 13.6 h. This range of scales has been arbitrarily identified as a kind of “inertial range”, say a region were a linear scaling exists between log S _{ r } ^{(p)} and log r, and the scaling exponents have been calculated as the slope of these curves. However, structure functions of order p ≤ 20 were determined on the basis of only about 4500 data points. Nevertheless the scaling was found to be quite in agreement with that found in ordinary fluid flows. Although the data might be in agreement with the randomβ model, from a theoretical point of view Carbone (1993, (1994b) showed that normalized scaling exponents ζ_{ p }/ζ_{4} calculated by Burlaga (1991a) would be better fitted by using a pmodel derived from the Kraichnan phenomenology (Kraichnan, (1965; Carbone, (1993), and considering the parameter μ ≃ 0.77. The same author (Burlaga, 1991b) investigated the multifractal structure of the interplanetary magnetic field near 25 AU and analyzed positive defined fields as magnetic field strength, temperature, and density using the multifractal machinery of dissipation fields (Paladin and Vulpiani, 1987; Meneveau, (1991). Burlaga (1991c) showed that intermittent events observed in corotating streams at 1 AU should be described by a multifractal geometry. Even in this case the number of points used was very low to assure the reliability of highorder moments.
Marsch and Liu (1993) investigated the structure of intermittency of the turbulence observed in the inner heliosphere by using Helios 2 data. They analyzed both bulk velocity and Alfvén speed to calculate structure functions in the whole range 40.5 s (the instrument resolution) up to 24 h to estimate the pth order scaling exponents. Note that also in this analysis the number of data points used was too small to assure a reliability for order p = 20 structure functions as reported by Marsch and Liu (1993). From the analysis analogous to Burlaga (1991a), authors found that anomalous scaling laws are present. A comparison between fast and slow streams at two heliocentric distances, namely 0.3 AU and 1 AU, allows authors to conjecture a scenario for high speed streams were Alfvénic turbulence, originally selfsimilar (or poorly intermittent) near the Sun, “... loses its selfsimilarity and becomes more multifractal in nature” (Marsch and Liu, (1993), which means that intermittent corrections increase from 0.3 AU to 1 AU. No such behavior seems to occur in the slow solar wind. From a phenomenological point of view, Marsch and Liu (1993) found that data can be fitted with a piecewise linear function for the scaling exponents ζ_{ p }, namely a βmodel ζ_{pp} = 3 − D +p(D−2)/3, where D ≃ 3 for p ≤ 6 and D ≃ 2.6 for p > 6. Authors say that “We believe that we see similar indications in the data by Burlaga, who still prefers to fit his whole ζ_{ p } dataset with a single fit according to the nonlinear random βmodel”. We like to comment that the impression by Marsch and Liu (1993) is due to the fact that the number of data points used was very small. As a matter of fact, only structure functions of order p ≤ 4 are reliably described by the number of points used by Burlaga (1991a).
However, the data analyses quoted above, which in some sense present some contradictory results, are based on high order statistics which is not supported by an adequate number of data points and the range of scales, where scaling laws have been recovered, is not easily identifiable. To overcome these difficulties Carbone et al. (1996a) investigated the behavior of the normalized ratios ζ_{ p }/ζ_{3} through the ESS procedure described above, using data coming from lowspeed streams measurements of Helios 2 spacecraft. Using ESS the whole range covered by measurements is linear, and scaling exponent ratios can be reliably calculated. Moreover, to have a dataset with a high number of points, authors mixed in the same statistics data coming from different heliocentric distances (from 0.3 AU up to 1 AU). This is not correct as far as fast wind fluctuations are taken into account, because, as found by Marsch and Liu (1993) and Bruno et al. (2003b), there is a radial evolution of intermittency. Results showed that intermittency is a real characteristic of turbulence in the solar wind, and that the curve ζ_{ p }/ζ_{3} is a nonlinear function of p as soon as values of p ≤ 6 are considered.
Analysis of scaling exponents of pth order structure functions has been performed using different spacecraft datasets of Ulysses spacecraft. Horbury et al. (1995a) and Horbury et al. (1995c) investigated the structure functions of magnetic field as obtained from observations recorded between 1.7 and 4 AU, and covering a heliographic latitude between 40° and 80° south. By investigating the spectral index of the second order structure function, they found a decrease with heliocentric distance attributed to the radial evolution of fluctuations. Further investigations (see, e.g., Ruzmaikin et al., (1995) were obtained using structure functions to study the Ulysses magnetic field data in the range of scales 1 ≤ r ≤ 32 min. Ruzmaikin et al. (1995). showed that intermittency is at work and developed a bifractal model to describe Alfvénic turbulence They found that intermittency may change the spectral index of the second order structure function and this modifies the calculation of the spectral index (Carbone, (1994a.). Ruzmaikin et al. (1995). found that polar Alfvénic turbulence should be described by a Kraichnan phenomenology (Kraichnan, (1965). However, the same data can be fitted also with a fluidlike scaling law (Tu et al., (1996) and, due to the relatively small amount of data, it is difficult to decide, on the basis of the second order structure function, which scaling relation describes appropriately intermittency in the solar wind.
In a further paper Carbone et al. (1995b) provided evidence for differences in the ESS scaling laws between ordinary fluid flows and solar wind turbulence Through the analysis of different datasets collected in the solar wind and in ordinary fluid flows, it was shown that normalized scaling exponents ζ_{ p }/ζ_{3} are the same as far as p ≤ 8 are considered. This indicates a kind of universality in the scaling exponents for the velocity structure functions. Differences between scaling exponents calculated in ordinary fluid flows and solar wind turbulence are confined to highorder moments. Nevertheless, the differences found in the datasets were related to different kind of singular structures in the model described by Equation (65). Solar wind data can be fitted by that model as soon as the most intermittent structures are assumed to be planar sheets C = 1 and m = 4, that is a Kraichnan scaling is used. On the contrary, ordinary fluid flows can be fitted only when C = 2 and m = 3, that is, structures are filaments and the Kolmogorov scaling have been used. However it is worthwhile to remark that differences have been found for highorder structure functions, just where measurements are unreliable.
9.2 Probability distribution functions
As said in Section 7.2 the statistics of turbulent flows can be characterized by the PDF of field differences over varying scales. At large scales PDFs are Gaussian, while tails become higher than Gaussian (actually, PDFs decay as exp[−δZ _{ℓ} ^{±} ]) at smaller scales.
Marsch and Tu (1994) started to investigate the behavior of PDFs of fluctuations against scales and they found that PDFs are rather spiky at small scales and quite Gaussian at large scales. The same behavior have been obtained by SorrisoValvo et al. (1999, (2001) who investigated Helios 2 data for both velocity and magnetic field.
The values of the parameters σ_{0}, μ, and γ, in the fit of λ^{2}(τ) (see Equation (69) as a kernel for the scaling behavior of PDFs. FW and SW refer to fast and slow wind, respectively, as obtained from the Helios 2 spacecraft, by collecting in a single dataset all periods.
parameter  B field (SW)  V field (SW)  B field (FW)  V field (FW) 

σ_{0}  0.90 ± 0.05  0.95 ± 0.05  0.85 ± 0.05  0.90 ± 0.05 
μ  0.75 ± 0.03  0.38 ± 0.02  0.90 ± 0.03  0.54 ± 0.03 
γ  0.18 ± 0.03  0.20 ± 0.04  0.19 ± 0.02  0.44 ± 0.05 
10 Turbulent Structures
The nonlinear energy cascade towards smaller scales accumulates fluctuations only in relatively small regions of space, where gradients become singular. As a rather different point of view (see Farge, (1992) these regions can be viewed as localized zones of fluid where phase correlation exists, in some sense coherent structures. These structures, which dominate the statistics of small scales, occur as isolated events with a typical lifetime greater than that of stochastic fluctuations surrounding them. The idea of a turbulence in the solar wind made by a mixture of structures convected by the wind and stochastic fluctuations is not particularly new (see, e.g., Tu and Marsch, (1995a). However, these largescale structures cannot be considered as intermittent structures at all scales. Structures continuously appear and disappear apparently in a random fashion, at some random location of fluid, and carry a great quantity of energy of the flow. In this framework intermittency can be considered as the result of the occurrence of coherent (nonGaussian) structures at all scales, within the sea of stochastic Gaussian fluctuations.
This point of view is the result of data analysis of scaling laws of turbulent fluctuations made by using wavelets filters (see Appendix C) instead of the usual Fourier transform. Unlike the Fourier basis, wavelets allow a decomposition both in time and frequency (or space and scale). In analyzing intermittent structures it is useful to introduce a measure of local intermittency, as for example the Local Intermittency Measure (LIM) introduced by Farge (1992) (see Appendix C).
 i.
i. Some of the structures are the well known onedimensional current sheets, characterized by pressure balance and almost constant density and temperature. When a minimum variance analysis is made on the magnetic field near the structure, it can be seen that the most variable component of the magnetic field changes sign. This component is perpendicular to the average magnetic field, the third component being zero. An interesting property of these structures is that the correlation between velocity and magnetic field within them is opposite with respect to the rest of fluctuations. That is, when they occur during Alfvénic periods velocity and magnetic field correlation is low; on the contrary, during nonAlfvénic periods the correlation of structure increases.
 ii.
ii. A different kind of structures looks like a shock wave. They can be parallel shocks or slowmode shocks. In the first case they are observed on the radial component of the velocity field, but are also seen on the magnetic field intensity, proton temperature, and density. In the second case they are characterized by a very low value of the plasma β parameter, constant pressure, anticorrelation between density and proton temperature, no magnetic fluctuations, and velocity fluctuations directed along the average magnetic field.
However, Salem et al. (2009), as already anticipated in Section 3.1.1, demonstrated that a monofractal can be recovered and intermittency eliminated simply by subtracting a small subset of the events at small scales.
Given a turbulent time series, as derived in the solar wind, a very interesting statistics can be made on the time separation between the occurrence of two consecutive structures. Let us consider a signal, for example u(t) or b(t) derived from solar wind, and let us define the wavelets set w_{ s }(r, t) as the set which captures, at time t, the occurrence of structures at the scale r. Then define the waiting times δt, as that time between two consecutive structures at the scale r, that is, between w_{ s }(r, t) and w_{ s }(r, t + δt). The PDFs of waiting times P(δt) are reported in Figure 82. As it can be seen, waiting times are distributed according to a power law P(δt) ~ δt^{−β} extended over at least two decades. This property is very interesting, because this means that the underlying process for the energy cascade is nonPoissonian. Waiting times occurring between isolated Poissonian events, must be distributed according to an exponential function. The power law for P(δt) represents the asymptotic behavior of a LLevy function with characteristic exponent α = β − 1. This describes selfaffine processes and are obtained from the central limit theorem by relaxing the hypothesis that the variance of variables is finite. The power law for waiting times we found is a clear evidence that longrange correlation (or in some sense “memory”) exists in the underlying cascade process.
On the other hand, Bruno et al. (2001), analyzing the statistics of the occurrence of waiting times of magnetic field intensity and wind speed intermittent events for a short time interval within the trailing edge of a high velocity stream, found a possible Poissonianlike behavior with a characteristic time around 30 min for both magnetic field and wind speed. These results are to be compared with previous estimates of the occurrence of interplanetary discontinuities performed by Tsurutani and Smith (1979), who found a waiting time around 14 min. In addition, Bruno et al. (2001), taking into account the wind speed and the orientation of the magnetic field vector at the site of the observation, in the hypothesis of spherical expansion, estimated the corresponding size at the Sun surface that resulted to be of the order of the photospheric structures estimated also by Thieme et al. (1989). Obviously, the Poissonian statistics found by these authors does not agree with the clear power law shown in Figure 82. However, Bruno et al. (2001) included intermittent events found at all scales while results shown in Figure 82 refer to waiting times between intermittent events extracted at the smallest scale, which results to be about an order of magnitude smaller than the time resolution used by Bruno et al. (2001). A detailed study on this topic would certainly clarify possible influences on the waiting time statistics due to the selection of intermittent events according to the corresponding scale.
In the same study by Bruno et al. (2001), these authors analyzed in detail an event characterized by a strong intermittent signature in the magnetic field intensity. A comparative study was performed choosing a closeby time interval which, although intermittent in velocity, was not characterized by strong magnetic intermittency. This time interval was located a few hours apart from the previous one. These two intervals are indicated in Figure 96 by the two vertical boxes labeled 1 and 2, respectively. Wind speed profile and magnetic field magnitude are shown in the first two panels. In the third panel, the blue line refers to the logarithmic value of the magnetic pressure P_{ m }, here indicated by P_{ B }; the red line refers to the logarithmic value of the thermal pressure P_{ k }, here indicated by P_{ K } and the black line refers to the logarithmic value of the total pressure P_{tot}, here indicated by P_{T} = P_{ B } + P_{ K }, including an average estimate of the electrons and αs contributions. Magnetic field intensity residuals, obtained from the LIM technique, are shown in the bottom panel. The first interval is characterized by strong magnetic field intermittency while the second one is not. In particular, the first event corresponds to a relatively strong field discontinuity which separates two regions characterized by a different bulk velocity and different level of total pressure. While kinetic pressure (red trace) does not show any major jump across the discontinuity but only a light trend, magnetic pressure (blue trace) clearly shows two distinct levels.
A minimum variance analysis further reveals the intrinsic different nature of these two intervals as shown in Figure 97 where original data have been rotated into the field minimum variance reference system (see Appendix D.1) where maximum, intermediate and minimum variance components are identified by λ_{3}, λ_{2}, and λ_{1}, respectively. Moreover, at the bottom of the column we show the hodogram on the maximum variance plane λ_{3} − λ_{2}, as a function of time on the vertical axis.
Within the second interval, magnetic field intensity is rather constant and the three components do not show any particular fluctuation, which could resemble any sort of rotation. In other words, the projection on the maximum variance plane does not show any coherent path. Even in this case, these fluctuations happen to be in a plane almost perpendicular to the average field direction since the angle between this direction and the minimum variance direction is about 9.3°.
These differences in the dynamics of the orientation of the field vector can be appreciated running the two animations behind Figures 99 and 100. Although the data used for these movies do not exactly correspond to the same time intervals analyzed in Figure 96, they show the same dynamics that the field vector has within intervals #1 and #2. In particular, the animation corresponding to Figure 99 represents interval #2 while, Figure 100 represents interval #1.
10.1 On the statistics of magnetic field directional fluctuations
Interesting enough is to look at the statistics of the angular jumps relative to the orientation of the magnetic field vector. Studies of this kind can help to infer the relevance of modes and advected structures within MHD turbulent fluctuations. Bruno et al. (2004) found that PDFs of interplanetary magnetic field vector angular displacements within high velocity streams can be reasonably fitted by a double lognormal distribution, reminiscent of multiplicative processes following turbulence evolution. As a matter of fact, the multiplicative cascade notion was introduced by Kolmogorov into his statistical theory (Kolmogorov, (1941, (1991, (1962) of turbulence as a phenomenological framework to accommodate extreme behavior observed in real turbulent fluids.
Using the same methodology, Li et al. (2008) also studied fluctuations in the Earth’s magnetotail to highlight the absence of similar structures and to conclude that most of those advected structures observed in the solar wind must be of solar origin.
10.2 Radial evolution of intermittency in the ecliptic
Bruno et al. (2003a) investigated the radial evolution of intermittency in the inner heliosphere, using the behavior of the flatness of the PDF of magnetic field and velocity fluctuations as a function of scale. As a matter of fact, probability distribution functions of fluctuating fields affected by intermittency become more and more peaked at smaller and smaller scales. Since the peakedness of a distribution is measured by its flatness factor, they studied the behavior of this parameter at different scales to estimate the degree of intermittency of their time series, as suggested by Frisch (1995).

Magnetic field fluctuations are more intermittent than velocity fluctuations.

Compressive fluctuations are more intermittent than directional fluctuations.

Slow wind intermittency does not show appreciable radial dependence

Fast wind intermittency, for both magnetic field and velocity, clearly increases with distance

Magnetic and velocity fluctuations have a rather Gaussian behavior at large scales, as expected, regardless of type of wind or heliocentric distance.
Moreover, they also found that the intermittency of the components rotated into the mean field reference system (see Appendix D.1) showed that the most intermittent component of the magnetic field is the one along the mean field, while the other two show a similar level of intermittency within the associated uncertainties. Finally, with increasing the radial distance, the component along the mean field becomes more and more intermittent with respect to the transverse components. These results agree with conclusions drawn by Marsch and Tu (1994) who, analyzing fast and slow wind at 0.3 AU in Solar Ecliptic (SE hereafter) coordinate system, found that the PDFs of the fluctuations of transverse components of both velocity and magnetic fields, constructed for different time scales, were appreciably more Gaussianlike than fluctuations observed for the radial component, which resulted to be more and more spiky for smaller and smaller scales.
However, at odds with results by Bruno et al. (2003a), Tu et al. (1996) could not establish any radial dependence due to the fact that their analysis was performed in the SE reference system instead of the mean field reference system as in the analysis of Bruno et al. (2003a). As a matter of fact, the mean field reference system is a more natural reference system where to study magnetic field fluctuations.
The reason is that components normal to the mean field direction are more influenced by Alfvénic fluctuations and, as a consequence, their fluctuations are more stochastic and less intermittent. This effect largely reduces during the radial excursion mainly because in the SE reference system crosstalking between different components is artificially introduced. As a matter of fact, the presence of the large scale spiral magnetic field breaks the spatial symmetry introducing a preferential direction parallel to the mean field. The same Bruno et al. (2003b) showed that it was not possible to find a clear radial trend unless magnetic field data were rotated into this more natural reference system.
On the other hand, it looks more difficult to reconcile the radial evolution of intermittency found by Bruno et al. (2003b) and Marsch and Liu (1993) in fast wind with conclusions drawn by Tu et al. (1996), who stated that “Neither a clear radial evolution nor a clear anisotropy can be established. The value of P1 in highspeed and lowspeed wind are not prominent different.”. However, it is very likely that the conclusions given above are related with how to deal with the flat slope of the spectrum in fast wind near 0.3 AU. Tu et al. (1996) concluded, indeed: “It should be pointed out that the extended model cannot be used to analyze the intermittency of such fluctuations which have a flat spectrum. If the index of the power spectrum is near or less than unity ... P1 would be 0.5. However, this does not mean there is no intermittency. The model simply cannot be used in this case, because the structure function(1) does not represent the effects of intermittency adequately for those fluctuations which have a flat spectrum and reveal no clear scaling behavior”.
Bruno et al. (2003a) suggested that, depending on the type of solar wind sample and on the heliocentric distance, the observed scaling properties would change accordingly. In particular, as the radial distance increases, convected, coherent structures of the wind assume a more relevant role since the Alfvénic component of the fluctuations is depleted. This would be reflected in the increased intermittent character of the fluctuations. The coherent nature of the convected structures would contribute to increase intermittency while the stochastic character of the Alfvénic fluctuations would contribute to decrease it. This interpretation would also justify why compressive fluctuations are always more intermittent than directional fluctuations. As a matter of fact, coherent structures would contribute to the intermittency of compressive fluctuations and, at the same time, would also produce intermittency in directional fluctuations. However, since directional fluctuations are greatly influenced by Alfvénic stochastic fluctuations, their intermittency will be more or less reduced depending on the amplitude of the Alfvén waves with respect to the amplitude of compressive fluctuations.
The radial dependence of the intermittency behavior of solar wind fluctuations stimulated Bruno et al. (1999b) to reconsider previous investigations on fluctuations anisotropy reported in Section 3.1.4. These authors studied magnetic field and velocity fluctuations anisotropy for the same corotating, high velocity stream observed by Bavassano et al. (1982a) within the framework of the dynamics of nonlinear systems. Using the Local Intermittency Measure (Farge et al., (1990; Farge, (1992; Bruno et al., (1999b) were able to justify the controversy between results by Klein et al. (1991) in the outer heliosphere and Bavassano et al. (1982a) in the inner heliosphere. Exploiting the possibility offered by this technique to locate in space and time those events which produce intermittency, these authors were able to remove intermittent events and perform again the anisotropy analysis. They found that intermittency strongly affected the radial dependence of magnetic fluctuations while it was less effective on velocity fluctuations. In particular, after intermittency removal, the average level of anisotropy decreased for both magnetic and velocity field at all distances. Although magnetic fluctuations remained more anisotropic than their kinetic counterpart, the radial dependence was eliminated. On the other hand, the velocity field anisotropy showed that intermittency, although altering the anisotropic level of the fluctuations, does not markedly change its radial trend.
10.3 Radial evolution of intermittency at high latitude
Recently, Pagel and Balogh (2003), studied intermittency in the outer heliosphere using Ulysses observations at high heliographic latitude, well within high speed solar wind. In particular, these authors used Castaing distribution Castaing et al. (2001) to study the Probability Distribution Functions (PDF) of the fluctuations of magnetic field components (see Section 9.2 for description of Castaing distribution and related governing parameters definition λ and σ). They found that intermittency of small scales fluctuations, within the inertial range, increased with increasing the radial distance from the Sun as a consequence of the growth to larger scales of the inertial range.
As a matter of fact, using the scaling found by Horbury et al. (1996a) between the transition scale (the inverse of the frequency corresponding to the breakpoint in the magnetic field spectrum) R_{B} ~ r^{1.1±0.1}, Pagel and Balogh (2003), quantitatively evaluated how the top of the inertial range in their data should shift to larger time scales with increasing heliocentric distance Moreover, taking into account that inside the inertial range λ_{2} ~ τ−β ⟹ λ^{2} = aτ^{−β} and that the proposed scaling from Castaing et al. (2001) would be λ^{2} ~ const.(τ/T)^{−β}, we should expect that for τ = T the parameter λ^{2} = const.. Thus, these authors calculated σ^{2} and λσ^{2} at different heliocentric distances and made the hypothesis of a similar scaling for λ^{2} and σ^{2}, although this is not assured by the model. Figure 108 reports values of λ^{2} and σ^{2} vs. distance calculated for the top of the inertial range at that distance using the above procedure. The radial behavior shown in this figure suggests that there is no radial dependence for these parameters for all the three components (indicated by different symbols), as expected if the observed radial increase of intermittency in the inertial range is due to a broadening of the inertial range itself.
Pagel and Balogh (2002) focused also on the different intermittent level of magnetic field fluctuations during two fast latitudinal scans which happened to be during solar minimum the first one, and during solar maximum the second one. Their results showed a strong latitudinal dependence but were probably not, or just slightly, affected by radial dependence given the short heliocentric radial variations during these time intervals. They analyzed the anomalous scaling of the third order magnetic field structure functions looking at the value of the parameter μparticular, this last model was an obtained from the best fit performed using the pmodel (see Section 7.4). In a previous analysis of the same kind, but focalized on the first latitudinal scan, the same authors tested three intermittency models, namely: “lognormal”, “p” and “Ginfinity” models. In particular, this last model was an empirical model introduced by Pierrehumbert (1999), and Cho et al. (2000), and was not intended for turbulent systems. Anyhow, the best fits were obtained with the lognormal and Kolmogorovp model. These authors concluded that magnetic field components display a very high level of intermittency throughout minimum and maximum phases of solar cycle, and slow wind shows a lower level of intermittency compared with the Alfvénic polar flows. These results do not seem to agree with ecliptic observations (Marsch and Liu, (1993,; Bruno et al., (2003a) which showed that fast wind is generally less intermittent than slow wind not only for wind speed and magnetic field magnitude, but also for the components. At this point, since it has been widely recognized that low latitude fast wind collected within corotating streams and fast polar wind share many common turbulence features, they should be expected to have many similarities also as regards intermittency. Thus, it is possible that also in this case the reference system in which the analysis is performed plays some role in determining some of the results regarding the behavior of the components. In any case, further analyses should clarify the reasons for this discrepancy.
11 Solar Wind Heating by the Turbulent Energy Cascade
The Parker theory of solar wind (Parker, (1964) predicts an adiabatic expansion from the hot corona without further heating. For such a model, the proton temperature T(r) should decrease with the heliocentric distance r as T(r) ~ r^{−4/3}. The radial profile of proton temperature have been obtained from measurements by the Helios spacecraft at 0.3 AU (Marsch et al., (1982,; Marsch, (1983,; Schwenn, (1983,; Freeman, (1988,; Goldstein, (1996), up to 100 AU or more by Voyager and Pioneer spacecrafts (Gazis, (1984,; Gazis et al., (1994,; Richardson et al., (1995). These measurements show that the temperature decay is in fact considerably slower than expected. Fits of the radial temperature profile gave an effective decrease T ~ T_{0}(r_{0}/r)^{ξ} in the ecliptic plane, with the exponent ξ ∈ [0.7; 1], much smaller than the adiabatic case. Actually ξ ≃ 1 within 1 AU, while ξ flattens to ξ ≃ 0.7 beyond 30 AU, where pickup ions probably contribute significantly (Richardson et al., (1995,; Zank et al., (1996,; Smith et al., (2001b). These observations imply that some heating mechanism must be at work within the wind plasma to supply the energy required to slow down the decay. The nature of the heating process of solar wind is an open problem.
The primary process governing the solar wind heating is probably active locally in the wind. However, since collisions are very rare in the solar wind plasma, the usual viscous coefficients have no meaning, say energy must be transferred to very small scales before it can be efficiently dissipated, perhaps by kinetic processes. As a consequence, the presence of a turbulent energy flux is the crucial first step towards the understanding of solar wind heating (Coleman, (1968,; Tu and Marsch, (1995a) because, as said in Section 2.4, the turbulent energy cascade represents nothing but the way for energy to be efficiently dissipated in a highReynolds number flow.^{11} In other words, before to face the problem of what actually be the physical mechanisms responsible for energy dissipation, if we conjecture that these processes happens at small scales, the turbulent energy flux towards small scales must be of the same order of the heating rate.
11.1 Dissipative/dispersive range in the solar wind turbulence
As we saw in Section 8, the energy cascade in turbulence can be recognized by looking at Yaglom’s law. The presence of this law in the solar wind turbulence showed that an energy cascade is at work, thus transferring energy to small scales where it is dissipated by some mechanism. While, as we showed before, the inertial range of turbulence in solar wind can be described more or less in a fluid framework, the small scales dissipative region can be much more (perhaps completely) different. The main motivation for this is the fact that the collision length in the solar wind, as a rough estimate the thermal velocity divided by the collision frequency, results to be of the order of 1 AU. Then the solar wind behaves formally as a collisionless plasma, that is the usual viscous dissipation is negligible. At the same time, in a magnetized plasma there are a number of characteristic scales, then understanding the physics of the generation of the smallscale region of turbulence in solar wind is a challenging topic from the point of view of basic plasma physics. With smallscales we mean scales ranging between the ioncyclotron frequency f_{ci} = eB/m_{ i } (which in the solar wind at 1 AU is about f_{ci} ≃ 0.1 Hz), or the ion inertial length λ_{ i } = c/ω_{ pi }, and the electroncyclotron frequency. At these scales the usual MHD approximation could breaks down in favour of a more complex description of plasma where kinetic processes must take place.
It is evident that this quantity increases as the frequency becomes smaller, thus indicating the presence of intermittency. However the rate at which K(f) increases is pronounced above the ion cyclotron frequency, meaning that intermittency in the highfrequency range is much more effective than in the lowfrequency region. Recently, by analyzing a different dataset from Cluster spacecraft, Kiyani et al. (2009), using highorder statistics of magnetic differences, showed that the scaling exponents of structure functions, evaluated at small scales, are no more anomalous as the lowfrequency range, even if the situation is not so clear (Yordanova et al., (2008, (2009). This is a good example of absence of universality in turbulence, a topic which received renewed attention in the last years (Chapman et al., (2009; Lee et al., (2010; Matthaeus, (2009).
12 The Origin of the HighFrequency Region
How is the highfrequency region of the spectrum generated? This has become the urgent topic which must be addressed. Ghosh et al. (1996) appeals to change of invariants in controlling the flow of spectral energy transfer in the cascade process, and in this picture no dissipation is required to explain the steepening of the magnetic power spectrum. Furthermore it is believed that the highfrequency region is highly anisotropic, with a significant fraction of turbulent energy cascades mostly in the quasi 2D structures, perpendicular to the background magnetic field. How magnetic energy dissipated in the anisotropic energy cascade still remains an open question.
12.1 A dissipation range
As we have already said, in their analysis of Wind data Leamon et al. (1998), attribute the presence of the region at frequencies higher than the ioncyclotron frequency to a kind of dissipative range. Apart for the power spectrum, the authors examined the normalized reduced magnetic helicity σ(f), and they found an excess of negative values at high frequencies. Since this quantity is a measure of the spatial handedness of the magnetic field (Moffatt, (1978) and can be related to the polarization in the plasma frame once the direction propagation direction is known (Smith et al., (1983), the above observations should be consistent with the ioncyclotron damping of Alfvén waves. Using a reference system relative to the mean magnetic field direction e_{ B } and radial direction e_{ R } as (e_{ B } × e_{ R }, e_{ B } × (e_{ B } × e_{ R }), e_{ B }), they conclude that transverse fluctuations are less dominant than in the inertial range and the high frequency range is best described by a mixture of 46% slab waves and of 54% 2D geometry. Since in the lowfrequency range they found 11% and 89% respectively, the increased slab fraction my be explained by the preferential dissipation of oblique structures. Thermal particles interactions with the 2D slab component may be responsible for the formation of dissipative range, even if the situation seems to be more complicated. In fact they found that kinetic Alfvén waves propagating at large angles to the background magnetic field might be also consistent with the observations and may form some portion of the 2D component.
Recently the question of the increased anisotropy of the highfrequency region has been addressed by Perri et al. (2009) who investigated the scaling behavior of the eigenvalues of the variance matrix of magnetic fluctuations, which give information on the anisotropy due to different polarizations of fluctuations. The authors investigated data coming from Cluster spacecrafts when satellites orbited in front of the Earth’s parallel Bow Shock (Perri et al., (2009). Results indicates that magnetic turbulence in the highfrequency region is strongly anisotropic, the minimum variance direction being almost parallel to the background magnetic field at scales larger than the ion cyclotron scale. A very interesting result is the fact that the eigenvalues of the variance matrix have a strong intermittent behavior, with very high localized fluctuations below the ion cyclotron scale. This behavior, never investigated before, generates a crossscale effect in magnetic turbulence Indeed, PDFs of eigenvalues evolve with the scale, namely they are almost Gaussian above the ion cyclotron scale and become power laws at scales smaller than the ion cyclotron scale. As a consequence it is not possible to define a characteristic value (as the average value) for the eigenvalues of the variance matrix at small scales. Since the wavevector spectrum of magnetic turbulence is related to the characteristic eigenvalues of the variance matrix (Carbone et al., (1995a), the absence of a characteristic value means that a typical power spectrum at small scales cannot be properly defined. This is a feature which received little attention, and represents a further indication for the absence of universal characteristics of turbulence at small scales.
12.2 A dispersive range
13 Two Further Questions About SmallScale Turbulence
The most “conservative” way to describe the presence of a dissipative/dispersive region in the solar wind turbulence, as we reported before, is for example through the HallMHD model. While when dealing with large scale we can successfully approach the problem of turbulence by saying that some form of dissipation must exist at small scales, the dissipationless character of solar wind cannot be avoided when we deal with small scales. The full understanding of the physical mechanisms that allow the dissipation of energy in the absence of collisional viscosity would be a step of crucial importance in the problem of high frequency turbulence in space plasmas. Another fundamental question concerns the dispersive properties of smallscale turbulence beyond the spectral break. This last question has been reformulated by saying: what are the principal constituent modes of smallscale turbulence? This approach explicitly assumes that smallscale fluctuations in solar wind can be described through a weak turbulence framework. In other words, a dispersion relation, namely a precise relationship between the frequency ω and the wavevector k, is assumed.
As it is well known from basic plasma physics, linear theory for homogeneous, collisionless plasma yields three kind of modes at and below the proton cyclotron frequency Ω_{ p }. At wavevectors transverse to the background magnetic field and Ω_{ p } > ω_{ r } (being ω_{ r } the real part of the frequency of fluctuation), two modes are present, namely a lefthand polarized Alfvén cyclotron mode and a righthand polarized magnetosonic mode. A third ionacoustic (slow) mode exists but is damped, except when T_{ e } ≫ T_{ p }, which is not common in solar wind turbulence At quasiperpendicular propagation the Alfvénic branch evolves into Kinetic Alfvén Waves (KAW), while magnetosonic modes may propagate at Ω_{ p } ≪ ω_{ r } as whistler modes. As the wavevector becomes oblique to the background magnetic field both modes develop a nonzero magnetic compressibility where parallel fluctuations becomes important. There are two distinct scenarios for the subsequent energy cascade of KAW and whistlers (Gary and Smith, (2009).
13.1 Whistler modes scenario
This scenario involves a twomode cascade process, both Alfvénic and magnetosonic modes which are only weakly damped as the plasma β ≤ 1, transfer energy to quasiperpendicular propagating wavevectors. The KAW are damped by Landau damping which is proportional to k _{⊥} ^{2} , so that they cannot contribute to the formation of dispersive region (unless for fluctuations propagating along the perpendicular direction). Even lefthand polarized Alfvén modes at quasiparallel propagation suffer for proton cyclotron damping at scales k_{∥} ~ ω_{ p }/c and do not contribute. Quasiparallel magnetosonic modes are not damped at the above scale, so that a weak cascade of righthand polarized fluctuations can generate a dispersive region of whistler modes (Stawicki et al., (2001; Gary and Borovsky, (2004, (2008; Goldstein et al., (1994). The cascade of weakly damped whistler modes has been reproduced through electron MHD numerical simulations (Biskamp et al., (1996, (1999; Wareing and Hollerbach, (2009; Cho and Lazarian, (2004) and ParticleinCell (PIC) codes (Gary et al., (2008; Saito et al., (2008).
13.2 Kinetic Alfvén waves scenario
In this scenario (Howes, (2008; Schekochihin et al., (2009) longwavelength Alfvénic turbulence transfer energy to quasiperpendicular propagation for the primary turbulent cascade up to the thermal proton gyroradius where fluctuations are subject to the proton Landau damping. The remaining fluctuation energy continues the cascade to small scales as KAW at quasiperpendicular propagation and at frequencies ω_{ r } < Ω_{ p } Bale et al. (2005); Sahraoui et al. (2009)). Fluctuations are completely damped via electron Landau resonance at wavelength of the order of the electron gyroradius. This scenario has been observed through gyrokinetic numerical simulations Howes et al. (2008b)), where the spectral breakpoint k_{⊥} ~ Ω_{ p }/u_{ th } (being u_{ th } the proton thermal speed) has been observed.
13.3 Where does the fluidlike behavior break down in solar wind turbulence?
Up to now spacecraft observations do not allow us to unambiguously distinguish between both previous scenarios. As stated by Gary and Smith (2009) at our present level of understanding of linear theory, the best we can say is that quasiparallel whistlers, quasiperpendicular whistlers, and KAW all probably could contribute to dispersion range turbulence in solar wind. Thus, the critical question is not which mode is present (if any exists in a nonlinear, collisionless medium as solar wind), but rather, what are the conditions which favor one mode over the others. On the other hand, starting from observations, we cannot rule out the possibility that strong turbulence rather than “modes” are at work to account for the highfrequency part of the magnetic energy spectrum. One of the most striking observations of smallscale turbulence is the fact that the electric field is strongly enhanced after the spectral break (Bale et al., (2005)). This means that turbulence at small scales is essentially electrostatic in nature, even if weak magnetic fluctuations are present. The enhancement of the electrostatic part has been viewed as a strong indication for the presence of KAW, because gyrokinetic simulations show the same phenomenon Howes et al. (2008b)). However, as pointed out by Matthaeus et al. (2008)) (see also the Reply by Howes et al., (2008a) to the comment by Matthaeus et al., (2008)), the enhancement of electrostatic fluctuations can be well reproduced by HallMHD turbulence, without the presence of KAW modes. Actually, the enhancement of the electric field turns out to be a statistical property of the inviscid Hall MHD (Servidio et al., 2008), that is in the absence of viscous and dissipative terms the statistical equilibrium ensemble of Hall MHD equations in the wavevectors space is build up with an enhancement of the electric field at large wavevectors. This represents a thermodynamic equilibrium property of equations, and has little to do with a nonequilibrium turbulent cascade^{13}. This would means that the enhancement of the electrostatic part of fluctuations cannot be seen as a proof firmly establishing that KAW are at work in the dispersive region.
One of the most peculiar possibility from the Cluster spacecraft was the possibility to separate the time domain from the space domain, using the tetrahedral formation of the four spacecrafts which form the Cluster mission (Escoubet et al., (2001)). This allows us to obtain a 3D wavevector spectrum and the possibility to identify the actual dispersion relation of solar wind turbulence, if any exists, at small scales. This can be made by using the kfiltering technique which is based on the strong assumption of planewave propagation (Glassmeier et al., (2001)). Of course, due to the relatively small distances between spacecrafts, this cannot be applied to largescale turbulence.
Apart for the spectral break identified by Leamon et al. (1998), a new break has been identified in the solar wind turbulence using highfrequency Cluster data, at about few tens of Hz. In fact, Cluster data at the burst mode can reach the characteristic electron inertial scale λ_{ e } and the electron Larmor radius ρ_{ e }. Using FluxGate Magnetometer and Spatiotemporal Analysis of Field Fluctuations experiment/search coil, Sahraoui et al. (2009)) showed that the turbulent spectrum changes shape at wavevectors of about kρ_{ e } ~ kλ_{ e } ≃ 1. This result, which perhaps identify the occurrence of a dissipative range in solar wind turbulence, has been obtained in the upstream solar wind magnetically connected to the bow shock. However, in these studies the plasma β was of the order of β ≃ 1, thus not allowing the separation between both scales. Alexandrova et al. (2009)), using three instruments onboard Cluster spacecrafts operating in different frequency ranges, resolved the spectrum up to 300 Hz. They confirmed the presence of the highfrequency spectral break at about kρ_{ e } ~ [0.1, 1] and, what is mainly interesting, they fitted this part of the spectrum through an exponential decay ~ exp[− √kρ_{ e }], thus indicating the onset of dissipation.
The 3D spectral shape reveals poor surprise, that is the energy distribution exhibits anisotropic features characterized by a prominently extended structure perpendicular to the mean magnetic field preferring the ecliptic north direction and also by a moderately extended structure parallel to the mean field (Narita et al., (2010)). Results of the 3D energy distribution suggest the dominance of quasi 2D turbulence toward smaller spatial scales, overall symmetry to changing the sign of the wave vector (reflectional symmetry) and absence of spherical and axial symmetry. This last was one of the main hypothesis for the Maltese Cross (Matthaeus et al., (1990)), even if bias due to satellite fly through can generate artificial deviations from axisymmetry (Turner et al., (2011)).
The above discussed papers shed some “darkness” on the scenario of small scales solar wind turbulence as made by “modes”, or at least they indicate that solar wind turbulence, at least at small scales, is far from universality. As a further stroke of the grey brush, Perri et al. (2011)) simply calculated the frequency of the spectral break as a function of radial distances from the Sun. In fact, since plasma parameters, and in particular the magnetic field intensity, changes when going towards large radial distances, the frequency break should change accordingly. They used Messenger data, as far as the inner heliosphere is concerned, and Ulysses data for outer heliosphere. Data from 0.5 AU, up to 5 AU, are summarized in Figure 2 of Perri et al. (2011)). While the characteristic frequencies of plasma lower going to higher radial distances, the position of the spectral break remains constant over all the interval of distances investigated. That is the observed highfrequency spectral break seems to be independent of the distance from the Sun, and then of both the ioncyclotron frequency and the proton gyroradius. So, where does the fluidlike behavior break down in solar wind turbulence?
13.4 What physical processes replace “dissipation” in a collisionless plasma?
As we said before, the understanding of the smallscale termination of the turbulent energy cascade in collisionless plasmas is nowadays one of the outstanding unsolved problem in space plasma physics. In the absence of collisional viscosity and resistivity the dynamics of small scales is kinetic in nature and must be described by the kinetic theory of plasma. The identification of the physical mechanism that “replaces” dissipation in the collisionless solar wind plasma and establishes a link between the macroscopic and the microscopic scales would open new scenarios in the study of the turbulent heating in space plasmas. This problem is yet in its infancy. Kinetic theory is known since long time from plasma physics, the interested reader can read the excellent review by Marsch (2006)). However, it is restricted mainly to linear theoretical arguments. The fast technological development of supercomputers gives nowadays the possibility of using kinetic Eulerian Vlasov codes that solve the Vlasov.Maxwell equations in multidimensional phase space The only limitation to the “dream” of solving 3D3V problems (3D in real space and 3D in velocity space) resides in the technological development of fast enough solvers. The use of almost noiseless codes is crucial and allows for the first time the possibility of analyzing kinetic nonlinear effects as the nonlinear evolution of particles distribution function, nonlinear saturation of Landau damping, etc. Of course, faster numerical way to solve the dissipation issue in collisionless plasmas might consist in using intermediate gyrokinetic descriptions (Brizard and Hahm, (2007)) based on a gyrotropy and strong anisotropy assumptions k_{∥} ≪ k_{⊥}.
As we said before, observations of smallscale turbulence showed the presence of a significant level of electrostatic fluctuations (Gurnett and Anderson, (1977); Gurnett and Frank, (1978); Gurnett et al., (1979); Bale et al., (2005)). Old observations of plasma wave measurements on the Helios 1 and 2 spacecrafts (Gurnett and Anderson, (1977); Gurnett and Frank, (1978); Gurnett et al., (1979)) have revealed the occurrence of electric field wavelike turbulence in the solar wind at frequencies between the electron and ion plasma frequencies. Wavelength measurements using the IMP 6 spacecraft provided strong evidence for the presence of electric fluctuations which were identified as ion acoustic waves which are Dopplershifted upward in frequency by the motion of the solar wind (Gurnett and Frank, (1978)). Comparison of the Helios results showed that the ion acoustic wavelike turbulence detected in interplanetary space has characteristics essentially identical to those of bursts of electrostatic turbulence generated by protons streaming into the solar wind from the earth’s bow shock (Gurnett and Frank, (1978); Gurnett et al., (1979)). Gurnett and Frank (1978)) observed that in a few cases of Helios data, ion acoustic wave intensities are enhanced in direct association with abrupt increases in the anisotropy of the solar wind electron distribution. This relationship strongly suggests that the ion acoustic wavelike structures detected by Helios far from the earth are produced by an electron heat flux instability or by protons streaming into the solar wind from the earth’s bow shock. Further evidences (Marsch, (2006)) revealed the strong association between the electrostatic peak and nonthermal features of the velocity distribution function of particles like temperature anisotropy and generation of accelerated beams.
Araneda et al. (2008) using Vlasov kinetic theory and onedimensional ParticleinCell hybrid simulations provided a novel explanation of the bursts of ionacoustic activity occurring in the solar wind. These authors studied the effect on the proton velocity distributions in a lowβ plasma of compressible fluctuations driven by the parametric instability of Alfvéncyclotron waves. Simulations showed that fieldaligned proton beams are generated during the saturation phase of the waveparticle interaction, with a drift speed which is slightly greater than the Alfvén speed. As a consequence, the main part of the distribution function becomes anisotropic due to phase mixing. This observation is relevant, because the same anisotropy is typically observed in the velocity distributions measured in the fast solar wind (Marsch, (2006).
In recent papers, Valentini et al. (2008) and Valentini and Veltri (2009) used hybrid Vlasov. Maxwell model where ions are considered as kinetic particles, while electrons are treated as a fluid. Numerical simulations have been obtained in 1D3V phase space (1D in the physical space and 3D in the velocity space) where a turbulent cascade is triggered by the nonlinear coupling of circularly lefthand polarized Alfvén waves, in the perpendicular plane and in parallel propagation, at plasmaβ of the order of unity. Numerical results show that energy is transferred to short scales in longitudinal electrostatic fluctuations of the acoustic form. The numerical dispersion relation in the k − ω plane displays the presence of two branches of electrostatic waves. The upper branch, at higher frequencies, consists of ionacoustic waves while the new lower frequency branch consists of waves propagating with a phase speed of the order of the ion thermal speed. This new branch is characterized by the presence of a plateau around the thermal speed in the ion distribution function, which is a typical signature of the nonlinear saturation of waveparticle interaction process.
Numerical simulations show that energy should be “dissipated” at small scales through the generation of an ionbeam in the velocity distribution function as a consequence of the trapping process and the nonlinear saturation of Landau damping, which results in bursts of electrostatic activity. Whether or not this picture, which seems to be confirmed by recent numerical simulations (Araneda et al., (2008; Valentini et al., (2008; Valentini and Veltri, (2009), represents the final fate of the real turbulent energy cascade observed at macroscopic scales, requires further investigations. Available measurements in the interplanetary space, even using Cluster spacecrafts, do not allow analysis at typical kinetic scales.
14 Conclusions and Remarks
Now that the reader finally reached the conclusions, hoping that he was so patient to read the whole paper, we suggest him to go back for a moment to the List of Contents, not to start all over again, but just to take a look at the various problems that have been briefly touched by this review. He will certainly realize how complex is the phenomenon of turbulence in general and, in particular, in the solar wind. Almost four decades of observations and theoretical efforts have not yet been sufficient to fully understand how this natural and fascinating phenomenon really works in the solar wind. We certainly are convinced that we cannot think of a single mechanism able to reproduce all the details we have directly observed since physical boundary conditions favor or inhibit different generation mechanisms, like for instance, velocityshear or parametric decay, depending on where we are in the heliosphere.
On the other hand, there are some aspects which we believe are at the basis of turbulence generation and evolution like: a) we do need nonlinear interactions to develop the observed Kolmogorovlike spectrum; b) in order to have nonlinear interactions we need to have inward modes and/or convected structures which the majority of the modes can interact with; c) outward and inward modes can be generated by different mechanisms like velocity shear or parametric decay; d) convected structures actively contribute to turbulent development of fluctuations and can be of solar origin or locally generated.
In particular, ecliptic observations have shown that what we call Alfvénic turbulence, mainly observed within high velocity streams, tends to evolve towards the more “standard” turbulence that we mainly observe within slow wind regions, i.e., a turbulence characterized by e^{+} ~ e^{−}, an excess of magnetic energy, and a Kolmogorovlike spectral slope. Moreover, the presence of a well established “background” spectrum already at short heliocentric distances and the low Alfvénicity of the fluctuations suggest that within slow wind turbulence is mainly due to convected structures frozen in the wind which may well be the remnants of turbulent processes already acting within the first layers of the solar corona. In addition, velocity shear, whenever present, seems to have a relevant role in driving turbulence evolution in lowlatitude solar wind.
Polar observations performed by Ulysses, combined with previous results in the ecliptic, finally allowed to get a comprehensive view of the Alfvénic turbulence evolution in the 3D heliosphere, inside 5 AU. However, polar observations, when compared with results obtained in the ecliptic, do not appear as a dramatic break. In other words, the polar evolution is similar to that in the ecliptic, although slower. This is a middle course between the two opposite views (a nonrelaxing turbulence, due to the lack of velocity shear, or a quick evolving turbulence, due to the large relative amplitude of fluctuations) which were popular before the Ulysses mission. The process driving the evolution of polar turbulence still is an open question although parametric decay might play some role. As a matter of fact, simulations of nonlinear development of the parametric instability for largeamplitude, broadband Alfvénic fluctuations have shown that the final state resembles values of σ_{c} not far from solar wind observations, in a state in which the initial Alfvénic correlation is partially preserved. As already observed in the ecliptic, polar Alfvénic turbulence appears characterized by a predominance of outward fluctuations and magnetic fluctuations. As regards the outward fluctuations, their dominant character extends to large distances from the Sun. At low solar activity, with the polar wind filling a large fraction of the heliosphere, the outward fluctuations should play a relevant role in the heliospheric physics. Relatively to the imbalance in favor of the magnetic energy, it does not appear to go beyond an asymptotic value. Several ways to alter the balance between kinetic and magnetic energy have been proposed (e.g., 2D processes, propagation in a nonuniform medium, and effect of magnetic structures, among others). However, convincing arguments to account for the existence of such a limit have not yet been given, although promising results from numerical simulations seem to be able to qualitatively reproduce the final imbalance in favor of the magnetic energy.
Definitely, the relatively recent adoption of numerical methods able to highlight scaling laws features hidden to the usual spectral methods, allowed to disclose a new and promising way to analyze turbulent interplanetary fluctuations. Interplanetary space is now looked at as a natural wind tunnel where scaling properties of the solar wind can be studied on scales of the order of (or larger than) 109 times than laboratory scales. Within this framework, intermittency represents an important topic in both theoretical and observational studies. Intermittency properties have been recovered via very promising models like the MHD shell models, and the nature of intermittent events has finally been disclosed thanks to new numerical techniques based on wavelet transforms. Moreover, similar techniques have allowed to tackle the problem of identify the spectral anisotropic scaling although no conclusive and final analyses have been reported so far. In addition, recent studies on intermittency of magnetic field and velocity vector fluctuations, together with analogous analyses on magnitude fluctuations, contributed to sketch a scenario in which propagating stochastic Alfvénic fluctuations and advected structures, possibly flux tubes embedded in the wind, represent the main ingredients of interplanetary turbulence The varying predominance of one of the two species, waves or structures would make the observed turbulence more or less intermittent. However, the fact that we can make measurements just at one point of this natural wind tunnel represented by the solar wind does not allow us to discriminate temporal from spatial phenomena. As a consequence, we do not know whether these advected structures are somehow connected to the complicated topology observed at the Sun surface or can be considered as byproduct of chaotic developing phenomena. Comparative studies based on the intermittency phenomenon within fast and slow wind during the wind expansion would suggest a solar origin for these structures which would form a sort of turbulent background frozen in the wind. As a matter of fact, intermittency in the solar wind is not limited to the dissipation range of the spectrum but abundantly extends orders of magnitude away from dissipative scales, possibly into the inertial range which can be identified taking into account all the possible caveats related to this problem and briefly reported in this review. This fact introduces serious differences between hydrodynamic turbulence and solar wind MHD turbulence, and the same “intermittency” assumes a different intrinsic meaning when observed in interplanetary turbulence In practice, coherent structures observed in the wind are at odds with filaments or vortices observed in ordinary fluid turbulence since these last ones are dissipative structures continuously created and destroyed by turbulent motion.
Smallscale turbulence, namely observations of turbulent fluctuations at frequencies greater than say 0.1 Hz. revealed a rich and yet poorly understood physics, mainly related to the big problem of dissipation in a dissipationless plasma. Data analysis received a strong impulse from the Cluster spacecrafts, thus revealing a few number of well established and not contradictory observations, as the presence of a double spectral breaks. However, the interpretation of the presence of a power spectrum at small scales is far from being clear and a number of contradictory interpretations can be found in literature. Numerical simulations, based on VlasovMaxwell, gyrokinetic and PIC codes, have been made possible due to the increasingly power of computers. They indicated some possible interpretation of the highfrequency part of the turbulent spectrum, but unfortunately the interpretation is not unequivocal. The study of highfrequency part of the turbulent spectrum is a rapidly growing field of research, here we reported the up to date state of the art, while a more complete, systematic and thoughtout analysis of the wide literature will be done in a future version of the paper.
As a final remark, we would like to point out that we tried to start writing a particular point of view on the turbulence in the solar wind. We apologize for the lack of some aspects of the phenomenon at hand which can be found in the existing literature. There are still several topics which we did not discuss in this revised version of our review.

recent (nonshell) turbulent modeling;

simulation of turbulence in the expanding solar wind;

numerical simulations of anisotropic turbulence;

a deeper view on VlasovMaxwell and gyrokinetic approaches.
Fortunately, we are writing a Living Review paper and mistakes and/or omissions will be adequately fixed in the next version also with the help of all our colleagues, whom we strongly encourage to send us comments and/or different points of view on particularly interesting topics which we have not yet taken into account or discussed properly.
Footnotes
 1.
This concept will be explained better in the next sections.
 2.
A fluid particle is defined as an infinitesimal portion of fluid which moves with the local velocity. As usual in fluid dynamics, infinitesimal means small with respect to large scale, but large enough with respect to molecular scales.
 3.
 4.
These authors were the first ones to use physical technologies and methodologies to investigate turbulent flows from an experimental point of view. Before them, experimental studies on turbulence were motivated mainly by engineering aspects.
 5.
We can use a different definition for the third invariant H(t), for example a quantity positive defined, without the term (−1)n and with α = 2. This can be identified as the surrogate of the square of the vector potential, thus investigating a kind of 2D MHD. In this case, we obtain a shell model with λ = 2, a = 5/4, and c = .1/3. However, this model does not reproduce the inverse cascade of the square of magnetic potential observed in the true 2D MHD equations.
 6.
We have already defined fluctuations of a field as the difference between the field itself and its average value. This quantity has been defined as ..... Here, the differences Δψ_{ℓ} of the field separated by a distance ℓ represents characteristic fluctuations at the scale ℓ, say characteristic fluctuations of the field across specific structures (eddies) that are present at that scale. The reader can realize the difference between both definitions.
 7.
To be precise, it is worth remarking again that there are no convincing arguments to identify as inertial range the intermediate range of frequencies where the observed spectral properties are typical of fully developed turbulence. From a theoretical point of view here the association “intermediate range” ≃ “inertial range” is somewhat arbitrary. Really an operative definition of inertial range of turbulence is the range of scales ℓ where relation (42) (for fluid flows) or (41) (for MHD flows) is verified.
 8.
Since the solar wind moves at supersonic speed V_{sw}, the usual Taylor’s hypothesis is verified, and we can get information on spatial scaling laws ℓ by using time differences τ = ℓ/V_{sw}.
 9.
Note that, according to the occurrence of the Yaglom’s law, that is a thirdorder moment is different from zero, the fluctuations at a given scale in the inertial range must present some nonGaussian features. From this point of view the calculation of structure functions with the absolute value is unappropriate because in this way we risk to cancel out nonGaussian features. Namely we symmetrize the probability density functions of fluctuations. However, in general, the number of points at disposal is much lower than required for a robust estimate of odd structure functions, even in usual fluid flows. Then, as usually, we will obtain structure functions by taking the absolute value, even if some care must be taken in certain conclusions which can be found in literature.
 10.
The lognormal model is derived by using a multiplicative process, where random variable generates the cascade. Then, according to the Central Limit Theorem, the process converges to a lognormal distribution of finite variance. The logLévy model is a modification of the lognormal model. In such case, the Central Limit Theorem is used to derive the limit distribution of an infinite sum of random variables by relaxing the hypothesis of finite variance usually used. The resulting limit function is a Lévy function.
 11.
For a discussion on nonturbulent mechanism of solar wind heating cf. Tu and Marsch (1995a).
 12.
Of course, this is based on classical turbulence. As said before, in the solar wind the dissipative term is unknown, even if it might happens at very small kinetic scales.
 13.
It is worthwhile to remark that a turbulent fluid flows is out of equilibrium, say the cascade requires the injection of energy (input) and a dissipation mechanism (output), usually lying on well separated scales, along with a transfer of energy. Without input and output, the nonlinear term of equations works like an energy redistribution mechanism towards an equilibrium in the wave vectors space. This generates an equilibrium energy spectrum which should in general be the same as that obtained when the cascade is at work (cf., e.g., Frisch et al., (1975). However, even if the turbulent spectra could be anticipated by looking at the equilibrium spectra, the physical mechanisms are different. Of course, this should also be the case for the Hall MHD.
Notes
Acknowledgments
Writing a large review paper is not an easy task and it would not have been possible to accomplish this goal without having a good interaction with our colleagues, whom we have been working with in our Institutions. To this regard, we like to acknowledge the many discussions (more or less “heated”) we had with and the many advices and comments we had from all of them, particularly from B. Bavassano and P. Veltri. We also like to acknowledge the use of plasma and magnetic field data from Helios spacecraft to freshly produce some of the figures shown in the present review. In particular, we like to thank H. Rosenbauer and R. Schwenn, PIs of the plasma experiment, and F. Mariani and N.F. Ness, PIs of the second magnetic experiment on board Helios. We thank A. Pouquet, H. Politano, and V. Antoni for the possibility we had to compare solar wind data with both highresolution numerical simulations and laboratory plasmas. Finally, we owe special thanks to E. Marsch and S.K. Solanki for giving us the opportunity to write this review.
Finally, we greatly appreciate the great effort made by an anonymous referee to review such a long paper. We like to thank her/him for her/his meticulous, efficient, and competent analysis of this first updated version of our review which helped us to make a better and more readable document.
Supplementary material
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