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.
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.
What does turbulence stand for?
The word turbulent is used in the everyday experience to indicate something which is not regular. In Latin the word turba means something confusing or something which does not follow an ordered plan. A turbulent boy, in all Italian schools, is a young fellow who rebels against ordered schemes. Following the same line, the behavior of a flow which rebels against the deterministic rules of classical dynamics is called turbulent. Even the opposite, namely a laminar motion, derives from the Latin word lámina, which means stream or sheet, and gives the idea of a regular streaming motion. Anyhow, even without the aid of a laboratory experiment and a Latin dictionary, we experience turbulence every day. It is relatively easy to observe turbulence and, in some sense, we generally do not pay much attention to it (apart when, sitting in an airplane, a nice lady asks us to fasten our seat belts during the flight because we are approaching some turbulence!). Turbulence appears everywhere when the velocity of the flow is high enough^{Footnote 1}, for example, when a flow encounters an obstacle (cf., e.g., Figure 1) in the atmospheric flow, or during the circulation of blood, etc. Even charged fluids (plasma) can become turbulent. For example, laboratory plasmas are often in a turbulent state, as well as natural plasmas like the outer regions of stars. Living near a star, we have a big chance to directly investigate the turbulent motion inside the flow which originates from the Sun, namely the solar wind. This will be the main topic of the present review.
Turbulence that we observe in fluid flows appears as a very complicated state of motion, and at a first sight it looks (apparently!) strongly irregular and chaotic, both in space and time. The only dynamical rule seems to be the impossibility to predict any future state of the motion. However, it is interesting to recognize the fact that, when we take a picture of a turbulent flow at a given time, we see the presence of a lot of different turbulent structures of all sizes which are actively present during the motion. The presence of these structures was well recognized long time ago, as testified by the beautiful pictures of vortices observed and reproduced by the Italian genius Leonardo da Vinci, as reported in the textbook by Frisch (1995). Figure 2 shows, as an example, one picture from Leonardo which can be compared with Figure 3 taken from a typical experiment on a turbulent jet.
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.
When we look at a flow at two different times, we can observe that the general aspect of the flow has not changed appreciably, say vortices are present all the time but the flow in each single point of the fluid looks different. We recognize that the gross features of the flow are reproducible but details are not predictable. We have to use a statistical approach to turbulence, just as it is done to describe stochastic processes, even if the problem is born within the strange dynamics of a deterministic system!
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.
Turbulence became an experimental science since Osborne Reynolds who, at the end of 19th century, observed and investigated experimentally the transition from laminar to turbulent flow. He noticed that the flow inside a pipe becomes turbulent every time a single parameter, a combination of the viscosity coefficient η, a characteristic velocity U, and length L, would increase. This parameter Re = ULρ/η (ρ is the mass density of the fluid) is now called the Reynolds number. At lower Re, say Re ≤ 2300, the flow is regular (that is the motion is laminar), but when Re increases beyond a certain threshold of the order of Re ≃ 4000, the flow becomes turbulent. As Re increases, the transition from a laminar to a turbulent state occurs over a range of values of Re with different characteristics and depending on the details of the experiment. In the limit Re → ∞ the turbulence is said to be in a fully developed turbulent state. The original pictures by Reynolds are shown in Figure 7.
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.
As it is well documented in these figures, the main feature of fully developed turbulence is the chaotic character of the time behavior. Said differently, this means that the behavior of the flow is unpredictable. While the details of fully developed turbulent motions are extremely sensitive to triggering disturbances, average properties are not. If this was not the case, there would be little significance in the averaging process. Predictability in turbulence can be recast at a statistical level. In other words, when we look at two different samples of turbulence, even collected within the same medium, we can see that details look very different. What is actually common is a generic stochastic behavior. This means that the global statistical behavior does not change going from one sample to the other. The idea that fully developed turbulent flows are extremely sensitive to small perturbations but have statistical properties that are insensitive to perturbations is of central importance throughout this review. Fluctuations of a certain stochastic variable ψ are defined here as the difference from the average value δψ = ψ−ψ, where brackets mean some averaging process. Actually, the method of taking averages in a turbulent flow requires some care. We would like to recall that there are, at least, three different kinds of averaging procedures that may be used to obtain statisticallyaveraged properties of turbulence The space averaging is limited to flows that are statistically homogeneous or, at least, approximately homogeneous over scales larger than those of fluctuations. The ensemble averages are the most versatile, where average is taken over an ensemble of turbulent flows prepared under nearly identical external conditions. Of course, these flows are not completely identical because of the large fluctuations present in turbulence Each member of the ensemble is called a realization. The third kind of averaging procedure is the time average, which is useful only if the turbulence is statistically stationary over time scales much larger than the time scale of fluctuations. In practice, because of the convenience offered by locating a probe at a fixed point in space and integrating in time, experimental results are usually obtained as time averages. The ergodic theorem (Halmos, (1956) assures that time averages coincide with ensemble averages under some standard conditions (see Appendix B).
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.
Since fully developed turbulence involves a hierarchy of scales, a large number of interacting degrees of freedom are involved. Then, there should be an asymptotic statistical state of turbulence that is independent on the details of the flow. Hopefully, this asymptotic state depends, perhaps in a critical way, only on simple statistical properties like energy spectra, as much as in statistical mechanics equilibrium where the statistical state is determined by the energy spectrum (Huang, (1987). Of course, we cannot expect that the statistical state would determine the details of individual realizations, because realizations need not to be given the same weight in different ensembles with the same loworder statistical properties.
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
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.
The NavierStokes equation and the Reynolds number
Equations which describe the dynamics of real incompressible fluid flows have been introduced by ClaudeLouis Navier in 1823 and improved by George G. Stokes. They are nothing but the momentum equation based on Newton’s second law, which relates the acceleration of a fluid particle^{Footnote 2} to the resulting volume and body forces acting on it. These equations have been introduced by Leonhard Euler, however, the main contribution by Navier was to add a friction forcing term due to the interactions between fluid layers which move with different speed. This term results to be proportional to the viscosity coefficients η and ξ and to the variation of speed. By defining the velocity field u(r, t) the kinetic pressure p and the density ρ, the equations describing a fluid flow are the continuity equation to describe the conservation of mass
the equation for the conservation of momentum
and an equation for the conservation of energy
where s is the entropy per mass unit, T is the temperature, and χ is the coefficient of thermoconduction. An equation of state closes the system of fluid equations.
The above equations considerably simplify if we consider the incompressible fluid, where ρ = const. so that we obtain the NavierStokes (NS) equation
where the coefficient ν = η/ρ is the kinematic viscosity. The incompressibility of the flow translates in a condition on the velocity field, namely the field is divergencefree, i.e., ∇·u = 0. This condition eliminates all highfrequency sound waves and is called the incompressible limit. The nonlinear term in equations represents the convective (or substantial) derivative. Of course, we can add on the right hand side of this equation all external forces, which eventually act on the fluid parcel.
We use the velocity scale U and the length scale L to define dimensionless independent variables, namely r = r’L (from which ∇ = ∇’/L) and t = t’(L/U), and dependent variables u = u’U andp = p’U^{2}ρ. Then, using these variables in Equation (4), we obtain
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.
The coupling between a charged fluid and the magnetic field
Magnetic fields are ubiquitous in the Universe and are dynamically important. At high frequencies, kinetic effects are dominant, but at frequencies lower than the ion cyclotron frequency, the evolution of plasma can be modeled using the MHD approximation. Furthermore, dissipative phenomena can be neglected at large scales although their effects will be felt because of nonlocality of nonlinear interactions. In the presence of a magnetic field, the Lorentz force j × B, where j is the electric current density, must be added to the fluid equations, namely
and the Joule heat must be added to the equation for energy
where σ is the conductivity of the medium, and we introduced the viscous stress tensor
An equation for the magnetic field stems from the Maxwell equations in which the displacement current is neglected under the assumption that the velocity of the fluid under consideration is much smaller than the speed of light. Then, using
and the Ohm’s law for a conductor in motion with a speed u in a magnetic field
we obtain the induction equation which describes the time evolution of the magnetic field
together with the constraint ∇ · B = 0 (no magnetic monopoles in the classical case).
In the incompressible case, where ∇ · u = 0, MHD equations can be reduced to
and
Here P_{tot} is the total kinetic P_{k} = nkT plus magnetic pressure P_{m} = B^{2}/8π, divided by the constant mass density ρ. Moreover, we introduced the velocity variables b = B/√πρ and the magnetic diffusivity η.
Similar to the usual Reynolds number, a magnetic Reynolds number R_{m} can be defined, namely
where c_{A} = B_{0}/√4πρ is the Alfvén speed related to the largescale B_{0} magnetic field B_{0}. This number in most circumstances in astrophysics is very large, but the ratio of the two Reynolds numbers or, in other words, the magnetic Prandtl number P_{m} = ν/η can differ widely. In absence of dissipative terms, for each volume V MHD equations conserve the total energy E(t)
the crosshelicity H_{c}(t), which represents a measure of the degree of correlations between velocity and magnetic fields
and the magnetic helicity H(t), which represents a measure of the degree of linkage among magnetic flux tubes
where b = ∇ × a.
The change of variable due to Elsäasser (1950), say z^{±} = u ± b’, where we explicitly use the background uniform magnetic field b’ = b + c_{A} (at variance with the bulk velocity, the largest scale magnetic field cannot be eliminated through a Galilean transformation), leads to the more symmetrical form of the MHD equations in the incompressible case
where 2ν^{±} = ν±η are the dissipative coefficients, and F^{±} are eventual external forcing terms. The relations ∇ · z^{±} = 0 complete the set of equations. On linearizing Equation (15) and neglecting both the viscous and the external forcing terms, we have
which shows that z^{−}(x − c_{A}t) describes Alfvénic fluctuations propagating in the direction of B_{0}, and z^{+}(x + c_{A}t) describes Alfvénic fluctuations propagating opposite to B_{0}. Note that MHD Equations (15) have the same structure as the NavierStokes equation, the main difference stems from the fact that nonlinear coupling happens only between fluctuations propagating in opposite directions. As we will see, this has a deep influence on turbulence described by MHD equations.
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.
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.
The nonlinear energy cascade
The basic properties of turbulence, as derived both from the NavierStokes equation and from phenomenological considerations, is the legacy of A. N. Kolmogorov (Frisch, (1995).^{Footnote 3} Phenomenology is based on the old picture by Richardson who realized that turbulence is made by a collection of eddies at all scales. Energy, injected at a length scale L, is transferred by nonlinear interactions to small scales where it is dissipated at a characteristic scale ℓ_{D}, the length scale where dissipation takes place The main idea is that at very large Reynolds numbers, the injection scale L and the dissipative scale ℓ_{D} are completely separated. In a stationary situation, the energy injection rate must be balanced by the energy dissipation rate and must also be the same as the energy transfer rate ε measured at any scale ℓ within the inertial range ℓ_{D} ≪ ℓ ≪ L. From a phenomenological point of view, the energy injection rate at the scale L is given by ∈_{D} ~ U^{2}/τ_{L}, where τ_{L} is a characteristic time for the injection energy process, which results to be τ_{L} ~ L/U At the same scale L the energy dissipation rate is due to ∈_{D} ~ U^{2}/τ_{D}, where τ_{D} is the characteristic dissipation time which, from Equation (4), can be estimated to be of the order of τ_{D} ~ L^{2}/ν. As a result, the ratio between the energy injection rate and dissipation rate is
that is, the energy injection rate at the largest scale L is Retimes the energy dissipation rate. In other words, in the case of large Reynolds numbers, the fluid system is unable to dissipate the whole energy injected at the scale L. The excess energy must be dissipated at small scales where the dissipation process is much more efficient. This is the physical reason for the energy cascade.
Fully developed turbulence involves a hierarchical process, in which many scales of motion are involved. To look at this phenomenon it is often useful to investigate the behavior of the Fourier coefficients of the fields. Assuming periodic boundary conditions the αth component of velocity field can be Fourier decomposed as
where k = 2πn/L and n is a vector of integers. When used in the NavierStokes equation, it is a simple matter to show that the nonlinear term becomes the convolution sum
where Mαβγ(k) = −ik_{β}(δ_{αγ} − k_{α} − k_{β}/k^{2}) (for the moment we disregard the linear dissipative term).
MHD equations can be written in the same way, say by introducing the Fourier decomposition for Elsäasser variables
and using this expression in the MHD equations we obtain an equation which describes the time evolution of each Fourier mode. However, the divergenceless condition means that not all Fourier modes are independent, rather k · z^{±}(k, t) = 0 means that we can project the Fourier coefficients on two directions which are mutually orthogonal and orthogonal to the direction of k, that is,
with the constraint that k · e^{(a)}(k) = 0. In presence of a background magnetic field we can use the well defined direction B_{0}, so that
Note that in the linear approximation where the Elsäasser variables represent the usual MHD modes, z ^{±}_{1} (k, t) represent the amplitude of the Alfvén mode while z ^{±}_{2} (k, t) represent the amplitude of the incompressible limit of the magnetosonic mode. From MHD Equations (15) we obtain the following set of equations:
The coupling coefficients, which satisfy the symmetry condition A_{abc} (k, p, q) = −A_{bac}(p, k, q), are defined as
and the sum in Equation (19) is defined as
where δ_{k,p+q} is the Kronecher’s symbol. Quadratic nonlinearities of the original equations correspond to a convolution term involving wave vectors k, p and q related by the triangular relation p = k−q. Fourier coefficients locally couple to generate an energy transfer from any pair of modes p and q to a mode k = p + q.
The pseudoenergies E^{±}(t) are defined as
and, after some algebra, it can be shown that the nonlinear term of Equation (19) conserves separately E^{±}(t). This means that both the total energy E(t) = E^{+} + E^{−} and the crosshelicity E_{c}(t) = E^{+}−E^{−}, say the correlation between velocity and magnetic field, are conserved in absence of dissipation and external forcing terms.
In the idealized homogeneous and isotropic situation we can define the pseudoenergy tensor, which using the incompressibility condition can be written as
brackets being ensemble averages, where q^{±}(k) is an arbitrary odd function of the wave vector k and represents the pseudoenergies spectral density. When integrated over all wave vectors under the assumption of isotropy
where we introduce the spectral pseudoenergy E^{±}(k, t) = 4πk^{2}q^{±}(k, t). This last quantity can be measured, and it is shown that it satisfies the equations
We use ν = η in order not to worry about coupling between + and − modes in the dissipative range. Since the nonlinear term conserves total pseudoenergies we have
so that, when integrated over all wave vectors, we obtain the energy balance equation for the total pseudoenergies
This last equation simply means that the time variations of pseudoenergies are due to the difference between the injected power and the dissipated power, so that in a stationary state
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
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.
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.
The Landau transition scenario is, however, untenable because incommensurate frequencies cannot exist without coupling between them. Ruelle and Takens (1971) proposed a new mathematical model, according to which after few, usually three, Hopf bifurcations the flow becomes suddenly chaotic. In the phase space this state is characterized by a very intricate attracting subset, a strange attractor. The flow corresponding to this state is highly irregular and strongly dependent on initial conditions. This characteristic feature is now known as the butterfly effect and represents the true definition of deterministic chaos. These authors indicated as an example for the occurrence of a strange attractor the old strange time behavior of the Lorenz’s model. The model is a paradigm for the occurrence of turbulence in a deterministic system, it reads
where x(t), y(t), and z(t) represent the first three modes of a Fourier expansion of fluid convective equations in the Boussinesq approximation, P_{r} is the Prandtl number, b is a geometrical parameter, and R is the ratio between the Rayleigh number and the critical Rayleigh number for convective motion. The time evolution of the variables x(t), y(t), and z(t) is reported in Figure 12. A reproduction of the Lorenz butterfly attractor, namely the projection of the variables on the plane (x, z) is shown in Figure 13. A few years later, Gollub and Swinney (1975) performed very sophisticated experiments,^{Footnote 4} concluding that the transition to turbulence in a flow between corotating cylinders is described by the Ruelle and Takens (1971) model rather than by the Landau scenario.
After this discovery, the strange attractor model gained a lot of popularity, thus stimulating a large number of further studies on the time evolution of nonlinear dynamical systems. An enormous number of papers on chaos rapidly appeared in literature, quite in all fields of physics, and transition to chaos became a new topic. Of course, further studies on chaos rapidly lost touch with turbulence studies and turbulence, as reported by Feynman et al. (1977), still remains ... the last great unsolved problem of the classical physics. Furthermore, we like to cite recent theoretical efforts made by Chian and coworkers (Chian et al., (1998, (2003) related to the onset of Alfvénic turbulence These authors, numerically solved the derivative nonlinear Schrödinger equation (Mjølhus, (1976; Ghosh and Papadopoulos, (1987) which governs the spatiotemporal dynamics of nonlinear Alfvén waves, and found that Alfvénic intermittent turbulence is characterized by strange attractors. Note that, the physics involved in the derivative nonlinear Schrödinger equation, and in particular the spatiotemporal dynamics of nonlinear Alfvén waves, cannot be described by the usual incompressible MHD equations. Rather dispersive effects are required. At variance with the usual MHD, this can be satisfied by requiring that the effect of ion inertia be taken into account. This results in a generalized Ohm’s law by including a (j̲ × B̲)term, which represents the compressible Hall correction to MHD, say the socalled compressible HallMHD model.
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.
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.
In the following, we will refer to the MHD shell model as the FSGC model. The shell model can be built up through four different steps:

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 ^{−1}_{n} . That is, the fields have the same scalings as field differences, for example Z ^{±}_{n} ~ Z^{±}(x + ℓ_{n}) − Z^{±}(x) ~ ℓ ^{h}_{n} 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 pseudoenergies
$$E^ \pm (t) = \frac{1} {2}\sum\limits_n {\left {Z_n^ \pm } \right^2 },$$((22a))that means the conservation of both the total energy and the crosshelicity:
$$\begin{array}{*{20}c} {E(t) = \frac{1} {2}\sum\limits_n {\left {u_n } \right^2 + \left {b_n } \right^2 ;} } & {H_c (t) = \sum\limits_n {2\Re e(u_n b_n^* )} } \\ \end{array},$$((22b))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
$$H(t) = \sum\limits_n {\left {  1} \right^n \frac{{\left {b_n } \right^2 }} {{k_n^\alpha }}},$$((23))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).
After some algebra, taking into account both the dissipative and forcing terms, FSGC model can be written as
where
where^{Footnote 5} λ = 2, a = 1/2, and c = 1/3. In the following, we will consider only the case where the dissipative coefficients are the same, i.e., ν = μ.
The phenomenology of fully developed turbulence: Fluidlike case
Here we present the phenomenology of fully developed turbulence, as far as the scaling properties are concerned. In this way we are able to recover a universal form for the spectral pseudoenergy in the stationary case. In real space a common tool to investigate statistical properties of turbulence is represented by field increments Δz ^{±}_{ℓ} (r) = [z^{±}(r + ℓ) − z^{±}(r)] · e, being e the longitudinal direction. These stochastic quantities represent fluctuations^{Footnote 6} across eddies at the scale ℓ. The scaling invariance of MHD equations (cf. Section 2.3), from a phenomenological point of view, implies that we expect solutions where Δz ^{±}_{ℓ} ~ ℓ^{h}. All the statistical properties of the field depend only on the scale ℓ, on the mean pseudoenergy dissipation rates ε^{±}, and on the viscosity ν. Also, ε^{±} is supposed to be the common value of the injection, transfer and dissipation rates. Moreover, the dependence on the viscosity only arises at small scales, near the bottom of the inertial range. Under these assumptions the typical pseudoenergy dissipation rate per unit mass scales as ε^{±} ~ (Δz ^{±}_{ℓ} ±)^{2}/t ^{±}_{ℓ} . The time t ^{±}_{ℓ} associated with the scale . is the typical time needed for the energy to be transferred on a smaller scale, say the eddy turnover time t ^{±}_{ℓ} ~ ℓ/Δz ^{∓}_{ℓ} , so that
When we conjecture that both Δz^{±} fluctuations have the same scaling laws, namely Δz^{±} ~ ℓ^{h} we recover the Kolmogorov scaling for the field increments
Usually, we refer to this scaling as the K41 model (Kolmogorov, (1941, (1991; Frisch, (1995). Note that, since from dimensional considerations the scaling of the energy transfer rate should be ε^{±} ~ ℓ^{1−3h}, h = 1/3 is the choice to guarantee the absence of scaling for ε^{±}.
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 ^{(p)}_{ℓ} = 〈Δu ^{p}_{ℓ} 〉. These quantities, in the inertial range, behave as a power law S ^{(p)}_{ℓ} ~ 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 ^{(p)}_{ℓ} ~ ℓ^{p/3}. Then ξ_{p} = p/3 results to be a linear function of the order p.
When we assume the scaling law Δz ^{±}_{ℓ} ~ ℓ^{h}, we can compute the highorder moments of the structure functions for increments of the Elsäasser variables, namely 〈(Δz ^{±}_{ℓ} )^{p}〉 ~ ℓ^{ξ}_{p}, thus obtaining a linear scaling ξ_{p} = p/3, similar to usual fluid flows. For Gaussianly distributed fields, a particular role is played by the secondorder moment, because all moments can be computed from S ^{(2)}_{ℓ} . It is straightforward to translate the dimensional analysis results to Fourier spectra. The spectral property of the field can be recovered from S ^{(2)}_{ℓ} , say in the homogeneous and isotropic case
where k ~ 1/ℓ is the wave vector, so that in the inertial range where Equation (42) is verified
The Kolmogorov spectrum (see Equation (27)) is largely observed in all experimental investigations of turbulence, and is considered as the main result of the K41 phenomenology of turbulence (Frisch, (1995). However, spectral analysis does not provide a complete description of the statistical properties of the field, unless this has Gaussian properties. The same considerations can be made f.o[.r the spectral pseudoenergies E^{±}(k), which are related to the 2nd order structure functions 〈[±z ^{±}_{ℓ} ]^{2}〉.
The phenomenology of fully developed turbulence: Magneticallydominated case
The phenomenology of the magneticallydominated case has been investigated by Iroshnikov (1963) and Kraichnan (1965), then developed by Dobrowolny et al. (1980b) to tentatively explain the occurrence of the observed Alfvénic turbulence, and finally by Carbone (1993) and Biskamp (1993) to get scaling laws for structure functions. It is based on the Alfvén effect, that is, the decorrelation of interacting eddies, which can be explained phenomenologically as follows. Since nonlinear interactions happen only between opposite propagating fluctuations, they are slowed down (with respect to the fluidlike case) by the sweeping of the fluctuations across each other. This means that ε^{±} ~ (Δz ^{±}_{ℓ} )^{2}/T ^{±}_{ℓ} but the characteristic time T ^{±}_{ℓ} required to efficiently transfer energy from an eddy to another eddy at smaller scales cannot be the eddyturnover time, rather it is increased by a factor t ^{±}_{ℓ} /t_{A} (t_{A} ~ ℓ/c_{A} < t ^{±}_{ℓ} is the Alfvén time), so that T ^{±}_{ℓ} ~ (t ^{±}_{ℓ} )^{2}/t_{A}. Then, immediately
This means that both ± modes are transferred at the same rate to small scales, namely ∈^{+} ~ ∈^{−} ~ ∈, and this is the conclusion drawn by Dobrowolny et al. (1980b). In reality, this is not fully correct, namely the Alfvén effect yields to the fact that energy transfer rates have the same scaling laws for ± modes but, we cannot say anything about the amplitudes of ε^{+} and ε^{−} (Carbone, (1993). Using the usual scaling law for fluctuations, it can be shown that the scaling behavior holds ∈ → λ^{1−4h}ε’. Then, when the energy transfer rate is constant, we found a scaling law different from that of Kolmogorov and, in particular,
Using this phenomenology the highorder moments of fluctuations are given by S ^{(p)}_{∓} ~ ℓ^{p/4}. Even in this case, ξ_{p} = p/4 results to be a linear function of the order p. The pseudoenergy spectrum can be easily found to be
This is the IroshnikovKraichnan spectrum. However, in a situation in which there is a balance between the linear Alfvén time scale or wave period, and the nonlinear time scale needed to transfer energy to smaller scales, the energy cascade is indicated as critically balanced (Goldreich and Sridhar, (1995). In these conditions, it can be shown 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° while, the same scaling would follow f^{−2} in case θ_{B} = 0° and the spectrum would also have a smaller energy content than in the other case.
Some exact relationships
So far, we have been discussing about the inertial range of turbulence What this means from a heuristic point of view is somewhat clear, but when we try to identify the inertial range from the spectral properties of turbulence, in general the best we can do is to identify the inertial range with the intermediate range of scales where a Kolmogorov’s spectrum is observed. The often used identity inertial range ≃ intermediate range, is somewhat arbitrary. In this regard, a very important result on turbulence, due to Kolmogorov (1941, (1991), is the socalled “4/5law” which, being obtained from the NavierStokes equation, is “... one of the most important results in fully developed turbulence because it is both exact and nontrivial” (cf. Frisch, (1995). As a matter of fact, Kolmogorov analytically derived the following exact relation for the third order structure function of velocity fluctuations:
where r is the sampling direction, ℓ is the corresponding scale, and ∈ is the mean energy dissipation per unit mass, assumed to be finite and nonvanishing.
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).
Yaglom’s law for MHD turbulence
To obtain a general law we start from the incompressible MHD equations. If we write twice the MHD equations for two different and independent points x_{i} and x_{i}’ = x_{i} + ℓ_{i}, by substraction we obtain an equation for the vector differences Δz ^{±}_{i} = (z ^{±}_{i} )’ − z ^{±}_{i} . Using the hypothesis of independence of points x_{i}’ and x_{i} with respect to derivatives, namely ∂_{i}(z ^{±}_{i} )’ = ∂_{i}’z ^{±}_{j} = 0 (where ∂_{i}’ represents derivative with respect to x_{i}’), we get
(ΔP = P_{tot}’ − P_{tot}). We look for an equation for the secondorder correlation tensor 〈Δz ^{±}_{i} Δz ^{±}_{j} 〉 related to pseudoenergies. Actually the more general thing should be to look for a mixed tensor, namely 〈Δz ^{±}_{i} Δz ^{∓}_{j} 〉, taking into account not only both pseudoenergies but also the time evolution of the mixed correlations 〈z ^{+}_{i} z ^{−}_{j} 〉 and 〈z ^{−}_{i} z ^{+}_{j} 〉. However, using the DIA closure by Kraichnan, it is possible to show that these elements are in general poorly correlated (Veltri, (1980). Since we are interested in the energy cascade, we limit ourselves to the most interesting equation that describes correlations about Alfvénic fluctuations of the same sign. To obtain the equations for pseudoenergies we multiply Equations (31) by Δz ^{±}_{j} , then by averaging we get
where we used the hypothesis of local homogeneity and incompressibility. In Equation (32) we defined the average dissipation tensor
The first and second term on the r.h.s. of the Equation (32) represent respectively a tensor related to largescales inhomogeneities
and the tensor related to the pressure term
Furthermore, In order not to worry about couplings between Elsäasser variables in the dissipative terms, we make the usual simplifying assumption that kinematic viscosity is equal to magnetic diffusivity, that is ν^{±} = ν^{∓} = ν. Equation (32) is an exact equation for anisotropic MHD equations that links the secondorder complete tensor to the thirdorder mixed tensor via the average dissipation rate tensor. Using the hypothesis of global homogeneity the term Λ_{ij} = 0, while assuming local isotropy Π_{ij} = 0. The equation for the trace of the tensor can be written as
where the various quantities depends on the vector ℓ_{α}. Moreover, by considering only the trace we ruled out the possibility to investigate anisotropies related to different orientations of vectors within the secondorder moment. It is worthwhile to remark here that only the diagonal elements of the dissipation rate tensor, namely ∈ ^{±}_{ii} are positive defined while, in general, the offdiagonal elements ∈ ^{±}_{ij} are not positive. For a stationary state the Equation (36) can be written as the divergenceless condition of a quantity involving the thirdorder correlations and the dissipation rates
from which we can obtain the Yaglom’s relation by projecting Equation (37) along the longitudinal ℓ_{α} = ℓe_{r} direction. This operation involves the assumption that the flow is locally isotropic, that is fields depends locally only on the separation ℓ, so that
The only solution that is compatible with the absence of singularity in the limit ℓ → 0 is
which reduces to the Yaglom’s law for MHD turbulence as obtained by Politano and Pouquet (1998) in the inertial range when ν → 0
Finally, in the fluidlike case where z ^{+}_{i} = z ^{−}_{i} = u_{i} we obtain the usual Yaglom’s law for fluid flows
which in the isotropic case, where 〈Δu ^{3}_{ℓ} 〉 = 3〈Δu_{ℓ}Δ_{u} ^{2}_{y} 〉 = 3〈Δu_{ℓ}Δu ^{2}_{z} 〉 (Monin and Yaglom, (1975), immediately reduces to the Kolmogorov’s law
(the separation ℓ has been taken along the streamwise xdirection).
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).
From Equation (37), by defining ΔZ ^{±}_{i} = Δu_{i} ± Δb_{i} we immediately obtain the two equations
where we defined the energy fluctuations ΔE = Δu_{i}^{2} + Δb_{i}^{2} and the correlation fluctuations ΔC = Δu_{i}Δb_{i}. In the same way the quantities ∈_{E} = (∈ ^{+}_{ii} + ∈ ^{−}_{ii} )/2 and ∈_{C} = (∈ ^{+}_{ii} − ∈ ^{−}_{ii} /2 represent the energy and correlation dissipation rate, respectively. By projecting once more on the longitudinal direction, and assuming vanishing viscosity, we obtain the Yaglom’s law written in terms of velocity and magnetic fluctuations
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).
A first attempt to include density fluctuations in the framework of fluid turbulence was due to Lighthill (1955). He pointed out that, in a compressible energy cascade, the mean energy transfer rate per unit volume ∈_{V} ~ ρu^{3}/ℓ should be constant in a statistical sense (u being the characteristic velocity fluctuations at the scale ℓ), thus obtaining the scaling relation u ~ (ℓ/ρ)^{1/3}. Fluctuations of a densityweighted velocity field u ≡ ρ^{1/3}v should thus follow the usual Kolmogorov scaling u^{3} ~ ℓ. The same phenomenological arguments can be introduced in MHD turbulence Carbone et al. (2009a) by considering the pseudoenergy dissipation rates per unit volume ∈ ^{±}_{V} = ρ∈ ^{±}_{ii} and introducing densityweighted Elsäasser fields, defined as w^{±} ≡ ρ^{1/3}z^{±}. A relation equivalent to the Yaglomtype relation (40)
(C is some constant assumed to be of the order of unit) should then hold for the densityweighted increments Δw^{±}. Relation W ^{±}_{ℓ} reduces to Y ^{±}_{ℓ} in the case of constant density, allowing for comparison between the Yaglom’s law for incompressible MHD flows and their compressible counterpart. Despite its simple phenomenological derivation, the introduction of the density fluctuations in the Yaglomtype scaling (47) should describe the turbulent cascade for compressible fluid (or magnetofluid) turbulence Even if the modified Yaglom’s law (47) is not an exact relation as (40), being obtained from phenomenological considerations, the law for the velocity field in a compressible fluid flow has been observed in numerical simulations, the value of the constant C results negative and of the order of unity (Padoan et al., (2007; Kowal and Lazarian, (2007).
Yaglom’s law in the shell model for MHD turbulence
As far as the shell model is concerned, the existence of a cascade towards small scales is expressed by an exact relation, which is equivalent to Equation (41). Using Equations (24), the scalebyscale pseudoenergy budget is given by
The second and third terms on the right hand side represent, respectively, the rate of pseudoenergy dissipation and the rate of pseudoenergy injection. The first term represents the flux of pseudoenergy along the wave vectors, responsible for the redistribution of pseudoenergies on the wave vectors, and is given by
Using the same assumptions as before, namely: i) the forcing terms act only on the largest scales, ii) the system can reach a statistically stationary state, and iii) in the limit of fully developed turbulence, ν → 0, the mean pseudoenergy dissipation rates tend to finite positive limits ∈^{±}, it can be found that
This is an exact relation which is valid in the inertial range of turbulence Even in this case it can be used as an operative definition of the inertial range in the shell model, that is, the inertial range of the energy cascade in the shell model is defined as the range of scales k_{n}, where the law from Equation (49) is verified.
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.
Turbulence in the ecliptic
When dealing with laboratory turbulence it is important to know all the aspects of the experimental device where turbulent processes take place in order to estimate related possible effects driven or influenced by the environment. In the solar wind, the situation is, in some aspects, similar although the plasma does not experience any confinement due to the “experimental device”, which would be represented by free interplanetary space However, it is a matter of fact that the turbulent state of the wind fluctuations and the subsequent radial evolution during the wind expansion greatly differ from fast to slow wind, and it is now well accepted that the macrostructure convected by the wind itself plays some role (see reviews by Tu and Marsch, (1995a; Goldstein et al., (1995b).
Fast solar wind originates from the polar regions of the Sun, within the open magnetic field line regions identified by coronal holes. Beautiful observations by SOHO spacecraft (see animation of Figure 14) have localized the birthplace of the solar wind within the intergranular lane, generally where three or more granules get together. Clear outflow velocities of up to 10 km s^{−1} have been recorded by SOHO/SUMER instrument (Hassler et al., (1999).
Slow wind, on the contrary, originates from the equatorial zone of the Sun. The slow wind plasma leaks from coronal features called “helmets”, which can be easily seen protruding into the Sun’s atmosphere during a solar eclipse (see Figure 15). Moreover, plasma emissions due to violent and abrupt phenomena also contribute to the solar wind in these regions of the Sun. An alternative view is that both high and lowspeed winds come from coronal holes (defined as open field regions) and that the wind speed at 1 AU is determined by the rate of fluxtube expansion near the Sun as firstly suggested by Levine et al. (1977) (Wang and Sheeley Jr, (1990; Bravo and Stewart, (1997; Arge and Pizzo, (2000; Poduval and Zhao, (2004; Whang et al., (2005, see also:) and/or by the location and strength of the coronal heating (Leer and Holzer, (1980; Hammer, (1982; Hollweg, (1986; Withbroe, (1988; Wang, (1993, (1994; Sandbaek et al., (1994; Hansteen and Leer, (1995; Cranmer et al., (2007).
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 18 focuses on a region centered on day 75, recognizable in Figure 16, when the s/c was at approximately 0.7 AU from the Sun. Slow wind on the lefthand side of the plot, fast wind on the right hand side, and the stream interface in between, can be clearly seen. This is a sort of canonical situation often encountered in the ecliptic, within the inner heliosphere, during solar activity minimum. Typical solar wind parameters, like proton number density ρ_{p} proton temperature T_{p}, magnetic field intensity _{B}, azimuthal angle Φ, and elevation angle Θ are shown in the panels below the wind speed profile. A quick look at the data reveals that fast wind is less dense but hotter than slow wind. Moreover, both proton number density and magnetic field intensity are more steady and, in addition, the bottom two panels show that magnetic field vector fluctuates in direction much more than in slow wind. This last aspect unravels the presence of strong Alfvénic fluctuations which act mainly on magnetic field and velocity vector direction, and are typically found within fast wind (Belcher and Davis Jr, (1971; Belcher and Solodyna, (1975). The region just ahead of the fast wind, namely the stream interface, where dynamical interaction between fast and slow wind develops, is characterized by compressive effects which enhance proton density, temperature and field intensity. Within slow wind, a further compressive region precedes the stream interface but it is not due to dynamical effects but identifies the heliospheric current sheet, the surface dividing the two opposite polarities of the interplanetary magnetic field. As a matter of fact, the change of polarity can be noted within the first half of day 73 when the azimuthal angle Φ rotates by about 180°. Detailed studies (Bavassano et al., (1997) based on interplanetary scintillations (IPS) and insitu measurements have been able to find a clear correspondence between the profile of pathintegrated density obtained from IPS measurements and insitu measurements by Helios 2 when the s/c was around 0.3 AU from the Sun.
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.
Spectral properties
First evidences of the presence of turbulent fluctuations were showed by Coleman (1968), who, using Mariner 2 magnetic and plasma observations, investigated the statistics of interplanetary fluctuations during the period August 27  October 31, 1962, when the spacecraft orbited from 1.0 to 0.87 AU. At variance with Coleman (1968), Barnes and Hollweg (1974) analyzed the properties of the observed lowfrequency fluctuations in terms of simple waves, disregarding the presence of an energy spectrum. Here we review the gross features of turbulence as observed in space by Mariner and Helios spacecraft. By analyzing spectral densities, Coleman (1968) concluded that the solar wind flow is often turbulent, energy being distributed over an extraordinarily wide frequency range, from one cycle per solar rotation to 0.1 Hz. The frequency spectrum, in a range of intermediate frequencies [2 × 10^{−5} −2.3 × 10^{−3}], was found to behave roughly as f^{−1.2}, the difference with the expected Kraichnan f^{−1.5} spectral slope was tentatively attributed to the presence of highfrequency transverse fluctuations resulting from plasma gardenhose instability (Scarf et al., (1967). Waves generated by this instability contribute to the spectrum only in the range of frequencies near the proton cyclotron frequency and would weaken the frequency dependence relatively to the Kraichnan scaling. The magnetic spectrum obtained by Coleman (1968) is shown in Figure 20.
Spectral properties of the interplanetary medium have been summarized by Russell (1972), who published a composite spectrum of the radial component of magnetic fluctuations as observed by Mariner 2, Mariner 4, and OGO 5 (see Figure 21). The frequency spectrum so obtained was divided into three main ranges: i) up to about 10^{−4} Hz the spectral slope is about 1/f; ii) at intermediate frequencies 10^{−4} ≤ f ≤ 10^{−1} Hz a spectrum which roughly behaves as f^{3/2} has been found; iii) the highfrequency part of the spectrum, up to 1 Hz, behaves as 1/f^{2}. The intermediate range^{Footnote 7} of frequencies shows the same spectral properties as that introduced by Kraichnan (1965) in the framework of MHD turbulence It is worth reporting that scatter plots of the values of the spectral index of the intermediate region do not allow us to distinguish between a Kolmogorov spectrum f^{−5/3} and a Kraichnan spectrum f^{−3/2} (Veltri, (1980).
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.
It is worth mentioning that this difference was first observed by Salem (2000) years before, but, at that time, the accuracy of the data was questioned Salem et al. (2009). Thus, to corroborate previous results, Salem et al. (2009) investigated anomalous scaling and intermittency effects of both magnetic field and solar wind velocity fluctuations in the inertial range using WIND data. These authors used a wavelet technique for a systematic elimination of intermittency effects on spectra and structure functions in order to recover the actual scaling properties in the inertial range. They found that magnetic field and velocity fluctuations exhibit a welldefined, although different, monofractal behavior, following a Kolmogorov −5/3 scaling and a IroshnikovKraichnan −3/2 scaling, respectively. These results are clearly opposite to the expected scaling for kinetic and magnetic fluctuations which should follow Kolmogorov and Kraichnan scaling, respectively (see Section 2.8). However, as remarked by Roberts (2007), Voyager observations of the velocity spectrum have demonstrated a likely asymptotic state in which the spectrum steepens towards a spectral index of −5/3, finally matching the magnetic spectrum and the theoretical expectation of Kolmogorov turbulence Moreover, the same authors examined Ulysses spectra to determine if the Voyager result, based on a very few sufficiently complete intervals, were correct. Preliminary results confirmed the −5/3 slope for velocity fluctuations at ~5 AU from the Sun in the ecliptic.
Figure 23, taken from Roberts (2007), shows the evolution of the spectral index during the radial excursion of Ulysses. These authors examined many intervals in order to develop a more general picture of the spectral evolution in various conditions, and how magnetic and velocity spectra differ in these cases. The general trend shown in Figure 23 is towards −5/3 as the distance increases. Lower values are due to the highly Alfvénic fast polar wind while higher values, around 2, are mainly due to the jumps at the stream fronts as previously shown by Roberts (2007). Thus, the discrepancy between magnetic and velocity spectral slope is only temporary and belongs to the evolutionary phase of the spectra towards a well developed Kolmogorov like turbulence spectrum.
Horbury et al. (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.
Experimental evaluation of Reynolds number in the solar wind
Properties of solar wind fluctuations have been widely studied in the past, relying on the “frozenin approximation” (Taylor, (1938). The hypothesis at the basis of Taylor’s approximation is that, since large integral scales in turbulence contain most of the energy, the advection due to the smallest turbulent scales fluctuations can be disregarded and, consequently, the advection of a turbulent field past an observer in a fixed location is considered solely due to the larger scales. In experimental physics, this hypothesis allows time series measured at a single point in space to be interpreted as spatial variations in the mean flow being swept past the observer. However, the canonical way to establish the presence of spatial structures relies in the computation of twopoint single time measurements. Only recently, the simultaneous presence of several spacecraft sampling solar wind parameters allowed to correlate simultaneous insitu observations in two different observing locations in space Matthaeus et al. (2005) and Weygand et al. (2007) firstly evaluated the twopoint correlation function using simultaneous measurements of interplanetary magnetic field from the Wind, ACE, and Cluster spacecraft. Their technique allowed to compute for the first time fundamental turbulence parameters previously determined from single spacecraft measurements. In particular, these authors evaluated the correlation scale λ_{C} and the Taylor microscale λ_{T} which allow to determine empirically the effective magnetic Reynolds number.
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.
The Taylor scale λ_{T} and the correlation length λ_{C}, as indicated in Figure 26, can be obtained from the twopoint correlation function being the former the radius of curvature of the Correlation function at the origin and the latter the scale at which turbulent fluctuation are no longer correlated. Thus, λ_{T} can be obtained from from Taylor expansion of the two point correlation function for r → 0 (Tennekes and Lumely, (1972):
where r is the spacecraft separation and R(r) = 〈b(x) · b(x + r). is the autocorrelation function computed along the x direction for the fluctuating field b(x). On the other hand, the correlation length λ_{C} can be obtained integrating the normalized correlation function along a chosen direction of integration ξ:
At this point, following Batchelor (1970) it is possible to obtain the effective magnetic Reynolds number:
Figure 27 shows estimates of the correlation function from ACEWind for separation distances 20 − 350 R_{E} and two sets of Cluster data for separations 0.02 − 0.04 R_{E} and 0.4 − 1.2 R_{E}, respectively.
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 ^{eff}_{m} which resulted to be of the order of 2.3 × 10^{5}, as illustrated in Figure 28.
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.
Helios 2 power density spectra were obtained from the trace of the spectral matrix of magnetic field fluctuations, and belong to the same corotating stream observed on day 49, at a heliocentric distance of 0.9 AU, on day 75 at 0.7 AU and, finally, on day 104 at 0.3 AU. Ulysses spectra, constructed in the same way as those of Helios 2, were taken at 1.4 and 4.8 AU during the ecliptic phase of the orbit. Observations at 4.8 AU refer to the end of 1991 (fast wind period started on day 320, slow wind period started on day 338) while observations taken at 1.4 AU refer to fast wind observed at the end of August of 2007, starting on day 241:12.
While the spectral index of slow wind does not show any radial dependence, being characterized by a single Kolmogorov type spectral index, fast wind is characterized by two distinct spectral slopes: about −1 within low frequencies and about a Kolmogorov like spectrum at higher frequencies. These two regimes are clearly separated by a knee in the spectrum often referred to as “frequency break”. As the wind expands, the frequency break moves to lower and lower frequencies so that larger and larger scales become part of the 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).
Figure 30 shows the frequency location of the spectral breaks observed in the lefthandside panel of Figure 29 as a function of heliocentric distance The radial distribution of these 5 points suggests that the frequency break moves at lower and lower frequencies during the wind expansion following a powerlaw of the order of R^{−1.5}. Previous results, obtained for long data sets spanning hundreds of days and inevitably mixing fast and slow wind, were obtained by Matthaeus and Goldstein (1986) who found the breakpoint around 10 h at 1 AU, and Klein et al. (1992) who found that the breakpoint was near 16 h at 4 AU. Obviously, the frequency location of the breakpoint provided by these early determinations is strongly affected by the fact that mixing fast and slow wind would shift the frequency break to lower frequencies with respect to solely fast wind. In any case, this frequency break is strictly related to the correlation length (Klein, (1987) and the shift to lower frequency, during the wind expansion, is consistent with the growth of the correlation length observed in the inner (Bruno and Dobrowolny, (1986) and outer heliosphere (Matthaeus and Goldstein, (1982a). Analogous behavior for the low frequency shift of the spectral break, similar to the one observed in the ecliptic, has been reported by Horbury et al. (1996a) studying the rate of turbulent evolution over the Sun’s poles. These authors used Ulysses magnetic field observations between 1.5 and 4.5 AU selecting mostly undisturbed, high speed polar flows. They found a radial gradient of the order of R^{−1.1}, clearly slower than the one reported in Figure 30 or that can be inferred from results by Bavassano et al. (1982b) confirming that the turbulence evolution in the polar wind is slower than the one in the ecliptic, as qualitatively predicted by Bruno (1992), because of the lack of large scale stream shears. However, these results will be discussed more extensively in in Section 4.1.
However, the phenomenology described above only apparently resembles hydrodynamic turbulence where the large eddies, below the frequency break, govern the whole process of energy cascade along the spectrum (Tu and Marsch, (1995b). As a matter of fact, when the relaxation time increases, the largest eddies provide the energy to be transferred along the spectrum and dissipated, with a decay rate approximately equal to the transfer rate and, finally, to the dissipation rate at the smallest wavelengths where viscosity dominates. Thus, we expect that the energy containing scales would loose energy during this process but would not become part of the turbulent cascade, say of the inertial range. Scales on both sides of the frequency break would remain separated. Accurate analysis performed in the solar wind (Bavassano et al., (1982b; Marsch and Tu, (1990b; Roberts, (1992) have shown that the low frequency range of the solar wind magnetic field spectrum radially evolves following the WKB model, or geometrical optics, which predicts a radial evolution of the power associated with the fluctuations ~ r^{−3}. Moreover, a steepening of the spectrum towards a Kolmogorov like spectral index can be observed. On the contrary, the same insitu observations established that the radial decay for the higher frequencies was faster than ~r^{−3} and the overall spectral slope remained unchanged. This means that the energy contained in the largest eddies does not decay as it would happen in hydrodynamic turbulence and, as a consequence, the largest eddies cannot be considered equivalent to the energy containing eddies identified in hydrodynamic turbulence So, this low frequency range is not separated from the inertial range but becomes part of it as the turbulence ages. These observations cast some doubts on the applicability of hydrodynamic turbulence paradigm to interplanetary MHD turbulence A theoretical help came from adopting a local energy transfer function (Tu et al., (1984; Tu, (1987a,b, (1988), which would take into account the nonlinear effects between eddies of slightly differing wave numbers, together with a WKB description which would mainly work for the large scale fluctuations. This model was able to reproduce the displacement of the frequency break with distance by combining the linear WKB law and a model of nonlinear coupling besides most of the features observed in the magnetic power spectra P(f, r) observed by Bavassano et al. (1982b). In particular, the concept of the “frequency break”, just mentioned, was pointed out for the first time by Tu et al. (1984) who, developing the analytic solution for the radially evolving power spectrum P(f, r) of fluctuations, obtained a critical frequency “f_{c}” such that for frequencies f ≪ f_{c}, P(f, r) ∝ f^{−1} and for f≫ f_{c}, P(f, r) ∝ f^{−1.5}.
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).
Sonnerup and Cahill (1967) introduced the minimum variance analysis which consists in determining the eigenvectors of the matrix
where i and j denote the components of magnetic field along the axes of a given reference system. The statistical properties of eigenvalues approximately satisfy the following statements:

One of the eigenvalues of the variance matrix is always much smaller than the others, say 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.
A different approach to anisotropic fluctuations in solar wind turbulence have been made by Bigazzi et al. (2006) and SorrisoValvo et al. (2006, (2010b). In these studies the full tensor of the mixed secondorder structure functions has been used to quantitatively measure the degree of anisotropy and its effect on smallscale turbulence through a fit of the various elements of the tensor on a typical function (SorrisoValvo et al., (2006). Moreover three different regions of the nearEarth space have been studied, namely the solar wind, the Earth’s foreshock and magnetosheath showing that, while in the undisturbed solar wind the observed strong anisotropy is mainly due to the largescale magnetic field, near the magnetosphere other sources of anisotropy influence the magnetic field fluctuations (SorrisoValvo et al., (2010b).
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).
To quantify anisotropy in the distribution of wave vectors k for a given dynamical variable Q(k, t) (namely the energy, crosshelicity, etc.), it is useful to introduce the parameter
(Shebalin et al., (1983; Carbone and Veltri, (1990), where the average of a given quantity g(k) is defined as
For a spectrum with wave vectors perpendicular to B_{0} we have a spectral anisotropy Ω = 90°, while for an isotropic spectrum Ω = 45°. Numerical simulations in 2D configuration by Shebalin et al. (1983) confirmed the occurrence of anisotropy, and found that anisotropy increases with the Reynolds number. Unfortunately, in these old simulations, the Reynolds numbers used are too small to achieve a well defined spectral anisotropy. Carbone and Veltri (1990) started from the spectral equations obtained through the Direct Interaction Approximation closure by Veltri et al. (1982), and derived a shell model analogous for the anisotropic MHD turbulence Of course the anisotropy is oversimplified in the model, in particular the Alfvén time is assumed isotropic. However, the model was useful to investigate spectral anisotropy at very high Reynolds numbers. The phenomenological anisotropic spectrum obtained from the model, for both pseudoenergies obtained through polarizations a = 1, 2 defined through Equation (18), can be written as
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).
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.
Correlation lengths in the solar wind generally increase with heliocentric distance (Matthaeus and Goldstein, (1982b; Bruno and Dobrowolny, (1986), suggesting that large scale correlations are built up during the wind expansion. This kind of evolution is common to both fast and slow wind as shown in Figure 32, where we can observe the behavior of the B_{z} correlation function for fast and slow wind at 0.3 and 0.9 AU.
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).
Bieber et al. (1996) formulated an observational test to distinguish the slab (Alfvénic) from the 2D component within interplanetary turbulence These authors assumed a mixture of transverse fluctuations, some of which have wave vectors perpendicular k ⊥ B_{0} and polarization of fluctuations δB(k_{⊥}) perpendicular to both vectors (2D geometry with k ∥ ≃ 0), and some parallel to the mean magnetic field k ∥ B_{0}, the polarization of fluctuations δB(k_{∥}) being perpendicular to the direction of B_{0} (slab geometry with k_{⊥} ≃ 0). The magnetic field is then rotated into the same mean field coordinate system used by Belcher and Davis Jr (1971) and Belcher and Solodyna (1975), where the ycoordinate is perpendicular to both B_{0} and the radial direction, while the xcoordinate is perpendicular to B_{0} but with a component also in the radial direction. Using that geometry, and defining the power spectrum matrix as
it can be found that, assuming axisymmetry, a twocomponent model can be written in the frequency domain
where the anisotropic energy spectrum is the sum of both components:
Here f is the frequency, C_{s} is a constant defining the overall spectrum amplitude in wave vector space, U_{w} is the bulk solar wind speed and ψ is the angle between B_{0} and the wind direction. Finally, r is the fraction of slab components and (1 − r) is the fraction of 2D components.
The ratio test adopted by these authors was based on the ratio between the reduced perpendicular spectrum (fluctuations ⊥ 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}.
Going back from the correlation tensor to the power spectrum is a complicated technical problem. However, Carbone et al. (1995a) derived a description of the observed anisotropy in terms of a model for the threedimensional energy spectra of magnetic fluctuations. The divergenceless of the magnetic field allows to decompose the Fourier amplitudes of magnetic fluctuations in two independent polarizations: The first one I^{[1]}(k) corresponds, in the weak turbulence theory, to the Alfvénic mode, while the second polarization I^{[2]}(k) corresponds to the magnetosonic mode. By using only the hypothesis that the medium is statistically homogeneous and some algebra, authors found that the energy spectra of both polarizations can be related to the twopoints correlation tensor and to the variance matrix. Through numerical simulations of the shell model (see later in the review) it has been shown that the anisotropic energy spectrum can be described in the inertial range by a phenomenological expression
where k_{i} are the Cartesian components of the wave vector k, and C_{s}, ℓ ^{[s]}_{i} , and μ_{s} (s = 1, 2 indicates both polarizations; i = x, y, z) are free parameters. In particular, C_{s} gives information on the energy content of both polarizations, ℓ ^{[s]}_{i} represent the spectral extensions along the direction of a given system of coordinates, and μ_{s} are two spectral indices.
A fit to the eigenvalues of the variance matrix allowed Carbone et al. (1995a) to fix the free parameters of the spectrum for both polarizations. They used data from Bavassano et al. (1982a) who reported the values of 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 ^{[1]}_{x} > ℓ ^{[1]}_{y} ≫ ℓ_{z}[1], indicating that wave vectors k ∥ B_{0} largely dominate. Concerning polarization [2], it can be found that ℓ_{x}[2] ≫ ℓ ^{[2]}_{y} ≃ ∓ ^{[2]}_{z} , 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).
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.
Based on these results, Goldstein et al. (1991) were able to reproduce the distribution of the percentage of occurrence of values of σ_{m}(f) adopting a model where the magnitude of the magnetic field was allowed to vary in a random way and the tip of the vector moved near a sphere. By this way they showed that the interplanetary magnetic field helicity measurements were inconsistent with the previous idea that fluctuations were randomly circularly polarized at all scales and were also magnitude preserving.
However, evidence for circular polarized MHD waves in the high frequency range was provided by Polygiannakis et al. (1994), who studied interplanetary magnetic field fluctuations from various datasets at various distances ranging from 1 to 20 AU. They also concluded that the difference between left and 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).
Mininni et al. (2003a) performed the first direct numerical simulations of turbulent Hall dynamo. They showed that the Hall current can have strong effects on turbulent dynamo action, enhancing or even suppressing the generation of the largescale magnetic energy. These authors injected a weak magnetic field at small scales in a system kept in a stationary regime of hydrodynamic turbulence and followed the exponential growth of magnetic energy due to the dynamo action. This evolution can be seen in Figure 35 in the same format described for Figure 40, shown in Section 3.1.9. Now, the forcing is applied at wave number k_{force} = 10 in order to give enough room for the inverse cascade to develop. The fluid is initially in a strongly turbulent regime as a result of the action of the external force at wave number k_{force} = 10. An initial magnetic fluctuation is introduced at t = 0 at k_{seed} = 35. The magnetic energy starts growing exponentially fast and, when the saturation is reached, the magnetic energy is larger than the kinetic energy. Notably, it is much larger at the largest scales of the system (i.e., k = 1). At these large scales, the system is very close to a magnetostatic equilibrium characterized by a forcefree configuration.
Alfvén correlations as incompressive turbulence
In a famous paper, Belcher and Davis Jr (1971) showed that a strong correlation exists between velocity and magnetic field fluctuations, in the form
where the sign of the correlation is given by the sign[−k · B_{0}], being k the wave vector and B_{0} the background magnetic field vector. These authors showed that in about 25 d of data from Mariner 5, out of the 160 d of the whole mission, fluctuations were described by Equation (59), and the sign of the correlation was such to indicate always an outward sense of propagation with respect to the Sun. Authors also noted that these periods mainly occur within the trailing edges of highspeed streams. Moreover, in the regions where Equation (59) is verified to a high degree, the magnetic field magnitude is almost constant (B^{2} ~ const.).
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.
Alfvénic modes are not the only low frequency plasma fluctuations allowed by the MHD equations but they certainly are the most frequent fluctuations observed in the solar wind. The reason why other possible propagating modes like the slow sonic mode and the fast magnetosonic mode cannot easily be found, besides the fact that the eigenvectors associated with these modes are not directly identifiable because they necessitate prior identification of wavevectors, contrary to the simple Alfvénic eigenvectors, depends also on the fact that these compressive modes are strongly damped in the solar wind shortly after they are generated (see Section 6). On the contrary, Alfvén fluctuations, which are difficult to be damped because of their incompressive nature, survive much longer and dominate solar wind turbulence Nevertheless, there are regions where Alfvén correlations are much stronger like the trailing edge of fast streams, and regions where these correlations are weak like intervals of slow wind (Belcher and Davis Jr, (1971; Belcher and Solodyna, (1975). However, the degree of Alfvén correlations unavoidably fades away with increasing heliocentric distance, although it must be reported that there are cases when the absence of strong velocity shears and compressive phenomena favor a high Alfvén correlation up to very large distances from the Sun (Roberts et al., (1987a; see Section 5.1).
Just to give a qualitative quick example about Alfvénic correlations in fast and slow wind, we show in Figure 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.
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, this argument forecasts an Alfvén ratio r_{A} ≠ 1 but, it does not say whether it would be larger or smaller than “1”, i.e., we could also have a final excess of kinetic energy.
Similar unbalance between magnetic and kinetic energy has recently been found in numerical simulations by Mininni et al. (2003a), already cited in Section 3.1.7. These authors studied the effect of a weak magnetic field at small scales in a system kept in a stationary regime of hydrodynamic turbulence In these conditions, the dynamo action causes the initial magnetic energy to grow exponentially towards a state of quasi equipartition between kinetic and magnetic energy. This simulation was aiming to provide more insights on a microscopic theory of the alphaeffect, which is responsible to convert part of the toroidal magnetic field on the Sun back to poloidal to sustain the cycle. However, when the simulation saturates, the unbalance between kinetic and magnetic energy reminds the conditions in which the Alfvén ratio is found in interplanetary space Results from the above study can be viewed in the animation of Figure 40. At very early time the fluid is in a strongly turbulent regime as a result of the action of the external force at wave number k_{force} = 3. An initial magnetic fluctuation is introduced at t = 0 at k_{seed} = 35. The magnetic energy starts growing exponentially fast and, when the simulation reaches the saturation stage, the magnetic power spectrum exceeds the kinetic power spectrum at large wave numbers (i.e., k > k_{force}), as also observed in Alfvénic fluctuations of the solar wind (Bruno et al., (1985; Tu and Marsch, (1990a) as an asymptotic state (Roberts et al., (1987a,b; Bavassano et al., (2000b) of turbulence
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.
Lately, Bavassano and Bruno (2000) performed a threefluid analysis of solar wind Alfvénic fluctuations in the inner heliosphere, in order to evaluate the effect of disregarding the multifluid nature of the wind on the factor relating velocity and magnetic field fluctuations. It is well known that converting magnetic field fluctuations into Alfvén units we divide by the factor F_{p} = (4πM_{p}N_{p})^{1/2}. However, fluctuations in velocity tend to be smaller than fluctuations in Alfvén units. In Figure 41 we show scatter plots between the zcomponent of the Alfvén velocity and the proton velocity fluctuations. The zdirection has been chosen as the same of V_{p}×B, where V_{p} is the proton bulk flow velocity and B is the mean field direction. The reason for such a choice is due to the fact that this direction is the least affected by compressive phenomena deriving from the wind dynamics. These results show that although the correlation coefficient in both cases is around −0.95, the slope of the best fit straight line passes from 1 at 0.29 AU to a slope considerably different from 1 at 0.88 AU.
Belcher and Davis Jr (1971) suggested that this phenomenon had to be ascribed to the presence of α particles and to an anisotropy in the thermal pressure. Moreover, taking into account the multifluid nature of the solar wind, the dividing factor should become F = F_{p}F_{i}F_{a}, where F_{i} would take into account the presence of other species besides protons, and F_{a} would take into account the presence of pressure anisotropy P_{∥} ≠ P_{⊥}, where ∥ and ⊥ refer to the background field direction. In particular, following Bavassano and Bruno (2000), the complete expressions for F_{i} and F_{i} are
and
where the letter “s” stands for the sth species, being U_{s} = V_{s} − V its velocity in the center of mass frame of reference V_{s} is the velocity of the species “s” in the s/c frame and V = (Σ_{s}M_{s}N_{s}V_{s})/(Σ_{s}M_{s}N_{s}) is the velocity of the center of mass.
Bavassano and Bruno (2000) analyzed several time intervals within the same corotating high velocity stream observed at 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).
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.
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.
Spectral power associated with outward (right panel) and inward (left panel) Alfvénic fluctuations, based on Helios 2 observations in the inner heliosphere, are concisely reported in Figure 43. The e^{−} spectrum, if we exclude the high frequency range of the spectrum relative to fast wind at 0.4 AU, shows an average power law profile with a slope of −1.64, consistent with Kolmogorov’s scaling. The lack of radial evolution of e^{−} spectrum brought Tu and Marsch (1990a) to name it “the background spectrum” of solar wind turbulence
Quite different is the behavior of e^{+} spectrum. Close to the Sun and within fast wind, this spectrum appears to be flatter at low frequency and steeper at high frequency. The overall evolution is towards the “background spectrum” by the time the wind reaches 0.8 AU.
In particular, Figure 43 tells us that the radial evolution of the normalized crosshelicity has to be ascribed mainly to the radial evolution of e^{+} rather than to both Alfvénic fluctuations (Tu and Marsch, (1990a). In addition, Figure 44, relative to the Elsässer ratio r_{E}, shows that the hourly frequency range, up to ~ 2 × 10^{−3} Hz, is the most affected by this radial evolution.
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.
However, low values of the normalized crosshelicity can also be associated with a particular type of incompressive events, which Tu and Marsch (1991) called Magnetic Field Directional Turnings or MFDT. These events, found within slow wind, were characterized by very low values of δ_{c} close to zero and low values of the Alfvén ratio, around 0.2. Moreover, the spectral slope of e^{+}, e^{−} and the power associated with the magnetic field fluctuations was close to the Kolmogorov slope. These intervals were only scarcely compressive, and short period fluctuations, from a few minutes to about 40 min, were nearly pressure balanced. Thus, differently from what had previously been observed by Bruno et al. (1989), who found low values of crosshelicity often accompanied by compressive events, these MFDTs were mainly incompressive. In these structures most of the fluctuating energy resides in the magnetic field rather than velocity as shown in Figure 47 taken from Tu and Marsch (1991). It follows that the amplitudes of the fluctuating Alfvénic fields δz^{±} result to be comparable and, consequently, the derived parameter σ_{c} → 0. Moreover, the presence of these structures would also be able to explain the fact that r_{A} < 1. Tu and Marsch (1991) suggested that these fluctuations might derive from a special kind of magnetic structures, which obey the MHD equations, for which (B · ∇)B = 0, field magnitude, proton density, and temperature are all constant. The same authors suggested the possibility of an interplanetary turbulence mainly made of outwardly propagating Alfvén waves and convected structures represented by MFDTs. In other words, this model assumed that the spectrum of e^{−} would be caused by MFDTs. The different radial evolution of the power associated with these two kind of components would determine the radial evolution observed in both σ_{c} and r_{A}. Although the results were not quantitatively satisfactory, they did show a qualitative agreement with the observations.
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).
These results, shown in the left panels of Figure 48, highlight the presence of a radial evolution of the fluctuations towards a doublepeaked distribution during the expansion of the solar wind. The relative analysis has been performed on a corotating fast stream observed by Helios 2 at three different heliocentric distances over consecutive solar rotations (see Figure 16 and related text). Closer to the Sun, at 0.3 AU, the distribution is well centered around σ_{r} = 0 and σ_{c} = 1, suggesting that Alfvénic fluctuations, outwardly propagating, dominate the scenario. By the time the wind reaches 0.7 AU, the appearance of a tail towards negative values of σ_{r} and lower values of σ_{c} indicates a partial loss of the Alfvénic character in favor of fluctuations characterized by a stronger magnetic energy content. This clear tendency ends up with the appearance of a secondary peak by the time the wind reaches 0.88 AU. This new family of fluctuations forms around σ_{r} = −1 and σ_{c} = 0. The values of σ_{r} and σ_{c} which characterize this new population are typical of MFDT structures described by Tu and Marsch (1991). Together with the appearance of these fluctuations, the main peak characterized by Alfvén like fluctuations looses much of its original character shown at 0.3 AU. The yellow straight line that can be seen in the left panels of Figure 48 would be the linear relation between σ_{r} and σ_{c} in case fluctuations were made solely by Alfvén waves outwardly propagating and advected MFDTs (Tu and Marsch, (1991) and it would replace the canonical, quadratic relation σ ^{2}_{r} + σ ^{2}_{c} ≤ 1 represented by the yellow circle drawn in each panel. However, the yellow dashed line shown in the left panels of Figure 48 does not seem to fit satisfactorily the observed distributions.
Quite different is the situation within slow wind, as shown in the right panels of Figure 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.
On the nature of Alfvénic fluctuations
The Alfvénic nature of outward modes has been widely recognized through several frequency decades up to periods of the order of several hours in the s/c rest frame (Bruno et al., (1985). Conversely, the nature of those fluctuations identified by δd^{−}, called “inward Alfvén modes”, is still not completely clear. There are many clues which would suggest that these fluctuations, especially in the hourly frequencies range, have a nonAlfvénic nature. Several studies on this topic in the low frequency range have suggested that structures convected by the wind could well mimic nonexistent inward propagating modes (see the review by Tu and Marsch, (1995a). However, other studies (Tu et al., (1989b) have also found, in the high frequency range and within fast streams, a certain anisotropy in the components which resembles the same anisotropy found for outward modes. So, these observations would suggest a close link between inward modes at high frequency and outward modes, possibly the same nature.
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).
An alternative generation mechanism was proposed by Tu et al. (1989b) based on the parametric decay of e^{+} in high frequency range (Galeev and Oraevskii, (1963). This mechanism is such that large amplitude Alfvénic waves, unstable to perturbations of random field intensity and density fluctuations, would decay into two secondary Alfvénic modes propagating in opposite directions and a soundlike wave propagating in the same direction of the pump wave. Most of the energy of the mother wave would go into the soundlike fluctuation and the backward propagating Alfvénic mode. On the other hand, the production of e^{−} modes by parametric instability is not particularly fast if the plasma β ~ 1, like in the case of solar wind (Goldstein, (1978; Derby, (1978), since this condition slows down the growth rate of the instability. It is also true that numerical simulations by Malara et al. (2000, (2001a, (2002), and Primavera et al. (2003) have shown that parametric decay can still be thought as a possible mechanism of local production of turbulence within the polar wind (see Section 4). However, the strong correlation between e^{+} and e^{−} profiles found only within the highest frequency bands would support this mechanism and would suggest that e^{−} modes within these frequency bands would have an Alfvénic nature. Another feature shown in Figure 50 that favors these conclusions is the fact that both δz^{+} and δz^{−} keep the direction of their minimum variance axis aligned with the background magnetic field only within the fast wind, and exclusively within the highest frequency bands. This would not contradict the view suggested by Barnes (1981). Following this model, the majority of Alfvénic fluctuations propagating in one direction have the tip of the magnetic field vector randomly wandering on the surface of half a sphere of constant radius, and centered along the ambient field B_{∘}. In this situation the minimum variance would be oriented along B_{∘}, although this would not represent the propagation direction of each wave vector which could propagate even at large angles from this direction. This situation can be seen in the right hand panel of Figure 98 of Section 10, which refers to a typical Alfvénic interval within fast wind. Moreover, δz^{+} fluctuations show a persistent anisotropy throughout the fast stream since the minimum variance axis remains quite aligned to the background field direction. This situation downgrades only at the very low frequencies where θ^{+}, the angle between the minimum variance direction of δz^{+} and the direction of the ambient magnetic field, starts wandering between 0° and 90°. On the contrary, in slow wind, since Alfvénic modes have a smaller amplitude, compressive structures due to the dynamic interaction between slow and fast wind or, of solar origin, push the minimum variance direction to larger angles with respect to B_{。}, not depending on the frequency range.
In a way, we can say that within the stream, both θ^{+} and θ^{−}, the angle between the minimum variance direction of δz^{−} and the direction of the ambient magnetic field, show a similar behavior as we look at lower and lower frequencies. The only difference is that θ^{−} reaches higher values at higher frequencies than θ^{+}. This was interpreted (Bruno et al., (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.
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.
During solar maximum (look at the upper right panel of Figure 51) the situation dramatically changes and the equatorial wind extends to higher latitudes, to the extent that there is no longer difference between polar and equatorial wind.
Evolving turbulence in the polar wind
Ulysses observations gave us the possibility to test whether or not we could forecast the turbulent evolution in the polar regions on the basis of what we had learned in the ecliptic. We knew that, in the ecliptic, velocity shear, parametric decay, and interaction of Alfvénic modes with convected structures (see Sections 3.2.1, 5.1) all play some role in the turbulent evolution and, before Ulysses reached the polar regions of the Sun, three possibilities were given:

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.
Figure 52 shows the hourly correlation coefficient for the transverse components of magnetic and velocity fields as Ulysses climbs to the south pole and during the fast latitude scanning that brought the s/c from the south to the north pole of the Sun in just half a year. While the equatorial phase of Ulysses journey is characterized by low values of the correlation coefficients, a gradual increase can be noticed starting at half of year 1993 when the s/c starts to increase its heliographic latitude from the ecliptic plane up to 80.2° south, at the end of 1994. Not only the degree of δb − δv correlation resembled Helios observations but also the spectra of these fluctuations showed characteristics which were very similar to those observed in the ecliptic within fast wind like the spectral index of the components, that was found to be flat at low frequency and more Kolmogorovlike at higher frequencies (Smith et al., (1995). Balogh et al. (1995) and Forsyth et al. (1996) discussed magnetic fluctuations in terms of latitudinal and radial dependence of their variances. Similarly to what had been found within fast wind in the ecliptic (Mariani et al., (1978; Bavassano et al., (1982b; Tu et al., (1989b; Roberts et al., (1992), variance of magnetic magnitude was much less than the variance associated with the components. Moreover, transverse variances had consistently higher values than the one along the radial direction and were also much more sensitive to latitude excursion, as shown in Figure 53. In addition, the level of the normalized hourly variances of the transverse components observed during the ecliptic phase, right after the compressive region ahead of corotating interacting regions, was maintained at the same level once the s/c entered the pure polar wind. Again, these observations showed that the fast wind observed in the ecliptic was coming from the equatorward extension of polar coronal holes.
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.
They calculated the spectral index at different frequencies from the scaling of the second order structure function (see Section 7 and papers by Burlaga, (1992a,b; Marsch and Tu, (1993a; Ruzmaikin et al., (1995; and Horbury et al., (1996b) since the spectral scaling α is related to the scaling of the structure function s by the following relation: α = s+1 (Monin and Yaglom, (1975). Horbury et al. (1996a), studying variations of the spectral index with frequency for polar turbulence, found that there are two frequency ranges where the spectral index is rather steady. The first range is around 10^{−2} Hz with a spectral index around ∡5/3, while the second range is at very low frequencies with a spectral index around −1. This last range is the one where Goldstein et al. (1995a) found the best example of Alfvénic fluctuations. Similarly, ecliptic studies found that the best Alfvénic correlations belonged to the hourly, low frequency regime (Bruno et al., (1985).
Horbury et al. (1995a) presented an analysis of the high latitude magnetic field using a fractal method. Within the solar wind context, this method has been described for the first time by Burlaga and Klein (1986) and Ruzmaikin et al. (1993), and is based on the estimate of the scaling of the length function L(τ) with the scale τ. This function is closely related to the first order structure function and, if statistical selfsimilar, has scaling properties L(τ) ~ τ^{ℓ}, where ℓ is the scaling exponent. It follows that L(τ) is an estimate of the amplitude of the fluctuations at scale τ, and the relation that binds L(τ) to the variance of the fluctuations (δB)^{2} ~ τ^{s(2)} is:
where N(τ) represents the number of points at scale τ and scales like τ^{−1}. Since the power density spectrum fW(f) is related to (δB)^{2} through the relation fW(f) ~ (δB)^{2}, if W(f) ~ f^{−α}, then s(2) = α − 1, and, as a consequence α = 2ℓ + 3 (Marsch and Tu, (1996). Thus, it results very easy to estimate the spectral index at a given scale or frequency, without using spectral methods but simply computing the length function.
Results in Figure 55 show the existence of two different regimes, one with a spectral index around the Kolmogorov scaling extending from 10^{1.5} to 10^{3} s and, separated by a clear breakpoint at scales of 10^{3} s, a flatter and flatter spectral exponent for larger and larger scales. These observations were quite similar to what had been observed by Helios 2 in the ecliptic, although the turbulence state recorded by Ulysses resulted to be more evolved than the situation seen at 0.3 AU and, perhaps, more similar to the turbulence state observed around 1 AU, as shown by Marsch and Tu (1996). These authors compared the spectral exponents, estimated using the same method of Horbury et al. (1995a), from Helios 2 magnetic field observations at two different heliocentric distances: 0.3 and 1.0 AU. The comparison with Ulysses results is shown in Figure 56 where it appears rather clear that the slope of the B_{z} spectrum experiences a remarkable evolution during the wind expansion between 0.3 and 4 AU. Obviously, this comparison is meaningful in the reasonable hypothesis that fluctuations observed by Helios 2 at 0.3 AU are representative of outof theecliptic solar wind (Marsch and Tu, 1996). This figure also shows that the degree of spectral evolution experienced by the fluctuations when observed at 4 AU at high latitude, is comparable to Helios observations at 1 AU in the ecliptic. Thus, the spectral evolution at high latitude is present although quite slower with respect to the ecliptic.
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 ^{2}_{r} , α = 3.45 ± 0.09 for σ ^{2}_{T} , α = 3.37 ± 0.09 for σ ^{2}_{N} , and α = 2.48 ± 0.14 for σ ^{2}_{B} . 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.
Horbury and Balogh (2001) presented a detailed comparison between Ulysses and Helios observations about the evolution of magnetic field fluctuations in highspeed solar wind. Ulysses results, between 1.4 and 4.1 AU, were presented as wave number dependence of radial and latitudinal power scaling. The first results of this analysis showed (Figure 3 of their work) a general decrease of the power levels with solar distance, in both magnetic field components and magnitude fluctuations. In addition, the power associated with the radial component was always less than that of the transverse components, as already found by Forsyth et al. (1996). However, Horbury and Balogh (2001), supposing a possible latitude dependence, performed a multiple linear regression of the type:
where w is the power density integrated in a given spectral band, r is the radial distance and θ is the heliolatitude (0° at the equator). Moreover, the same procedure was applied to spectral index estimates α of the form α = A_{α} + B_{α} log_{10} r + C_{α} sin θ. Results obtained for B_{p}, C_{p}, _{B}_{α}, C_{α} are shown in Figure 58.
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).
Although most of these results support previous conclusions obtained for the ecliptic turbulence, the negative value of the latitudinal power trend that starts within the second range, is unexpected. As a matter of fact, moving towards more Alfvén regions like the polar regions, one would perhaps expect a positive latitudinal trend similarly to what happens in the ecliptic when moving from slow to fast wind.
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.
The same authors investigated the anisotropy of these fluctuations as a function of radial and latitudinal excursion. Their results, reported in Figure 59, show that, at 2.5 AU, the lowest compressibility is recorded within the hourly frequency band (third and part of the fourth band), which has been recognized as the most Alfvénic frequency range. The anisotropy of the components confirms that the power associated with the transverse components is larger than that associated with the radial one, and this difference slightly tends to decrease at higher wave numbers.
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.
It is interesting to notice that the spectral slope of the spectrum of the components for Helios 1 is slightly higher than that of Ulysses, suggesting a slower radial evolution of turbulence in the polar wind (Bruno, (1992; Bruno and Bavassano, (1992). However, the faster spectral evolution at low latitudes does not lead to strong differences between the spectra.
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.
The Alfvén ratio (bottom panels of Figure 62) has values around 0.5 for frequencies higher than roughly 10.5 Hz, with no much evolution between 2 and 4 AU. A result similar to what was originally obtained in the ecliptic at about 1 AU (Martin et al., (1973; Belcher and Solodyna, (1975; Solodyna et al., (1977; Neugebauer et al., (1984; Bruno et al., (1985; Marsch and Tu, (1990a; Roberts et al., (1990). The low frequency extension of r_{A⊥} together with σ_{c⊥}, where the subscript ⊥ indicates that these quantities are calculated from the transverse components only, was interpreted by the authors as due to the sampling of Alfvénic features in longitude rather than to a real presence of Alfvénic fluctuations. However, by the time Ulysses reaches to 4 AU, σ_{c⊥} has strongly decreased as expected while r_{A⊥} gets closer to 1, making the situation less clear. Anyhow, these results suggest that the situation at 2 AU and, even more at 4 AU, can be considered as an evolution of what Helios 2 recorded in the ecliptic at shorter heliocentric distance Ulysses observations at 2 AU resemble more the turbulence conditions observed by Helios at 0.9 AU rather than at 0.3 AU.
Bavassano et al. (2000a) studied in detail the evolution of the power e^{+} and e^{−} associated with outward δz^{+} and inward δz^{−} Alfvénic fluctuations, respectively. The study referred to the polar regions, during the wind expansion between 1.4 and 4.3 AU. These authors analyzed 1 h variances of δz^{±} and found two different regimes, as shown in Figure 63. Inside 2.5 AU outward modes e^{+} decrease faster than inward modes e^{−}, in agreement with previous ecliptic observations performed within the trailing edge of corotating fast streams (Bruno and Bavassano, (1991; Tu and Marsch, (1990b; Grappin et al., (1989). Beyond this distance, the radial gradient of e^{−} becomes steeper and steeper while that of e^{+} remains approximately unchanged. This change in e^{−} is rather fast and both species keep declining with the same rate beyond 2.5 AU. The radial dependence of e^{+} between r^{−1.39} and r^{−1.48}, reported by Bavassano et al. (2000a), indicate a radial decay faster than r^{−1} predicted by WKB approximation. This is in agreement with the analysis performed by Forsyth et al. (1996) using magnetic field observations only.
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.
On the other hand, also the Alfvén ratio r_{A} shows a clear radial dependence that stops at about the same limit distance of 2.5 AU. In this case, r_{A} constantly decreases from ~ 0.4 at 1.4 AU to ~ 0.25 at 2.5 AU, slightly fluctuating around this value for larger distances. A different interpretation of these results was offered by Grappin (2002). For this author, since Ulysses has not explored the whole threedimensional heliosphere, solar wind parameters experience different dependencies on latitude and distance which would result in the same radial distance variation along Ulysses trajectory as claimed in Bavassano’s works. Another interesting feature observed in polar turbulence is unraveled by Figure 64 from Bavassano et al. (1998, 2000b). The plot shows 2D histograms of normalized crosshelicity and normalized residual energy (see Appendix B.3.1 for definition) for different heliospheric regions (ecliptic wind, midlatitude wind with strong velocity gradients, polar wind). A predominance of outward fluctuations (positive values of σ_{c}) and of magnetic fluctuations (negative values of σ_{r}) seems to be a general feature. It results that the most Alfvénic region is the one at high latitude and at shorter heliocentric distances. However, in all the panels there is always a relative peak at σ_{c} ≃ 0 and σ_{r} ≃ −1, which might well be due to magnetic structures like the MFDT found by Tu and Marsch (1991) in the ecliptic.
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.
As already discussed in the previous section, Horbury and Balogh (2001), using Ulysses data from the northern polar pass, evaluated the dependence of magnetic field power levels on solar distance and latitude using a multiple regression analysis based on Equation (60). In the Alfvénic range, the latitudinal coefficient “C” for power in field components was appreciably different from 0 (around 0.3). However, this analysis was limited to magnetic field fluctuations alone and cannot be transferred sic et simpliciter to Alfvénic turbulence In their analysis, Bavassano et al. (2002b) used the first southern and northern polar passes and removed from their dataset all intervals with large gradients in plasma velocity, and/or plasma density, and/or magnetic field magnitude, as already done in Bavassano et al. (2000a). As a matter of fact, the use of Elsässer variables (see Appendix B.3.1) instead of magnetic field, and of selected data samples, leads to very small values of the latitudinal coefficient as shown in Figure 65, where different contributions are plotted with different colors and where the top panel refers to the same dataset used by Horbury and Balogh (2001), while the bottom panel refers to a dataset omnicomprehensive of south and north passages free of strong compressive events (Bavassano et al., (2000a). Moreover, the latitudinal effect appears to be very weak also for the data sample used by Horbury and Balogh (2001), although this is the sample with the largest value of the “C” coefficient.
A further argument in favor of radial vs. latitudinal dependence is represented by the comparison of the radial gradient of e^{+} in different regions, in the ecliptic and in the polar wind. These results, shown in Figure 66, provide the radial slopes for e^{+} (red squares) and e^{+} (blue diamonds) in different regions. The first three columns (labeled EQ) summarize ecliptic results obtained with different values of an upper limit (TBN) for relative fluctuations of density and magnetic intensity. The last two columns (labeled POL) refer to the results for polar turbulence (north and south passes) outside and inside 2.6 AU, respectively. A general agreement exists between slopes in ecliptic and in polar wind with no significant role left for latitude, the only exception being e^{+} in the region inside 2.6 AU. The behavior of the inward component cannot be explained by a simple power law over the range of distances explored by Ulysses. Moreover, a possible latitudinal effect has been clearly rejected by the results from a multiple regression analysis performed by Bavassano et al. (2002a) similar to that reported above for e^{+}.
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).
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.
The above simulations by Roberts et al. (1991) were successively implemented with a com pressive pseudospectral code (Ghosh and Matthaeus, (1990) which provided evidence that, during this turbulence evolution, clear correlations between magnetic field magnitude and density fluctuations, and between z^{−} and density fluctuations should arise. However, such a clear correlation, byproduct of the nonlinear evolution, was not found in solar wind data (Marsch and Tu, (1993b; Bruno et al., (1996). Moreover, their results did not show the flattening of e^{−} spectrum at higher frequency, as observed by Helios (Tu et al., (1989b). As a consequence, velocity shear alone cannot explain the whole phenomenon, other mechanisms must also play a relevant role in the evolution of interplanetary turbulence
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.
Grappin et al. (1992) observed that the solar wind expansion increases the lengths normal to the radial direction, thus producing an effect similar to a kind of inverse energy cascade. This effect perhaps should be able to compete with the turbulent cascade which transfers energy to small scales, thus stopping the nonlinear interactions. In absence of nonlinear interactions, the natural tendency towards an increase of σ_{c} is stopped. These inferences have been corroborated by further studies like those by Grappin and Velli (1996) and Goldstein and Roberts (1999). A numerical model treating the evolution of e^{+} and e^{−}, including parametric decay of e^{+}, was presented by Marsch and Tu (1993a). The parametric decay source term was added in order to reproduce the decreasing crosshelicity observed during the wind expansion. As a matter of fact, the cascade process, when spectral equations for both e^{+} and e^{−} are included and solved selfconsistently, can only steepen the spectra at high frequency. Results from this model, shown in Figure 68, partially reproduce the observed evolution of the normalized crosshelicity. While the radial evolution of e^{+} is correctly reproduced, the behavior of e^{−} shows an overproduction of inward modes between 0.6 and 0.8 AU probably due to an overestimation of the strength of the pumpwave. However, the model is applied to the situation observed by Helios at 0.3 AU where a rather flat e^{−} spectrum already exists.
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.
In addition, Umeki and Terasawa (1992) studying the nonlinear evolution of a largeamplitude incoherent Alfvén wave via 1D magnetohydrodynamic simulations, reported that while in a low beta plasma (B ≈ 0.2) the growth of backscattered Alfvén waves, which are opposite in helicity and propagation direction from the original Alfvén waves, could be clearly detected, in a high beta plasma (B ≈ 2) there was no production of backscattered Alfvén waves. Consequently, although numerical results obtained by Malara et al. (2001b) are very encouraging, the high beta plasma (B ≈ 2), characteristic of fast polar wind at solar minimum, plays against a relevant role of parametric instability in developing solar wind turbulence as observed by Ulysses. However, these simulations do remain an important step forward towards the understanding of turbulent evolution in the polar wind until other mechanisms will be found to be active enough to justify the observations shown in Figure 63.
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.
On the nature of compressive turbulence
Considerable efforts, both theoretical and observational, have been made in order to disclose the nature of compressive fluctuations. It has been proposed (Montgomery et al., (1987; Matthaeus and Brown, (1988; Zank et al., (1990; Zank and Matthaeus, (1990; Matthaeus et al., (1991; Zank and Matthaeus, (1992) that most of compressive fluctuations observed in the solar wind could be accounted for by the Nearly Incompressible (NI) model. Within the framework of this model, (Montgomery et al., (1987) showed that a spectrum of small scale density fluctuations follows a k^{−5/3} when the spectrum of magnetic field fluctuations follows the same scaling. Moreover, it was showed (Matthaeus and Brown, (1988; Zank and Matthaeus, (1992) that if compressible MHD equations are expanded in terms of small turbulent sonic Mach numbers, pressure balanced structures, Alfvénic and magnetosonic fluctuations naturally arise as solutions and, in particular, the RMS of small density fluctuations would scale like M^{2}, being M = δu/C_{s} the turbulent sonic Mach number, δu the RMS of velocity fluctuations and C_{s} the sound speed. In addition, if heat conduction is allowed in the approximation, temperature fluctuations dominate over magnetic and density fluctuations, temperature and density are anticorrelated and would scale like M. However, in spite of some examples supporting this theory (Matthaeus et al., (1991 reported 13% of cases satisfied the requirements of NItheory), wider statistical studies, conducted by Tu and Marsch (1994), Bavassano et al. (1995) and Bavassano and Bruno (1995), showed that NI theory is not applicable sic et simpliciter to the solar wind. The reason might be in the fact that interplanetary medium is highly inhomogeneous because of the presence of an underlying structure convected by the wind. As a matter of fact, Thieme et al. (1989) showed evidence for the presence of time intervals characterized by clear anticorrelation between kinetic pressure and magnetic pressure while the total pressure remained fairly constant. These pressure balance structures were for the first time observed by Burlaga and Ogilvie (1970) for a time scale of roughly one to two hours. Later on, Vellante and Lazarus (1987) reported strong evidence for anticorrelation between field intensity and proton density, and between plasma and field pressure on time scales up to 10 h. The anticorrelation between kinetic and magnetic pressure is usually interpreted as indicative of the presence of a pressure balance structure since slow magnetosonic modes are readily damped (Barnes, (1979).
These features, observed also in their dataset, were taken by Thieme et al. (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.
These structures were finally related to the fine raylike structures or plumes associated with the underlying chromospheric network and interpreted as the signature of interplanetary flowtubes. The estimated dimension of these structures, back projected onto the Sun, suggested that they overexpand in the solar wind. In addition, Grappin et al. (2000) simulated the evolution of Alfvén waves propagating within such pressure equilibrium ray structures in the framework of global Eulerian solar wind approach and found that the compressive modes in these simulations are very much reduced within the ray structures, which indeed correspond to the observational findings (Buttighoffer et al., (1995, (1999).
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”.
A spectral analysis performed by Marsch and Tu (1993a) in the frequency range 6×10^{−3} – 6×10^{−6} showed that the nature and intensity of compressive fluctuations systematically vary with the stream structure. They concluded that compressive fluctuations are a complex superposition of magnetoacoustic fluctuations and pressure balance structures whose origin might be local, due to stream dynamical interaction, or of coronal origin related to the flow tube structure. These results are shown in Figure 74 where the correlation coefficient between number density n and total pressure P_{tot} (indicated with the symbols p_{T} in the figure), and between kinetic pressure P_{k} and magnetic pressure P_{m} (indicated with the symbols p_{k} and p_{b}, respectively) is plotted for both Helios s/c relatively to fast wind. Positive values of correlation coefficients C(n, p_{T}) and C(p_{k}, p_{b}) identify magnetosonic waves, while positive values of C(n, p_{T}) and negative values of C(p_{k}, p_{b}) identify pressure balance structures. The purest examples of each category are located at the upper left and right corners.
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.
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.
In Figure 77 scatter plots of total pressure vs. density fluctuations are used to test a model by Tu and Marsch (1994), based on the hypothesis that the compressive fluctuations observed in solar wind are mainly due to a mixture of pressurebalanced structures (PBS) and fast magnetosonic waves (W).Waves can only contribute to total pressure fluctuations while both waves and pressurebalanced structures may contribute to density fluctuations. A tunable parameter in the model is the relative PBS/W contribution to density fluctuations α. Straight lines in Figure 77 indicate the model predictions for different values of α. It is easily seen that for all polar wind samples the great majority of experimental data fall in the α > 1 region. Thus, pressurebalanced structures appear to play a major role with respect to magnetosonic waves. This is a feature already observed by Helios in the ecliptic wind (Tu and Marsch, (1994), although in a less pronounced way. Different panels of Figure 77 refer to different heliocentric distances within the polar wind. Namely, going from P1 to P4 is equivalent to move from 1.4 to 4 AU. A comparison between these panels indicates that the observed distribution tends to shift towards higher values of α (i.e., pressurebalanced structures become increasingly important), which probably is a radial distance effect.
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.
It is clearly seen that for all the polar wind samples no clear trend emerges in the data. Thus, NIMHD effects do not seem to play a relevant role in driving the polar wind fluctuations. This confirms previous results in the ecliptic by Helios in the inner heliosphere (Bavassano et al., (1995; Bavassano and Bruno, (1995) and by Voyagers in the outer heliosphere (Matthaeus et al., (1991). It is worthy of note that, apart from the lack of NI trends, the experimental data from Ulysses, Voyagers, and Helios missions in all cases exhibit quite similar distributions. In other words, for different heliospheric regions, solar wind regimes, and solar activity conditions, the behavior of the compressive fluctuations in terms of relative density fluctuations and turbulent Mach numbers seems almost to be an invariant feature.
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.
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.
It has been suggested that disturbances of the mean magnetic field intensity and plasma density act destructively on δV − δB correlation. Bruno and Bavassano (1993) analyzed the loss of the Alfvénic character of interplanetary fluctuations in the inner heliosphere within the low frequency part of the Alfvénic range, i.e., between 2 and 10 h. Figure 79, from their work, shows the wind speed profile, σ_{c}, the correlation coefficients, phase and coherence for the three components (see Appendix B.2.1), the angle between magnetic field and velocity minimum variance directions, and the heliocentric distance Magnetic field sectors were rectified (see Appendix B.3) and magnetic field and velocity components were rotated into the magnetic field minimum variance reference system (see Appendix D). Although the three components behave in a similar way, the most Alfvénic ones are the two components Y and Z transverse to the minimum variance component X. As a matter of fact, for an Alfvén mode we would expect a high δV − δB correlation, a phase close to zero for outward waves and a high coherence Moreover, it is rather clear that the most Alfvénic intervals are located within the trailing edges of high velocity streams. However, as the radial distance increases, the Alfvénic character of the fluctuations decreases and the angle Θ_{bu} increases. The same authors found that high values of Θ_{bu} are associated with low values of σ_{c} and correspond to the most compressive intervals. They concluded that the depletion of the Alfvénic character of the fluctuations, within the hourly frequency range, might be driven by the interaction with static structures or magnetosonic perturbations able to modify the homogeneity of the background medium on spatial scales comparable to the wavelength of the Alfvénic fluctuations. A subsequent paper by Klein et al. (1993) showed that the δV − δB decoupling increases with the plasma β, suggesting that in regions where the local magnetic field is less relevant, compressive events play a major role in this phenomenon.
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.
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^{Footnote 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.
Let us consider the pth moment of both absolute values^{Footnote 9} of velocity fluctuations R_{p}(τ) = 〈Δu_{τ}^{p}〉 and magnetic fluctuations S_{p}(τ) = 〈Δb_{τ}^{p}〉, also called pth order structure function in literature (brackets being time average). Here we use magnetic fluctuations across structures at intervals τ calculated by using the magnetic field intensity. Typical structure functions of magnetic field fluctuations, for two different values of p, for both a slow wind and a fast wind at 0.9 AU, are shown in Figures 80. The magnetic field we used is that measured by Helios 2 spacecraft. Structure functions calculated for the velocity fields have roughly the same shape. Looking at these Figures the typical scaling features of turbulence can be observed. Starting from low values at small scales, the structure functions increase towards a region where S_{p} → const. at the largest scales. This means that at these scales the field fluctuations are uncorrelated. A kind of “inertial range”, that is a region of intermediate scales τ where a power law can be recognized for both
is more or less visible only for the slow wind. In this range correlations exists, and we can obtain the scaling exponents ζ_{p} and ξ_{p} through a simple linear fit.
Since as we have seen, Yaglom’s law is observed only in some few samples, the inertial range in the whole solar wind is not well defined. A look at Figure 80 clearly shows that we are in a situation similar to a lowReynolds number fluid flow. In order to compare scaling exponents of the solar wind turbulent fluctuations with other experiments, it is perhaps better to try to recover exponents using the Extended SelfSimilarity (ESS), introduced some time ago by Benzi et al. (1993), and used here as a tool to determine relative scaling exponents. In the fluidlike case, the thirdorder structure function can be regarded as a generalized scaling using the inverse of Equation (42) or of Equation (41) (Politano et al., (1998). Then, we can plot the pth order structure function vs. the thirdorder one to recover at least relative scaling exponents ζ_{p}/ζ_{3} and ζ_{p}/ξ_{3} (61). Quite surprisingly (see Figure 81), we find that the range where a power law can be recovered extends well beyond the inertial range, covering almost all the experimental range. In the fluid case the scaling exponents which can be obtained through ESS at low or moderate Reynolds numbers, coincide with the scaling exponents obtained for high Reynolds, where the inertial range is very well defined Benzi et al. (1993). This is due to the fact that, since by definition ζ_{3} = 1 in the inertial range (Frisch, (1995), whatever its extension might be. In our case scaling exponents obtained through ESS can be used as a surrogate, since we cannot be sure that an inertial range exists.
It is worthwhile to remark (as shown in Figure 81) that we can introduce a general scaling relation between the qth order velocity structure function and the qth order structure function, with a relative scaling exponent α_{p}(q). It has been found that this relation becomes an exact relation
when the velocity structure functions are normalized to the average velocity within each period used to calculate the structure function (Carbone et al., (1996a). This is very interesting because it implies (Carbone et al., (1996a) that the above relationship is satisfied by the following probability distribution function, if we assume that odd moments are much smaller than the even ones:
That is, for each scale τ the knowledge of the relative scaling exponents α_{p}(q) completely determines the probability distribution of velocity differences as a function of a single parameter S_{p}(τ).
Relative scaling exponents, calculated by using data coming from Helios 2 at 0.9 AU, are reported in Table 1. As it can be seen, two main features can be noted:

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.
In Table 1 we show scaling exponents up to the sixth order. Actually, a question concerns the validation of highorder moments estimates, say the maximum value of the order p which can be determined with a finite number of points of our dataset. As the value of p increases, we need an increasing number of points for an optimal determination of the structure function (RuíTennekes). Anomalous scaling laws are generated by rare and intense events due to singularities in the gradients: the higher their intensity the more rare these events are. Of course, when the data set has a finite extent, the probability to get singularities stronger than a certain value approaches zero. In that case, scaling exponents ζ_{p} of order higher than a certain value become linear functions of p. Actually, the structure function S_{p}(τ) depends on the probability distribution function PDF(Δu_{τ}) through
and, the function S_{p} is determined only when the integral converges. As p increases, the function F_{p}(δu_{τ}) = Δu ^{p}_{τ} PDF(Δu_{τ}) becomes more and more disturbed, with some spikes, so that the integral becomes more and more undefined, as can be seen for example in Figure 1 of the paper by Dudok de Wit (2004). A simple calculation (Dudok de Wit, (2004) for the maximum value of the order p_{m} which can reliably be estimated with a given number N of points in the dataset, gives
the empirical criterion p_{m} ≃ log N. Structure functions of order p > p_{m} cannot be determined accurately.
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.
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.
Since, as we have just seen, turbulence is characterized by scaling laws, this must be a signature of selfsimilarity for fluctuations. Let us see what this means. Let us consider fluctuations at two different scales, namely Δz ^{±}_{ℓ} and Δz ^{±}_{λℓ} . Their ratio Δz ^{±}_{λℓ} /Δz ^{±}_{ℓ} depends only on the value of h, and this should imply that fluctuations are selfsimilar. This means that PDFs are related through
Let us consider the normalized variables
When h is unique or in a pure selfsimilar situation, PDFs are related through P(y ^{±}_{ℓ} ) = PDF(y ^{±}_{λℓ} ), say by changing scale PDFs coincide.
The PDFs relative to the normalized magnetic fluctuations δb_{τ} = Δb_{τ}/〈Δb ^{2}_{τ} 〉^{1/2}, at three different scales τ, are shown in Figure 82. It appears evident that the global selfsimilarity in real turbulence is broken. PDFs do not coincide at different scales, rather their shape seems to depend on the scale τ. In particular, at large scales PDFs seem to be almost Gaussian, but they become more and more stretched as τ decreases. At the smallest scale PDFs are stretched exponentials. This scaling dependence of PDFs is a different way to say that scaling exponents of fluctuations are anomalous, or can be taken as a different definition of intermittency. Note that the wings of PDFs are higher than those of a Gaussian function. This implies that intense fluctuations have a probability of occurrence higher than that they should have if they were Gaussianly distributed. Said differently, intense stochastic fluctuations are less rare than we should expect from the point of view of a Gaussian approach to the statistics. These fluctuations play a key role in the statistics of turbulence The same statistical behavior can be found in different experiments related to the study of the atmosphere (see Figure 83) and the laboratory plasma (see Figure 84).
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.
The behavior we have just described is at the heart of the multifractal approach to turbulence (Frisch, (1995). In that description of turbulence, even if the small scales of fluid flow cannot be globally selfsimilar, selfsimilarity can be reintroduced as a local property. In the multifractal description it is conjectured that turbulent flows can be made by an infinite set of points S_{h}(r), each set being characterized by a scaling law ΔZ ^{±}_{ℓ} ~ ℓ^{h(r)}, that is, the scaling exponent can depend on the position r. The usual dimension of that set is then not constant, but depends on the local value of h, and is quoted as D(h) in literature. Then, the probability of occurrence of a given fluctuation can be calculated through the weight the fluctuation assumes within the whole flow, i.e.,
and the pth order structure function is immediately written through the integral over all (continuous) values of . weighted by a smooth function μ(h) ~ 0(1), i.e.,
A moment of reflection allows us to realize that in the limit ℓ → 0 the integral is dominated by the minimum value (over .) of the exponent and, as shown by Frisch (1995), the integral can be formally solved using the usual saddlepoint method. The scaling exponents of the structure function can then be written as
In this way, the departure of ζ_{p} from the linear Kolmogorov scaling and thus intermittency, can be characterized by the continuous changing of D(h) as h varies. That is, as p varies we are probing regions of fluid where even more rare and intense events exist. These regions are characterized by small values of h, that is, by stronger singularities of the gradient of the field.
Owing to the famous Landau footnote on the fact that fluctuations of the energy transfer rate must be taken into account in determining the statistics of turbulence, people tried to interpret the nonlinear energy cascade typical of turbulence theory, within a geometrical framework. The old Richardson’s picture of the turbulent behavior as the result of a hierarchy of eddies at different scales has been modified and, as realized by Kraichnan (1974), once we leave the idea of a constant energy cascade rate we open a “Pandora’s box” of possibilities for modeling the energy cascade. By looking at scaling laws for Δz ^{±}_{ℓ} and introducing the scaling exponents for the energy transfer rate 〈∈ ^{p}_{ℓ} ~ r^{τ}_{p}, it can be found that ζ_{p} = p/m + τ_{p/m} (being m = 3 when the Kolmogorovlike phenomenology is taken into account, or m = 4 when the IroshnikovKraichnan phenomenology holds). In this way the intermittency correction are determined by a cascade model for the energy transfer rate. When τ_{p} is a nonlinear function of p, the energy transfer rate can be described within the multifractal geometry (see, e.g., Meneveau, (1991, and references therein) characterized by the generalized dimensions D_{p} = 1 − τ_{p}/(p − 1) (Hentschel and Procaccia, (1983). The scaling exponents of the structure functions are then related to D_{p} by
The correction to the linear scaling p/m is positive for p < m, negative for p > m, and zero for p = m. A fractal behavior where D_{p} = const. < 1 gives a linear correction with a slope different from 1/m.
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.
The randomβ model (Benzi et al., (1984), a multifractal modification of the βmodel, can be derived by invoking that the spacefilling factor for the fragments at a given scale in the energy cascade is not fixed, but is given by a random variable β. The probability of occurrence of a given β is assumed to be a bimodal distribution where the eddies fragmentation process generates either spacefilling eddies with probability ξ or planar sheets with probability (1 − ξ) (for conservation 0 ≤ ξ ≤ 1). It can be found that
where the free parameter ξ can be fixed through a fit on the data.
The pmodel (Meneveau, (1991; Carbone, (1993) consists in an eddies fragmentation process described by a twoscale Cantor set with equal partition intervals. An eddy at the scale ℓ, with an energy derived from the transfer rate ∈_{r}, breaks down into two eddies at the scale ℓ/2, with energies μ∈_{r} and (1 − μ)∈_{r}. The parameter 0.5 ≤ μ ≤ 1 is not defined by the model, but is fixed from the experimental data. The model gives
In the model by She and Leveque (see, e.g., She and Leveque, (1994; Politano and Pouquet, (1998) one assumes an infinite hierarchy for the moments of the energy transfer rates, leading to ∈ ^{(p+1)}_{r} ~ [∈ ^{(p)}_{r} ]^{β}[∈ ^{(∞)}_{r} ]^{1−β}, and a divergent scaling law for the infiniteorder moment ∈ ^{(∞)}_{r} ~ r^{−x}, which describes the most singular structures within the flow. The model reads
The parameter C = x/(1 − β) is identified as the codimension of the most singular structures. In the standard MHD case (Politano and Pouquet, (1995) x = β = 1/2, so that C = 1, that is, the most singular dissipative structures are planar sheets. On the contrary, in fluid flows C = 2 and the most dissipative structures are filaments. The large p behavior of the pmodel is given by ζ_{p} ~ (p/m) log_{2}(1/μ) + 1, so that Equations (64, 65) give the same results providing μ ≃ 2^{−x}. As shown by Carbone et al. (1996b) all models are able to capture intermittency of fluctuations in the solar wind. The agreement between the curves ζ_{p} and normalized scaling exponents is excellent, and this means that we realistically cannot discriminate between the models we reported above. The main problem is that all models are based on a conjecture which gives a curve ζ_{p} as a function of a single free parameter, and that curve is able to fit the smooth observed behavior of ζ_{p}. Statistics cannot prove, just disprove. We can distinguish between the fractal model and multifractal models, but we cannot realistically distinguish among the various multifractal models.
A model for the departure from selfsimilarity
Besides the idea of selfsimilarity underlying the process of energy cascade in turbulence, a different point of view can be introduced. The idea is to characterize the behavior of the PDFs through the scaling laws of the parameters, which describe how the shape of the PDFs changes when going towards small scales. The model, originally introduced by Castaing et al. (2001), is based on a multiplicative process describing the cascade. In its simplest form the model can be introduced by saying that PDFs of increments δZ ^{±}_{ℓ} , at a given scale, are made as a sum of Gaussian distributions with different widths σ = 〈(δZ ^{±}_{ℓ} )^{2}〉^{1/2}. The distribution of widths is given by G_{λ}(σ), namely
In a purely selfsimilar situation, where the energy cascade generates only a trivial variation of σ with scales, the width of the distribution G_{λ}(σ) is zero and, invariably, we recover a Gaussian distribution for P(δZ ^{±}_{ℓ} ). On the contrary, when the cascade is not strictly selfsimilar, the width of G_{λ}(σ) is different from zero and the scaling behavior of the width λ^{2} of G_{λ}(σ) can be used to characterize intermittency.
Intermittency properties recovered via a shell model
Shell models have remarkable properties which closely resemble those typical of MHD phenomena (Gloaguen et al., (1985; Biskamp, (1994; Giuliani and Carbone (1998; Plunian et al., (2012). However, the presence of a constant forcing term always induces a dynamical alignment, unless the model is forced appropriately, which invariably brings the system towards a state in which velocity and magnetic fields are strongly correlated, that is, where Z ^{±}_{n} ≠ = 0 and Z ^{∓}_{n} ≠ = 0. When we want to compare statistical properties of turbulence described by MHD shell models with solar wind observations, this term should be avoided. It is possible to replace the constant forcing term by an exponentially timecorrelated Gaussian random forcing which is able to destabilize the Alfvénic fixed point of the model (Giuliani and Carbone (1998), thus assuring the energy cascade. The forcing is obtained by solving the following Langevin equation:
where μ(t) is a Gaussian stochastic process δcorrelated in time 〈μ(t)μ(t’) = 2Dδ(t’ − t). This kind of forcing will be used to investigate statistical properties.
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 ^{−3/2}_{n} , 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.
Intermittency in the shell model is due to the time behavior of shell variables. It has been shown (Okkels, (1997) that the evolution of GOY model consists of short bursts traveling through the shells and long period of oscillations before the next burst arises. In Figures 89 and 90 we report the time evolution of the real part of both velocity variables u_{n}(t) and magnetic variables b_{n}(t) at three different shells. It can be seen that, while at smaller k_{n} variables seems to be Gaussian, at larger k_{n} variables present very sharp fluctuations in between very low fluctuations.
The time behavior of variables at different shells changes the statistics of fluctuations. In Figure 91 we report the probability distribution functions P(δu_{n}) and P(δB_{n}), for different shells n, of normalized variables
where Re indicates that we take the real part of u_{n} and b_{n}. Typically we see that PDFs look differently at different shells: At small k_{n} fluctuations are quite Gaussian distributed, while at large k_{n} they tend to become increasingly nonGaussian, by developing fat tails. Rare fluctuations have a probability of occurrence larger than a Gaussian distribution. This is the typical behavior of intermittency as observed in usual fluid flows and described in previous sections.
The same phenomenon gives rise to the departure of scaling laws of structure functions from a Kolmogorov scaling. Within the framework of the shell model the analogous of structure functions are defined as
For MHD turbulence it is also useful to report mixed correlators of the flux variables, i.e.,
Scaling exponents have been determined from a least square fit in the inertial range 3 ≤ n ≤ 12. The values of these exponents are reported in Table 4. It is interesting to notice that, while scaling exponents for velocity are the same as those found in the solar wind, scaling exponents for the magnetic field found in the solar wind reveal a more intermittent character. Moreover, we notice that velocity, magnetic and Elsässer variables are more intermittent than the mixed correlators and we think that this could be due to the cancelation effects among the different terms defining the mixed correlators.
Time intermittency in the shell model generates rare and intense events. These events are the result of the chaotic dynamics in the phasespace typical of the shell model (Okkels, (1997). That dynamics is characterized by a certain amount of memory, as can be seen through the statistics of waiting times between these events. The distributions P(δt) of waiting times is reported in the bottom panels of Figures 91, at a given shell n = 12. The same statistical law is observed for the bursts of total dissipation (Boffetta et al., (1999).
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.
In order to avoid variations of the solar activity and ecliptic disturbances (like slow wind sources, coronal mass ejections, ecliptic current sheet, and so on), and mainly mixing between fast and slow wind, SorrisoValvo et al. (2007) used high speed polar wind data measured by the Ulysses spacecraft. In particular, authors analyze the first seven months of 1996, when the heliocentric distance slowly increased from 3 AU to 4 AU, while the heliolatitude decreased from about 55° to 30°. The thirdorder mixed structure functions have been obtained using 10days moving averages, during which the fields can be considered as stationary. A linear scaling law, like the one shown in Figure 92, has been observed in a significant fraction of samples in the examined period, with a linear range spanning more than two decades. The linear law generally extends from few minutes up to 1 day or more, and is present in about 20 periods of a few days in the 7 months considered. This probably reflects different regimes of driving of the turbulence by the Sun itself, and it is certainly an indication of the nonstationarity of the energy injection process. According to the formal definition of inertial range in the usual fluid flows, authors attribute to the range where Yaglom’s law appear the role of inertial range in the solar wind turbulence (SorrisoValvo et al., (2007). This range extends on scales larger than the usual range of scales where a Kolmogorov relation has been observed, say up to about few hours (cf. Figure 25).
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.
As far as the scaling law (47) is concerned, Carbone et al. (2009a) found that a linear scaling for W ^{±}_{ℓ} as defined in (47), appears almost in all Ulysses dataset. In particular, the linear scaling for W ^{±}_{ℓ} is verified even when there is no scaling at all for Y ^{±}_{ℓ} (40). In particular, it has been observed (Carbone et al., (2009a) that a linear scaling for W ^{+}_{ℓ} . appears in about half the whole signal, while W ^{−}_{ℓ} displays scaling on about a quarter of the sample. The linear scaling law generally extends on about two decades, from a few minutes up to one day or more, as shown in Figure 93. At variance to the incompressible case, the two fluxes W ^{±}_{ℓ} coexist in a large number of cases. The pseudoenergies dissipation rates so obtained are considerably larger than the relative values obtained in the incompressible case. In fact it has been found that on average ∈^{+} ≃ 3 × 10^{3} J/Kg s. This result shows that the nonlinear energy cascade in solar wind turbulence is considerably enhanced by density fluctuations, despite their small amplitude within the Alfvénic polar turbulence Note that the new variables Δw ^{±}_{i} are built by coupling the Elsässer fields with the density, before computing the scaledependent increments. Moreover, the thirdorder moments are very sensitive to intense field fluctuations, that could arise when density fluctuations are correlated with velocity and magnetic field. Similar results, but with a considerably smaller effect, were found in numerical simulations of compressive MHD (Mac Low and Klessen, (2004).
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).