The Solar Wind as a Turbulence Laboratory
Abstract
In this review we will focus on a topic of fundamental importance for both plasma physics and astrophysics, namely the occurrence of large-amplitude low-frequency 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, Ulysses’ high latitude observations and new numerical approaches to the problem, based on the dynamics of complex systems, brought new important insights which helped to better understand how turbulent fluctuations behave in the solar wind. In particular, numerical simulations within the realm of magnetohydrodynamic (MHD) turbulence theory unraveled what kind of physical mechanisms are at the basis of turbulence generation and energy transfer across the spectral domain of the fluctuations. In other words, the advances reached in these past years in the investigation of solar wind turbulence now offer a rather complete picture of the phenomenological aspect of the problem to be tentatively presented in a rather organic way.
1 Introduction
The whole heliosphere is permeated by the solar wind, a supersonic and super-Alfvénic 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 high-amplitude 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). Because of the presence of a strong magnetic field carried by the wind, low-frequency fluctuations in the solar wind are usually described within a magnetohydrodynamic (MHD, hereafter) benchmark (Kraichnan, 1965; Biskamp, 1993; Tu and Marsch, 1995a; Biskamp, 2003). 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, high-energy 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 in-situ 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 high-latitude 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 near-equatorial and polar turbulence in order to provide the reader with a rather complete view of the low-frequency turbulence phenomenon in the 3D heliosphere.
New interesting insights in the theory of turbulence derive from the point of view which considers a turbulent flow as a complex system, a sort of benchmark for the theory of dynamical systems. The theory of chaos received the fundamental impulse just through the theory of turbulence developed by Ruelle and Takens (1971) who, criticizing the old theory of Landau and Lifshitz (1971), were able to put the numerical investigation by Lorenz (1963) in a mathematical framework. Gollub and Swinney (1975) set up accurate experiments on rotating fluids confirming the point of view of Ruelle and Takens (1971) who showed that a strange attractor in the phase space of the system is the best model for the birth of turbulence. This gave a strong impulse to the investigation of the phenomenology of turbulence from the point of view of dynamical systems (Bohr et al., 1998). For example, the criticism by Landau leading to the investigation of intermittency in fully developed turbulence was worked out through some phenomenological models for the energy cascade (cf. Frisch, 1995). Recently, turbulence in the solar wind has been used as a big wind tunnel to investigate scaling laws of turbulent fluctuations, multifractals models, etc. The review by Tu and Marsch (1995a) contains a brief introduction to this important argument, which was being developed at that time relatively to the solar wind (Burlaga, 1993; Carbone, 1993; Biskamp, 1993, 2003; Burlaga, 1995). The reader can convince himself that, because of the wide range of scales excited, space plasma can be seen as a very big laboratory where fully developed turbulence can be investigated not only per se, rather as far as basic theoretical aspects are concerned.
Turbulence is perhaps the most beautiful unsolved problem of classical physics, the approaches used so far in understanding, describing, and modeling turbulence are very interesting even from a historic point of view, as it clearly appears when reading, for example, the book by Frisch (1995). History of turbulence in interplanetary space is, perhaps, even more interesting since its knowledge proceeds together with the human conquest of space. Thus, whenever appropriate, we will also introduce some historical references to show the way particular problems related to turbulence have been faced in time, both theoretically and technologically. Finally, since turbulence is a phenomenon visible everywhere in nature, it will be interesting to compare some experimental and theoretical aspects among different turbulent media in order to assess specific features which might be universal, not limited only to turbulence in space plasmas. In particular, we will compare results obtained in interplanetary space with results obtained from ordinary fluid flows on Earth, and from experiments on magnetic turbulence in laboratory plasmas designed for thermonuclear fusion.
1.1 What does turbulence stand for?
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 restore a statistical approach to turbulence, just like random or stochastic processes, even if the problem is born within the strange dynamics of a deterministic system!
1.2 Dynamics vs. statistics
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 means some averages. Actually, the method of taking averages in a turbulent flow requires some care. We would like to remind that there are, at least, three different kinds of averaging procedures that may be used to obtain statistically-averaged 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 nearly 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 13).
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 low-order 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.
2 Equations and Phenomenology
In this section, we present the basic equations that are used to describe charged fluid flows, and the basic phenomenology of low-frequency 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). Plasma is seen as a continuous collisional medium so that all quantities are functions of space r and time t. Apart for the required quasi-neutrality, 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 non-relativistic approximation. Then, MHD equations can be derived as shown in the following sections.
2.1 The Navier-Stokes equation and the Reynolds number
The Reynolds number Re = UL/ν is evidently the only parameter of the fluid flow. This defines a Reynolds number similarity for fluid flows, namely fluids with the same value of the Reynolds number behaves in the same way. Looking at Equation (5) it can be realized that the Reynolds number represents a measure of the relative strength between the non-linear convective term and the viscous term in Equation (4). The higher Re, the more important the non-linear term is in the dynamics of the flow. Turbulence is a genuine result of the non-linear dynamics of fluid flows.
2.2 The coupling between a charged fluid and the magnetic field
It is worthwhile to remark that in the classical hydrodynamics, dissipative processes are defined through three coefficients, namely two viscosities and one thermoconduction coefficient. In the hydromagnetic case the coefficients considerably increase. Apart from few additional electrical coefficients, we have a large-scale (background) magnetic field B_{0}. This makes the MHD equations intrinsically anisotropic. Furthermore, the stress tensor (8) is deeply modified by the presence of a magnetic field B_{0}, in that kinetic viscous coefficients must depend on the magnitude and direction of the magnetic field (Braginskii, 1965). This has a strong influence on the determination of the Reynolds number.
2.3 Scaling features of the equations
The scaled Euler equations are the same as Equations (4, 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 Equation (4) own scaling properties (Frisch, 1995), that is, there exists a class of solutions which are invariate under scaling transformations. Introducing a length scale ℓ, it is straightforward to verify that the scaling transformations ℓ → λℓ′ and u → λ^{h}u′ (A is a scaling factor and h is a scaling index) leave invariate 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 ν is in general very low, we expect that ℓ_{D} is very small. Actually, there exists a simple relationship for the scaling of ℓ_{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 p → λ^{2}(β−^{h})ρ′. This means that velocity and magnetic variables have different scalings, say h ≠ β, only when the scaling for the density is taken into account. In the incompressible case, we cannot distinguish between scaling laws for velocity and magnetic variables.
2.4 The non-linear energy cascade
The basic properties of turbulence, as derived both from the Navier-Stokes equation and from phenomenological considerations, is the legacy of A.N. Kolmogorov^{3} (Frisch, 1995). 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 non-linear 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 e measured at any scale ℓ within the inertial range ℓ_{D} ≪ ℓ ≪ L.
Looking at Equation (16), we see that the role played by the non-linear term is that of a redistribution of energy among the various wave vectors. This is the physical meaning of the non-linear energy cascade of turbulence.
2.5 The inhomogeneous case
Equations (16) 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 large-scale magnetic field. A set of equations for the cross-correlation functions of both Elsässer 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 two-scale separation, and ii) small-scale fluctuations are represented as a kind of stochastic process (Tu and Marsch, 1996). These equations look quite complicated, and just a comparison based on order-of-magnitude estimates can be made between them and solar wind observations (Tu and Marsch, 1996).
2.6 Dynamical system approach to turbulence
In the limit of fully developed turbulence, when dissipation goes to zero, an infinite range of scales are excited, that is, energy lies over all available wave vectors. Dissipation takes place at a typical dissipation length scale which depends on the Reynolds number Re through ℓ_{D} ∼ LRe^{−3/4} (for a Kolmogorov spectrum E(k) ∼ k^{−5/3}). In 3D numerical simulations the minimum number of grid points necessary to obtain information on the fields at these scales is given by N ∼ (L/ℓ_{D})^{3} ∼ Re^{9/4}. This rough estimate shows that a considerable amount of memory is required when we want to perform numerical simulations with high Re. At the 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 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 issue. This can be achieved by introducing turbulence models which are investigated using tools of dynamical system theory (Bohr et al., 1998). Dynamical models, then, represent minimal set of ordinary differential equations that can mimic the gross features of energy cascade in 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 flow, becoming a paradigm for the study of chaotic systems.
Up to the Lorenz’s chaotic model, studies on the birth of turbulence dealt with linear and, very rarely, with weak non-linear evolution of external disturbances. The first physical model of laminar-turbulent 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 serie of infinite Hopf bifurcations at fixed values of the Reynolds number. Each subsequent bifurcation adds a new incommensurate frequency to the flow which, in some sense, is obliged to become turbulent because the infinite number of degrees of freedom involved.
After this discovery, the strange attractor model gained a lot of popularity, thus stimulating a great number of further studies on the time evolution of non-linear 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. 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. On the other hand, 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 non-linear Schrödinger equation (Mjolhus, 1976; Ghosh and Papadopoulos, 1987) which governs the spatio-temporal dynamics of non-linear Alfvén waves, and found that Alfvénic intermittent turbulence is characterized by strange attractors. Turbulence can evolve via two distinct routes: Pomeau-Manneville intermittency and crisis-induced intermittency. Both types of chaotic transitions follow episodic switching between different temporal behaviors. In one case (Pomeau-Manneville) the behavior of the magnetic fluctuations evolve from nearly periodic to chaotic while, in the other case the behavior intermittently assumes weakly chaotic or strongly chaotic features.
2.7 Shell models for turbulence cascade
Since numerical simulations, in some cases, cannot be used, simple dynamical systems can be introduced to investigate, for example, statistical properties of turbulent flows which can be compared with observations.
- 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 inter-shell 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 Elsässer variables \(Z_{n}^{\pm}(t)=u_{n}\pm b_{n}(t)\). 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 \(\ell_{n}\sim k_{n}^{-1}\). That is, the fields have the same scalings as field differences, for example \(Z_{n}^{\pm}\sim\vert Z^{\pm}(x+\ell_{n})-Z^{\pm}(x)\vert\sim\ell_{n}^{h}\) in fully developed turbulence. In this way, the possibility to describe spatial behavior within the model is ruled out. We can only get, from a dynamical shell model, time series for shell variables at a given k_{n}, and we loose the fact that turbulence is a typical temporal and spatial complex phenomenon.
- c)
Introduce a dynamical model which describes non-linear evolution:
Looking at Equation (15) a model must have quadratic non-linearities among opposite variables \(Z_{n}^{\pm}{t}\) and \(Z_{n}^{\mp}{t}\), 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 pseudo-energiesthat means the conservation of both the total energy and the cross-helicity:$${E^ \pm}(t) = {1 \over 2}\sum\limits_n {\vert Z_n^ \pm {\vert ^2},}$$where ℜe indicates the real part of the product \(u_{n}b_{n}^{\ast}\). As we said before, shell models cannot describe spatial geometry of non-linear interactions in turbulence, so that we loose the possibility of distinguishing between two-dimensional and three-dimensional 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$$E(t) = {1 \over 2}\sum\limits_n {\vert {u_n}{\vert ^2}} + \vert {b_n}{\vert ^2}\;;\;{H_c}(t) = \sum\limits_n 2 \Re e({u_n}b_n^{\ast}),$$which can be dimensionally identified as the magnetic helicity when α = 1, so that the shell model so obtained is able to mimic a kind of 3D MHD turbulence (Giuliani and Carbone, 1998).$$H(t) = \sum\limits_n {{{(- 1)}^n}} {{\vert {b_n}{\vert ^2}} \over {k_n^\alpha}},$$(19)
2.8 The phenomenology of fully developed turbulence: Fluid-like case
In the real space turbulence properties can be described using either the probability distribution functions (PDFs hereafter) of increments, or the longitudinal structure functions, which represents nothing but the higher order moments of the field. Disregarding the magnetic field, in a purely fully developed fluid turbulence, this is defined as \(S_{\ell}^{(p)}=\langle\delta u_{\ell}^{2}\rangle\). These quantities, in the inertial range, behave as a power law \(S_{\ell}^{(p)}\sim \ell^{\xi_{p}}\), 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 (22)), it is straightforward to compute the scaling laws \(S_{\ell}^{(p)}\sim \ell^{p/3}\). Then ξ_{p} = p/3 results to be a linear function of the order p.
2.9 The phenomenology of fully developed turbulence: Magnetically-dominated case
2.10 Some exact relationships
3 Early Observations of MHD Turbulence in the Ecliptic
Here we briefly present the history, since the first Mariner missions during the’ 60s, 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. We have a much larger laboratory but, up to now, we have made only single point measurements although the magnetospheric ESA-Cluster project has shown all the advantages of performing 3D measurements in space.
3.1 Turbulence in the ecliptic
When dealing with laboratory turbulence it is important to know all the aspects of the experimental device where turbulent processes take place in order to estimate related possible effects driven or influenced by the environment. In the solar wind, the situation is, in some aspects, similar although the plasma does not experience any confinement due to the “experimental device”, which would be represented by free interplanetary space. However, it is a matter of fact that the turbulent state of the wind fluctuations and the subsequent radial evolution during the wind expansion greatly differ from fast to slow wind, and it is now well accepted that the macrostructure convected by the wind itself plays some role (see reviews by Tu and Marsch, 1995a; Goldstein et al., 1995b).
However, this situation greatly changes during different phases of the solar activity cycle. Polar coronal holes, which during the maximum of activity are limited to small and not well defined regions around the poles, considerably widen up during solar minimum, reaching the equatorial regions (Forsyth et al., 1997; Forsyth and Breen, 2002; Balogh et al., 1999). This new configuration produces an alternation of fast and slow wind streams in the ecliptic plane, the plane where most of the spacecraft operate and record data. During the expansion, a dynamical interaction between fast and slow wind develops, generating the so called “stream interface”, a thin region ahead of the fast stream characterized by strong compressive phenomena.
3.1.1 Spectral properties
Then, as far as the solar wind turbulence concerns we do not think we should long discuss here whether or not solar wind developed turbulence be represented by f^{−5/3} or f^{−3/2}, since observations showed that the slope is usually around f^{−1.6} (Bavassano et al., 1982b; Tu and Marsch, 1995a) which, irony of fate, is just between the two cited values. Although we prefer to postpone to a future version of the present paper a detailed discussion on this topic and the related inertial range of solar wind fluctuations, it is worth citing that Tu et al. (1989c) already discussed this problem on the basis of Tu’s model (Tu, 1988), using a variable ratio of the inward to outward Alfvénic energy as determined by observations of normalized cross-helicity. These values were then used to find the cascade constant that determines the level of the energy spectrum. The value they found for this constant resulted to be very close to the value observed in ordinary fluid turbulence, assuming that the correspondence between fluid and magnetofluid theories is reached by imposing zero cross-helicity for the MHD turbulence.
As a final comment, the situation of spectral indices determination in MHD turbulence is not changed since the’ 70s (cf. Carbone and Pouquet, 2005), numerical simulations deal with MHD flows of moderate Reynolds numbers and an inertial range is scarcely observed. The debate, after thirty years, is always open and contributions are welcome.
3.1.2 Evidence for non-linear interactions
These power density spectra were obtained from the trace of the spectral matrix of magnetic field fluctuations, and belong to the same corotating stream observed by Helios 2 on day 49, at a heliocentric distance of 0.9 AU, on day 75 at 0.7 AU and, finally, on day 104 at 0.3 AU. All the spectra are characterized by two distinct spectral slopes: about −1 within low frequencies and about a Kolmogorov like spectrum at higher frequencies. These two regimes are clearly separated by a knee in the spectrum often referred to as “frequency break”. As the wind expands, the frequency break moves to lower and lower frequencies so that larger and larger scales become part of the Kolmogorov-like turbulence spectrum, i.e., of what we will indicate as “inertial range” (see discussion at the end of the previous section). Thus, the power spectrum of solar wind fluctuations is not solely function of frequency f, i.e., P(f), but it also depends on heliocentric distance r, i.e., P (f) → P (f, r).
Matthaeus and Goldstein (1986) found the breakpoint around 10 h at 1 AU, and Klein et al. (1992) found that the breakpoint was near 16 h at 4 AU. This frequency break is strictly related to the correlation length (Klein, 1987) and the shift to lower frequency, during the wind expansion, is consistent with the growth of the correlation length observed in the inner (Bruno and Dobrowolny, 1986) and outer heliosphere (Matthaeus and Goldstein, 1982a). This phenomenology only apparently resembles hydrodynamic turbulence where the large eddies, below the frequency break, govern the whole process of energy cascade along the spectrum (Tu and Marsch, 1995b). As a matter of fact, when the relaxation time increases, the largest eddies provide the energy to be transferred along the spectrum and dissipated, with a decay rate approximately equal to the transfer rate and, finally, to the dissipation rate at the smallest wavelengths where viscosity dominates. Thus, we expect that the energy containing scales would loose energy during this process but would not become part of the turbulent cascade, say of the inertial range. Scales on both sides of the frequency break would remain separated. Accurate analysis performed in the solar wind (Bavassano et al., 1982b; Marsch and Tu, 1990b; Roberts, 1992) have shown that the low frequency range of the solar wind magnetic field spectrum radially evolves following the WKB model, or geometrical optics, which predicts a radial evolution of the power associated with the fluctuations ∼ r^{−3}. Moreover, a steepening of the spectrum towards a Kolmogorov like spectral index can be observed. On the contrary, the same in-situ observations established that the radial decay for the higher frequencies was faster than ∼ r^{−3} and the overall spectral slope remained unchanged. This means that the energy contained in the largest eddies does not decay as it would happen in hydrodynamic turbulence and, as a consequence, the largest eddies cannot be considered equivalent to the energy containing eddies identified in hydrodynamic turbulence. So, this low frequency range is not separated from the inertial range but becomes part of it as the turbulence ages. These observations cast some doubts on the applicability of hydrodynamic turbulence paradigm to interplanetary MHD turbulence. A theoretical help came from adopting a local energy transfer function (Tu et al., 1984; Tu, 1987a,b, 1988), which would take into account the non-linear effects between eddies of slightly differing wave numbers, together with a WKB description which would mainly work for the large scale fluctuations. This model was able to reproduce most of the features observed in the magnetic power spectra P(f, r) observed by Bavassano et al. (1982b). In particular, the concept of the “frequency break”, just mentioned, was pointed out for the first time by Tu et al. (1984) who, developing the analytic solution for the radially evolving power spectrum P(f, r) of fluctuations, obtained a critical frequency “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}. In addition, their model was the first model able to explain the decreasing of the “break frequency” with increasing heliocentric distance.
3.1.3 Fluctuations anisotropy
Interplanetary magnetic field (IMF) and velocity fluctuations are rather anisotropic as for the first time observed by Belcher and Davis Jr (1971), Belcher and Solodyna (1975), Chang and Nishida (1973), Burlaga and Turner (1976), Solodyna and Belcher (1976), Parker (1980), Bavassano et al. (1982a), Tu et al. (1989a), and Marsch and Tu (1990a). Moreover, this feature can be better observed if fluctuations are rotated into the minimum variance reference system (see Appendix 15).
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 minimum-variance direction (see Appendix 15.1 for more details). This indicates that, at least locally, the magnetic fluctuations are confined in a plane perpendicular to the minimum-variance direction.
In the plane perpendicular to 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.
A systematic analysis for both magnetic and velocity fluctuations was performed by Klein et al. (1991, 1993) between 0.3 and 10 AU. These studies showed that magnetic field and velocity minimum variance directions are close to each other within fast wind and mainly clustered around the local magnetic field direction. The effects of expansion are such to separate field and velocity minimum variance directions. While magnetic field fluctuations keep their minimum variance direction loosely aligned with the mean field direction, velocity fluctuations tend to have their minimum variance direction oriented along the radial direction. The depleted alignment to the background magnetic field would suggest a smaller anisotropy of the fluctuations. As a matter of fact, Klein et al. (1991) found that the degree of anisotropy, which can be defined as the ratio between the power perpendicular to and that along the minimum variance direction, decreases with heliocentric distance in the outer heliosphere.
At odds with these conclusions were the results by Bavassano et al. (1982a) who showed that the ratio 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 9.1), performed by Bruno et al. (1999b), found a reason for this discrepancy.
3.1.4 Simulations of anisotropic MHD
In the presence of a DC background magnetic field 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; Carbone and Veltri, 1990). The main effect produced by the presence of the background field is to generate an anisotropic distribution of wave vectors as a consequence of the dependence of the characteristic time for the non-linear coupling on the angle between the wave vector and the background field. This effect can be easily understood if one considers the MHD equation. Due to the presence of a term (B_{0} · ∇)z^{±}, which describes the convection of perturbations in the average magnetic field, the non-linear interactions between Alfvénic fluctuations are weakened, since convection decorrelates the interacting eddies on a time of the order (k · 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).
Authors showed that spectral anisotropy is different within the three ranges of turbulence. Wave vectors perpendicular to B_{0} are present in the spectrum, but when the process of energy transfer generates a strong anisotropy (at small times), a competing process takes place which redistributes the energy over all wave vectors. The dynamical balance between these tendencies fixes the value of the spectral anisotropy 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 10^{5}. This value corresponds to a rather low value for the ratio between parallel and perpendicular correlation lengthsℓ_{∥}/ℓ_{⊥} ≃ 2, too small with respect to the observed valueℓ_{∥}/ℓ_{⊥} ≥ 10. This suggests that the non-linear dynamical evolution of an initially isotropic spectrum of turbulence is perhaps not sufficient to explain the observed anisotropy. Recent numerical simulations confirmed these results (Oughton et al., 1994).
3.1.5 Fluctuations correlation length and the Maltese Cross
The correlation time, as defined in Appendix 12, estimates how much an element of our time series x(t) at time t_{1} depends on the value assumed by x(t) at time t_{0}, being t_{1} = t_{0} + δt. This concept can be transferred from the time domain to the space domain if we adopt the Taylor hypothesis and, consequently, we can talk about spatial scales.
Moreover, the fast wind correlation functions decrease much faster than those related to slow wind. This behavior reflects 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.
This figure has been obtained under the hypothesis of cylindrical symmetry. Real determination of the correlation function could be obtained only in the positive quadrant, and the whole plot was then made by mirroring these results on the remaining three quadrants. The iso-contour lines show contours mainly elongated along the ambient field direction or perpendicular to it. Alfvénic fluctuations with k ∥ B_{0} contribute to contours elongated parallel to r_{⊥}. Fluctuations in the two-dimensional turbulence limit (Montgomery, 1982) contribute to contours elongated parallel to r_{∥}. This two-dimensional turbulence is characterized for having both the wave vector k and the perturbing field 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.
The ratio test adopted by these authors was based on the ratio between the reduced perpendicular spectrum (fluctuations ⊥ to the mean field and solar wind flow direction) and the reduced quasi-parallel spectrum (fluctuations ⊥ to the mean field and in the plane defined by the mean field and the flow direction). This ratio, expected to be 1 for slab turbulence, resulted to be ∼ 1.4 for fluctuations within the inertial range, consistent with 74% of 2D turbulence and 26% of slab. A further test, the anisotropy test, evaluated how the spectrum should vary with the angle between the mean magnetic field and the flow direction of the wind. The measured slab spectrum should decrease with the field angle while the 2D spectrum should increase, depending on how these spectra project on the flow direction. The results from this test were consistent with with 95% of 2D turbulence and 5% of slab. In other words, the slab turbulence due to Alfvénic fluctuations would be a minor component of interplanetary MHD turbulence. A third test derived from Mach number scaling associated with the nearly incompressible theory (Zank and Matthaeus, 1992), assigned the same fraction ∼ 80% to the 2D component. However, the data base for this analysis was derived from Helios magnetic measurements, and all data were recorded near times of solar energetic particle events. Moreover, the quasi totality of the data belonged to slow solar wind (Wanner and Wibberenz, 1993) and, as such, this analysis cannot be representative of the whole phenomenon of turbulence in solar wind. As a matter of fact, using Ulysses observations, Smith (2003) found that in the polar wind the percentage of slab and 2D components is about the same, say the high latitude slab component results unusually higher as compared with ecliptic observations.
Successive theoretical works by Ghosh et al. (1998a,b) in which they used compressible models in large variety of cases was able to obtain, in some cases, parallel and perpendicular correlations similar to those obtained in the solar wind. However, they concluded that the “Maltese” cross does not come naturally from the turbulent evolution of the fluctuations but it strongly depends on the initial conditions adopted when the simulation starts. It seems that individual existence of these correlations in the initial data represents an unavoidable constraint. Moreover, they also stressed the importance of time-averaging since the interaction between slab waves and transverse pressure-balanced magnetic structures causes the slab turbulence to evolve towards a state in which a two-component correlation function emerges during the process of time averaging.
The presence of two populations, i.e., a slab-like and a quasi-2D like, was also inferred by Dasso et al. (2003). These authors computed the reduced spectra of the normalized cross-helicity and the Alfvén ratio from ACE dataset. These parameters, calculated for different intervals of the angle θ between the flow direction and the orientation of the mean field B_{0}, showed a remarkable dependence on θ.
The geometry used in these analyses assumes that the energy spectrum in the rest frame of the plasma is axisymmetric and invariant for rotations about the direction of B_{0}. Even if these assumption are good when we want to translate results coming from 2D numerical simulations to 3D geometry, these assumptions are quite in contrast with the observational fact that the eigenvalues of the variance matrix are different, namely λ_{3} ≠ λ_{2}.
A fit to the eigenvalues of the variance matrix allowed Carbone et al. (1995a) to fix the free parameters of the spectrum for both polarizations. They used data from Bavassano et al. (1982a) who reported the values of λ_{i} at five wave vectors calculated at three heliocentric distances, selecting periods of high correlation (Alfvénic periods) using magnetic field measured by the Helios 2 spacecraft. They found that the spectral indices of both polarizations, in the range 1.1 ≤ µ_{1} ≤ 1.3 and 1.46 ≤ µ_{2} ≤ 1.8 increase systematically with increasing distance from the Sun, the polarization [2] spectra are always steeper than the corresponding polarization [1] spectra, while polarization [1] is always more energetic than polarization [2]. As far as the characteristic lengths are concerned, it can be found that \(\ell_{x}^{[1]} > \ell_{y}^{[1]} \gg \ell_{z}[1]\), indicating that wave vectors k ∥ B_{0} largely dominate. Concerning polarization [2], it can be found that \(\ell_{x}[2]\gg \ell_{y}^{[2]}\simeq \ell_{z}^{[2]}\), indicating that the spectrum I^{[2]} (k) is strongly flat on the plane defined by the directions of B_{0} and the radial direction. Within this plane, the energy distribution does not present any relevant anisotropy.
Let us compare these results with those by Matthaeus et al. (1990), the comparison being significant as far as the plane yz is taken into account. The decomposition of Carbone et al. (1995a) in two independent polarizations is similar to that of Matthaeus et al. (1990), a contour plot of the trace of the correlation tensor Fourier transform T(k) = I^{[1]} (k) + I^{[2]} (k) on the plane (k_{y}; k_{z}) shows two populations of fluctuations, with wave vectors nearly parallel and nearly perpendicular to B_{0}, respectively. The first population is formed by all the polarization [1] fluctuations and by the fluctuations with k ∥ B_{0} belonging to polarization [2]. The latter fluctuations are physically indistinguishable from the former, in that when k is nearly parallel to B_{0}, both polarization vectors are quasi-perpendicular to B_{0}. On the contrary, the second population is almost entirely formed by fluctuations belonging to polarization [2]. While it is clear that fluctuations with k nearly parallel to 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 of 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. To conclude, it is worth reporting that a model like that discussed here, that is a superposition of fluctuations with both slab and 2D components, has been used to describe turbulence in the Jovian magnetosphere (Saur et al., 2002, 2003).
3.1.6 Magnetic helicity
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 right hand 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 cross-helicity σ_{c} it is possible to infer the sense of polarization of the fluctuations. As a matter of fact, a positive cross-helicity indicates an Alfvén mode propagating outward, while a negative cross-helicity indicates a mode propagating inward. On the other hand, we know that a positive magnetic-helicity indicates a right hand polarized mode, while a negative magnetic-helicity indicates a left hand polarized mode. Thus, since the sense of polarization depends on the propagating direction with respect to the observer, σ_{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, any time magnetic helicity and cross-helicity are available from measurements in a super-Alfvénic flow, it is possible to infer the rest frame polarization of the fluctuations from a single point measurements, assuming the validity of the slab geometry.
The high variability of σ_{m}, observable in Voyager’s data (see Figure 27), was equally observed in Helios 2 data in the inner heliosphere (Bruno and Dobrowolny, 1986). The authors of this last work computed the difference (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).
3.1.7 Alfvénic correlations as uncompressive turbulence
Alfvén modes are not the only low frequency plasma fluctuations allowed by the MHD equations but they certainly are the most frequent fluctuations observed in the solar wind. The reason why other possible propagating modes like the slow sonic mode and the fast magnetosonic mode cannot easily be found, depends on the fact that these compressive modes are strongly damped in the solar wind shortly after they are generated (see Section 6). On the contrary, Alfvénic fluctuations, which are difficult to be damped because of their uncompressive nature, survive much longer and dominate solar wind turbulence. Nevertheless, there are regions where Alfvénic correlations are much stronger like the trailing edge of fast streams, and regions where these correlations are weak like intervals of slow wind (Belcher and Davis Jr, 1971; Belcher and Solodyna, 1975). However, the degree of Alfvénic correlations unavoidably fades away with increasing heliocentric distance, although it must be reported that there are cases when the absence of strong velocity shears and compressive phenomena favor a high Alfvénic correlation up to very large distances from the Sun (Roberts et al., 1987a; see Section 5.1).
The discovery of Alfvénic correlations in the solar wind stimulated fundamental remarks by Kraichnan (1974) who, following previous theoretical works by Kraichnan (1965) and Iroshnikov (1963), showed that the presence of a strong correlation between velocity and magnetic fluctuations renders non-linear transfer to small scales less efficient than for the Navier-Stokes equations, leading to a turbulent behavior which is different from that described by Kolmogorov (1941). In particular, when Equation (36) is exactly satisfied, non-linear interactions in MHD turbulent flows cannot exist. This fact introduces a problem in understanding the evolution of MHD turbulence as observed in the interplanetary space. Both a strong correlation between velocity and magnetic fluctuations and a well defined turbulence spectrum (Figures 23, 30) are observed, and the existence of the correlations is in contrast with the existence of a spectrum which in turbulence is due to a non-linear energy cascade. Dobrowolny et al. (1980b) started to solve the puzzle on the existence of Alfvénic turbulence, say the presence of predominately outward propagation and the fact that MHD turbulence with the presence of both Alfvénic modes present will evolve towards a state where one of the mode disappears. However, a lengthy debate based on whether the highly Alfvénic nature of fluctuations is what remains of the turbulence produced at the base of the corona or the solar wind itself is an evolving turbulent magnetofluid, has been stimulating the scientific community for quite a long time.
3.1.8 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.
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), analysing 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, as pointed out by Grappin et al. (1991), the magnetic energy excess can be explained as a competing action between the “Alfvén effect” (Kraichnan, 1965), which would work towards equipartition, and the non-linear terms (Grappin et al., 1983). 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.
However, when the two-fluid effect, such as the Hall current and the electron pressure (Mininni et al., 2003b), is included in the simulation, the dynamo can work more efficiently and the final stage of the simulation is towards equipartition between kinetic and magnetic energy.
On the other hand, Marsch and Tu (1993a) analyzed several intervals of interplanetary observations to look for a linear relationship between the mean electromotive force ε =< δVδB >, generated by the turbulent motions, and the mean magnetic field B_{0}, as predicted by simple dynamo theory (Krause and Rädler, 1980). Although sizable electromotive force was found in interplanetary fluctuations, these authors could not establish any simple linear relationship between B_{0} and ε.
Bavassano and Bruno (2000) analyzed several time intervals within the same corotating high velocity stream observed at 0.3 and 0.9 AU and performed the analysis using the new factor “F” to express magnetic field fluctuations in Alfvén units, taking into account the presence of α particles and electrons, besides the protons. However, the correction resulted to be insufficient to bring back to “1” the slope of the δV_{Pz} − δV_{Az} relationship shown in the right panel of Figure 34. 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 multi-fluid 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 33 taken from numerical simulations by Mininni et al. (2003a) or, it is due to the increased presence of magnetic structures like MFDT (Tu and Marsch, 1993).
3.2 Turbulence studied via Elsässer variables
The Alfvénic character of solar wind fluctuations, especially within corotating high velocity streams, suggests to use the Elsässer variables (Appendix 13.3) to separate the “outward” from the “inward” contribution to turbulence. These variables, used in theoretical studies by Dobrowolny et al. (1980a), Dobrowolny et al. (1980b), Veltri et al. (1982), Marsch and Mangeney (1987), and Zhou and Matthaeus (1989), were for the first time used in interplanetary data analysis by Grappin et al. (1990) and Tu et al. (1989b). In the following, we will describe and discuss several differences between “outward” and “inward” modes, but the most important one is about their origin. As a matter of fact, the existence of the Alfvénic critical point implies that only “outward” propagating waves of solar origin will be able to escape from the Sun. “Inward” waves, being faster than the wind bulk speed, will precipitate back to the Sun if they are generated before this point. The most important implication due to this scenario is that “inward” modes observed beyond the Alfvénic point cannot have a solar origin but they must have been created locally by some physical process. Obviously, for the other Alfvénic component, both solar and local origins are still possible.
3.2.1 Ecliptic scenario
Early studies by Belcher and Davis Jr (1971), performed on magnetic field and velocity fluctuations recorded by Mariner 5 during its trip to Venus in 1967, already suggested that the majority of the Alfvénic fluctuations are characterized by an “outward” sense of propagation, and that the best regions where to observe these fluctuations are the trailing edge of high velocity streams. Moreover, Helios spacecraft, repeatedly orbiting around the Sun between 0.3 to 1 AU, gave the first and unique opportunity to study the radial evolution of turbulence (Bavassano et al., 1982b; Denskat and Neubauer, 1983). Successively, when Elsässer variables were introduced in the analysis (Grappin et al., 1989), it was finally possible not only to evaluate the “inward” and “outward” Alfvénic contribution to turbulence but also to study the behavior of these modes as a function of the wind speed and radial distance from the Sun.
This configuration vanishes when we pass to the slow wind where both spectra have almost equivalent power density and follow the Kolmogorov slope. This behavior, for the first time reported by Grappin et al. (1990), is commonly found within corotating high velocity streams, although much more clearly expressed at shorter heliocentric distances, as shown below.
Quite different is the behavior of e^{+}spectrum. Close to the Sun and within fast wind, this spectrum appears to be flatter at low frequency and steeper at high frequency. The overall evolution is towards the “background spectrum” by the time the wind reaches 0.8 AU.
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).
3.2.2 On the nature of Alfvénic fluctuations
The Alfvénic nature of outward modes has been widely recognized through several frequency decades up to periods of the order of several hours in the s/c rest frame (Bruno et al., 1985). Conversely, the nature of those fluctuations identified by δz^{−}, called “inward Alfvén modes”, is still not completely clear. There are many clues which would suggest that these fluctuations, especially in the hourly frequencies range, have a non-Alfvénic nature. Several studies on this topic in the low frequency range have suggested that structures convected by the wind could well mimic non-existent inward propagating modes (see the review by Tu and Marsch, 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.
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 Alfvén waves, unstable to perturbations of random field intensity and density fluctuations, would decay into two secondary Alfvén modes propagating in opposite directions and a sound-like wave propagating in the same direction of the pump wave. Most of the energy of the mother wave would go into the sound-like fluctuation and the backward propagating Alfvén mode. On the other hand, the production of e^{−}modes by parametric instability is not particularly fast if the plasma β ∼ 1, like in the case of solar wind (Goldstein, 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, 2001, 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 42 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 89 of Section 9, 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^{+} 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^{−} 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, sound-like perturbations (Marsch and Tu, 1993a). On the other hand, low frequency turbulence would be made of outward Alfvénic fluctuations and static convected structures representing the inhomogeneities of the background medium.
4 Observations of MHD Turbulence in the Polar Wind
During solar maximum (look at the upper right panel of Figure 43) 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.
4.1 Evolving turbulence in the polar wind
- i.
Alfvénic turbulence would have not relaxed towards standard turbulence because the large scale velocity shears would have been much less relevant (Grappin et al., 1991);
- ii.
since the magnetic field would be smaller far from the ecliptic, at large heliocentric distances, even small shears would lead to an isotropization of the fluctuations and produce a turbulent cascade faster than the one observed at low latitudes, and the subsequent evolution would take less (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 cross-helicity (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., 2001, 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 43), 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 stream-stream dynamical interaction effects (Bruno et al., 1985) and strong velocity shears (Roberts et al., 1987a). At high latitude, the relative absence of enhanced dynamical interaction between flows at different speed and, as a consequence, the absence of strong velocity shears favors the survival of these extremely low frequency Alfvénic fluctuations for larger heliocentric excursions.
Horbury et al. (1995c) and Forsyth et al. (1996) showed that the interplanetary magnetic field fluctuations observed by Ulysses continuously evolve within the fast polar wind, at least out to 4 AU. Since this evolution was observed within the polar wind, rather free of co-rotating 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.
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 15 to learn about the RTN reference system). The variance along the radial direction was computed as \(\sigma_{R}^{2} = < B_{R}^{2} > - {{< B_{R} >}^{2}}\) 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 \(\sigma_{{\rm r}}^{2}\), α = 3.45 ± 0.09 for \(\sigma_{T}^{2}\), α = 3.37 ± 0.09 for \(\sigma_{N}^{2}\), and α = 2.48 ± 0.14 for \(\sigma_{B}^{2}\). Thus, for hourly variances, the power associated with the components showed a radial dependence stronger than the one predicted by the WKB approximation, which would provide α = 3. These authors also showed that including data between 3 and 4 AU, corresponding to intervals characterized by compressional features mainly due to high latitude CMEs, they would obtain less steep radial gradients, much closer to a WKB type. These results suggested that compressive effects can feed energy at the smallest scales, counteracting dissipative phenomena and mimicking a WKB-like behavior of the fluctuations. However, they concluded that for lower frequencies, below the frequency break point, fluctuations do follow the WKB radial evolution.
On the basis of variations of spectral index and radial and latitudinal dependencies, these authors were able to identify four wave number ranges as indicated by the circled numbers in the top panel of Figure 50. 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 scale-shift 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 totally unexpected. 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 over-expansion of the polar coronal hole at higher latitudes. In addition, within the fourth range, field magnitude fluctuations radially decrease less rapidly than the fluctuations of the components, but do not show significant latitudinal variations. Finally, the smaller spectral index reveals that the high frequency range of the field magnitude spectrum shows a flattening.
As already shown by Horbury et al. (1995b), around the 5 min range, magnetic field fluctuations are transverse to the mean field direction the majority of the time. The minimum variance direction lies mainly within an angle of about 26° from the average background field direction and fluctuations are highly anisotropic, such that the ratio between perpendicular to parallel power is about 30. Since during the observations reported in Horbury and Balogh (2001) and Horbury and Tsurutani (2001) the mean field resulted to be radially oriented most of the time, the radial minimum variance direction at short time scales is an effect induced by larger scales behavior.
Anyhow, radial and latitudinal anisotropy trends tend to disappear for higher frequencies. In the mean time, interesting enough, there is a strong radial increase of magnetic field compression (top panel of Figure 51), 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 pick-up ions was not satisfactory in all cases.
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.
4.2 Polar turbulence studied via Elsässer variables
Anyhow, results relative to the normalized cross-helicity σ_{c} (see Figure 54) 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 54) 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 for the first time obtained by Bruno et al. (1985), Marsch and Tu (1990a), and Roberts et al. (1990) in the ecliptic at about 1 AU. The low frequency extension of r_{A⊥} together with σ_{c⊥} is probably due to the fact that Ulysses was longitudinally sampling Alfvénic fluctuations and has been considered by these authors not really indicative of the existence of such low frequency 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 even 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.
This different radial behavior is readily seen in the radial plot of the Elsässer ratio r_{E} shown in the top panel of the left column of Figure 55. 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.
In a successive paper, Bavassano et al. (2002a) tested whether or not the radial dependence observed in e^{±} was to be completely ascribed to the radial expansion of the wind or possible latitudinal dependencies also contributed to the turbulence evolution in the polar wind.
5 Numerical Simulations
Numerical simulations currently represent one of the main source of information about non-linear evolution of fluid flows. The actual super-computers are now powerful enough to simulate equations (NS or MHD) that describe turbulent flows with Reynolds numbers of the order of 10^{4} in two-dimensional configurations, or 10^{3} in three-dimensional 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 by Velli et al. (1989, 1990). Even with this far too pessimistic point of view, used here solely as a few words of caution, simulations in some cases were able to reproduce some phenomena observed in the solar wind.
Nevertheless, numerical simulations have been playing a key role, and will continue to do so in our seeking an understanding of turbulent flows. Numerical simulations allows us to get information that cannot be obtained in laboratory. For example, high resolution numerical simulations provide information at every point on a grid and, for some times, about basic vector quantities and their derivatives. The number of degree of freedom required to resolve the smaller scales is proportional to a power of the Reynolds number, say to Re^{9/4}, although the dynamically relevant number of modes may be much less. Then one of the main challenge remaining is how to handle and analyze the huge data files produced by large simulations (of the order of Terabytes). Actually a lot of papers appeared in literature on computer simulations related to MHD turbulence. The interested reader can look at the book by Biskamp (1993) and the reviews by Pouquet (1993, 1996).
5.1 Local production of Alfvénic turbulence in the ecliptic
The discovery of the strong correlation between velocity and magnetic field fluctuations has represented the motivation for some MHD numerical simulations, aimed to confirm the conjecture by Dobrowolny et al. (1980b). The high level of correlation seems to be due to a kind of self-organization (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 cross-helicity and E the total energy of the flow (see Appendix 13.1), 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 cross-helicity. 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.
These results, although obtained from simulations performed using 2D incompressible spectral and pseudo-spectral 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 cross-helicity 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 compressive pseudo-spectral 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, by-product of the non-linear 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 δ|B| ≠ 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 tanh-like 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 non-linear interactions. In absence of non-linear interactions, the natural tendency towards an increase of σ_{c} is stopped.
5.2 Local production of Alfvénic turbulence at high latitude
An interesting solution to the radial behavior of the minority modes might be represented by local generation mechanisms, like parametric decay (Malara et al., 2001; 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 non-linear 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, non-linear MHD equations in a one-dimensional configuration using a pseudo-spectral numerical code. The simulation starts with a non-monochromatic, 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, non-linear 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, non-linear 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.
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.
In addition, Umeki and Terasawa (1992) studying the non-linear evolution of a large-amplitude incoherent Alfvén wave via 1D magnetohydrodynamic simulations, reported that while in a low beta plasma (β ≈ 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 (β ≈ 2) there was no production of backscattered Alfvén waves. Consequently, although numerical results obtained by Malara et al. (2001) are very encouraging, the high beta plasma (β ≈ 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 55.
6 Compressive Turbulence
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.
Spectral index is around −5/3 for all the spectra in slow wind while, fast wind spectral index is around −5/3 for k < 4 · 10^{−4} km^{−1} and slightly less steep for larger wave numbers.
6.1 On the nature of compressive Turbulence
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 = δυ/C_{s} the turbulent sonic Mach number, δυ 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 NI-theory), 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 anti-correlation 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 anti-correlation between field intensity and proton density, and between plasma and field pressure on time scales up to 10 h. The anti-correlation 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 structures were finally related to the fine ray-like structures or plumes associated with the underlying cromospheric network and interpreted as the signature of interplanetary flow-tubes. The estimated dimension of these structures, back projected onto the Sun, suggested that they over-expand in the solar wind.
The idea of filamentary structures in the solar wind dates back to Parker (1963), followed by other authors like McCracken and Ness (1966), Siscoe et al. (1968), and more recently re-proposed in literature with new evidences (see Section 9). 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 “spaghetti-model”.
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 non-linear 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 quasi-static 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 λ is replaced by a new adiabatic index υ’ = (c_{P} − c)/(c_{V} − c) where c is the specific heat of the polytropic variation, and c_{P} and c_{V} 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 (υ′ = 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 total-pressure balanced structures where the equilibrium was assured by the thermal component only, perhaps tiny flow tubes like those described by Thieme et al. (1989) and Tu and Marsch (1994). Adiabatic changes were probably related to magnetosonic waves excited by contiguous flow tubes (Tu and Marsch, 1994). Proton temperature changes at almost constant density were preferentially found in fast wind, close to the Sun. These regions were characterized by values of B and N remarkable stable and by strong Alfvénic fluctuations (Bruno et al., 1985). Thus, they suggested that these temperature changes could be remnants of thermal features already established at the base of the corona.
Thus, the polytropic index offers a very simple way to identify basic properties of solar wind fluctuations, provided that the magnetic field does not play a major role.
6.2 Compressive turbulence in the polar wind
Compressive fluctuations in high latitude solar wind have been extensively studied by Bavassano et al. (2004) looking at the relationship between different parameters of the solar wind and comparing these results with predictions by existing models.
It is clearly seen that for all the polar wind samples no clear trend emerges in the data. Thus, NI-MHD 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, nearly-incompressible (see Section 6.1) effects do not seem to play any relevant role. Thus, polar observations do not show major differences when compared with ecliptic observations in fast wind, the only possible difference being a major role of pressure balanced structures.
6.3 The effect of compressive phenomena on Alfvénic correlations
A lack of δV − δB correlation does not strictly indicate a lack of Alfvénic fluctuations since a superposition of both outward and inward oriented fluctuations of the same amplitude would produce a very low correlation as well. In addition, the rather complicated scenario we observe at the base of the corona, where complicated 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 due to the ubiquitous Alfvénic and other MHD propagating modes but also to an underlying structure convected by the wind, not necessarily characterized by Alfvén-like correlations. Moreover, dynamical interactions between fast and slow wind, built up during the expansion, contribute to increase the compressibility of the medium.
7 A Natural Wind Tunnel
The solar wind has been used as a wind tunnel by L.F. Burlaga who, at the beginning of the’ 90s, started to investigate anomalous fluctuations (Burlaga, 1991a,b,c, 1995) as observed by measurements in the outer heliosphere by the Voyager spacecraft. In 1991 E. Marsch, in a review on solar wind turbulence given at the Solar Wind Seven conference (Marsch, 1992), 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, page 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 low-frequency solar wind turbulence on the other side, has produced a lot of papers, and is now mature enough to be tentatively presented in a more organic way.
7.1 Scaling exponents of structure functions
The phenomenology of turbulence developed by Kolmogorov (1941) deals with some statistical hypotheses for fluctuations. The famous footnote remark by Landau (Landau and Lifshitz, 1971) pointed out a defect in the Kolmogorov theory, namely the fact that the theory does not take proper account of spatial fluctuations of local dissipation rate (Frisch, 1995). This led different authors to investigate the features related to scaling laws of fluctuations and, in particular, to investigate the departure from the Kolmogorov’s linear scaling of the structure functions (cf. Section 2.8). An up-to-date 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 high-order moments of stochastic variables defined as variations of fields separated by a time^{8} interval r. First of all, it is worthwhile to remark that scaling laws and, in particular, the exact relation (26) 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_{r} = u(t + r) − 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_{r} = B(t + r) − 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.
- i.
There is a significant departure from the Kolmogorov linear scaling, that is, real scaling exponents are anomalous and seem to be non-linear 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 magnetic field is more intermittent than velocity field. The same difference is observed between the velocity field and a passive scalar (in our case the temperature) in ordinary fluid flows (Ruíz-Chavarría et al., 1995). That is the magnetic field, as long as intermittency properties are concerned, behaves like a passive field.
Scaling exponents for velocity ζ_{p} and magnetic ζ_{p} variables calculated through ESS. Errors represent the standard deviations of the linear fitting. The data used comes from a turbulent sample of slow wind at 0.9 AU from Helios 2 spacecraft. As a comparison we show the normalized scaling exponents of structure functions calculated in a wind tunnel on Earth (Ruíz-Chavarría et al., 1995) for velocity and temperature. The temperature is a passive scalar in this experiment.
p | ζ_{p} | ξ_{p} | u(t) (fluid) | T(t) (fluid) |
---|---|---|---|---|
1 | 0.37 ± 0.06 | 0.56 ± 0.06 | 0.37 | 0.61 |
2 | 0.70 ± 0.05 | 0.83 ± 0.05 | 0.70 | 0.85 |
3 | 1.00 | 1.00 | 1.00 | 1.00 |
4 | 1.28 ± 0.02 | 1.14 ± 0.02 | 1.28 | 1.12 |
5 | 1.54 ± 0.03 | 1.25 ± 0.03 | 1.54 | 1.21 |
6 | 1.79 ± 0.05 | 1.35 ± 0.05 | 1.78 | 1.40 |
Normalized scaling exponents ξ_{p}/ξ_{3} for radial magnetic fluctuations in a laboratory plasma, as measured at different distances r/R (R ≃ 0.45 cm being the minor radius of the torus in the experiment) from the external wall. Errors represent the standard deviations of the linear fitting. Scaling exponents have been obtained using the ESS.
p | r/R = 0.96 | r/R = 0.93 | r/R = 0.90 | r/R = 0.86 |
---|---|---|---|---|
1 | 0.39 ± 0.01 | 0.38 ± 0.01 | 0.37 ± 0.01 | 0.36 ± 0.01 |
2 | 0.74 ± 0.01 | 0.73 ± 0.02 | 0.71 ± 0.01 | 0.70 ± 0.01 |
3 | 1.00 | 1.00 | 1.00 | 1.00 |
4 | 1.20 ± 0.02 | 1.24 ± 0.02 | 1.27 ± 0.01 | 1.28 ± 0.01 |
5 | 1.32 ± 0.03 | 1.41 ± 0.03 | 1.51 ± 0.03 | 1.55 ± 0.03 |
6 | 1.38 ± 0.04 | 1.50 ± 0.04 | 1.71 ± 0.03 | 1.78 ± 0.04 |
Normalized scaling exponents ξ_{p}/ξ_{3} for Alfvénic, velocity, and magnetic fluctuations obtained from data of high resolution 2D MHD numerical simulations. Scaling exponents have been calculated from spatial fluctuations; different times, in the statistically stationary state, have been used to improve statistics. The scaling exponents have been calculated by ESS using Equation (27) as characteristic scale rather than the third-order structure function (cf. Politano et al., 1998b for details).
p | Z^{+} | Z^{−} | υ | B |
---|---|---|---|---|
1 | 0.36 ± 0.06 | 0.56 ± 0.06 | 0.37 ± 0.01 | 0.46 ± 0.02 |
2 | 0.70 ± 0.05 | 0.83 ± 0.05 | 0.70 ± 0.01 | 0.78 ± 0.01 |
3 | 1.00 | 1.00 | 1.00 | 1.00 |
4 | 1.28 ± 0.02 | 1.14 ± 0.02 | 1.28 ± 0.02 | 1.18 ± 0.02 |
5 | 1.53 ± 0.03 | 1.25 ± 0.03 | 1.54 ± 0.03 | 1.31 ± 0.03 |
6 | 1.79 ± 0.05 | 1.35 ± 0.05 | 1.78 ± 0.05 | 1.40 ± 0.03 |
7.2 Probability density functions and self-similarity of fluctuations
The presence of scaling laws for fluctuations is a signature of the presence of self-similarity 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.
7.3 What is intermittent in the solar wind turbulence? The multifractal approach
7.4 Fragmentation models for the energy transfer rate
Cascade models can be organized 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 p-th order structure function.
7.5 A model for the departure from self-similarity
7.6 Intermittency properties recovered via a shell model
Scaling exponents for velocity and magnetic variables, Elsässer variables, and fluxes. Errors on \(\beta_{p}^{\pm}\) are about one order of magnitude smaller than the errors shown.
p | ζ_{p} | η_{p} | \(\zeta_{p}^{+}\) | \(\zeta_{p}^{-}\) | \(\beta_{p}^{+}\) | \(\beta_{p}^{-}\) |
---|---|---|---|---|---|---|
1 | 0.36 ± 0.01 | 0.35 ± 0.01 | 0.35 ± 0.01 | 0.36 ± 0.01 | 0.326 | 0.318 |
2 | 0.71 ± 0.02 | 0.69 ± 0.03 | 0.70 ± 0.02 | 0.70 ± 0.03 | 0.671 | 0.666 |
3 | 1.03 ± 0.03 | 1.01 ± 0.04 | 1.02 ± 0.04 | 1.02 ± 0.04 | 1.000 | 1.000 |
4 | 1.31 ± 0.05 | 1.31 ± 0.06 | 1.30 ± 0.05 | 1.32 ± 0.06 | 1.317 | 1.323 |
5 | 1.57 ± 0.07 | 1.58 ± 0.08 | 1.54 ± 0.07 | 1.60 ± 0.08 | 1.621 | 1.635 |
6 | 1.80 ± 0.08 | 1.8 ± 0.10 | 1.79 ± 0.09 | 1.87 ± 0.10 | 1.91 | 1.94 |
Time intermittency in the shell model generates rare and intense events. These events are the result of the chaotic dynamics in the phase-space 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 83, at a given shell n = 12. The same statistical law is observed for the bursts of total dissipation (Boffetta et al., 1999).
8 Intermittency Properties in the 3D Heliosphere: Taking a Look at the Data
In this chapter, we present a reasoned look at the main aspect of what has been reported in literature about the problem of intermittency in the solar wind turbulence. In particular, we present results from data analysis.
8.1 Structure functions
Apart from the earliest investigations on the fractal structure of magnetic field as observed in interplanetary space (Burlaga and Klein, 1986), the starting point for the investigation of intermittency in the solar wind dates back to 1991, when Burlaga (1991a) started to look at the scaling of the bulk velocity fluctuations at 8.5 AU using Voyager 2 data. This author found that anomalous scaling laws for structure functions could be recovered in the range 0.85 ≤ r ≤ 13.6 h. This range of scales has been arbitrarily identified as a kind of “inertial range”, say a region were a linear scaling exists between log \(S_{r}^{(p)}\) and log r, and the scaling exponents have been calculated as the slope of these curves. However, structure functions of order p ≤ 20 were determined on the basis of only about 4500 data points. Nevertheless the scaling was found to be quite in agreement with that found in ordinary fluid flows. Although the data might be in agreement with the random-β model, from a theoretical point of view Carbone (1993, 1994b) showed that normalized scaling exponents ζ_{p}/ζ_{4} calculated by Burlaga (1991a) would be better fitted by using a p-model derived from the Kraichnan phenomenology (Kraichnan, 1965; Carbone, 1993), and considering the parameter μ ≃.77. The same author (Burlaga, 1991b) investigated the multifractal structure of the interplanetary magnetic field near 25 AU and analyzed positive defined fields as magnetic field strength, temperature, and density using the multifractal machinery of dissipation fields (Paladin and Vulpiani, 1987; Meneveau, 1991). Burlaga (1991c) showed that intermittent events observed in corotating streams at 1 AU should be described by a multifractal geometry. Even in this case the number of points used was very low to assure the reliability of high-order moments.
Marsch and Liu (1993) investigated the structure of intermittency of the turbulence observed in the inner heliosphere by using Helios 2 data. They analyzed both bulk velocity and Alfvén speed to calculate structure functions in the whole range 40.5 s (the instrument resolution) up to 24 h to estimate the p-th order scaling exponents. Note that also in this analysis the number of data points used was too small to assure a reliability for order p = 20 structure functions as reported by Marsch and Liu (1993). From the analysis analogous to Burlaga (1991a), authors found that anomalous scaling laws are present. A comparison between fast and slow streams at two heliocentric distances, namely 0.3 AU and 1 AU, allows authors to conjecture a scenario for high speed streams were Alfvénic turbulence, originally self-similar (or poorly intermittent) near the Sun, “… loses its self-similarity and becomes more multifractal in nature” (Marsch and Liu, 1993), which means that intermittent corrections increase from 0.3 AU to 1 AU. No such behavior seems to occur in the slow solar wind. From a phenomenological point of view, Marsch and Liu (1993) found that data can be fitted with a piecewise linear function for the scaling exponents ζ_{p}, namely a β-model ζ_{p} = 3 − D + p(D − 2)/3, where D ≃ 3 for p ≤ 6 and D ≃2.6 for p > 6. Authors say that “We believe that we see similar indications in the data by Burlaga, who still prefers to fit his whole ζ_{p} dataset with a single fit according to the non-linear random β-model.” We like to comment that the impression by Marsch and Liu (1993) is due to the fact that the number of data points used was very small. As a matter of fact, only structure functions of order p ≤ 4 are reliably described by the number of points used by Burlaga (1991a).
However, the data analyses quoted above, which in some sense present some contradictory results, are based on high order statistics which is not supported by an adequate number of data points and the range of scales, where scaling laws have been recovered, is not easily identifiable. To overcome these difficulties Carbone et al. (1996a) investigated the behavior of the normalized ratios ζ_{p}/ζ_{3} through the ESS procedure described above, using data coming from low-speed streams measurements of Helios 2 spacecraft. Using ESS the whole range covered by measurements is linear, and scaling exponent ratios can be reliably calculated. Moreover, to have a dataset with a high number of points, authors mixed in the same statistics data coming from different heliocentric distances (from 0.3 AU up to 1 AU). This is not correct as far as fast wind fluctuations are taken into account, because, as found by Marsch and Liu (1993) and Bruno et al. (2003b), there is a radial evolution of intermittency. Results showed that intermittency is a real characteristic of turbulence in the solar wind, and that the curve ζ_{p}/ζ_{3} is a non-linear function of p as soon as values of p ≤ 6 are considered.
Marsch et al. (1996) for the first time investigated the geometrical and scaling properties of the energy flux along the turbulent cascade and dissipation rate of kinetic energy. They showed the multifractal nature of the dissipation field and estimated, for the first time in solar wind MHD turbulence, the associated singularity spectrum which resulted to be very similar to those obtained for ordinary fluid turbulence (Meneveau and Sreenivasan, 1987b). They also estimated the energy dissipation rate for time scales of 10^{2} s to be around 5.4 × 10^{−16} erg cm^{−3} s^{−1}. This value was similar to the theoretical heating rate required in the model by Tu (1988) with Alfvén waves to explain the radial temperature dependence observed in fast solar wind.
Analysis of scaling exponents of p-th order structure functions has been performed using different spacecraft datasets of Ulysses spacecraft. Horbury et al. (1995a) and Horbury et al. (1995c) investigated the structure functions of magnetic field as obtained from observations recorded between 1.7 and 4 AU, and covering a heliographic latitude between 40° and 80° south. By investigating the spectral index of the second order structure function, they found a decrease with heliocentric distance attributed to the radial evolution of fluctuations. Further investigations (see, e.g., Ruzmaikin et al., 1995) were obtained using structure functions to study the Ulysses magnetic field data in the range of scales 1 ≤ r ≤ 32 min. Ruzmaikin et al. (1995) showed that intermittency is at work and developed a bi-fractal model to describe Alfvénic turbulence. They found that intermittency may change the spectral index of the second order structure function and this modifies the calculation of the spectral index (Carbone, 1994a). Ruzmaikin et al. (1995) found that polar Alfvénic turbulence should be described by a Kraichnan phenomenology (Kraichnan, 1965). However, the same data can be fitted also with a fluid-like scaling law (Tu et al., 1996) and, due to the relatively small amount of data, it is difficult to decide, on the basis of the second order structure function, which scaling relation describes appropriately intermittency in the solar wind.
In a further paper Carbone et al. (1995b) provided evidence for differences in the ESS scaling laws between ordinary fluid flows and solar wind turbulence. Through the analysis of different datasets collected in the solar wind and in ordinary fluid flows, it was shown that normalized scaling exponents ζ_{p}/ζ_{3} are the same as far as p ≤ 8 are considered. This indicates a kind of universality in the scaling exponents for the velocity structure functions. Differences between scaling exponents calculated in ordinary fluid flows and solar wind turbulence are confined to high-order moments. Nevertheless, the differences found in the datasets were related to different kind of singular structures in the model described by Equation (41). Solar wind data can be fitted by that model as soon as the most intermittent structures are assumed to be planar sheets C = 1 and m = 4, that is a Kraichnan scaling is used. On the contrary, ordinary fluid flows can be fitted only when C = 2 and m = 3, that is, structures are filaments and the Kolmogorov scaling have been used. However it is worthwhile to remark that differences have been found for high-order structure functions, just where measurements are unreliable.
8.2 Probability distribution functions
As said in Section 7.2 the statistics of turbulent flows can be characterized by the PDF of field differences over varying scales. At large scales PDFs are Gaussian, while tails become higher than Gaussian (actually, PDFs decay as \({\rm exp}[-\delta Z_{\ell}^{\pm}]\) at smaller scales.
Marsch and Tu (1994) started to investigate the behavior of PDFs of fluctuations against scales and they found that PDFs are rather spiky at small scales and quite Gaussian at large scales. The same behavior have been obtained by Sorriso-Valvo et al. (1999, 2001) who investigated Helios 2 data for both velocity and magnetic field.
The values of the parameters σ_{0}, μ and γ, in the fit of λ^{2}(τ) (see Equation (45) as a kernel for the scaling behavior of PDFs. FW and SW refer to fast and slow wind, respectively, as obtained from the Helios 2 spacecraft, by collecting in a single dataset all periods.
parameter | B field (SW) | V fiele (SW) | B field (FW) | V field (FW) |
---|---|---|---|---|
σ_{0} | 0.90 ± 0.05 | 0.95 ± 0.05 | 0.85 ± 0.05 | 0.90 ± 0.05 |
μ | 0.75 ± 0.03 | 0.38 ± 0.02 | 0.90 ± 0.03 | 0.54 ± 0.03 |
γ | 0.18 ± 0.03 | 0.20 ± 0.04 | 0.19 ± 0.02 | 0.44 ± 0.05 |
The same analysis has been repeated by Forman and Burlaga (2003). These authors used 64 s averages of radial solar wind speed reported by the SWEPAM instrument on the ACE spacecraft, increments have been calculated over a range of lag times from 64 s to several days. From the PDF obtained through the Equation (45) authors calculated the structure functions and compared the free parameters of the model with the scaling exponents of the structure functions. Then a fit on the scaling exponents allows to calculate the values of λ^{2} and σ_{0}. Once these parameters have been calculated, the whole PDF is evaluated. The same authors found that the PDFs do not precisely fit the data, at least for large values of the moment order. Interesting enough, Forman and Burlaga (2003) investigated the behavior of PDFs when different kernels G_{λ}(σ), derived from different cascade models, are taken into account in Equation (42). They discussed the physical content of each model, concluding that a cascade model derived from lognormal or log-Lévy theories^{10}, modified by self-organized criticality proposed by Schertzer et al. (1997), seems to avoid all problems present in other cascade models.
9 Turbulent Structures
The non-linear energy cascade towards smaller scales accumulates fluctuations only in relatively small regions of space, where gradients become singular. As a rather different point of view (see Farge, 1992) these regions can be viewed as localized zones of fluid where phase correlation exists, in some sense coherent structures. These structures, which dominate the statistics of small scales, occur as isolated events with a typical lifetime greater than that of stochastic fluctuations surrounding them. The idea of a turbulence in the solar wind made by a mixture of structures convected by the wind and stochastic fluctuations is not particularly new (see, e.g., Tu and Marsch, 1995a). However, these large-scale structures cannot be considered as intermittent structures at all scales. Structures continuously appear and disappear apparently in a random fashion, at some random location of fluid, and carry a great quantity of energy of the flow. In this framework intermittency can be considered as the result of the occurrence of coherent (non-Gaussian) structures at all scales, within the sea of stochastic Gaussian fluctuations.
This point of view is the result of data analysis of scaling laws of turbulent fluctuations made by using wavelets filters (see Appendix 14) instead of the usual Fourier transform. Unlike the Fourier basis, wavelets allow a decomposition both in time and frequency (or space and scale). In analyzing intermittent structures it is useful to introduce a measure of local intermittency, as for example the Local Intermittency Measure (LIM) introduced by Farge (1992) (see Appendix 14).
- i.
Some of the structures are the well known one-dimensional current sheets, characterized by pressure balance and almost constant density and temperature. When a minimum variance analysis is made on the magnetic field near the structure, it can be seen that the most variable component of the magnetic field changes sign. This component is perpendicular to the average magnetic field, the third component being zero. An interesting property of these structures is that the correlation between velocity and magnetic field within them is opposite with respect to the rest of fluctuations. That is, when they occur during Alfvénic periods velocity and magnetic field correlation is low; on the contrary, during non-Alfvénic periods the correlation of structure increases.
- ii.
A different kind of structures looks like a shock wave. They can be parallel shocks or slow-mode shocks. In the first case they are observed on the radial component of the velocity field, but are also seen on the magnetic field intensity, proton temperature, and density. In the second case they are characterized by a very low value of the plasma β parameter, constant pressure, anti-correlation between density and proton temperature, no magnetic fluctuations, and velocity fluctuations directed along the average magnetic field.
Given a turbulent time series, as derived in the solar wind, a very interesting statistics can be made on the time separation between the occurrence of two consecutive structures. Let us consider a signal, for example u(t) or b(t) derived from solar wind, and let us define the wavelets set w_{s}(r, t) as the set which captures, at time t, the occurrence of structures at the scale r. Then define the waiting times δt, as that time between two consecutive structures at the scale r, that is, between w_{s}(r, t) and w_{s}(r, t + δt). The PDFs of waiting times P(δt) are reported in Figure 74. As it can be seen, waiting times are distributed according to a power law P(δt) ∼ δt^{−β} extended over at least two decades. This property is very interesting, because this means that the underlying process for the energy cascade is non-Poissonian. Waiting times occurring between isolated Poissonian events, must be distributed according to an exponential function. The power law for P(δt) represents the asymptotic behavior of a Lévy function with characteristic exponent α = β − 1. This describes self-affine processes and are obtained from the central limit theorem by relaxing the hypothesis that the variance of variables is finite. The power law for waiting times we found is a clear evidence that long-range correlation (or in some sense “memory”) exists in the underlying cascade process.
On the other hand, Bruno et al. (2001), analyzing the statistics of the occurrence of waiting times of magnetic field intensity and wind speed intermittent events for a short time interval within the trailing edge of a high velocity stream, found a possible Poissonian-like behavior with a characteristic time around 30 min for both magnetic field and wind speed. These results recalled previous estimates of the occurrence of interplanetary discontinuities performed by Tsurutani and Smith (1979), who found a waiting time around 14 min. In addition, Bruno et al. (2001), taking into account the wind speed and the orientation of the magnetic field vector at the site of the observation, in the hypothesis of spherical expansion, estimated the corresponding size at the Sun surface that resulted to be of the order of the photospheric structures estimated also by Thieme et al. (1989). Obviously, the Poissonian statistics found by these authors does not agree with the clear power law shown in Figure 74. However, Bruno et al. (2001) included intermittent events found at all scales while results shown in Figure 74 refer to waiting times between intermittent events extracted at the smallest scale, which results to be about an order of magnitude smaller than the time resolution used by Bruno et al. (2001). A detailed study on this topic would certainly clarify possible influences on the waiting time statistics due to the selection of intermittent events according to the corresponding scale.
The good correlation existing between magnetic and velocity variations for both time intervals highlights the presence of Alfvénic fluctuations. However, only within the first interval the magnetic field vector describes an arc-like structure larger than 90° on the maximum variance plane (see rotation from A to B on the 3D graph at the bottom of the left column in Figure 88) in correspondence with the time interval identified, in the profile of the magnetic field components, by the green color. At this location, the magnetic field intensity shows a clear discontinuity, B[λ_{3}] changes sign, B[λ_{2}] shows a hump whose maximum is located where the previous component changes sign and, finally, B[λ_{1}] keeps its value close to zero across the discontinuity. Velocity fluctuations are well correlated with magnetic field fluctuations and, in particular, the minimum variance component V[λ_{1}] has the same value on both sides of the discontinuity, approximately 350 km s^{−1}, indicating that there is no mass flux through the discontinuity. During this interval, which lasts about 26 min, the minimum variance direction lies close to the background magnetic field direction at 11.9° so that the arc is essentially described on a plane perpendicular to the average background magnetic field vector. However, additional although smaller and less regular arc-like structures can be recognized on the maximum variance plane λ_{2} − λ_{3}, and they tend to cover the whole 2π.
Within the second interval, magnetic field intensity is rather constant and the three components do not show any particular fluctuation, which could resemble any sort of rotation. In other words, the projection on the maximum variance plane does not show any coherent path. Even in this case, these fluctuations happen to be in a plane almost perpendicular to the average field direction since the angle between this direction and the minimum variance direction is about 9.3°.
Within the first time interval, the magnetic field vector experiences for some time small displacements around a given direction in space and then it suddenly performs a much larger displacement towards another direction in space, about which it starts to wander again. This process keeps on going several times within this time interval. In particular, the thick green line extending from label A to label B refers to the arc-like discontinuity shown in Figure 88, which is also the largest directional variation within this time interval. Within the second interval, the vector randomly fluctuates in all direction and, as a consequence, both the 3D trajectory and its projection on the maximum variance plane do not show any large empty spot. In practice, the second time interval, although longer, is similar to any sub-interval corresponding to one of the trajectory patches recognizable in the left hand side panel. As a matter of fact, selecting a single patch from the first interval and performing a minimum variance analysis, the maximum variance plane would result to be perpendicular to the local average magnetic field direction and the tip of the vector would randomly fluctuate in all directions. The first interval can be seen as a collection of several sub-intervals similar to interval # 2 characterized by different field orientations and, possibly, intensities. Thus, magnetic field intermittent events mark the border between adjacent intervals populated by stochastic Alfvénic fluctuations.
The border between these flux tubes can be a tangential discontinuity where the total pressure on both sides of the discontinuity is in equilibrium or, as in the case of interval #1, the discontinuity is located between two regions not in pressure equilibrium. If the observer moves across these tubes he will record the patchy configuration shown in Figure 91 relative to interval #1. Within each flux tube he will observe a local average field direction and the magnetic field vector would mainly fluctuate on a plane perpendicular to this direction. Moving to the next tube, the average field direction would rapidly change and magnetic vector fluctuations would cluster around this new direction. Moreover, if we imagine a situation with many flux-tubes, each one characterized by a different magnetic field intensity, moving across them would possibly increase the intermittent level of the fluctuations. On the contrary, moving along a single flux tube, the same observer would constantly be in the situation typical of interval #2, which is mostly characterized by a rather constant magnetic field intensity and directional stochastic fluctuations mainly on a plane quasi perpendicular to the average magnetic field direction. In such a situation, magnetic field intensity fluctuations would not increase their intermittency.
9.1 Radial evolution of intermittency in the ecliptic
Marsch and Liu (1993) investigated for the first time solar wind scaling properties in the inner heliosphere. These authors provided some insights on the different intermittent character of slow and fast wind, on the radial evolution of intermittency, and on the different scaling characterizing the three components of velocity. In particular, they found that fast streams were less intermittent than slow streams and the observed intermittency showed a weak tendency to increase with heliocentric distance. They also concluded that the Alfvénic turbulence observed in fast streams starts from the Sun as self-similar but then, during the expansion, decorrelates becoming more multifractal. This evolution was not seen in the slow wind, supporting the idea that turbulence in fast wind is mainly made of Alfvén waves and convected structures (Tu and Marsch, 1993), as already inferred by looking at the radial evolution of the level of cross-helicity in the solar wind (Bruno and Bavassano, 1991).
Bruno et al. (2003a) investigated the radial evolution of intermittency in the inner heliosphere, using the behavior of the flatness of the PDF of magnetic field and velocity fluctuations as a function of scale. As a matter of fact, probability distribution functions of fluctuating fields affected by intermittency become more and more peaked at smaller and smaller scales. Since the peakedness of a distribution is measured by its flatness factor, they studied the behavior of this parameter at different scales to estimate the degree of intermittency of their time series, as suggested by Frisch (1995).
In particular, vector field, like velocity and magnetic field, encompasses two distinct contributions, a compressive one due to intensity fluctuations that can be expressed as δ|B(t, τ)| = |B(t + τ)| − |B(t)|, and a directional one due to changes in the vector orientation \(\delta{\bf B}(t,\tau)=\sqrt{\sum_{i=x,y,z}(B_{i}(t+\tau)-B_{i}(t))^{2}}\). Obviously, relation δB(t, τ) takes into account also compressive contributions, and the expression δB(t, τ)| ≥ |δ|B(t, τ)∥ is always true.
Magnetic field fluctuations are more intermittent than velocity fluctuations.
Compressive fluctuations are more intermittent than directional fluctuations.
Slow wind intermittency does not show appreciable radial dependence.
Fast wind intermittency, for both magnetic field and velocity, clearly increases with distance.
Magnetic and velocity fluctuations have a rather Gaussian behavior at large scales, as expected, regardless of type of wind or heliocentric distance.
Moreover, they also found that the intermittency of the components rotated into the mean field reference system (see Appendix 15.1) showed that the most intermittent component of the magnetic field is the one along the mean field, while the other two show a similar level of intermittency within the associated uncertainties. This different behavior is then enhanced for larger heliocentric distances. These results agree with conclusions drawn by Marsch and Tu (1994) who, analyzing fast and slow wind at 0.3 AU, found that the PDFs of the fluctuations of transverse components of both velocity and magnetic fields, constructed for different time scales, were appreciably more Gaussian-like than fluctuations observed for the radial component, which resulted to be more and more spiky for smaller and smaller scales. However, this difference between radial and transverse components seemed to vanish with increasing heliocentric distance, and Tu et al. (1996) could not establish a clear radial trend or anisotropy. These results might be reconciled with conclusions by Bruno et al. (2003b) if the analysis by Tu et al. (1996) was repeated in the mean field reference system. The reason is that components normal to the mean field direction are more influenced by Alfvénic fluctuations and, as a consequence, their fluctuations are more stochastic and less intermittent. This effect largely reduces during the radial excursion mainly because the Solar Ecliptic (SE) reference system is not the most appropriate one for studying magnetic field fluctuations, and a cross-talking between different components is artificially introduced. As a matter of fact, the presence of the large scale spiral magnetic field breaks the spatial symmetry introducing a preferential direction parallel to the mean field. The same Bruno et al. (2003b) showed that it was not possible to find a clear radial trend unless magnetic field data were rotated into this more natural reference system.
On the other hand, it looks more difficult to reconcile the radial evolution of intermittency found by Bruno et al. (2003b) and Marsch and Liu (1993) in fast wind with conclusions drawn by Tu et al. (1996), who stated that “Neither a clear radial evolution nor a clear anisotropy can be established. The value of P1 in high-speed and low-speed wind are not prominent different.”. However, it is very likely that the conclusions given above are related with how to deal with the flat slope of the spectrum in fast wind near 0.3 AU. Tu et al. (1996) concluded, indeed: “It should be pointed out that the extended model cannot be used to analyze the intermittency of such fluctuations which have a flat spectrum. If the index of the power spectrum is near or less than unity … P1 would be 0.5. However, this does not mean there is no intermittency. The model simply cannot be used in this case, because the structure function (1) does not represent the effects of intermittency adequately for those fluctuations which have a flat spectrum and reveal no clear scaling behavior”.
Bruno et al. (2003a) concluded that the two major ingredients of interplanetary MHD fluctuations are compressive fluctuations due to a sort of underlying, coherent structure convected by the wind, and stochastic Alfvénic fluctuations propagating in the wind. Depending on the type of solar wind sample and on the heliocentric distance, the observed scaling properties would change accordingly. In particular, the same authors suggested that, as the radial distance increases, convected, coherent structures of the wind assume a more relevant role since the Alfvénic component of the fluctuations is depleted. This would be reflected in the increased intermittent character of the fluctuations. The coherent nature of the convected structures would contribute to increase intermittency while the stochastic character of the Alfvénic fluctuations would contribute to decrease it. This interpretation would also justify why compressive fluctuations are always more intermittent than directional fluctuations. As a matter of fact, coherent structures would contribute to the intermittency of compressive fluctuations and, at the same time, would also produce intermittency in directional fluctuations. However, since directional fluctuations are greatly influenced by Alfvénic stochastic fluctuations, their intermittency will be more or less reduced depending on the amplitude of the Alfvénic waves with respect to the amplitude of compressive fluctuations.
The radial dependence of the intermittency behavior of solar wind fluctuations stimulated Bruno et al. (1999b) to reconsider previous investigations on fluctuations anisotropy reported in Section 3.1.3. These authors studied magnetic field and velocity fluctuations anisotropy for the same corotating, high velocity stream observed by Bavassano et al. (1982a) within the framework of the dynamics of non-linear systems. Using the Local Intermittency Measure (Farge et al., 1990; Farge, 1992), Bruno et al. (1999b) were able to justify the controversy between results by Klein et al. (1991) in the outer heliosphere and Bavassano et al. (1982a) in the inner heliosphere. Exploiting the possibility offered by this technique to locate in space and time those events which produce intermittency, these authors were able to remove intermittent events and perform again the anisotropy analysis. They found that intermittency strongly affected the radial dependence of magnetic fluctuations while it was less effective on velocity fluctuations. In particular, after intermittency removal, the average level of anisotropy decreased for both magnetic and velocity field at all distances. Although magnetic fluctuations remained more anisotropic than their kinetic counterpart, the radial dependence was eliminated. On the other hand, the velocity field anisotropy showed that intermittency, although altering the anisotropic level of the fluctuations, does not markedly change its radial trend.
9.2 Radial evolution of intermittency at high latitude
Recently, Pagel and Balogh (2003) studied intermittency in the outer heliosphere using Ulysses observations at high heliographic latitude, well within high speed solar wind. In particular, these authors used Castaing distribution Castaing et al. (2001) to study the Probability Density Functions (PDF) of the fluctuations of magnetic field components. They found that intermittency of small scales fluctuations, within the inertial range, increased with increasing the radial distance from the Sun as a consequence of the growth to larger scales of the inertial range.
They also found that, in the RTN reference system, transverse magnetic field components exhibit less Gaussian behavior with respect to the radial component. This result should be compared with results from similar studies by Marsch and Tu (1994) and Bruno et al. (2003b) who, studying the radial evolution of intermittency in the ecliptic, found that the components transverse to the local magnetic field direction, are the most Gaussian ones. Probably, the above discrepancy depends totally on the reference system adopted in these different studies and it would be desirable to perform a new comparison between high and low latitude intermittency in the mean-field reference system.
Pagel and Balogh (2002) focused also on the different intermittent level of magnetic field fluctuations during two fast latitudinal scans which happened to be during solar minimum the first one, and during solar maximum the second one. Their results showed a strong latitudinal dependence but were probably not, or just slightly, affected by radial dependence given the short heliocentric radial variations during these time intervals. They analyzed the anomalous scaling of the third order magnetic field structure functions looking at the value of the parameter μ obtained from the best fit performed using the p-model (see Section 8.1). In a previous analysis of the same kind, but focused on the first latitudinal scan, the same authors tested three intermittency models, namely: “lognormal”, “p” and “G-infinity” models. In particular, this last model was an empirical model introduced by Pierrehumbert (1999) and Cho et al. (2000) and was not intended for turbulent systems. Anyhow, the best fits were obtained with the lognormal and Kolmogorov-p model. These authors concluded that magnetic field components display a very high level of intermittency throughout minimum and maximum phases of solar cycle, and slow wind shows a lower level of intermittency compared with the Alfvénic polar flows. These results do not seem to agree with ecliptic observations (Marsch and Liu, 1993; Bruno et al., 2003a) which showed that fast wind is generally less intermittent than slow wind not only for wind speed and magnetic field magnitude, but also for the components. At this point, since it has been widely recognized that low latitude fast wind collected within corotating streams and fast polar wind share many common turbulence features, they should be expected to have many similarities also as regards intermittency. Thus, it is possible that also in this case the reference system in which the analysis is performed plays some role in determining some of the results regarding the behavior of the components. In any case, further analyses should clarify the reasons for this discrepancy.
10 Conclusions and Remarks
Now that the reader finally reached the conclusions, hoping that he was so patient to read the whole paper, we suggest him to go back for a moment to the List of Contents, not to start all over again, but just to take a look at the various problems that have been briefly touched by this review. He will certainly realize how complex is the phenomenon of turbulence in general and, in particular, in the solar wind. Almost four decades of observations and theoretical efforts have not yet been sufficient to fully understand how this natural and fascinating phenomenon really works in the solar wind. We certainly are convinced that we cannot think of a single mechanism able to reproduce all the details we have directly observed since physical boundary conditions favor or inhibit different generation mechanisms, like for instance, velocity-shear or parametric decay, depending on where we are in the heliosphere.
On the other hand, there are some aspects which we believe are at the basis of turbulence generation and evolution like: a) we do need non-linear interactions to develop the observed Kolmogorov-like spectrum; b) in order to have non-linear interactions we need to have inward modes and/or convected structures which the majority of the modes can interact with; c) outward and inward modes can be generated by different mechanisms like velocity shear or parametric decay; d) convected structures actively contribute to turbulent development of fluctuations and can be of solar origin or locally generated.
In particular, ecliptic observations have shown that what we call Alfvénic turbulence, mainly observed within high velocity streams, tends to evolve towards the more “standard” turbulence that we mainly observe within slow wind regions, i.e., a turbulence characterized by e^{+} ∼ e^{−}, an excess of magnetic energy, and a Kolmogorov-like spectral slope. Moreover, the presence of a well established “background” spectrum already at short heliocentric distances and the low Alfvénicity of the fluctuations suggest that within slow wind turbulence is mainly due to convected structures frozen in the wind which may well be the remnants of turbulent processes already acting within the first layers of the solar corona. In addition, velocity shear, whenever present, seems to have a relevant role in driving turbulence evolution in low-latitude solar wind.
Polar observations performed by Ulysses, combined with previous results in the ecliptic, finally allowed to get a comprehensive view of the Alfvénic turbulence evolution in the 3D heliosphere, inside 5 AU. However, polar observations, when compared with results obtained in the ecliptic, do not appear as a dramatic break. In other words, the polar evolution is similar to that in the ecliptic, although slower. This is a middle course between the two opposite views (a non-relaxing turbulence, due to the lack of velocity shear, or a quick evolving turbulence, due to the large relative amplitude of fluctuations) which were popular before the Ulysses mission. The process driving the evolution of polar turbulence still is an open question although parametric decay might play some role. As a matter of fact, simulations of non-linear development of the parametric instability for large-amplitude, broadband Alfvénic fluctuations have shown that the final state resembles values of σ_{c} not far from solar wind observations, in a state in which the initial Alfvénic correlation is partially preserved. As already observed in the ecliptic, polar Alfvénic turbulence appears characterized by a predominance of outward fluctuations and magnetic fluctuations. As regards the outward fluctuations, their dominant character extends to large distances from the Sun. At low solar activity, with the polar wind filling a large fraction of the heliosphere, the outward fluctuations should play a relevant role in the heliospheric physics. Relatively to the imbalance in favor of the magnetic energy, it does not appear to go beyond an asymptotic value. Several ways to alter the balance between kinetic and magnetic energy have been proposed (e.g., 2D processes, propagation in a non-uniform medium, and effect of magnetic structures, among others). However, convincing arguments to account for the existence of such a limit have not yet been given, although promising results from numerical simulations seem to be able to qualitatively reproduce the final imbalance in favor of the magnetic energy.
Definitely, the relatively recent adoption of numerical methods able to highlight scaling laws features hidden to the usual spectral methods, allowed to disclose a new and promising way to analyze turbulent interplanetary fluctuations. Interplanetary space is now looked at as a natural wind tunnel where scaling properties of the solar wind can be studied on scales of the order of (or larger than) 10^{9} times than laboratory scales. Within this framework, intermittency represents an important topic in both theoretical and observational studies. Intermittency properties have been recovered via very promising models like the MHD GOY-model, and the nature of intermittent events has finally been disclosed thanks to new numerical techniques based on wavelet transforms. Moreover, similar techniques have allowed to tackle the problem of identifying either Kraichnan or Kolmogorov scaling although no conclusive and final analyses have been reported so far. In addition, recent studies on intermittency of magnetic field and velocity vector fluctuations, together with analogous analyses on magnitude fluctuations, contributed to sketch a scenario in which propagating stochastic Alfvénic fluctuations and convected structures, possibly flux tubes embedded in the wind, represent the main ingredients of interplanetary turbulence. The relative predominance of one or the other contribution would make the observed turbulence more or less intermittent. However, the fact that we can make measurements just in one point of this natural wind tunnel represented by the solar wind does not allow us to discriminate temporal from spatial phenomena. As a consequence, we do not know whether these convected structures are somehow connected to the complicated topology observed at the Sun surface or can be considered as by-product of chaotic developing phenomena. Comparative studies based on the intermittency phenomenon within fast and slow wind during the wind expansion would suggest a solar origin for these structures which would form a sort of turbulent background frozen in the wind. As a matter of fact, intermittency in the solar wind is not limited to the dissipation range of the spectrum but abundantly extends orders of magnitude away from dissipative scales, possibly into the inertial range which can be identified taking into account all the possible caveats related to this problem and briefly reported in this review. This fact introduces serious differences between hydrodynamic turbulence and solar wind MHD turbulence, and the same “intermittency” assumes a different intrinsic meaning when observed in interplanetary turbulence. In practice, coherent structures observed in the wind are at odds with filaments or vortexes observed in ordinary fluid turbulence since these last ones are dissipative structures continuously created and destroyed by turbulent motion.
As a final remark, we would like to point out that we tried to start writing a particular point of view on the turbulence in the solar wind. We apologize for the lack of some aspects of the phenomenon at hand which can be found in the existing literature. There are several topics which we did not discuss in this first version of our review like recent (non-shell) turbulence modeling, simulation of turbulence in the expanding solar wind, uses of turbulence in coronal heating models, multispacecraft observations, and the kinetic approach to the dissipation of turbulence. Fortunately, we are writing a Living Review paper and mistakes and/or omissions will be adequately fixed in the next version also with the help of all our colleagues whom we strongly encourage to send us comments and/or different points of view on particularly interesting topics which we have not yet taken into account or discussed properly.
Footnotes
- 1.
This concept will be explained better in the next sections.
- 2.
A fluid particle is defined as an infinitesimal portion of fluid which moves with the local velocity. As usual in luid dynamics, infinitesimal means small with respect to large scale, but large enough with respect to molecular scales.
- 3.
- 4.
These authors were the first ones to use physical technologies and methodologies to investigate turbulent flows from an experimental point of view. Before them, experimental studies on turbulence were motivated mainly by engineering aspects.
- 5.
We can use a different definition for the third invariant H(t), for example a quantity positive defined, without the term (−1)^{n} and with α = 2. This can be identified as the surrogate of the square of the vector potential, thus investigating a kind of 2D MHD. In this case, we obtain a shell model with λ = 2, a = 5/4, and c = −1/3. However, this model does not reproduce the inverse cascade of the square of magnetic potential observed in the true 2D MHD equations.
- 6.
We have already defined fluctuations of a field as the difference between the field itself and its average value. This quantity has been defined as δψ. Here, the differences δψ_{ℓ} of the field separated by a distance ℓ represents characteristic fluctuations at the scale ℓ, say characteristic fluctuations of the field across specific structures (eddies) that are present at that scale. The reader can realize the difference between both definitions.
- 7.
To be precise, it is worth remarking that there are no convincing arguments to identify as inertial range the intermediate range of frequencies where the observed spectral properties are typical of fully developed turbulence. From a theoretical point of view, here the association “intermediate range” ≃ “inertial range” is somewhat arbitrary as it can be inferred from the short discussion given in Section 2.10.
- 8.
Since the solar wind moves at supersonic speed V_{sw}, the usual Taylor’s hypothesis is verified, and we can get information on spatial (ℓ) scaling laws by using time differences r = ℓ/V_{sw}.
- 9.
It is worthwhile to remark that neither the fluid relation (26) nor its MHD counterpart (27) are satisfied in the solar wind. Namely there is not any extended range of scales, from which we can derive scaling exponents, where the above relations which formally define the inertial range are verified. Here we are in a situation similar to a low-Reynolds number fluid flow.
- 10.
The log-Lévy model is a modification of the lognormal model. In such case, the central limit theorem is used to derive the limit distribution of an infinite sum of stochastic variables by relaxing the hypothesis of finite variance usually used. The resulting limit function is a Lévy function.
Notes
Acknowledgments
Writing a large review paper is not an easy task and it would not have been possible to accomplish this goal without having a good interaction with our colleagues, whom we have been working with in our Institutions. To this regard, we like to acknowledge the many discussions (more or less “heated”) we had with and the many advices and comments we had from all of them, particularly from B. Bavassano and P. Veltri. We also like to acknowledge the use of plasma and magnetic field data from Helios spacecraft to freshly produce some of the figures shown in the present review. In particular, we like to thank H. Rosenbauer and R. Schwenn, PIs of the plasma experiment, and F. Mariani and N.F. Ness, PIs of the second magnetic experiment on board Helios. We thank A. Pouquet, H. Politano, and V. Antoni for the possibility we had to compare solar wind data with both high-resolution numerical simulations and laboratory plasmas. Finally, we ought special thanks to E. Marsch and S.K. Solanki for giving us the opportunity to write this review.
Supplementary material
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