"The Solar Wind as a Turbulence Laboratory"
Roberto Bruno and Vincenzo Carbone 
1 Introduction
1.1 What does turbulence stand for?
1.2 Dynamics vs. statistics
2 Equations and Phenomenology
2.1 The Navier–Stokes equation and the Reynolds number
2.2 The coupling between a charged fluid and the magnetic field
2.3 Scaling features of the equations
2.4 The non-linear energy cascade
2.5 The inhomogeneous case
2.6 Dynamical system approach to turbulence
2.7 Shell models for turbulence cascade
2.8 The phenomenology of fully developed turbulence: Fluid-like case
2.9 The phenomenology of fully developed turbulence: Magnetically-dominated case
2.10 Some exact relationships
2.11 Yaglom’s law for MHD turbulence
2.12 Density-mediated Elsässer variables and Yaglom’s law
2.13 Yaglom’s law in the shell model for MHD turbulence
3 Early Observations of MHD Turbulence in the Ecliptic
3.1 Turbulence in the ecliptic
3.2 Turbulence studied via Elsässer variables
4 Observations of MHD Turbulence in the Polar Wind
4.1 Evolving turbulence in the polar wind
4.2 Polar turbulence studied via Elsässer variables
5 Numerical Simulations
5.1 Local production of Alfvénic turbulence in the ecliptic
5.2 Local production of Alfvénic turbulence at high latitude
6 Compressive Turbulence
6.1 On the nature of compressive turbulence
6.2 Compressive turbulence in the polar wind
6.3 The effect of compressive phenomena on Alfvénic correlations
7 A Natural Wind Tunnel
7.1 Scaling exponents of structure functions
7.2 Probability distribution functions and self-similarity of fluctuations
7.3 What is intermittent in the solar wind turbulence? The multifractal approach
7.4 Fragmentation models for the energy transfer rate
7.5 A model for the departure from self-similarity
7.6 Intermittency properties recovered via a shell model
8 Observations of Yaglom’s Law in Solar Wind Turbulence
9 Intermittency Properties in the 3D Heliosphere: Taking a Look at the Data
9.1 Structure functions
9.2 Probability distribution functions
10 Turbulent Structures
10.1 On the statistics of magnetic field directional fluctuations
10.2 Radial evolution of intermittency in the ecliptic
10.3 Radial evolution of intermittency at high latitude
11 Solar Wind Heating by the Turbulent Energy Cascade
11.1 Dissipative/dispersive range in the solar wind turbulence
12 The Origin of the High-Frequency Region
12.1 A dissipation range
12.2 A dispersive range
13 Two Further Questions About Small-Scale Turbulence
13.1 Whistler modes scenario
13.2 Kinetic Alfvén waves scenario
13.3 Where does the fluid-like behavior break down in solar wind turbulence?
13.4 What physical processes replace “dissipation” in a collisionless plasma?
14 Conclusions and Remarks
A Some Characteristic Solar Wind Parameters
B Tools to Analyze MHD Turbulence in Space Plasmas
B.1 Statistical description of MHD turbulence
B.2 Spectra of the invariants in homogeneous turbulence
B.3 Introducing the Elsässer variables
C Wavelets as a Tool to Study Intermittency
D Reference Systems
D.1 Minimum variance reference system
D.2 The mean field reference system
E On-board Plasma and Magnetic Field Instrumentation
E.1 Plasma instrument: The top-hat
E.2 Measuring the velocity distribution function
E.3 Computing the moments of the velocity distribution function
E.4 Field instrument: The flux-gate magnetometer
F Spacecraft and Datasets

7 A Natural Wind Tunnel

The solar wind has been used as a wind tunnel by Burlaga who, at the beginning of the 1990s, started to investigate anomalous fluctuations (Burlaga, 1991aJump To The Next Citation Point,bJump To The Next Citation Point,cJump To The Next Citation Point, 1995) as observed by measurements in the outer heliosphere by the Voyager spacecraft. In 1991, MarschJump To The Next Citation Point, in a review on solar wind turbulence given at the Solar Wind Seven conference, underlined the importance of investigating scaling laws in the solar wind and we like to report his sentence: “The recent work by Burlaga (1991aJump To The Next Citation Point,bJump To The Next Citation Point) opens in my mind a very promising avenue to analyze and understand solar wind turbulence from a new theoretical vantage point. …This approach may also be useful for MHD turbulence. Possible connections between intermittent turbulence and deterministic chaos have recently been investigated …We are still waiting for applications of these modern concepts of chaos theory to solar wind MHD fluctuations.” (cf. Marsch, 1992, p. 503). A few years later Carbone (1993Jump To The Next Citation Point) 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 (1941Jump To The Next Citation Point) 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, 1995Jump To The Next Citation Point). 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 (1995Jump To The Next Citation Point).

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 time8 interval τ. First of all, it is worthwhile to remark that scaling laws and, in particular, the exact relation (40View Equation) which defines the inertial range in fluid flows, is valid for longitudinal (streamwise) fluctuations. In common fluid flows the Kolmogorov linear scaling law is compared with the moments of longitudinal velocity differences. In the same way for the solar wind turbulence we investigate the scaling behavior of Δu τ = u(t + τ) − u(t), where u (t) represents the component of the velocity field along the radial direction. As far as the magnetic differences are concerned Δb τ = B (t + τ) − B (t), we are free for different choices and, in some sense, this is more interesting from an experimental point of view. We can use the reference system where B(t) represents the magnetic field projected along the radial direction, or the system where B (t) represents the magnetic field along the local background magnetic field, or B (t) represents the field along the minimum variance direction. As a different case we can simply investigate the scaling behavior of the fluctuations of the magnetic field intensity.

Let us consider the p-th moment of both absolute values9 of velocity fluctuations R (τ) = ⟨|Δu |p⟩ p τ and magnetic fluctuations S (τ) = ⟨|Δb |p⟩ p τ, also called p-th order structure function in literature (brackets being time average). Here we use magnetic fluctuations across structures at intervals τ calculated by using the magnetic field intensity. Typical structure functions of magnetic field fluctuations, for two different values of p, for both a slow wind and a fast wind at 0.9 AU, are shown in Figures 80View Image. The magnetic field we used is that measured by Helios 2 spacecraft. Structure functions calculated for the velocity fields have roughly the same shape. Looking at these Figures the typical scaling features of turbulence can be observed. Starting from low values at small scales, the structure functions increase towards a region where Sp → const. at the largest scales. This means that at these scales the field fluctuations are uncorrelated. A kind of “inertial range”, that is a region of intermediate scales τ where a power law can be recognized for both

p ζp Rp (τ) = ⟨|Δu τ|⟩ ∼ τ Sp(τ) = ⟨|Δb τ|p⟩ ∼ τξp (60 )
is more or less visible only for the slow wind. In this range correlations exists, and we can obtain the scaling exponents ζ p and ξ p through a simple linear fit.
View Image

Figure 80: Structure functions for the magnetic field intensity Sn (r) for two different orders, n = 3 and n = 5, for both slow wind and fast wind, as a function of the time scale r. Data come from Helios 2 spacecraft at 0.9 AU.

Since as we have seen, Yaglom’s law is observed only in some few samples, the inertial range in the whole solar wind is not well defined. A look at Figure 80View Image clearly shows that we are in a situation similar to a low-Reynolds number fluid flow. In order to compare scaling exponents of the solar wind turbulent fluctuations with other experiments, it is perhaps better to try to recover exponents using the Extended Self-Similarity (ESS), introduced some time ago by Benzi et al. (1993Jump To The Next Citation Point), and used here as a tool to determine relative scaling exponents. In the fluid-like case, the third-order structure function can be regarded as a generalized scaling using the inverse of Equation (41View Equation) or of Equation (40View Equation) (Politano et al., 1998Jump To The Next Citation Point). Then, we can plot the p-th order structure function vs. the third-order one to recover at least relative scaling exponents ζ ∕ζ p 3 and ξ ∕ξ p 3 (60View Equation). Quite surprisingly (see Figure 81View Image), we find that the range where a power law can be recovered extends well beyond the inertial range, covering almost all the experimental range. In the fluid case the scaling exponents which can be obtained through ESS at low or moderate Reynolds numbers, coincide with the scaling exponents obtained for high Reynolds, where the inertial range is very well defined Benzi et al. (1993). This is due to the fact that, since by definition ζ = 1 3 in the inertial range (Frisch, 1995Jump To The Next Citation Point), whatever its extension might be. In our case scaling exponents obtained through ESS can be used as a surrogate, since we cannot be sure that an inertial range exists.

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Figure 81: Structure functions Sn (r) for two different orders, n = 3 and n = 5, for both slow wind and high wind, as a function of the fourth-order structure function S4(r). Data come from Helios 2 spacecraft at 0.9 AU.

It is worthwhile to remark (as shown in Figure 81View Image) that we can introduce a general scaling relation between the q-th order velocity structure function and the p-th order structure function, with a relative scaling exponent αp (q). It has been found that this relation becomes an exact relation

S (r) = [S (r)]αp(q), q p

when the velocity structure functions are normalized to the average velocity within each period used to calculate the structure function (Carbone et al., 1996aJump To The Next Citation Point). This is very interesting because it implies (Carbone et al., 1996aJump To The Next Citation Point) that the above relationship is satisfied by the following probability distribution function, if we assume that odd moments are much smaller than the even ones:

∫ ∞ ikΔu ∑∞ (ik )2q αp(2q) P DF (Δu τ) = dk e τ --------[Sp(τ )] . (61 ) −∞ q=0 2π(2q)!
That is, for each scale τ the knowledge of the relative scaling exponents αp(q) completely determines the probability distribution of velocity differences as a function of a single parameter Sp (τ ).

Relative scaling exponents, calculated by using data coming from Helios 2 at 0.9 AU, are reported in Table 1. As it can be seen, two main features can be noted:

  1. 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 ζp∕ζ3 > p∕3 for p < 3, while ζp∕ζ3 < p∕3 for p > 3. The same behavior can be observed for ξp∕ξ3. 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, 1995Jump To The Next Citation Point). Turbulence in the solar wind is intermittent, just like its fluid counterpart on Earth.
  2. The degree of intermittency is measured through the distance between the curve ζ ∕ζ p 3 and the linear scaling p∕3. It can be seen that the magnetic field is more intermittent than the velocity field. The same difference is observed between the velocity field and a passive scalar (in our case the temperature) in ordinary fluid flows (Ruíz-Chavarría et al., 1995Jump To The Next Citation Point). That is the magnetic field, as long as intermittency properties are concerned, has the same scaling laws of a passive field. Of course this does not mean that the magnetic field plays the same role as a passive field. Statistical properties are in general different from dynamical properties.

Table 1: Scaling exponents for velocity ζp and magnetic ξp 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 1.38

In Table 1 we show scaling exponents up to the sixth order. Actually, a question concerns the validation of high-order moments estimates, say the maximum value of the order p which can be determined with a finite number of points of our dataset. As the value of p increases, we need an increasing number of points for an optimal determination of the structure function (Tennekes and Wyngaard, 1972). Anomalous scaling laws are generated by rare and intense events due to singularities in the gradients: the higher their intensity the more rare these events are. Of course, when the data set has a finite extent, the probability to get singularities stronger than a certain value approaches zero. In that case, scaling exponents ζp of order higher than a certain value become linear functions of p. Actually, the structure function S (τ) p depends on the probability distribution function PDF (Δu ) τ through

∫ Sp(τ ) = Δupτ PDF (δu τ)dΔu τ
and, the function Sp is determined only when the integral converges. As p increases, the function F (δu ) = Δup PDF (Δu ) p τ τ τ becomes more and more disturbed, with some spikes, so that the integral becomes more and more undefined, as can be seen for example in Figure 1 of the paper by Dudok de Wit (2004Jump To The Next Citation Point). A simple calculation (Dudok de Wit, 2004) for the maximum value of the order pm which can reliably be estimated with a given number N of points in the dataset, gives the empirical criterion pm ≃ log N. Structure functions of order p > pm cannot be determined accurately.

Only few large structures are enough to generate the anomalous scaling laws. In fact, as shown by Salem et al. (2009Jump To The Next Citation Point), by suppressing through wavelets analysis just a few percentage of large structures on all scales, the scaling exponents become linear functions of p, respectively p∕4 and p∕3 for the kinetic and magnetic fields.

Table 2: Normalized scaling exponents ξp∕ξ3 for radial magnetic fluctuations in a laboratory plasma, as measured at different distances a ∕R (R ≃ 0.45 cm being the minor radius of the torus in the experiment) from the external wall. Errors represent the standard deviations of the linear fitting. Scaling exponents have been obtained using the ESS.
p a∕R = 0.96 a∕R = 0.93 a∕R = 0.90 a∕R = 0.86
1 0.39 ± 0.01 0.38 ± 0.01 0.37 ± 0.01 0.36 ± 0.01
2 0.74 ± 0.01 0.73 ± 0.02 0.71 ± 0.01 0.70 ± 0.01
3 1.00 1.00 1.00 1.00
4 1.20 ± 0.02 1.24 ± 0.02 1.27 ± 0.01 1.28 ± 0.01
5 1.32 ± 0.03 1.41 ± 0.03 1.51 ± 0.03 1.55 ± 0.03
6 1.38 ± 0.04 1.50 ± 0.04 1.71 ± 0.03 1.78 ± 0.04

As far as a comparison between different plasmas is concerned, the scaling exponents of magnetic structure functions, obtained from laboratory plasma experiments of a Reversed-Field Pinch at different distances from the external wall (Carbone et al., 2000Jump To The Next Citation Point) are shown in Table 2. In laboratory plasmas it is difficult to measure all the components of the vector field at the same time, thus, here we show only the scaling exponents obtained using magnetic field differences Br (t + τ) − Br (t) calculated from the radial component in a toroidal device where the z-axis is directed along the axis of the torus. As it can be seen, intermittency in magnetic turbulence is not so strong as it appears to be in the solar wind, actually the degree of intermittency increases when going toward the external wall. This last feature appears to be similar to what is currently observed in channel flows, where intermittency also increases when going towards the external wall (Pope, 2000).

Scaling exponents of structure functions for Alfvén variables, velocity, and magnetic variables have been calculated also for high resolution 2D incompressible MHD numerical simulations (Politano et al., 1998Jump To The Next Citation Point). In this case, we are freed from the constraint of the Taylor hypothesis when calculating the fluctuations at a given scale. From 2D simulations we recover the fields u (r,t) and b(r,t) at some fixed times. We calculate the longitudinal fluctuations directly in space at a fixed time, namely Δu ℓ = [u(r + ℓ,t) − u (r,t)] ⋅ ℓ∕ℓ (the same are made for different fields, namely the magnetic field or the Elsässer fields). Finally, averaging both in space and time, we calculate the scaling exponents through the structure functions. These scaling exponents are reported in Table 3. Note that, even in numerical simulations, intermittency for magnetic variables is stronger than for the velocity field.

Table 3: Normalized scaling exponents ξp∕ξ3 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 (40View Equation) as characteristic scale rather than the third-order structure function (cf. Politano et al., 1998, for details).
p + Z − Z v B
1 0.36 ± 0.06 0.56 ± 0.06 0.37 ± 0.01 0.46 ± 0.02
2 0.70 ± 0.05 0.83 ± 0.05 0.70 ± 0.01 0.78 ± 0.01
3 1.00 1.00 1.00 1.00
4 1.28 ± 0.02 1.14 ± 0.02 1.28 ± 0.02 1.18 ± 0.02
5 1.53 ± 0.03 1.25 ± 0.03 1.54 ± 0.03 1.31 ± 0.03
6 1.79 ± 0.05 1.35 ± 0.05 1.78 ± 0.05 1.40 ± 0.03

7.2 Probability distribution 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) = C ℓh, where the scaling exponent is h = − log λμ.

Since, as we have just seen, turbulence is characterized by scaling laws, this must be a signature of self-similarity for fluctuations. Let us see what this means. Let us consider fluctuations at two different scales, namely Δz ± ℓ and Δz ± λℓ. Their ratio Δz ± ∕Δz ± ∼ λh λℓ ℓ depends only on the value of h, and this should imply that fluctuations are self-similar. This means that PDFs are related through

± h ± P (Δz λℓ) = PDF (λ Δz ℓ ).

Let us consider the normalized variables

± y± = ----Δz-ℓ----. ℓ ⟨(Δz ±ℓ )2⟩1∕2

When h is unique or in a pure self-similar situation, PDFs are related through P (y±ℓ ) = PDF (y±λℓ), say by changing scale PDFs coincide.

The PDFs relative to the normalized magnetic fluctuations δb = Δb ∕⟨Δb2 ⟩1∕2 τ τ τ, at three different scales τ, are shown in Figure 82View Image. It appears evident that the global self-similarity in real turbulence is broken. PDFs do not coincide at different scales, rather their shape seems to depend on the scale τ. In particular, at large scales PDFs seem to be almost Gaussian, but they become more and more stretched as τ decreases. At the smallest scale PDFs are stretched exponentials. This scaling dependence of PDFs is a different way to say that scaling exponents of fluctuations are anomalous, or can be taken as a different definition of intermittency. Note that the wings of PDFs are higher than those of a Gaussian function. This implies that intense fluctuations have a probability of occurrence higher than that they should have if they were Gaussianly distributed. Said differently, intense stochastic fluctuations are less rare than we should expect from the point of view of a Gaussian approach to the statistics. These fluctuations play a key role in the statistics of turbulence. The same statistical behavior can be found in different experiments related to the study of the atmosphere (see Figure 83View Image) and the laboratory plasma (see Figure 84View Image).

View Image

Figure 82: Left panel: normalized PDFs for the magnetic fluctuations observed in the solar wind turbulence by using Helios data. Right panel: distribution function of waiting times Δt between structures at the smallest scale. The parameter β is the scaling exponent of the scaling relation PDF (Δt) ∼ Δt −β for the distribution function of waiting times.
View Image

Figure 83: Left panel: normalized PDFs of velocity fluctuations in atmospheric turbulence. Right panel: distribution function of waiting times Δt between structures at the smallest scale. The parameter β is the scaling exponent of the scaling relation −β PDF (Δt) ∼ Δt for the distribution function of waiting times. The turbulent samples have been collected above a grass-covered forest clearing at 5 m above the ground surface and at a sampling rate of 56 Hz (Katul et al., 1997).
View Image

Figure 84: Left panel: normalized PDFs of the radial magnetic field collected in RFX magnetic turbulence (Carbone et al., 2000). Right panel: distribution function of waiting times Δt between structures at the smallest scale. The parameter β is the scaling exponent of the scaling relation PDF (Δt) ∼ Δt −β for the distribution function of waiting times.

7.3 What is intermittent in the solar wind turbulence? The multifractal approach

Time dependence of Δu τ and Δb τ for three different scales τ is shown in Figures 85View Image and 86View Image, respectively. These plots show that, as τ becomes small, intense fluctuations become more and more important, and they dominate the statistics. Fluctuations at large scales appear to be smooth while, as the scale becomes smaller, intense fluctuations becomes visible. These dominating fluctuations represent relatively rare events. Actually, at the smallest scales, the time behavior of both Δu τ and Δb τ is dominated by regions where fluctuations are low, in between regions where fluctuations are intense and turbulent activity is very high. Of course, this behavior cannot be described by a global self-similar behavior. Allowing the scaling laws to vary with the region of turbulence we are investigating would be more convincing.

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Figure 85: Differences for the longitudinal velocity δu = u(t + τ ) − u (t) τ at three different scales τ, as shown in the figure.
View Image

Figure 86: Differences for the magnetic intensity Δbτ = B (t + τ) − B(t) at three different scales τ, as shown in the figure.

The behavior we have just described is at the heart of the multifractal approach to turbulence (Frisch, 1995Jump To The Next Citation Point). In that description of turbulence, even if the small scales of fluid flow cannot be globally self-similar, self-similarity can be reintroduced as a local property. In the multifractal description it is conjectured that turbulent flows can be made by an infinite set of points S (r) h, each set being characterized by a scaling law ± h(r) ΔZ ℓ ∼ ℓ, that is, the scaling exponent can depend on the position r. The usual dimension of that set is then not constant, but depends on the local value of h, and is quoted as D (h) in literature. Then, the probability of occurrence of a given fluctuation can be calculated through the weight the fluctuation assumes within the whole flow, i.e.,

± ± h P(ΔZ ℓ ) ∼ (ΔZ ℓ ) × volume occupied by fluctuations,

and the p-th order structure function is immediately written through the integral over all (continuous) values of h weighted by a smooth function μ(h) ∼ 0(1), i.e.,

∫ Sp(ℓ) = μ(h)(ΔZ ±)ph(ΔZ ±)3−D (h)dh. ℓ ℓ

A moment of reflection allows us to realize that in the limit ℓ → 0 the integral is dominated by the minimum value (over h) of the exponent and, as shown by Frisch (1995Jump To The Next Citation Point), the integral can be formally solved using the usual saddle-point method. The scaling exponents of the structure function can then be written as

ζp = minh [ph + 3 − D (h)].

In this way, the departure of ζ p from the linear Kolmogorov scaling and thus intermittency, can be characterized by the continuous changing of D (h) as h varies. That is, as p varies we are probing regions of fluid where even more rare and intense events exist. These regions are characterized by small values of h, that is, by stronger singularities of the gradient of the field.

Owing to the famous Landau footnote on the fact that fluctuations of the energy transfer rate must be taken into account in determining the statistics of turbulence, people tried to interpret the non-linear energy cascade typical of turbulence theory, within a geometrical framework. The old Richardson’s picture of the turbulent behavior as the result of a hierarchy of eddies at different scales has been modified and, as realized by Kraichnan (1974), once we leave the idea of a constant energy cascade rate we open a “Pandora’s box” of possibilities for modeling the energy cascade. By looking at scaling laws for Δz ± ℓ and introducing the scaling exponents for the energy transfer rate p τp ⟨𝜖ℓ⟩ ∼ r, it can be found that ζp = p∕m + τp∕m (being m = 3 when the Kolmogorov-like phenomenology is taken into account, or m = 4 when the Iroshnikov-Kraichnan phenomenology holds). In this way the intermittency correction are determined by a cascade model for the energy transfer rate. When τp is a non-linear function of p, the energy transfer rate can be described within the multifractal geometry (see, e.g., Meneveau, 1991Jump To The Next Citation Point, and references therein) characterized by the generalized dimensions Dp = 1 − τp∕(p − 1) (Hentschel and Procaccia, 1983). The scaling exponents of the structure functions are then related to Dp by

( p ) ζp = -- − 1 Dp ∕m + 1. m

The correction to the linear scaling p∕m is positive for p < m, negative for p > m, and zero for p = m. A fractal behavior where Dp = const.< 1 gives a linear correction with a slope different from 1∕m.

7.4 Fragmentation models for the energy transfer rate

Cascade models view turbulence as a collection of fragments at a given scale ℓ, which results from the fragmentation of structures at the scale ℓ′ > ℓ, down to the dissipative scale (Novikov, 1969). Sophisticated statistics are applied to obtain scaling exponents ζp for the p-th order structure function.

The starting point of fragmentation models is the old β-model, a “pedagogical” fractal model introduced by Frisch et al. (1978) to account for the modification of the cascade in a simple way. In this model, the cascade is realized through the conjecture that active eddies and non-active eddies are present at each scale, the space-filling factor for the fragments being fixed for each scale. Since it is a fractal model, the β-model gives a linear modification to ζ p. This can account for a fit on the data, as far as small values of p are concerned. However, the whole curve ζp is clearly nonlinear, and a multifractal approach is needed.

The random-β model (Benzi et al., 1984), a multifractal modification of the β-model, can be derived by invoking that the space-filling factor for the fragments at a given scale in the energy cascade is not fixed, but is given by a random variable β. The probability of occurrence of a given β is assumed to be a bimodal distribution where the eddies fragmentation process generates either space-filling eddies with probability ξ or planar sheets with probability (1 − ξ) (for conservation 0 ≤ ξ ≤ 1). It can be found that

p- [ p∕m −1] ζp = m − log2 1 − ξ + ξ2 , (62 )
where the free parameter ξ can be fixed through a fit on the data.

The p-model (Meneveau, 1991Jump To The Next Citation Point; Carbone, 1993Jump To The Next Citation Point) consists in an eddies fragmentation process described by a two-scale Cantor set with equal partition intervals. An eddy at the scale ℓ, with an energy derived from the transfer rate 𝜖 r, breaks down into two eddies at the scale ℓ∕2, with energies μ 𝜖 r and (1 − μ)𝜖r. The parameter 0.5 ≤ μ ≤ 1 is not defined by the model, but is fixed from the experimental data. The model gives

[ p∕m p∕m ] ζp = 1 − log2 μ + (1 − μ ) . (63 )

In the model by She and Leveque (see, e.g., She and Leveque, 1994; Politano and Pouquet, 1998) one assumes an infinite hierarchy for the moments of the energy transfer rates, leading to 𝜖(rp+1)∼ [𝜖(rp)]β[𝜖(r∞ )]1−β, and a divergent scaling law for the infinite-order moment 𝜖(r∞) ∼ r−x, which describes the most singular structures within the flow. The model reads

p [ ( x)p ∕m ] ζp = --(1 − x) + C 1 − 1 − -- . (64 ) m C
The parameter C = x∕ (1 − β ) is identified as the codimension of the most singular structures. In the standard MHD case (Politano and Pouquet, 1995) x = β = 1∕2, so that C = 1, that is, the most singular dissipative structures are planar sheets. On the contrary, in fluid flows C = 2 and the most dissipative structures are filaments. The large p behavior of the p-model is given by ζp ∼ (p∕m ) log2(1∕μ ) + 1, so that Equations (63View Equation, 64View Equation) give the same results providing μ ≃ 2 −x. As shown by Carbone et al. (1996b) all models are able to capture intermittency of fluctuations in the solar wind. The agreement between the curves ζp and normalized scaling exponents is excellent, and this means that we realistically cannot discriminate between the models we reported above. The main problem is that all models are based on a conjecture which gives a curve ζp as a function of a single free parameter, and that curve is able to fit the smooth observed behavior of ζp. Statistics cannot prove, just disprove. We can distinguish between the fractal model and multifractal models, but we cannot realistically distinguish among the various multifractal models.

7.5 A model for the departure from self-similarity

Besides the idea of self-similarity underlying the process of energy cascade in turbulence, a different point of view can be introduced. The idea is to characterize the behavior of the PDFs through the scaling laws of the parameters, which describe how the shape of the PDFs changes when going towards small scales. The model, originally introduced by Castaing et al. (2001Jump To The Next Citation Point), is based on a multiplicative process describing the cascade. In its simplest form the model can be introduced by saying that PDFs of increments δZ ± ℓ, at a given scale, are made as a sum of Gaussian distributions with different widths ± 21∕2 σ = ⟨(δZℓ )⟩. The distribution of widths is given by Gλ (σ ), namely

∫ ∞ ( ± 2) P(δZ ±) = 1-- G λ(σ)exp − (δZℓ-)- dσ-. (65 ) ℓ 2π 0 2 σ2 σ
In a purely self-similar situation, where the energy cascade generates only a trivial variation of σ with scales, the width of the distribution Gλ (σ ) is zero and, invariably, we recover a Gaussian distribution for P (δZℓ±). On the contrary, when the cascade is not strictly self-similar, the width of G λ(σ) is different from zero and the scaling behavior of the width λ2 of G λ(σ) can be used to characterize intermittency.

7.6 Intermittency properties recovered via a shell model

Shell models have remarkable properties which closely resemble those typical of MHD phenomena (Gloaguen et al., 1985Jump To The Next Citation Point; Biskamp, 1994Jump To The Next Citation Point; Giuliani and Carbone, 1998Jump To The Next Citation Point; Plunian et al., 2012Jump To The Next Citation Point). However, the presence of a constant forcing term always induces a dynamical alignment, unless the model is forced appropriately, which invariably brings the system towards a state in which velocity and magnetic fields are strongly correlated, that is, where Zn± ⁄= 0 and Z ∓n = 0. When we want to compare statistical properties of turbulence described by MHD shell models with solar wind observations, this term should be avoided. It is possible to replace the constant forcing term by an exponentially time-correlated Gaussian random forcing which is able to destabilize the Alfvénic fixed point of the model (Giuliani and Carbone, 1998Jump To The Next Citation Point), thus assuring the energy cascade. The forcing is obtained by solving the following Langevin equation:

dFn- = − Fn- + μ(t), (66 ) dt τ
where μ(t) is a Gaussian stochastic process δ-correlated in time ⟨μ (t)μ(t′)⟩ = 2D δ(t′ − t). This kind of forcing will be used to investigate statistical properties.
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Figure 87: We show the kinetic energy spectrum |un (t)|2 as a function of log2 kn for the MHD shell model. The full line refer to the Kolmogorov spectrum k −n2∕3.
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Figure 88: We show the magnetic energy spectrum |bn(t)|2 as a function of log2kn for the MHD shell model. The full line refer to the Kolmogorov spectrum k −n2∕3.

A statistically stationary state is reached by the system Gloaguen et al. (1985); Biskamp (1994Jump To The Next Citation Point); Giuliani and Carbone (1998); Plunian et al. (2012), with a well defined inertial range, say a region where Equation (48View Equation) is verified. Spectra for both the velocity |un(t)|2 and magnetic |bn(t)|2 variables, as a function of kn, obtained in the stationary state using the GOY MHD shell model, are shown in Figures 87View Image and 88View Image. Fluctuations are averaged over time. The Kolmogorov spectrum is also reported as a solid line. It is worthwhile to remark that, by adding a random term like iknB0 (t)Z±n to a little modified version of the MHD shell models (B0 is a random function with some statistical characteristics), a Kraichnan spectrum, say E (kn) ∼ k−n3∕2, where E (k ) n is the total energy, can be recovered (Biskamp, 1994; Hattori and Ishizawa, 2001). The term added to the model could represent the effect of the occurrence of a large-scale magnetic field.

Intermittency in the shell model is due to the time behavior of shell variables. It has been shown (Okkels, 1997Jump To The Next Citation Point) that the evolution of GOY model consists of short bursts traveling through the shells and long period of oscillations before the next burst arises. In Figures 89View Image and 90View Image we report the time evolution of the real part of both velocity variables un(t) and magnetic variables bn (t) at three different shells. It can be seen that, while at smaller kn variables seems to be Gaussian, at larger kn variables present very sharp fluctuations in between very low fluctuations.

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Figure 89: Time behavior of the real part of velocity variable u (t) n at three different shells n, as indicated in the different panels.
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Figure 90: Time behavior of the real part of magnetic variable b (t) n at three different shells n, as indicated in the different panels.

The time behavior of variables at different shells changes the statistics of fluctuations. In Figure 91View Image we report the probability distribution functions P (δun) and P (δBn ), for different shells n, of normalized variables

δu = ∘ℜe-(un)-- and δB = ∘ℜe-(bn)-, n ⟨|un |2⟩ n ⟨|bn|2⟩
where ℜe indicates that we take the real part of un and bn. Typically we see that PDFs look differently at different shells: At small kn fluctuations are quite Gaussian distributed, while at large kn they tend to become increasingly non-Gaussian, by developing fat tails. Rare fluctuations have a probability of occurrence larger than a Gaussian distribution. This is the typical behavior of intermittency as observed in usual fluid flows and described in previous sections.
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Figure 91: In the first three panels we report PDFs of both velocity (left column) and magnetic (right column) shell variables, at three different shells ℓn. The bottom panels refer to probability distribution functions of waiting times between intermittent structures at the shell n = 12 for the corresponding velocity and magnetic variables.

The same phenomenon gives rise to the departure of scaling laws of structure functions from a Kolmogorov scaling. Within the framework of the shell model the analogous of structure functions are defined as

± ⟨|un|p⟩ ∼ k−nξp; ⟨|bn|p⟩ ∼ k−nηp; ⟨|Z ±n |p⟩ ∼ k−nξp.
For MHD turbulence it is also useful to report mixed correlators of the flux variables, i.e.,
± p∕3 −β±p ⟨[Tn ] ⟩ ∼ kn .

Scaling exponents have been determined from a least square fit in the inertial range 3 ≤ n ≤ 12. The values of these exponents are reported in Table 4. It is interesting to notice that, while scaling exponents for velocity are the same as those found in the solar wind, scaling exponents for the magnetic field found in the solar wind reveal a more intermittent character. Moreover, we notice that velocity, magnetic and Elsässer variables are more intermittent than the mixed correlators and we think that this could be due to the cancelation effects among the different terms defining the mixed correlators.

Table 4: Scaling exponents for velocity and magnetic variables, Elsässer variables, and fluxes. Errors on β ± p are about one order of magnitude smaller than the errors shown.
p ζp ηp ξ+p ξ−p β+p βp−
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 91View Image, at a given shell n = 12. The same statistical law is observed for the bursts of total dissipation (Boffetta et al., 1999).

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