MAGNETIC VARIANCES AND PITCH-ANGLE SCATTERING TIMES UPSTREAM OF INTERPLANETARY SHOCKS

Understanding the transport of energetic particles and cosmic rays in the presence of magnetic turbulence is of great importance in order to identify the source and the acceleration mechanisms of energetic particles. Usually, diffusive regimes in which the mean square displacement grows linearly with time have been considered both parallel and perpendicular to the mean magnetic field (Jokipii 1966; Jokipii & Parker 1969; Matthaeus et al. 2003). However, nondiffusive transport regimes, characterized by a mean square displacement growing nonlinearly with time, 〈Δx²〉∝t^β, have also been reported (Klafter et al. 1996; Metzler & Klafter 2000, 2004). For astrophysical plasmas, these anomalous regimes have been found by numerical simulations of particle transport in a model magnetic turbulent field, and usually encompass subdiffusion, β < 1, for transport perpendicular to the mean magnetic field (Qin et al. 2002; Zimbardo et al. 2006; Shalchi & Kourakis 2007; Tautz 2010) and superdiffusion, β > 1, for transport parallel to the mean magnetic field (Zimbardo et al. 2006; Shalchi & Kourakis 2007; Pommois et al. 2007; Tautz 2010). The obtained transport regimes depend on several parameters, such as the turbulence level, the turbulence anisotropy, and the particle gyroradius with respect to the turbulence typical wavelengths (Pommois et al. 2007; Gkioulidou et al. 2007; Shalchi 2010; Hauff 2010; Zimbardo et al. 2012), so that a variety of different transport regimes can be expected for the propagation of energetic particles.

Recently, Perri & Zimbardo (2007, 2008) have found by analyzing energetic particle profiles measured by Ulysses that the electron transport upstream of corotating interaction region (CIR) shocks in the solar wind can be superdiffusive, while ion transport is found to be superdiffusive upstream of the termination shock crossing by Voyager 2 (Perri & Zimbardo 2009a). Very recently, superdiffusive transport for ions has been reported by Sugiyama & Shiota (2011) analyzing ACE data upstream of a coronal mass ejection (CME) driven interplanetary shock. It is also worth noticing that early indications of electron anomalous diffusion were given by the analysis of solar energetic electron profiles carried out by Lin (1974), where it is pointed out that transport regimes intermediate between normal diffusion and scatter-free propagation can be inferred from the analysis of impulsive nonrelativistic electron events. Anomalously large parallel mean-free paths were also indicated by Reames (1999) in connection with solar energetic particle events. Furthermore, the observation of the strahl component in the solar wind electron distribution functions also suggests that pitch-angle scattering is negligible (Vinas et al. 2010). These findings, both numerical and based on data analysis, call for a better understanding of the physics of superdiffusion for energetic particles in space plasmas (Giacalone 2011).

The technique developed by Perri & Zimbardo (2007, 2008) allows us to obtain information on the transport regime by analyzing the energetic particle flux profile J(x) upstream of a planar shock. For a normal diffusive regime, assuming a constant diffusion coefficient, this profile corresponds to an exponential decay, J(x) ∼ exp (− Vx/D) (e.g., Lee & Fisk 1982; Drury 1983; Lee 2005), where x is the distance upstream of the shock, V is the flow speed, and D is the diffusion coefficient along x. A fit of the observed profile as a function of x and V allows us to determine D. On the other hand, in the case of superdiffusion, the propagator of the transport process has power-law tails, P(x, t)∝|x|^−μ with 2 < μ < 3 for large |x|, at variance with the Gaussian propagator of normal diffusion, and it can be shown that the energetic particle flux upstream of the shock is an inverse power law, J(x) ∼ x^−γ with γ = μ − 2, for large enough x (Perri & Zimbardo 2007, 2008). A fit of the observed profile yields γ, which allows us to obtain the anomalous diffusion exponent as β = 4 − μ = 2 − γ.

In the shock events considered by Perri & Zimbardo (2007, 2008, 2009a, 2009b), energetic particle profiles are clearly better fitted by a power law than by an exponential profile. On the other hand, in the case of normal diffusion with a spatially varying diffusion coefficient, as envisaged for instance by Bell (1978), Giacalone (2004), and Lee (2005), the steepness of the exponential decay will decrease with the increase of the distance x from the shock, and this may resemble a power-law profile. Due to the substantial and unavoidable fluctuations of all quantities measured in the solar wind, it could be difficult to distinguish between a power-law profile (corresponding to superdiffusion) and a modified exponential profile with a spatially varying diffusion coefficient (corresponding to normal diffusion) by fitting the data. However, it is well known that parallel transport depends on the pitch-angle scattering rate, which depends on the turbulence energy at the resonant wavelengths (Jokipii 1966; Kennel & Petschek 1966). Therefore, it is possible to know whether the parallel transport properties (for instance, like the diffusion coefficient) vary with the distance from the shock by analyzing the magnetic field fluctuations upstream of the considered shocks. In this connection, we note that in the numerical simulation by Giacalone (2004), the normalized magnetic field variance upstream of the shock decreases by more than two orders of magnitude when going from the vicinity of the shock to the most upstream position reached by the shock-accelerated particles. In that case, the diffusion coefficient varies with x, and the nonexponential decay of energetic particles obtained in the numerical simulation can be explained within the framework of normal diffusion. Conversely, here we show that the observed magnetic field variances, upstream of those shocks for which superdiffusion has been reported, do not systematically vary with the shock distance, so that no decreasing trend of the magnetic variance is observed. Therefore, we argue that the observed power-law profiles of energetic particle fluxes represent actual superdiffusion. Furthermore, we analyze the probability distributions of the magnetic field variances and of the corresponding pitch-angle scattering times: it is shown for the first time that the pitch-angle scattering times have a broad distribution with a power-law tail, and this finding is discussed in connection with the statistical models of superdiffusion, which are based on the Lévy random walk.

Energetic particles observed upstream of interplanetary shocks are propagating both along and perpendicular to the average magnetic field, B₀, so that the mean square displacement upstream of the shock can be described as (e.g., Zank et al. 2006)

$$\begin{equation} \langle \Delta x^2 \rangle = \big\langle \Delta x_{\perp }^2 \big\rangle \sin ^2 \theta _{Bn} + \big\langle \Delta x_{\parallel }^2 \big\rangle \cos ^2 \theta _{Bn}, \end{equation} \tag{ 1 }$$

where ⊥ and ∥ refer to the direction with respect to the average magnetic field, and θ_Bn is the angle between the normal to the shock and B₀. Except when θ_Bn ≃ 90°, the contribution of parallel transport prevails, because even in the case of normal diffusion parallel transport is much faster than perpendicular transport. For instance, numerical simulations by Giacalone & Jokipii (1999) estimate D_∥/D_⊥ ∼ 10–100 for typical solar wind parameters, while Pommois et al. (2007) obtain D_∥/D_⊥ ∼ 100–1000, depending on the simulation parameters. Here, D_∥ (D_⊥) denotes the particle diffusion coefficient parallel (perpendicular) to B₀. For the considered shocks, the upstream angle between the magnetic field and the normal to the shock directions θ_Bn was always smaller than 70° (Balogh et al. 1995). Therefore, in the following discussion we will consider mainly parallel transport, this being the dominant contribution to Equation (1).

Parallel transport, further, depends on the rate of pitch-angle scattering, which causes random changes in the particle parallel velocity v_∥ = vcos α, where v is the particle speed and α is the pitch angle. The rate of pitch-angle scattering D^±_αα depends on the magnetic turbulence power in resonance with the particle gyromotion (Jokipii 1966; Kennel & Petschek 1966), and, within factors of the order of one, can be expressed as (Crooker et al. 1999)

$$\begin{equation} D_{\alpha \alpha }^{\pm } = \left(B_w \over B_0 \right)^2 \Omega _{\pm }, \end{equation} \tag{ 2 }$$

where plus or minus stands for ions or electrons, B_w is the magnetic field strength of the waves near resonance with the particles (Kennel & Petschek 1966), and Ω_± is the particle gyrofrequency times 2π. In practice, (B_w/B₀)² can be estimated by computing the magnetic field normalized variance (σ/B₀)² (see the text below) at the resonant wavelengths. Therefore, in this section, we concentrate on the spatial variations of the normalized variances, at a timescale close to resonance, as a function of the distance from the shock.

2.1. Data Analysis

The events analyzed in this work refer to those studied by Perri & Zimbardo (2008, 2009b) and for which superdiffusion has been observed for energetic electrons accelerated at CIRs and detected by the Ulysses spacecraft. A CIR forms when the fast solar wind coming from the coronal holes overtakes the slow solar wind; in such a case, an interaction region develops, where the two kinds of winds are separated by the so-called stream interface, which usually moves supersonically with respect to both the fast wind and the slow wind. A forward shock then forms in the slow wind, and a reverse shock (moving toward the Sun in the frame of the stream interface) in the fast wind. For the three CIR events of 1992 October 11, 1993 January 22, and 1993 May 10, Figures 1, 2, and 3, respectively, show the following quantities from top to bottom: the hourly resolution radial solar wind speed from the SWOOPS experiment (Bame et al. 1992), the magnetic field strength (data in the radial-tangential-normal reference frame from the MAG instrument; Balogh et al. 1992), the electron fluxes measured by LEFS 60/HI-SCALE in the energy channels 42–65 keV, 65–112 keV, 112–178 keV, and 178–290 keV (Lanzerotti et al. 1992), the 1 minute resolution total magnetic field normalized variance (black line), defined as σ²/B²₀ = ∑_{i = 1, 2, 3}σ²_i/B₀², where the index i runs over the magnetic field components (see below), along with the hourly averaged σ²/B²₀ (red line), and the hourly resolution azimuthal angle ϕ = arctan(B_T/B_R) defining the magnetic sectors, along with ψ which indicates the angle between the magnetic field and the radial direction. The CIR shock is indicated by the vertical dashed blue line, and the letters "D" and "U" in the top panels in each Figure stand for downstream and upstream, respectively. Note that the CIR shocks reported in Figures 1, 2, and 3 are reverse shocks, therefore the satellite detects the downstream, shocked, region first and subsequently the upstream, unshocked, plasma. For each event, data are presented for a time span consistent with that used by Perri & Zimbardo (2008, 2009b) to perform the fit of the energetic particle profiles. Those fits have allowed us to determine the superdiffusive transport of 42–290 keV electrons with the following exponents β for the time growth of the mean square displacement: β = 1.44–1.7 for the event of 1992 October 11, β = 1.08–1.19 for the event of 1993 January 22, and β = 1.15–1.38 for the event of 1993 May 10.

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For each magnetic field component B_i, the variance is defined as σ²_i = 〈(B_i − 〈B_i〉)²〉_T with T being the timescale for the average computation. In the present analysis, we consider T = 60 s; however, we point out that the timescale corresponding to electron resonance, i.e., to the electron gyroradius, for the energy channels here considered is somewhat shorter than one minute. For instance, considering 42–290 keV electrons in a field of 0.5 nT as typical of the region upstream of the shocks (see Figures 1, 2, and 3), we have an electron gyroradius ρ_e ≃ 1400–4100 km. Then, the timescale for spatial irregularities having a scale of 2ρ_e, in resonance with the energetic electrons, to be convected past the spacecraft in a fast wind flow with V_sw ≃ 750 km s⁻¹ is T = 3.7–10.9 s. However, in order to have a good statistics in the computation of the average magnetic field B₀ and of the variance σ², we have used the 1 s resolution MAG data over 60 s time windows (see also Crooker et al. 1999). Also, a cutoff for σ²/B²₀ < 10⁻⁵ has been used, in order to enforce the accuracy of data points (see the discussion in Section 3).

The plots displayed in Figures 1, 2, and 3 clearly show that the normalized variances, both at 60 s resolution and for 1 hr averages, do not exhibit a long-time trend; in particular, there is no decrease in magnitude as the distance from the shock increases. This is in contrast with the simulation of particle acceleration at shocks by Giacalone (2004), where the transverse-normalized variance upstream of the shock steadily decreases with the increase of the shock distance (see Figure 1 in Giacalone 2004). It is worth remarking that a possible change in the magnetic field-normalized variance could arise very close to the shock front, namely, for a few hours from the shock ramp; this is particularly evident in Figure 2, where a higher level in σ²/B²₀ can be seen after the shock on day 22. As discussed by Crooker et al. (1999), this increase of magnetic field variance can be explained by locally generated waves, possibly due to particle streaming, and the fluctuation level decreases to normal upstream values in about 12 hr. Indeed, after roughly 10–20 hr from the shock, σ²/B²₀ seems to oscillate "constantly" around an average value. Therefore, we can say that the magnetic fluctuations upstream of these shocks are quasi-stationary and then statistically homogeneous.

This evidence is crucial for assessing the behavior of the rate of pitch-angle scattering D^±_αα, according to Equation (2). Furthermore, note that in Perri & Zimbardo (2007, 2008, 2009b) the decays of particles' profiles as a function of the shock distance have been fitted from about 10–20 hr to ∼100 hr from the shock front in order to avoid possible strong variations in the magnetic field fluctuations, and thus in the particle transport, in the vicinity of the CIR. We also note that for the CME-driven shock considered by Sugiyama & Shiota (2011), for which ion superdiffusion was found, the magnetic field variance upstream of the shock averaged over four minutes is given in the bottom panel of their Figure 1: it is easily seen that in that case, too, no long-term trend in the variance is found. We remark that a constant average level in the normalized total magnetic field variance implies a constancy, as a function of the distance from the shock, of the diffusion coefficient. Therefore, a first conclusion from our analysis is that the power-law decay observed in the profiles of electrons accelerated at CIRs, as well as those of ions at CMEs, should not be related to a diffusive process with a spatially varying diffusion coefficient; superdiffusive transport characterized by power-law propagators yields a power-law decay in a statistically homogeneous medium, so that superdiffusion seems to be a more appropriate explanation for those observations.

Further insight into the properties of parallel transport can be obtained from the values of pitch-angle scattering times, defined as the inverse of the pitch-angle diffusion coefficient, τ = (D^±_αα)⁻¹. We have performed a statistical analysis, computing the probability density functions (PDFs) of τ. Figure 4 shows PDFs of τ for the three CIR events displayed in Figures 1, 2, and 3, and for a period of "quiet" fast solar wind observed by Ulysses on 1995 January 12–15, during the declining phase of solar cycle 22 at ∼1.5 AU (Perri & Balogh 2010); the latter period has been considered for comparison. The main panels report the PDFs of the scattering times (black stars-solid lines) in log–log scale computed within time intervals starting at ∼10 hr from the shock front, during which the normalized magnetic field variances remain statistically constant, as shown above. Gaussian distributions fitting the data (red solid lines) are also plotted for comparison. As can be seen, while the "core" of the distribution is well reproduced by a Gaussian, the tail of the PDFs is a power law in all of the three CIR events, and is fitted by a power law decaying as PDF(τ)∝τ^−2.4–τ^−3.8 (blue solid lines in Figure 4).

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We point out that the scattering time is obtained as the inverse of the pitch-angle scattering rate, which essentially depends on σ²/B²₀ (see Equation (2)). The computation of σ² is related to the fluctuations of the magnetic field components, i.e., δB_i = B_i − 〈B_i〉. Note that the sensitivity of the MAG instrument is 10⁻¹² T (see Balogh et al. 1992), while the typical magnitude of B₀ upstream of the shock is 0.5 × 10⁻⁹ T, see Figures 1–3. Therefore, for the present data sets, it is not reliable to determine relative magnetic fluctuations δB_i/B₀ smaller than 10⁻³, although they may be present, since they are below the sensitivity threshold. To avoid inaccurate values of σ² we have filtered the variance data under consideration, discarding data points for which σ²/B²₀ < 10⁻⁵. As a consequence, very small values of variance are undersampled, and so are very large values of the scattering time. This may be the origin of the knee in the power-law part of the PDFs for τ ≫ 10 s, rather evident for the events of 1992 October 11 and 1993 May 10. For this reason, we made the fit only on the upper part of the power-law tail for these two events (see Figure 4). We note that the problem of undersampling long scattering times is only minor for the analysis of the fast wind at 1.5 AU (lower right panel of Figure 4). Indeed, in that case B₀ ∼ 2 nT, so that smaller normalized magnetic variances and larger scattering times can be safely appreciated. Correspondingly, a more extended power-law tail with PDF(τ)∝τ^−2.5 is obtained.

The power-law distribution for large values of the scattering times can be associated with the burst-like time behavior which they exhibit within the analyzed intervals (see the plots in the insets of Figure 4). The abrupt and rare jumps in the τ values tend to deviate the distribution functions from being a standard Gaussian to a power law. A similar result has also been found in the pure fast solar wind data (bottom right panel): actually, in this case, only a few large amplitude bursts have been observed; this is caused by the fact that in the "quasi-stationary" fast wind, the magnetic field variances at short timescales are characterized by a lower degree of burstiness as compared to the slow wind (Perri & Balogh 2010).

Note that the distributions in Figure 4 are obtained from a statistical analysis of the scattering times in the asymptotic regimes, i.e., within time intervals during which both the electron profiles (Perri & Zimbardo 2007, 2008, 2009b) and the normalized magnetic field variances do not change anymore (as already indicated, this happens at about 10 hr from the shock event). It is worth mentioning that the obtained PDFs(τ) come from the scattering times of an ensemble of particles and not from the scattering times of a single particle (see Equation (4) below); therefore, they are the macroscopic result of the superposition of all the distributions of particle scattering times under the hypothesis of stationarity for those distributions during the given time interval.

The power-law PDFs of the scattering times imply that the probability of occurrence of very long τ and long range correlations, which means very low pitch-angle scattering, does not go exponentially to zero, as in the case of a Gaussian distribution, but is much higher than the one expected in the case of a normal diffusive process. We note that in the classical paper by Jokipii (1966), the parallel diffusion coefficient D_∥ is given as an integral over the turbulence power spectrum at resonance (Equation (28) in Jokipii 1966), and it is explained that that expression "assumes, of course, that there is enough scattering that the integrals are defined." If there is not enough scattering, then the integrals are diverging, and the divergence of the diffusion coefficient, i.e., of the velocity autocorrelation function, is precisely what is required to have superdiffusion (e.g., Hauff 2010; Zimbardo et al. 2012).

It is interesting to compare the obtained PDFs of scattering times with the free path probability distribution ψ(ℓ, τ) which characterizes a Lévy random walk in the case of superdiffusive propagation in the direction parallel to the magnetic field, i.e.,

$$\begin{equation} \psi (\ell,\tau) = A |\ell |^{-\mu } \delta (\ell - v_{\parallel } \tau) \,, \end{equation} \tag{ 3 }$$

valid for large |ℓ| (Klafter et al. 1987; Metzler & Klafter 2004). Here, A is a normalization constant, ℓ is the length of the free path (either positive or negative), v_∥ is the particle parallel velocity, and τ is the free path time duration, which for the parallel motion under consideration we identify with the pitch-angle scattering time. For an exponent μ of the free path length such that 2 < μ < 3, this probability distribution leads to superdiffusive transport with 〈Δx²〉∝t^β and with β = 4 − μ (Klafter et al. 1987, 1996). The probability of free path times, for large |ℓ| and large τ, can be obtained by integrating ψ(ℓ, τ) over the free path lengths, which gives

$$\begin{equation} \psi (\tau) = A |v_{\parallel }|^{-\mu } \tau ^{-\mu }. \end{equation} \tag{ 4 }$$

For a given particle energy, v_∥ ∼ v = const, so that a power-law probability distribution is obtained for large τ. It is very intriguing to note that the power-law exponents of the PDFs in Figure 4 are 2.4, 2.5, and 3.8, that is, they are often within the interval 2 < μ < 3 corresponding to superdiffusion; therefore, the power-law tails of the distribution of scattering times τ can give a physical explanation of particle superdiffusion in solar wind turbulence, according to the model of the Lévy random walk (Geisel 1995; Klafter et al. 1996; Zimbardo et al. 2012). However, while the PDFs(τ) in Figure 4 are derived from measurements in the spacecraft frame (at a given spacecraft position), ψ(τ) in Equation (4) refers to the distribution of scattering times along the particle random walk. Therefore, the two probability distributions are not the same, and further study is needed to explore more fully their mutual relationship.

An important concept related to the observed power-law PDFs is that there is not a typical value of the fluctuating quantity, i.e., of the scattering times, due to its broad distribution; nevertheless, here we consider the limiting values of the PDF power-law range of scattering times in order to discuss the corresponding free path lengths. From Figure 4, we can see that "limiting" scattering times range from the shoulder of the distributions, τ ≃ 10 s, to the end of the power-law range, τ ≃ 200 s. We now consider that the normalized variances have been computed on a timescale of T = 60 s in order to have a reliable statistics, as explained in Section 2, while the electron resonant timescale could be estimated to be in the T = 3.7–10.9 s range. Then, we can extrapolate the level of magnetic turbulence to the actual resonant electron scale by using the Kolmogorov spectral index in the inertial range, δB²_k∝k^−5/3, with k being the wave number. Let us assume, for reference, an electron resonant timescale of 6 s (i.e., ten times smaller than that used to compute σ²), so that the normalized variance level at this timescale is expected to be 10^5/3 ≃ 46.4 times smaller than that displayed in Figures 1, 2, and 3. Then, the limiting scattering times considered above would range from τ ≃ 460 s to τ ≃ 9300 s. If these times are multiplied by the speed of a 100 keV electron, the corresponding electron free path lengths would range from about 0.5 AU to 10 AU, that is, they are much larger than the so-called Palmer consensus (e.g., Reames 1999). Although an electron will not be sampling the same low level of fluctuations for such long distances, we can say that these estimates lend support to both superdiffusive and scatter-free propagation.

The scenario emerging from the present analysis is consistent with the occurrence of an anomalous, superdiffusive, process for energetic electrons accelerated at CIR shocks. The analysis of a quiet fast solar wind stream (bottom right panel in Figure 4) demonstrates that a broad distribution of scattering times decaying as a power law, as computed from Equation (2), is not uncommon throughout the interplanetary space. To be able to infer the particle transport properties, it would be interesting to carry out the analysis of magnetic field variances and of pitch-angle scattering times also in other space environments. For instance, the magnetic fluctuation properties at small scales change significantly when going from the "unperturbed" solar wind to the Earth's foreshock and then to the region beyond the Earth's bow shock, the magnetosheath (Perri et al. 2009), so that an investigation of these regions is in order.

The analysis presented in this paper has pointed out two important properties of the magnetic field fluctuations upstream of CIR shocks at the scales close to that of energetic electrons' resonance; these properties give support to the superdiffusive scenario envisaged by Perri & Zimbardo (2007, 2008, 2009a) in studying the upstream time profiles of particles accelerated at interplanetary shocks. In particular, average constant values of the normalized total magnetic field variances, showing no long distance trends, indicate that the interplanetary medium at few hours upstream of CIRs is statistically homogeneous. As a consequence, the diffusion coefficient, which is directly related to the normalized variance according to Equation (2), is not spatially varying and the inferred power-law profiles of energetic particles are rather indicative of a superdiffusive transport, which predicts power-law profiles even in a homogeneous medium.

In addition, the rate of particle pitch-angle scattering D^±_αα, which depends on the magnetic turbulence power in resonance with the particle gyromotion, has also been computed within the time intervals during which the normalized variance is roughly constant, and then the statistical behavior of the pitch-angle scattering times, i.e., τ = (D^±_αα)⁻¹, has been studied. In all the events considered, τ is very burst like, characterized by abrupt changes in its values over timescales of ∼ few minutes. This is reflected in the PDFs of the scattering times that are characterized by power-law tails (see Figure 4). This means that on average the process of pitch-angle diffusion for particles accelerated at CIRs deviates from a normal one and the scattering times have a non-negligible probability to assume very large values. In other words, the correlations in the particle parallel velocities do not vanish exponentially to zero but can exhibit a slower decay with respect to a Gaussian, standard transport process. This confirms the possibility of superdiffusion for energetic electrons, also in the light of the theoretical descriptions of superdiffusion based on the Lévy random walk. In particular, the slopes of the power-law tails of the PDFs of scattering times are consistent with those expected for superdiffusion. It is further worth mentioning that a coronal hole fast wind stream detected by Ulysses also shows a PDF(τ) decaying as a power law: this is due to the time evolution of the magnetic field variances, which also tend to fluctuate with large amplitude jumps, within very short timescales (Perri & Balogh 2010) in solar wind streams far from compressions. Therefore, the power-law distribution of scattering times could be a common property of the fast solar wind, and this should be taken into account in studying particle propagation throughout the heliosphere.

Finally, we note that understanding the physical origin of superdiffusion starting from the properties of magnetic fluctuations is important also to be able to predict the energetic particle transport properties for those systems for which in situ measurements are not possible, but for which turbulence features can be studied by methods like interstellar scintillation (e.g., Spangler 1999). Indeed, in a recent work we have shown that the hard spectral indices inferred for 1–10 GeV electrons at the Crab Nebula can be explained by shock acceleration if superdiffusive propagation is assumed (Perri & Zimbardo 2012). It is then important to understand the small-scale magnetic turbulence properties for pulsar wind nebulae, too.

This work was partially supported by the Italian Space Agency, contract ASI No. I/015/07/0 "Esplorazione del Sistema Solare," and by the Marie Curie "Geoplasmas" project No. 269198 of the European Union FP7 Program. S.P.'s research was supported by "Borsa Post-doc POR Calabria FSE 2007/2013 Asse IV Capitale Umano-Obiettivo Operativo M.2."