Intermittency of magnetic field turbulence

J. Plasma Physics (2015), vol. 81, 395810401 doi:10.1017/S0022377815000409

c Cambridge University Press 2015

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Intermittency of magnetic field turbulence: Astrophysical applications of in-situ observations Lev M. Zelenyi1 , Andrei M. Bykov2 Yury A. Uvarov2 and Anton V. Artemyev1 † 2

1 Space Research Institute (IKI), 117997, 84/32 Profsoyuznaya Str, Moscow, Russia A.F.Ioffe Physical-Technical Institute, St Petersburg 194021, also St Petersburg State Polytechnical University, Russia

(Received 11 October 2014; revised 23 March 2015; accepted 23 March 2015; first published online 22 April 2015)

We briefly review some aspects of magnetic turbulence intermittency observed in space plasmas. Deviation of statistical characteristics of a system (e.g. its high statistical momenta) from the Gaussian can manifest itself as domination of rare large intensity peaks often associated with the intermittency in the system’s dynamics. Thirty years ago, Zeldovich stressed the importance of the non-Gaussian appearance of the sharp values of vector and scalar physical parameters in random media as a factor of magnetic field amplification in cosmic structures. Magnetic turbulence is governing the behavior of collisionless plasmas in space and especially the physics of shocks and magnetic reconnections. Clear evidence of intermittent magnetic turbulence was found in recent in-situ spacecraft measurements of magnetic fields in the near-Earth and interplanetary plasma environments. We discuss the potentially promising approaches of incorporating the knowledge gained from spacecraft in-situ measurements into modern models describing plasma dynamics and radiation in various astrophysical systems. As an example, we discuss supernova remnants (SNRs) which are known to be the sources of energy, momentum, chemical elements, and high-energy cosmic rays (CRs) in galaxies. Supernova shocks accelerate charged particles to very high energies and may strongly amplify turbulent magnetic fields via instabilities driven by CRs. Relativistic electrons accelerated in SNRs radiate polarized synchrotron emission in a broad range of frequencies spanning from the radio to gamma-rays. We discuss the effects of intermittency of magnetic turbulence on the images of polarized synchrotron X-ray emission of young SNRs and emission spectra of pulsar wind nebula.

1. Introduction One of the main processes in numerous space plasma systems is charged particle interaction with turbulent magnetic fields. This process is responsible for charged particle acceleration and plasma heating in planetary magnetospheres, solar corona, and other astrophysical plasma systems. Moreover, scattering of high-energy charged particles on magnetic field fluctuations results in synchrotron radiation. Observations of such radiation often appear the only tool for a remote diagnostic of astrophysical plasmas. Efficiency of charged particle interaction with the magnetic turbulence is determined by the properties of the magnetic field and, in particular, its power spectra. However, † Email address for correspondence: [email protected]

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L. M. Zelenyi et al.

even for the fixed power of turbulence, the interaction of magnetic field fluctuations with charged particles can have different character depending on spatial and temporal distributions of fluctuations. Roughly speaking, charged particles are scattered and accelerated by Gaussian fluctuations in a manner, quite different from that of fluctuations, which follow substantially non-Gaussian distributions. For non-Gaussian distributions with the pronounced non-exponential tails, the time-intervals of almost stationary fields are mixed with short periods of very intense fluctuations. That property of fluctuations is often called intermittency. The papers by Zeldovich et al. (1985, 1987) were among the first studies devoted to the important role, which the intermittency can play in the behavior of dynamical systems with random fields. Today, many investigations confirm that these properties of magnetic turbulence could be critical for the description of charged particle interaction with an ensemble of magnetic field fluctuations. Interpretation of fine structures in X-ray synchrotron emission observed from distant astrophysical sources (Bykov et al. 2008, 2012) and direct in-situ measurements of magnetic fields in the near-Earth plasma environment (Burlaga 1991; Marsch and Tu 1997) indicate that the intermittency of the magnetic turbulence is rather a widespread property typical for various plasma systems. Investigations of distant astrophysical plasma systems and the near-Earth environments complement each other due to the peculiarities of the corresponding experimental techniques. In-situ spacecraft observations provide us with unique information on distributions of plasma particles and magnetic fields. However, such measurements are principally localized in space. Moreover, due to natural limitations of single-spacecraft experiments, one cannot even separate the temporal and spatial properties of the turbulence. To overcome this very serious limitation, many space missions conducted measurements at multiple spacecrafts (ISEE 1 and 2 (Ogilvie et al. 1977), Interball 1 and 2 (Zelenyi et al. 1997), Cluster (Escoubet et al. 2001)). The future multiscale magnetospheric mission (MMS, see Sharma and Curtis (2005)) will also include four almost identical satellites. On the other hand, remote observations of emissions radiated by accelerated particles in astrophysical systems cannot provide us with direct information about the internal plasma processes but rather give a chance to take a look on their manifestations in a global context. Many important properties of the turbulence responsible for the acceleration of radiating particles can be derived from such remote observations. However, the adequacy of our interpretation of emission from remote astrophysical systems critically depends on our models of these systems. Thus, it seems to be promising and important to discuss the possible implications of the important properties (e.g. intermittency) of magnetic turbulence observed in the near-Earth environment for various models of magnetic turbulence in astrophysical systems. Various ways of implementation of this approach and emerging problems are described below. 2. Astrophysical plasma systems with intermittent turbulence Nature blessed us with a wide variety of astrophysical plasma systems of various parameter ranges such as their characteristic sizes, plasma and magnetic pressures, β ratios, hydrodynamic flow and Alfvenic sound speeds, power and spectral distributions of magnetohydrodynamic turbulence, and so on. The direct in-situ studies of such systems are possible now only in the Solar system case where we can use various spacecraft measurements. Details and results of some in-situ measurements are discussed in Sec. 3.

Magnetic field turbulence

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In other cases, only indirect studies are possible which are based on the interpretation of images and spectral properties of considered objects combined with the results of numerical simulations. Instead of plasma fluxes and magnetic field values measured in direct spacecraft observations, in indirect measurements radiative properties are registered such as: spatial and spectral intensity and polarization. These properties are shaped in the course of radiation propagation along the line of sight including, in general, absorption, scattering, and (re)emission processes. Different mechanisms of emission can dominate in different environments. The synchrotron emission is the most informative one for studying magnetic field structures and turbulence since its efficiency and polarization strongly depend on the direction and amplitude of the magnetic field. Other mechanisms such as thermal or nonthermal bremsstrahlung or line emission can give information about the structure of hydrodynamic flows and positions of shocks and contact discontinuities, but not on the properties of magnetic fields.

2.1. Supernova remnants as systems with strong magnetic field turbulence SNRs are among the most powerful astrophysical objects with a non-thermal radiation extending from radio to gamma-rays. The non-thermal radiation is a signature of highly non-equilibrium plasma with power-law tails in particle distribution extended to many decades and strongly amplified turbulent magnetic fields. There are a few SNRs located nearby on a distance of a few kpc, which are strong and large enough to be studied at excellent angular and energy resolution with modern ground-based and space-borne observatories . Observations are made in a wide energy range from radio to the X-rays and even γ -rays. X-ray imaging of SNRs with superb angular resolution (about one arcsecond) of Chandra X-ray Observatory revealed thin filaments, stripes, and dots of synchrotron emission (see e.g. Vink and Laming 2003; Uchiyama et al. 2007; Eriksen et al. 2011; Tananbaum et al. 2014). Some of SNR images (e.g. RXJ 713.7-3946, SN 1006) are known to be dominated by synchrotron emission or have a strong clearly identified synchrotron component in their spectra. A power-law emission spectrum in a broad energy range and a shell-like geometry are interpreted as synchrotron emission of energetic electrons accelerated via Fermi mechanism in the vicinity of a collisionless shock, which is formed due to the interaction of the expanding SNR envelope with the interstellar medium. Fermi acceleration requires the presence of developed magnetic field turbulence. This turbulence allows energetic particles to scatter and diffuse in the upstream and downstream media, cross the shock back and forth several times. The existence of magnetic turbulence is indirectly confirmed by the presence of small-scale structures on the images of some SNRs. In some cases, these structures are variable with a characteristic time scale of about a year. This was noticed by Uchiyama et al. (2007) in the case of SNR RXJ 713.7-3946. The authors associated the variability with fast cooling of X-ray-emitting electrons and suggested that a milliGauss-level magnetic fields are needed to explain the X-ray data on RXJ 1713.7-3946. Another likely interpretation of the variable structures in the source images is their origin due to the intermittency of the synchrotron Xray emission, which requires a substantially lower level of the stochastic magnetic field in the supernova shell (Bykov et al. 2008, 2009). To model the images and spectra of SNRs with turbulent magnetic fields, numerical simulations are required.

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2.2. Numerical modeling of turbulence. Simulated synchrotron images in random magnetic fields Source image is a 2D structure while the source itself is a 3D object. When we study images, we should take the projection effect into account. So numerical simulations are necessary to study the properties of turbulence and its effect on synchrotron images. Bykov et al. (2009) studied the influence of magnetic turbulence on SNR images. The algorithm of magnetic field construction was based on that described in Giacalone and Jokipii (1999), but contained some modifications. The turbulent field was emulated by summation of a large number of stochastic harmonics with random phases. The amplitudes of harmonics were chosen in order to obtain a stochastic magnetic field with the desired spectral properties. Projections of the simulated magnetic field satisfy a normal distribution. The inhomogeneity of the electron distribution function in the vicinity of the shock was also taken into account. The distribution function was calculated in a frame of a kinetic model of diffuse shock acceleration. The simulated turbulent magnetic fields were used for modeling of intensity and polarization maps synchrotron emission. The final images contain clearly identified fine structures: clumps, filaments, dots. For model parameters typical for SNR, dots and small clumps are variable on an annual time scale. The time scale of the clumps is determined by the intermittent magnetic structures of sizes below the energy containing scale of CR-driven magnetic turbulence (which is about 0.1 pc), see for details Bykov et al. (2008). These structures are a direct consequence of an intermittent character of synchrotron radiation appearing due to nonlinear dependence of intensity on magnetic field amplitude. This holds even in the case when projections of stochastic magnetic field have a normal probability distribution. It was also shown that there is a difference in images obtained in simulations with different shapes of turbulent power spectrum (Fig. 1). In the case of the power-law spectrum index of magnetic field fluctuations γ = −2, large-scale structures were formed in emission maps, while in the case of γ = −1 small-scale features also manifest themselves in the images. The turbulent spectra in young SNR shells are determined by the magnetic field growth due to plasma instabilities which is balanced by nonlinear cascading and magnetic

Figure 1. Synchrotron X-ray images of a supernova shell at 5 keV simulated in the model of diffusive shock acceleration with turbulent magnetic field amplification. The simulations with the power-law spectrum of magnetic field fluctuations index γ = −1 are shown in panel A. The left image in the panel is the intensity map, in the middle image the polarized intensity is shown while the degree of polarization is indicated by the color bar in the right image. Images simulated for the power-law index of magnetic fluctuations γ = −2 are shown in panel B. (The figure adopted from Bykov et al. 2009.)


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Figure 2. Synchrotron X-ray emission images of supernova shell simulated with the effects of magnetic fluctuation spectra produced by anisotropic CR-current-driven instability without the spectral cascade along the mean magnetic field. The left panel shows the synchrotron X-ray intensity at 5 keV. The degree of polarization of the X-ray emission is shown in the right panel (the color bar shows the degree of polarization). The synchrotron stripes apparent in the simulated images may explain the similar structures observed by Chandra X-ray Observatory in Tycho’s SNR. (The figure is adopted from Bykov et al. 2011.)

field dissipation. Efficient mechanisms of the fluctuating magnetic field amplification are due to CR-driven instabilities in the vicinity of strong forward shocks. Strong amplification of the fluctuating magnetic field is due to both the resonant and nonresonant CR-driven instabilities (see e.g. Bell 2004; Schure et al. 2012; Bykov et al. 2013). Multiple scattering of charged particles by turbulent magnetic fields in the vicinity of a collisionless shock wave results in efficient particle acceleration (see e.g. Blandford and Eichler 1987; Schure et al. 2012). The anisotropy of accelerated particle distribution and CR pressure gradient upstream of the shock leads to the excitation of CR-driven instabilities producing a wide spectrum of magnetic fluctuations which become the scattering agents for CRs (Schure et al. 2012). The spectral shape of the amplified magnetic field fluctuations depends strongly on the efficiency of the spectral transfer along the turbulent spectrum. Nonlinear Monte Carlo simulations of the diffusive shock acceleration with magnetic field amplification revealed that in the case of an efficient Kolmogorov-type cascade the spectrum of CR-driven turbulence is a smooth power law of index somewhat flatter than −5/3 (see Bykov et al. 2014). On the contrary, if the spectral cascade along the mean magnetic field is forbidden as it is expected to be the case in the anisotropic weak MHD turbulence, the spectrum of magnetic fluctuations may contain a number of prominent peaks in the wavenumber space. These peaks may result in clear observational manifestations in synchrotron X-ray images of supernova shells (Bykov et al. 2011). The corresponding stripe-like highly polarized structures are better resolved near the cut-off energy where the flux of synchrotron radiation is most sensitive to magnetic field variations. Simulated synchrotron images for the peaked spectra of magnetic turbulence are shown in Fig. 2. They resemble the stripe-like X-ray structures observed in Tycho’s SNR (Eriksen et al. 2011). In these simulations, the amplified magnetic field in the downstream region was considered as a superposition of linearly polarized modes with random phases satisfying a power-law spectral distribution with a few isolated narrow peaks in wave-vector space as obtained in the nonlinear model of diffusive shock acceleration with CR-driven instabilities (Bykov et al. 2011).

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Figure 3. Simulated spectra of synchrotron radiation produced by a distribution of relativistic electrons in stochastic magnetic fields of the same r.m.s. value but with different higher moments. Two different probability distributions of magnetic field fluctuations with the same mean value were considered: Gaussian PDF (solid curve) and exponential PDF (dashed curve). Both Gaussian dispersion and the characteristic field of exponential PDF are equal to the mean field magnitude. The synchrotron radiation spectrum of the electrons in regular homogeneous field is presented by dot dashed curve for comparison. (The picture is adopted from Bykov et al. 2012.)

2.2.1. Numerical modeling of turbulence. Spectra and intermittency. In the case of a medium with the homogeneous amplitude of magnetic field in a standard model of the diffusive shock acceleration with Bohm diffusion coefficient, the maximal achievable photon energy does not dependent on the strength of magnetic field. This is not the case for a stochastic magnetic field. For stochastic conditions, a cut-off energy of the electron distribution function depends on the squared average of the magnetic field induction (because it is determined by the average synchrotron losses), while the emission strongly depends on the local field strength, which can be much higher. That is why an emission spectrum appears harder in this case. This is illustrated in Fig. 3 where it is apparent that the intermittent character of the turbulent magnetic field with different probability distribution functions of magnetic fluctuations with the same r.m.s. value results in huge differences in the radiation fluxes in the spectral cut-off regime. Recent discovery of the giant GeV flares in the Crab nebula (Abdo et al. 2011; Tavani et al. 2011; B¨ uhler and Blandford 2014) revealed an order of magnitude flux enhancements of GeV photon emission in the synchrotron cut-off regime. The flares may be understood if a reconstruction of the probability distribution function of magnetic fluctuation occurs from time to time downstream of the termination surface of the relativistic pulsar wind. In a numerical simulation of magnetic field with summation of a large number of random harmonics, the projection of magnetic field (say, on x-axis, Bx ) according to the central limit theorem is distributed by the normal law with the asymptotic ∼ exp −Bx2 /σx2 (no mean magnetic field is assumed). The synchrotron emission along a line of sight will be dependent on the value of the magnetic field transversal to 2 the line of sight (i.e. B⊥ = Bx + By2 ). The square of this field will be distributed according to χ 2 distribution with a single free parameter and will have exponential asymptotic for high values. In the case of power-law electron energy distribution


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Figure 4. Left panel: numerical probability distribution of the x -axis projection of simulated magnetic field (black solid curve) fitted with normal distribution (red-dashed curve). Right panel: numerical probability distribution of the B⊥2 = Bx2 + By2 (black solid curve) fitted with χ 2 distribution with one free parameter (red-dashed curve).

∼E −n , the dependence of the synchrotron emission on magnetic field is ∼B⊥(n+1)/2 in the intermediate energy range where this emission has a power-law dependence on frequency. It is clear that in this case the synchrotron emission distribution differs from the normal distribution. This effect is even more pronounced near and above the cut-off spectrum energy where there is much more stronger dependence on magnetic field than power law and synchrotron emission seems to have highly intermittent pattern (Fig. 4) In this case, a small region with high magnetic field can dominate the total emission from the whole emitting system. We have demonstrated above a possible appearance of the intermittency phenomena in SNRs where highly non-equilibrium states of plasmas and electromagnetic fields produced due to extreme energy release are observed. The quantitative interpretation of the observational data require new theoretical models which are based on our understanding of turbulence in collisionless plasmas. In this respect, the studies of plasma turbulence in the near-Earth environment and heliosphere where the direct space measurements are available are extremely useful to construct adequate models of astrophysical objects and we turn to a brief discussion of the studies. 3. Spacecraft observations in the near-Earth environment Electromagnetic field measurements onboard the numerous missions in the nearEarth environment give us a unique opportunity for investigation of properties of magnetic turbulence in the plasma systems in many senses qualitatively similar to astrophysical systems. Using two dimensionless parameters (ratio of plasma thermal pressure and magnetic field pressure β; ratio of plasma bulk speed and Alfven speed MA ), one can distinguish three main plasma regimes accessible for direct spacecraft investigations. In the solar wind, the beta parameter usually is β 6 1, while the Alfven Mach number is MA > 1 (or even 1). Hot plasma of planetary magnetospheres at some distance from the planet (where the planet dipole field becomes weak enough) is characterized by large plasma thermal pressure β ∈ [0.1, 100] and weak sporadic plasma flows MA < 1. In the inner planetary magnetospheres (or in the radiation

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belts), plasma pressure is negligibly small in comparison with magnetic field pressure β 1, while plasma flows are almost absent, MA → 0. There is a substantial number of spacecraft observations of magnetic turbulence in systems of all the three types (see reviews Bruno and Carbone 2005; Zimbardo et al. 2010; Meredith et al. 2012; Agapitov et al. 2013; Alexandrova et al. 2013; Zelenyi et al. 2014). In this section, we consider properties of magnetic turbulence measured in systems of the first two types: solar wind and planetary magnetospheres. Each subsection below is devoted to a certain property of turbulence described above in the context of astrophysical models. 3.1. Turbulent spectra Already the first investigations of magnetic field fluctuations in the solar wind have shown that the spectra can be approximated by the power-law function P ∼ f −γ with γ ∈ [−1, −2] for the low-frequency part f ∈ [10−4 , 10−1 ] Hz (Coleman 1968; Bruno and Carbone 2005). In Fig. 5(a), we show an example of magnetic field spectrum measured with MARINER 2 spacecraft in 1962. The magnetometers onboard modern spacecraft missions measure magnetic field fluctuations with higher accuracy and up to relatively high frequencies f ∼ 1000 Hz (search coil experiments allow measuring magnetic field fluctuations up to several thousand Hz). The example of magnetic field spectrum measured in the solar wind near the Earth is shown in Fig. 5(b). One can notice the significant difference of the spectrum slopes around 1 Hz. This is an intermediate value between two characteristic frequencies: the proton gyrofrequency and the frequency corresponding to the Doppler-shifted wavemode with the wavelength of the order of proton Larmor radius fρi = VSW /ρi (VSW is the solar wind velocity). The position of this spectrum break depends on the distance from the Sun (Fig. 5(c)). The main drivers of magnetic field fluctuations in the solar wind are local plasma instabilities (fire-hose, ion cyclotron, and mirror modes, see Bale et al. 2009). However, formation of current sheets and magnetic reconnection can also be important for generation of magnetic turbulence with ion scales (Gosling 2012; Greco et al. 2012). Partially, lower-frequency fluctuations can be generated at the region of solar wind formation and then transported by solar wind flow. Evolution of such fluctuations with the distance from the Sun is described by Milovanov and Zelenyi (1994a,b). In contrast, in the hot plasma of planetary magnetospheres, the local plasma currents and magnetic reconnection play the dominant role in formation of magnetic field turbulence. In this case, the low-frequency part of the spectrum of magnetic field fluctuations corresponds to the slow (MHD) oscillations of current sheets, while the high-frequency part is generated by small-scale magnetic field structures originated from the current sheet destruction in the course of reconnection (Zelenyi et al. 2014). The example of such spectrum is shown in Fig. 6. One can notice that the spectrum shapes for the magnetosphere and solar wind are similar and a spectrum break appears in the same frequency range (compare Figs 6 and 5(b)). Two main conclusions can be drawn from the data presented in Figs 5 and 6 for the modeling of charged particle scattering by magnetic turbulence in astrophysical systems (e.g. emission from SNRs). First of all, as discussed above, the fine structure of the synchrotron emission strongly depends on magnetic field spectrum (see Fig. 1). The clear difference of modeling results for indices γ = −1 and γ = −2 requires some theoretical (or/and experimental) motivation to choose this parameter correctly. Spacecraft observations in the solar wind suggest that γ ∈ [−1.6, −1.8] is a reasonable value for the low-frequency part of the spectrum. Moreover, the power at larger


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Figure 5. Spectra of magnetic field fluctuations: (a) collected by MARINER 2 spacecraft (figure is adopted from Coleman 1968), (b) collected by Cluster spacecraft for 42 min time interval at the distance ∼1.2 × 105 km from the Earth (few hours downstream of an interplanetary shock) (figure is adopted from Alexandrova et al. 2009), (c) collected by MESSENGER, Wind, and Ulysses spacecraft at different distance from the Sun (figure is adopted from Bruno and Trenchi 2014). In panel (b), characteristic frequencies are shown: fcα is the cyclotron frequency, fλα = VSW /λα , fρα = VSW /ρα where α = i, e for ions and electrons, λα = c/ωpα is the particle inertial length (ωpα is a plasma frequency calculated with mass of particles of type α), ρα is the particle gyroradius, VSW is solar wind velocity.

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Figure 6. Spectra of magnetic field fluctuations collected by Cluster spacecraft in the vicinity of the reconnection region (figure is adopted from Huanget al. 2012). Ranges of characteristic frequencies are shown: fλ = Vplasma /λp , fρ = Vplasma /ρp where λp = c/ωp is the proton inertial length, ρp is the proton gyroradius, Vplasma is plasma bulk velocity.

frequencies, i.e. for kinetic fluctuations, drops so strongly that these fluctuations seem to be unable to influence scattering of high-energy particles. The second potentially important conclusion is the universality of the low-frequency spectrum both for the cold solar wind and hot magnetospheric plasmas. The role of various plasma discontinuities and current sheets in formation of magnetic turbulence in astrophysical systems (e.g. SNRs) cannot be clarified at the present level of understanding. However, in-situ spacecraft observations demonstrate that such finely structured turbulence with many elements still relates to a simple power-law spectrum at the low-frequency range with the index γ ∼ −1.7 (see Fig. 6). Thus, the modeling results presented in Fig. 1 are expected to be relevant even for more complicated sources of turbulence. 3.2. Spectrum anisotropy The anisotropy of the solar wind turbulence on MHD scales (Alfvenic turbulence) is an intrinsic property of the turbulence formed by Alfven waves propagating in the media with a prescribed direction of motion (Dobrowolny et al. 1980; Goldreich and Sridhar et al. 1997; Galtier et al. 2000). The detailed review of modern experimental evidence of such anisotropy was recently published by Horbury et al. (2012). The important parameter for development of magnetic field fluctuations is the angle between the solar wind velocity and the background magnetic field. When the local magnetic field direction is parallel to the solar wind flow, the Fourier spectrum in the spacecraft frame has the spectral index ∼2. This can be interpreted as a signature of the presence of the ensemble of fluctuations distributed ∼k −2 in a field-parallel wave number k . Moreover, most of the power is contained in wave vectors at large angles to the local magnetic field and this component of the turbulence has a spectral index of ∼5/3 (Horbury et al. 2008). This spectrum anisotropy is weak for long wavelength range of the turbulence (kρi < 10−2 where ρi is ion thermal gyroradius), but becomes stronger for a small-scale turbulence (see Fig. 7). Multispacecraft observations and corresponding data analysis allowed us to determine that the spectra of fluctuations


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Figure 7. Spectra of magnetic field fluctuations parallel and perpendicular to the background magnetic field (figure is adopted from Wicks et al. 2010).

transverse to magnetic field is indeed harder than spectra of parallel fluctuations (e.g. Narita et al. 2006; Sahraoui et al. 2010). Another important effect related to the spectrum anisotropy is the dependence of fluctuation polarization (parallel or transverse to the background magnetic field) on the amplitude of the plasma flow. This is a common property for the solar wind and planetary magnetospheres. Fast plasma flows (for both systems velocities are larger than 300–500 km s−1 ) are more dominated by fluctuations with wave vectors quasiparallel to the local magnetic field, while slow plasma flows (for solar wind slow flows correspond to velocity amplitude