2.1. Measurements in space plasma

Space plasmas in the outer magnetosphere and SW have been investigated by a number of spacecraft: WIND/ACE (SW), INTERBALL-1, CLUSTER, DOUBLE STAR, GEOTAIL, POLAR, THEMIS etc. Measurements have allowed for the collection of data on the ion and electron moments (sampling rate up to 0.1–0.3 Hz), electric and magnetic fields (sampling rate 1–64 Hz) and energetic particles. Since 2011, the measurements have been made on the SPECTR-R spacecraft (Zelenyi et al. Reference Zelenyi, Zastenker, Petrukovich, Chesalin, Nazarov, Prokhorenko and Larionov2013) at a unique orbit with high apogee ${\sim}300\,000~\text{km}$ , sliding along the magnetospheric boundaries for up to a few days. Measurements by SPECTR-R provide data regarding ion flux, density, velocity and temperature from the solarward hemisphere, with record sampling over a broad frequency range up to 32 Hz and energetic particle distributions (see e.g. Savin et al. Reference Savin, Skalsky, Zelenyi, Avanov, Borodkova, Klimov, Lutsenko, Panov, Romanov and Smirnov2005–Reference Savin, Amata, Budaev, Zelenyi, Kronberg, Buechner, Safrankova, Nemecek, Blecki and Kozak2014, Zelenyi et al. Reference Zelenyi, Zastenker, Petrukovich, Chesalin, Nazarov, Prokhorenko and Larionov2013 and references therein). The magnetic field ( $B$ ) measurements rely on standard fluxgate magnetometers aboard scientific spacecraft (see e.g. Klimov et al. Reference Klimov, Romanov, Amata, Blecki, Buechner, Juchniewicz, Rustenbach, Triska, Woolliscroft and Savin1997). Faraday cups of various orientation are used for the measurement of the ion flux amplitude and its direction. Faraday cups on INTERBALL-1 and SPECTR-R were used to measure the ion flux with a sampling rate up to 32 Hz, which is comparable with magnetometer data sampling, see Zelenyi et al. (Reference Zelenyi, Zastenker, Petrukovich, Chesalin, Nazarov, Prokhorenko and Larionov2013) and references therein for details. For the other spacecraft, the data are provided by ion energy spectrometers (of the ‘top hat’ type), which give $n$ and $V$ time series with sampling rate less than 1 Hz (see e.g. Reme et al. Reference Reme, Bosqued, Sauvaud, Cros, Dandouras, Aoustin, Martz, Médale, Rouzaud and Möbius1997). The dynamic pressure ( ${\sim}nV^{2}$ ) is calculated from the measured parameters $n$ and $V$ . Signal-to-noise ratio is typically ${>}10$ for all plasma parameters measured in the outer magnetospheric space.

The plasma in the SW and at the magnetospheric boundaries are observed to be non-equilibrium and turbulent. Significant progress in studying the SW turbulence has been achieved in recent decades (see Chang, Tam & Wu Reference Chang, Tam and Wu2004; Hnat, Chapman & Rowlands Reference Hnat, Chapman and Rowlands2005; Marsh & Tu Reference Marsh and Tu2006; Macek Reference Macek2007; Matthaeus et al. Reference Matthaeus, Weygand, Chuychai, Dasso, Smith and Kivelson2008; Zastenker et al. Reference Zastenker and Zelenyi2011; Alexandrova et al. Reference Alexandrova, Saur, Lacombe, Mangeney, Mitchell, Schwartz and Robert2009; Riazantseva, Zastenker & Karavaev Reference Riazantseva, Zastenker, Karavaev and Maksmovic2010; Podesta & Gary Reference Podesta and Gary2011; Vörös Reference Vörös2011; Wicks et al. Reference Wicks, Horbury, Chen and Schekochihin2011; Chen et al. Reference Chen, Salem, Bonnell, Mozer and Bale2012, Reference Chen, Sorriso-Valvo, Safrankova and Nemecek2014; Dura et al. Reference Dura, Hnat, Robinson and Dendy2012; Kiyani et al. Reference Kiyani, Chapman, Sahraoui, Hnat, Fauvarque and Khotyaintsev2013; Zastenker et al. Reference Zastenker, Safrankova, Nemecek, Prech, Cermak, Vaverka, Komarek, Voita, Chesalin and Karimov2013; Osman et al. Reference Osman, Kiyani, Chapman and Hnat2014a ,Reference Osman, Matthaeus, Gosling, Greco, Servidio, Hnat, Chapman and Phan b ; Riazantseva et al. Reference Riazantseva, Budaev, Zelenyi, Zastenker, Pavlos, Safrankova, Nemecek, Prech and Nemec2015 and references therein). The most comprehensive study of SW turbulence properties are provided in the low-frequency range (see Bruno & Carbone Reference Bruno and Carbone2013 etc.). The majority of these papers discuss the power spectrum of the interplanetary magnetic field fluctuations. In numerous papers, it has been indicated that the slope of these spectrum nears $-5/3$ , corresponding to a Kolmogorov–Obuchov spectrum. The high-frequency magnetic field fluctuations have been well studied (see, e.g. Alexandrova et al. Reference Alexandrova, Saur, Lacombe, Mangeney, Mitchell, Schwartz and Robert2009; Yordanova et al. Reference Yordanova, Balogh, Noullez and von Steiger2009; Salem et al. Reference Salem, Howes, Sundkvist, Bale, Chaston, Chen and Mozer2012 etc.). The deviation of the velocity spectra from the magnetic field spectra in the SW has been discussed (see, e.g. Borovsky Reference Borovsky2008; Salem et al. Reference Salem, Howes, Sundkvist, Bale, Chaston, Chen and Mozer2012 etc.). The ion flux fluctuations measured on the SPECTR-R spacecraft (Riazantseva et al. Reference Riazantseva, Budaev, Zelenyi, Zastenker, Pavlos, Safrankova, Nemecek, Prech and Nemec2015) are shown in figure 3. The SW ion flux variations obey a multiscale character, which is discussed in numerous papers (see e.g. Zelenyi & Milovanov Reference Zelenyi and Milovanov2004; Budaev et al. Reference Budaev, Savin and Zelenyi2011; Riazantseva et al. Reference Riazantseva, Budaev, Zelenyi, Zastenker, Pavlos, Safrankova, Nemecek, Prech and Nemec2015 and references therein).

Figure 3.

(a) Typical signal of SW ion flux, BMSW measurements on SPECTR-R from 27-09-2011 23:00 to 28-09-2011 08:20 UT. (b) Power spectrum of ion flux (grey curve, black dots give average value in frequency with ${\rm\Delta}f/f\sim 20\,\%$ ) in SW, SPECTR-R 28-09-2011 03:08-03:25 UT. Solid black lines show the linear approximation by power laws with power exponents of P1 and P2. Above 10 Hz the device noise becomes comparable with the signal in SW and we do not use these data for analysis. $F_{break}$ marks the kink frequency where the average slope changes (‘is breaking’).

The spectrum of the ion flux fluctuations measured by SPECTR-R is shown in figure 3 (Riazantseva et al. Reference Riazantseva, Budaev, Zelenyi, Zastenker, Pavlos, Safrankova, Nemecek, Prech and Nemec2015). A spectral kink is typically observed in frequency spectra. The kink frequency, $F_{break}$ , is close to the low hybrid one. The Doppler shift effect has been discussed to explain the kinked frequency spectrum, for example a Kolmogorov-like $k$ -spectrum was observed by Doppler shifts, accounting for Cluster magnetic measurement above the cusp MP (Romanov, Zelenyi & Savin Reference Romanov, Zelenyi and Savin2012).

The distribution function of ion flux fluctuations is non-Gaussian, demonstrating heavy tails originating from the intermittent behaviour of the signal (see figure 4).

Figure 4.

An example of the PDF of ion flux variations on a scale 0.1 s (solid line) and the corresponding Gaussian fit (dashed line), SPECTR-R 28-09-2011 03:08–03:25 UT.

Multi-spacecraft simultaneous measurements allow comparison of turbulence spectra in the SW and at the magnetospheric boundaries, such as bow shock (BS) and MP. The ion flux power spectra in a quiet SW (obtained by the WIND spacecraft) is compared with data obtained by SPECTR-R in the TBL, figure 5(a): maxima of the spectra (e.g. at ${\sim}0.05~\text{mHz}$ , ${\sim}0.2{-}0.3$ and ${\sim}0.5~\text{mHz}$ , figure 5 a) are observed in the same frequency bandwidths. This allows for consideration of a link in the processes in the SW and TBL, including a cross-magnetospheric correlation (Savin et al. Reference Savin, Amata, Budaev, Zelenyi, Kronberg, Buechner, Safrankova, Nemecek, Blecki and Kozak2014). The spectra on figure 5(b) demonstrate rather good correspondence between BS/foreshock (most probable source region for the SPS/jets) and TBL at the magnetic obstacle boundary. The maxima in the low-frequency part and slopes in the high-frequency part of the spectra are observed downflow until the boundary of the geomagnetic tail (SPECTR-R, shadowed curve in figure 5 b, see Savin et al. Reference Savin, Amata, Zelenyi, Budaev, Consolini, Consolini, Treumann, Lucek, Safrankova and Nemecek2008, Reference Savin, Amata, Zelenyi, Lutsenko, Safrankova, Nemecek, Borodkova, Buechner, Daly and Kronberg2012, Reference Savin, Amata, Budaev, Zelenyi, Kronberg, Buechner, Safrankova, Nemecek, Blecki and Kozak2014).

Figure 5.

Power spectra of ion flux in SW, BS and TBL-simultaneous measurements on WIND, CLUSTER-4 and SPECTR-R on 20.03.12. (a) WIND (SW, lower curve) and SPECTR-R (TBL, upper curve), 13-21 UT, red line shows the power law fit with exponent of $-1.67$ , (vertical lines indicate local maxima); (b) Cluster-4 (SW/BS/MSH, thick violet curve) and SPECTR-R (TBL, shadowed), 13-18 UT.

In the magnetospheric boundary layers, the plasma jets (supermagnetosonic plasma streams, SPS, being the most powerful, see e.g. Budaev et al. (Reference Budaev, Savin and Zelenyi2011), Savin et al. (Reference Savin, Amata, Zelenyi, Lutsenko, Safrankova, Nemecek, Borodkova, Buechner, Daly and Kronberg2012–Reference Savin, Amata, Budaev, Zelenyi, Kronberg, Buechner, Safrankova, Nemecek, Blecki and Kozak2014) and references therein) are observed with a highly enhanced dynamic pressure ( $P_{dyn}$ ). At the magnetospheric boundary downtail, these deflected SPS are mixed with the laminar plasma flow around the MP, flowing with subsonic velocity at the dayside MSH area. The SPS are seen as spikes in the pressure evolution $P_{dyn}$ (detected by the spacecraft DOUBLE STAR, CLUSTER-4 and GEOTAIL) with an amplitude exceeding three standard deviations, see figure 6. Observations at the different locations demonstrated SPS in MSH with an average magnetosonic Mach number of 2.1. Statistics of the flux and dynamic pressure fluctuations have quite close features (Savin et al. Reference Savin, Amata, Budaev, Zelenyi, Kronberg, Buechner, Safrankova, Nemecek, Blecki and Kozak2014).

Figure 6.

Plasma jets with enhanced dynamic pressure observed on 27-03-2005 (Savin et al. Reference Savin, Amata, Budaev, Zelenyi, Kronberg, Buechner, Safrankova, Nemecek, Blecki and Kozak2014), (a–d) Dynamic pressure normalized by the SW pressure, $P_{dyn}/P_{SW}$ ; obtained by DOUBLE STAR, horizontal lines show the levels of 2 and 3 standard deviations used for jet identification; – dynamic pressure $P_{dyn}$ , (nP), obtained by Cluster-4; – CNO – channel of energetic particles ( ${>}274~\text{keV}$ , C $+$ , N $+$ and O $+$ , units – 1/( $\text{cm}^{2}$ sr s keV)) obtained by CLUSTER-4; – Dynamic pressure $P_{dyn}$ , (nP), obtained by GEOTAIL.

Maxima of the power spectra (see figure 5) can be analysed by using bicoherence, which is obtained by averaging the bispectrum over statistically equivalent realizations and normalizing the result. The bicoherence measures three-wave coupling and is only large when the phase between the wave at sum frequency $Fs=Fl+Fk$ and the other two is nearly constant over a significant number of realizations. The 2-D bicoherence graph is used to detect 3-wave interactions considering the 3-wave process with sum frequency $Fs$  ( $Fl$ is the vertical axis of figure 7, $Fk$ is the horizontal one). The bicoherence can be computed using a (continuous) wavelet transform (WT) instead of a Fourier transform, in order to improve the statistics (see details in van Milligen et al. Reference van Milligen, Sanchez, Estrada, Hidalgo, Brafias, Carreras and Garda1995; Savin et al. Reference Savin, Skalsky, Zelenyi, Avanov, Borodkova, Klimov, Lutsenko, Panov, Romanov and Smirnov2005). We assume cascade signatures e.g. for the horizontal maxima in figure 7, when at the sum frequency, $Fs=Fl+Fk$ , the bicoherence has a comparable value with that at the starting point ( $F_{1};F_{2}$ ). This implies that the wave at the sum frequency interacts in its turn with the same initial mode at frequency $F_{1}$ in the further 3-wave process: $F_{3}=F_{1}+F_{2}$ etc.

Figure 7.

Wavelet bicoherence of $P_{dyn}$ measured by DOUBLE STAR at 04-13 UT on 27-03-2005 (Savin et al. Reference Savin, Amata, Budaev, Zelenyi, Kronberg, Buechner, Safrankova, Nemecek, Blecki and Kozak2014).

Figure 7 demonstrates 3-wave interactions: at the low-frequency bandwidth there are local maxima points indicating sharply defined, locked frequencies on the bicoherence graph which are the result of a coupling of several resonance modes, with frequencies suggested by the cavity and surface modes’ theories. At the high-frequency bandwidth, the bicoherence graph has no discrete maxima. Such observation allows the following scenario of the excitation mechanism to be invoked: the SW fluctuations at 0.15–10 mHz excite several boundary modes; the BS and the BS foreshock (see figure 6) are likely a source of the energy. The coupling of these resonance boundary modes leads later on to a cascade process which forms a power-law spectrum in the high-frequency bandwidth (cf. figure 5). The resonance boundary modes modulate the penetration of the SW plasma through the flank MP due to SPS impacts, and they also modulate the O $+$ ion outflow from the magnetosphere into the SW (see figure 6 and Savin et al. Reference Savin, Amata, Budaev, Zelenyi, Kronberg, Buechner, Safrankova, Nemecek, Blecki and Kozak2014).

Taylor ‘frozen-in turbulence’ hypothesis. We here note the Eulerian viewpoints of turbulence and the Taylor ‘frozen-in turbulence’ hypothesis which is usually used in plasma measurements both in space and in laboratory devices. In space, the SW speed, $V_{sw}$ is much higher than the local MHD wave speeds. We can thus consider measurements at a set of times $t_{i}$ to be at a set of locations in the plasma given by $x_{i}=x_{0}-V_{sw}t_{i}$ This approximation, known as the Taylor ‘frozen-in turbulence’ hypothesis greatly simplifies the analysis of the data. The Taylor (Reference Taylor1938) hypothesis makes an assumption that the advection contributed by the turbulent circulations themselves must be small and that therefore, the advection of a field of turbulence past a fixed point can be taken as being mainly due to the larger, energy containing scales (the mean flow). Taylor proposed his frozen-in turbulence hypothesis in terms of Eulerian mean flow. Although it is only valid when the integral scales have sufficiently high power compared with the smaller scales, this hypothesis delivered a linkage between the Eulerian and Lagrangian viewpoints of turbulence. The Taylor ‘frozen-in turbulence’ hypothesis is widely used in neutral fluid measurements. Usually experimentalists performed Eulerian measurements based on time-resolved single-point measurements. Lagrangian measurements are challenging because they involve the tracking of particle trajectories, but such measurements are not now available in plasma. For a detailed introduction to the statistical mechanics of Eulerian and Lagrangian turbulence see Monin & Yaglom (Reference Monin and Yaglom1975) and Frish (Reference Frish1995). In MHD turbulence of interplanetary plasmas, the Lagrangian information has been extracted following the Taylor hypothesis, involving the magnetic flux frozen in a wide scale range. In the Earth’s magnetosheath, the plasma flow speed can be lower than the wave speed and therefore temporal changes at the spacecraft are due to a complex combination of the plasma moving over the spacecraft, and the turbulent fluctuations propagating in the plasma frame. Taylor’s hypothesis has been used to explain, e.g. the morphologies of jets in MSH (Savin et al. Reference Savin, Amata, Budaev, Zelenyi, Kronberg, Buechner, Safrankova, Nemecek, Blecki and Kozak2014). However, it remains a hypothesis. To describe intermittency, a problem of chaotic motion of individual fluid particles and mixing flow should be considered. To characterize the intermittency of developed turbulence in the Eulerian framework, one can consider mixing and chaotic motion (chaotic advection, when neighbouring trajectories separate exponentially fast with time) due to Lagrangian chaos phenomenon (see, e.g. Monin & Yaglom Reference Monin and Yaglom1975; Ottino Reference Ottino1989). Considering chaos in Lagrangian space, the exponential growth of separation between advected trajectories $l(t)=l_{0}\text{e}^{{\it\lambda}t}$ in time is assumed only for a separation $l\ll {\it\eta}$ , where ${\it\lambda}$ is the Lyapunov exponent, ${\it\eta}$ is the characteristic scale of the smallest Eulerian structures (i.e. dissipative eddies in plasma turbulence). In isotropic homogeneous 3-D turbulence in the inertial range ${\it\eta}\ll l\ll L$ the Richardson law $\langle l^{2}\rangle \sim t^{3}$ and standard diffusive regime $\langle l^{2}\rangle \sim t$ for very large separations $l\gg L$ are assumed. In order to treat intermittent turbulence keeping in the Eulerian framework, we may assume power law scaling $\langle l^{2}\rangle \sim t^{h}$ (i.e. index $h$ may depend on a scale) being more general than Richardson’s law. Such an approach corresponds to the multifractal formalism that is a bridge between the Eulerian and Lagrangian descriptions, see, e.g. Frish (Reference Frish1995). The intermittency can be described by the multifractal formalism (see below) exploiting Eulerian measurements in space and laboratory plasmas.

2.2. Measurements in fusion devices

Plasma turbulence has been measured in many fusion devices, starting with experiments on tokamak TMP in 1956 at Kurchatov Institute (see Nedospasov Reference Nedospasov1992). Experiments are made in fusion devices with different topology of magnetic field – tokamaks, helical devices and stellarators, reversed field pinch and linear devices. Topology of the magnetic field is taken into account when MHD stability and large-scale plasma dynamics are considered. A lot of experimental evidence has been observed now that plasma in fusion devices is turbulent and level of plasma fluctuations is increased in the peripheral zone of plasma discharge. Edge plasma turbulence is intermittent with an enhanced level of turbulent flux, leading to anomalously large cross-field plasma transport (see Introduction). Thus, the edge region of the plasma in fusion devices is highly interesting for the investigation of the nature of enhanced plasma transport linked with intermittency and the self-similarity of plasma turbulence.

Experimental measurements of edge plasma parameters are reviewed in Zweben et al. (Reference Zweben, Boedo, Grulke, Hidalgo, LaBombard, Maqueda, Scarin and Terry2007) and other related papers. In such experiments, plasma dynamics is characterized typically by moderate kinetic Reynolds numbers ( $\mathit{Re}\,<\,100$ ) and magnetic Reynolds numbers ( $\mathit{Re}_{m}<1000$ ), see Budaev et al. (Reference Budaev, Savin and Zelenyi2011).

The main fluctuating quantities of interest for edge plasma turbulence are the electron density $n$ , electron temperature $Te$ , the electrostatic potential ${\it\varphi}$ and the flow speed $\boldsymbol{v}$ . Electric (Langmuir) probes are used to measure plasma fluctuations in magnetic fusion devices. The technique of Langmuir probes is simply to insert one or more small biased electrodes inside the plasma and then to measure the fluctuations seen by these probes. Probe signals contain a large amount of data which can be interpreted in terms of the local $n$ , $Te$ , ${\it\varphi}$ within the frequency and length scales of ${\sim}1~\text{kHz}{-}1~\text{MHz}$ and ${\sim}0.1{-}10~\text{cm}$ . The limitation of probes (with size typically 1 mm) is that they must not significantly perturb the plasma or vice versa. In practice, this allows for the investigation of plasmas with up to $Te\sim 100~\text{eV}$ , which is a typical condition of edge plasma in fusion devices.

Accuracy of plasma parameter estimation depends on the assumptions made during probe data analysis. If the electron temperature fluctuations are neglected, the poloidal electric field $E$ and hence the radial drift $E\times B$ velocity (cross-field) of fluctuations, $v$ , can be calculated from measurements of the potential fluctuations with two separated Langmuir probe tips. The cross-field particle transport ${\it\Gamma}$ due to fluctuations can be calculated from such velocity estimation and density fluctuations with a third tip in between, ${\it\Gamma}=nv$ . The calculation of electric field and the radial $E\times B$ velocity assumes that the temperature and sheath drop are identical in both tips and that the intermittent objects are larger than the separation between the tips; the last assumption is justified later using beam emission spectroscopy data (see Boedo et al. Reference Boedo, Rudakov, Moyer, Krasheninnikov, Whyte, McKee, Tynan, Schaffer, Stangeby and West2001). In experiments, probe separation between two neighbouring probe tips is typically 3–5 mm which is sufficiently small. This separation influences the underestimation of Fourier components with wavelengths as small as twice the probe tip separation. Details on the possible errors introduced by neglect of the temperature fluctuations and finite probe tip separation is discussed by Endler (Reference Endler1999). Probe measurements of density and temperature have been successfully compared with non-perturbing measurements such as Thomson scattering (see Zweben et al. Reference Zweben, Boedo, Grulke, Hidalgo, LaBombard, Maqueda, Scarin and Terry2007).

Numerous Langmuir probe measurements were made on tokamaks: T-10 by Vershkov, Grashin & Chankin (Reference Vershkov, Grashin and Chankin1987), D-IIID by Boedo et al. (Reference Boedo, Rudakov, Moyer, Krasheninnikov, Whyte, McKee, Tynan, Schaffer, Stangeby and West2001), TEXTOR by Huber et al. (Reference Huber, Samm, Schweer and Mertens2005), JET by Gonçalves et al. (Reference Gonçalves, Hidalgo, Silva, Pedrosa and Erents2005), Tore Supra by Antar, Counsell & Yu (Reference Antar, Counsell and Yu2003), JT-60U by Asakura et al. (Reference Asakura, Sakurai, Shimada, Koide, Hosogane and Itami2000), Asdex-U by Endler et al. (1995), MAST by Antar, Counsell & Ahn (Reference Antar, Counsell and Ahn2005), on stellarators and helical devices: LHD by Ohno et al. (Reference Ohno, Masuzaki, Miyoshi, Takamura, Budaev, Morisaki, Ohyabu and Komori2006), L-2 by Skvortsova et al. (Reference Skvortsova, Korolev, Batanov, Petrov, Sarksyan, Kharchev, Shatalin, Laskul, Pavlov and Popov2006), TJ-II by Alonso et al. (Reference Alonso, Zweben, Carvalho, de Pablo, de la Cal, Hidalgo, Klinger, van Milligen, Maqueda and Pedrosa2006), on linear machines by Carter (Reference Carter2006), Chiu & Sen (Reference Chiu and Sen2000), on reversed field pinches by Antoni et al. (Reference Antoni, Cavazzana, Desideri, Martines, Serianni and Tramontin1998), on plasma device NAGDIS-II by Ohno et al. (Reference Ohno, Budaev, Furuta, Miyoshi and Takamura2004), PISCES by Schmitz et al. (Reference Schmitz, Lehmer, Chevalier and Tynan1990) and others fusion devices. The plasma fluctuations have been experimentally observed to be highly correlated and of constant amplitude along the magnetic field lines (over more than one toroidal turn in toroidal devices). Magnetic pickup coils are used to measure magnetic field.

Typical time traces of the density $n(t)$ fluctuations measured using Langmuir probes in the edge of fusion devices are shown in figure 8. Taylor’s hypothesis was used to make analyse this experimental data.

Figure 8.

(a) Time traces of density fluctuations (subtracted by mean value and normalized by standard deviation) in magnetized edge plasmas: the T-10 tokamak SOL plasma, NAGDIS-II attached plasma, LHD divertor plasma, JT-60U tokamak SOL plasma. (b) Typical waveform of high intensity bursts in T-10 tokamak and NAGDIS-II device. (c) Power spectrum $S(f)=|n(f)|^{2}$ of the density fluctuations. Solid line – power-law $1/f$ ; dashed line – a noise level in the signal. SOL plasma, T-10 tokamak.

The density, electric field (i.e. cross-field velocity) and cross-field particle flux fluctuations possess a high-frequency part and the peaks of total intensity (referred to as the bursts) caused by the intermittent structures. There exist fluctuations in each burst and maxima are separated by a time greater than the auto-correlation time. The characteristic time scale of the bursts is of ${\sim}40{-}200~{\rm\mu}\text{s}$ (figure 8 b). In many experiments, several scaling ranges with respect to the frequency are typically registered in the broadened frequency spectra of density and electric field fluctuation (see example on figure 8) with no $1/f$ behaviour. The typical value of the scaling exponent of the power spectra in the high-frequency range above ${\sim}100~\text{kHz}$ is in the range of ${\it\gamma}\approx 1.2{-}2.5$ . There are no monochromatic peaks in the broadened spectra. Thus, the process cannot be considered to consist of any coherent structures in isolation from the background turbulence. It should be noted that the frequency Fourier spectra of all observables (density, electric field and velocity, cross-field particle fluxes) are of decayed shape.

The spectrum of poloidal wavenumber, $k_{pol}$ , (integrated over all frequencies) broadens similarly to the shape of the frequency spectrum (integrated over all poloidal wavenumbers) as is expected from the Taylor hypothesis of frozen flux. In the edge plasma $k_{pol}$ is in the range ${\sim}0.1{-}5~\text{cm}^{-1}$ , corresponding to the scaling $\langle k_{pol}\rangle {\it\rho}_{i}\sim 0.02{-}0.1$ ( $\langle k_{pol}\rangle$ is averaged value, ${\it\rho}_{i}$ – local ion gyroradius), i.e. to the typical low-frequency turbulence range (see Zweben et al. Reference Zweben, Boedo, Grulke, Hidalgo, LaBombard, Maqueda, Scarin and Terry2007).

Turbulence properties regarding magnetic structure and magnetic curvature were studied in tokamaks TF-2 (Budaev et al. Reference Budaev, Bogomolov, Borovsky and Ivanov1990), T-10 (Kirnev et al. Reference Kirnev, Budaev, Grashin, Khimchenko and Sarytchev2005), HYBTOK-II (Budaev et al. Reference Budaev, Takamura, Kikuchi, Uesugi and Ohno2003) and the Large Helical Device (Ohno et al. Reference Ohno, Masuzaki, Miyoshi, Takamura, Budaev, Morisaki, Ohyabu and Komori2006). Turbulent fluxes, fluctuation level and turbulent spectra were measured in these toroidal plasma discharges at outboard and inboard sides (which are of different magnetic curvature) of the torus to reveal any differences in the property regarding magnetic curvature. Intermittent fluctuations were observed both at the outboard and inboard side of a toroidal plasma; they were identical statistically. These experimental observations have demonstrated that $B\boldsymbol{{\rm\nabla}}B$ drift due to magnetic curvature is likely not a dominant effect of intermittency origin in toroidal plasmas.

Intermittent plasma transport. Experimental results obtained in the last two decades have shown that intermittent turbulence of plasma in the edge of tokamaks, helical devices and linear devices have similar properties independent of which device the measurements are made in (see Budaev et al. Reference Budaev, Savin and Zelenyi2011). Cross-field plasma particle flux in the edge of fusion devices is inhomogeneous in time and space (see figure 9 a). Numerous experiments have shown that cross-field plasma transport in the edge plasma of fusion devices (tokamaks, helical and linear devices) cannot be described by classical diffusion with constant diffusion coefficient. A large fraction (up to ${\sim}50\,\%$ ) of cross-field plasma flux consists of large-scale coherent structure transport. Such cross-field transport in the literature is often described by an effective diffusion coefficient that can reach a value much larger than the Bohm coefficient (up to tens of magnitude). This is called anomalous diffusion (despite the fact that the process does not look like gas diffusion). Intermittent transport (radial and poloidal) was observed using different diagnostics (see review Boedo et al. Reference Boedo, Rudakov, Moyer, Krasheninnikov, Whyte, McKee, Tynan, Schaffer, Stangeby and West2001; D’Ippolito, Myra & Zweben Reference D’Ippolito, Myra and Zweben2011; Maqueda, Stotler & Zweben Reference Maqueda, Stotler and Zweben2011). The level of time-averaged flux originates not only from amplitudes of density and electric field fluctuations, but also a large contribution from the correlation between density $n(t)$ and drift velocity $v(t)$ (defined by electric field). The Fourier spectra of the flux is broadened. Detailed analysis of experimental data has shown the self-similarity of the plasma flux. A hierarchy of structures is seen, e.g. on the WT as tree-like structure (figure 9 b). This hierarchy of scales is evidence of fractality and long-range correlation. Such a feature is typical of a cascade process in developed turbulence and should be analysed by statistical methods.

Figure 9.

Cross-field plasma flux in the edge of the T-10 tokamak: (a) typical signal and (b) wavelet decomposition, white curve is the flux signal.

Numerous experimental evidence in edge plasmas of fusion devices have shown that the PDFs of the fluctuations (constructed as a histogram of the experimental signal – density, electric field or particle flux) are typically non-Gaussian, figure 10, and illustrate the intermittent nature of the process. Positively skewed PDFs are typically observed in the SOL and in the far-SOL region in fusion devices, e.g. on Tore Supra, Alcator C-Mod, MAST and PISCES (Antar et al. Reference Antar, Counsell and Yu2003). At the same time, a negatively skewed PDF (related to negative burst waveforms) was observed on the Large Helical Device in some edge plasma regions (Ohno et al. Reference Ohno, Masuzaki, Miyoshi, Takamura, Budaev, Morisaki, Ohyabu and Komori2006). Intermittent fluctuations (non-Gaussian PDFs) of the edge density, velocity and turbulent transport flux have been seen in many toroidal fusion devices (see Zweben et al. Reference Zweben, Boedo, Grulke, Hidalgo, LaBombard, Maqueda, Scarin and Terry2007).

Figure 10.

Typical PDFs of fluctuations in edge plasma. (a) Cross-field particle flux ${\it\Gamma}$ in the T-10 tokamak SOL by solid line. Experimental data are normalized by the standard deviation ${\it\sigma}$ . Gaussian fits of the PDF’s are shown in dashed-dotted line, the Lorentz distribution by dashed line. (b) Evolution of the PDF’s shape with minor radius, the SOL of the T-10 tokamak LCMS at $r=30~\text{cm}$ . (c) Density (blue thick dashed line) and velocity (red solid line) simultaneously measured in the same location in the T-10 tokamak SOL. Experimental data are normalized by the standard deviation value. Gaussian fits of the PDF’s are shown in dashed line.

It is worth stressing that both the density and velocity of the plasma fluctuations obey a non-Gaussian PDF of similar shape. This is illustrated in figure 10(c), where PDFs of density and velocity fluctuations measured by Langmuir probes in the same location (within size of 4 mm) are shown. Such observations are an argument in favour of treating turbulent transport intermittency as linked with both density and velocity intermittency.

There is no clear evidence that the PDF’s shape depends strictly on plasma parameters such as local plasma density, electron temperature, plasma gradients and particle flux value. An evolution of the PDF’s shape with minor radius is observed (see figure 10 b), e.g. from a near-Gaussian shape at LCFS to the strictly positively skewed at the far-SOL. To analyse the shape of the PDF in detail, the structure function should be investigated.

The fluctuations observed in tokamaks, helical devices and linear devices are characterized by scale-invariance (self-similarity) (see, e.g. Budaev et al. Reference Budaev, Ohno, Masuzaki, Morisaki, Komori and Takamura2008a ,Reference Budaev, Savin, Zelenyi, Ohno, Takamura and Amata b ). The self-similarity is illustrated by the hierarchy of scales on WT, figure 9(b). The observations of scale-invariance and power-laws are a manifestations of the fractality (self-similarity) of turbulence which can be characterised by a fractal dimension (or a set of dimensions). The fractal structure of edge plasma turbulence in fusion devices is very complicated. Depending on the accuracy of the description, different fractal indices can be used to characterize the scale invariance, such as the correlation dimension (fractal dimension) and Hurst exponent (see Hurst Reference Hurst1951; Grassberger & Procaccia Reference Grassberger and Procaccia1983). The correlation dimension of edge plasma turbulence in TEXTOR, T-10 tokamaks is in the range from ${\sim}6$ to 15 (Budaev et al. Reference Budaev, Fuchs, Ivanov and Samm1993, Reference Budaev, Savin and Zelenyi2011). This is a relatively low fractal dimension (in comparison with infinity for ‘white noise’). The Hurst exponent for edge plasma turbulence was estimated to be of 0.6–0.8 (on tokamaks DIII-D Carreras, Lynch & Zaslavsky (Reference Carreras, Lynch and Zaslavsky2001b ), Tore Supra (Antar et al. Reference Antar, Krasheninnikov, Devynck, Doerner, Hollmann, Boedo, Luckhardt and Conn2001), T-10 (Budaev & Khimchenko Reference Budaev and Khimchenko2007)) and demonstrates a tendency to increase toward the wall. Values $H>1/2$ corresponds to a long-range correlation and superdiffusion. The Hurst exponent describes the coarse grain structure of a fractal object. Detailed characterization is available by multifractal analysis.

3.1. Multifractal cascade process

A treatment of the turbulence problem in any analytic theory or direct numerical solution of the equations describing plasma turbulence face a fundamental problem: the cascade nature of the process, the number of degrees-of-freedom increases algebraically for increasingly small spatial scales. This is why cascade turbulence models are better suited for a description of the long range spatiotemporal scales. Within the cascade paradigm, many properties of such turbulence can be described, in particular, its multiscale and multifractal nature. By incorporating an anisotropic multiplicative cascade, it becomes possible to develop a multifractal turbulence model (see review Budaev et al. Reference Budaev, Savin and Zelenyi2011 and other related papers), i.e. to consider how the self-similarity depends on the local spatial scale.

The multifractal approach is a generalization of a self-similarity consideration in a system with rich statistical properties. It is appropriate to study both statistically homogeneous plasma turbulence (such as interplanetary SW or core tokamak plasmas) and intermittent plasma turbulence in the Earth’s magnetosphere and in edge of fusion devices. The multifractal approach is used to model a hierarchical turbulence cascade of a flow bounded by an upper limit (e.g. TBL). Such an approach reconsiders the cascade model K41 of Kolmogorov which represents a monofractal model with trivial self-similarity in the inertial range, supposing the inertial range is very large.

In the developed turbulence of incompressible fluids, Kolmogorov (Reference Kolmogorov1962) first formulated a hypothesis invoking some statistical independence in the cascading process, which led to the log-normal model for the rate of dissipation of turbulent kinetic energy. The next step of a generalization was the multifractal formalism (see Frish Reference Frish1995) proposed to quantify the boundary conditions influence on the turbulence statistical properties.

For multifractals, as far as the PDF of increments ${\it\delta}_{l}X(t)$ at different time scales $l$ are concerned, reshaping from ‘quasi-Gaussian’ at large scale $L$ (i.e. boundary scale) will occur, with fat tailed PDF’s at small scales. This transformation of the PDF’s for SW turbulence is illustrated in figure 11 where, the PDF’s for different time scales are plotted in logarithmic scale. The fat tails appear at the scale related to the ‘coarse’ time scale $L$ – a characteristics of multifractality. A typical scale for $L$ is of ${\sim}10{-}10^{4}~\text{s}$ for the Earth’s magnetosphere and of ${\sim}50{-}500~{\rm\mu}\text{s}$ for the edge plasma in fusion devices (see Budaev et al. Reference Budaev, Savin and Zelenyi2011). The multifractality property was registered in TBL of the Earth’s magnetosphere (Budaev et al. Reference Budaev, Savin and Zelenyi2011), in edge turbulence of tokamaks: Tore Supra (Antar et al. Reference Antar, Krasheninnikov, Devynck, Doerner, Hollmann, Boedo, Luckhardt and Conn2001), T-10 (Budaev et al. Reference Budaev, Ohno, Masuzaki, Morisaki, Komori and Takamura2008a ,Reference Budaev, Savin, Zelenyi, Ohno, Takamura and Amata b ), Hybtok-II (Budaev et al. Reference Budaev, Kikuchi, Uesugi and Takamura2004), CASTOR (Zajac et al. Reference Zajac, Budaev, Dufkova, Nanobashvili and Weinzettl2005), TCABR (Rodrigues Neto et al. Reference Rodrigues Neto, Guimarães-Filho, Caldas, Nascimento and Kuznetsov2008), MAST (Kono & Škorić Reference Kono and Škorić2010), JT-60U, heliotron LHD and a linear device NAGDIS-II (Budaev et al. Reference Budaev, Ohno, Masuzaki, Morisaki, Komori and Takamura2008a ,Reference Budaev, Savin, Zelenyi, Ohno, Takamura and Amata b ). The multifractal theory of energy transfer in the multiplicative cascade process can involve multifractal properties of the observables, see appendix A.

Figure 11.

Multifractal property of magnetized plasma, SW ion flux, SPECTR-R data, 20.03.2012 12UT. (a) Standardized estimated PDF’s of ion flux increments ${\it\delta}_{{\it\tau}}{\it\Gamma}(t)$ (normalized to its standard deviation ${\it\sigma}_{{\it\Gamma}}$ ) for different time scales (from top to bottom) ${\it\tau}=0.25$ , 1, 4, 16, 64, 256, 1024, 2048, 4096, 16 384 s (from top to bottom). Plots have been arbitrary shifted for illustration. (b) Third (skewness – by circles) and fourth (flatness – by squares) moments of the increment PDFs for different time scales ${\it\tau}$ .

Considering stochastic cascade models in the Eulerian framework, the intermittency can be described quantitatively in terms of a multifractal formalism that is a bridge between Eulerian and Lagrangian descriptions (see above discussion on Taylor ‘frozen-in turbulence’ hypothesis in § 2) The goal is to compute the multifractal spectra $D(h)$ starting from the equations of motion or from the experimental investigation of a turbulent process. Experimental observations of scaling ${\it\zeta}(q)$ and multifractal spectra $D(h)$ are crucial to understand how we can formulate the most general form of a multifractal random field which is consistent with the time and space scaling properties of the governing equations describing plasma turbulence.

A statistical description of plasma intermittency relates to the question how higher-order correlations influence the development of turbulence. In hydrodynamics, a well-known Kolmogorov (‘ $4/5$ ’) law relates the third-order structure function to the energy dissipation rate. Usually, it is stated with the assumptions of isotropy, homogeneity and time stationarity of the statistics of the velocity increments ${\it\delta}_{l}{\it\upsilon}$ . Kolmogorov’s third-order law (Kolmogorov Reference Kolmogorov1941a ,Reference Kolmogorov b ) requires that (certain) third-order moments, $S(q,l)=\langle |{\it\delta}_{l}{\it\upsilon}|^{q}\rangle ,q=3$ , are non-zero and proportional to the non-zero energy flux ${\it\varepsilon}$ ,

Derived from the Navier–Stokes equations (NSE) (see Frish Reference Frish1995) and valid in the inertial range, assure that the third-order structure function is unaffected by intermittency. It also requires adoption of the von Kármán–Howard–Monin hypothesis (von Kármán & Howarth Reference von Kármán and Howarth1938; Monin & Yaglom Reference Monin and Yaglom1975) that the rate of energy dissipation ${\it\varepsilon}$ approaches a constant non-zero value as the Reynolds number tends to infinity. The well-known von Kármán–Howard–Monin hierarchy of equations relate the evolution of the second-order correlations to third-order ones, the evolution of the third-order correlations to fourth order ones, and so on to infinite order.

For turbulent MHD fluctuations exact relations for third-order structure functions and their derivatives are considered in theoretical treatments (see Basu, Naji & Pandit Reference Basu, Naji and Pandit2014 and references therein, Politano & Pouquet Reference Politano and Pouquet1998a ,Reference Politano and Pouquet b ). It is a direct analog of the von Kármán–Howarth–Monin vector relation in anisotropic hydrodynamic turbulence. The quantities used in MHD treatment (e.g. by Politano & Pouquet Reference Politano and Pouquet1998a ,Reference Politano and Pouquet b ) are the plus and minus Elsässer vectors, $z^{\pm }={\it\upsilon}\pm b$ , where $b$ is the magnetic field in Alfven speed units, and their fluctuations over vector spatial differences. On this point, Chandrasekhar (Reference Chandrasekhar1951) noticed that the Elsässer variables have an ambiguous symmetry under parity. Considering von Kármán–Howard–Monin hypothesis, the prospects of a universal description of MHD turbulence are discussed in the literature (see Wan et al. Reference Wan, Oughton, Servidio and Matthaeus2012; Basu et al. Reference Basu, Naji and Pandit2014). Multifractal measures are appropriate quantities for such a description.

Exploiting Eulerian measurements (see discussion on Taylor ‘frozen-in turbulence’ hypothesis in § 2), the scaling ${\it\zeta}(q)$ and $D(h)$ are calculated from experimental time-series by using the WTMM method, which is very powerful in estimating the multifractal measures, see appendix A.

Multifractal spectra $D(h)$ estimated by the WTMM method (see appendix A) are plotted in figure 12 against the Hölder exponent $h^{\ast }=1+(h-h_{Dmax})$ (where $h_{Dmax}$ is the Hölder exponent at a maximum of the $D(h)$ spectrum). In this frame, the spectra are centred around 1. The experimental signals from space and laboratory plasmas are characterized by the broadened $D(h)$ spectra, being of a convex (bell-like) shape (figure 12). This is the typical shape of singularity spectra observed in multifractal fields (cf. examples in Harte Reference Harte2001). For a monofractal process, the $D(h)$ spectrum collapses to a point (an example is the data from SW and MSH outside the TBL in figure 12). The maximum of $D(h)$ is in the range of 0.8–1. A value close to 1 suggests that the experimental signal is singular almost everywhere. This implies that turbulent fluctuations are characterized by a rapid increase in amplitude. The broadness of the $D(h)$ singularity spectrum provides a measure of multifractality (deviation from the monofractal process, i.e. K41) and lies in the range of 0.5–1.2. This range is typical for the strong intermittency observed in numerical models and experiments in fluids and plasma turbulence (see Budaev et al. Reference Budaev, Savin and Zelenyi2011). The Hölder exponents exceed that of a Brownian signal with $h=1/3$ . A typical view of $h(q)$ is shown in figure 13.

Figure 12.

Multifractal spectra $D(h)$ versus the normalized Hölder exponent $h^{\ast }=1+(h-h_{Dmax})$ (centred around 1 by subtracting $h_{Dmax}$ ). TBL ion flux SPECTR-R data (SW SPECTR-R) 20.03.2012, magnetic field $B_{x}$ in the MSH outside the TBL (MSH $B_{x}$ ) INTERBALL-I, 19.06.1998, SW ion flux WIND data 20.03.2012, the ion flux in the Earth’s magnetosphere TBL (TBL ion flux), 29.03.1996; edge plasma turbulence in tokamak T-10 (T-10 SOL), near the last closed magnetic surface (T-10 LCFS); in NAGDIS-II for attached plasma (N-II attach) and for detached plasma (N-II detach).

Figure 13.

Hölder exponent $h(q)$ . Density fluctuations in SOL of T-10 tokamak (T-10 SOL), magnetic field $B_{x}$ in SW (SW $B_{x}$ ) from GEOTAIL data; ion flux in TBL (TBL ion flux) INTERBALL data 19.06.1998.

Waiting-time statistics. Another statistical approach is studying extreme events (bursts) in the signal. The waiting-time is defined as the time interval ${\rm\Delta}t$ between two consecutive maxima in a burst intensity selected above a threshold (of level crossing of a signal, e.g. of $2{-}3$ standard deviation in amplitude). The distribution $P({\rm\Delta}t)\sim ({\rm\Delta}t)^{{\it\gamma}}$ of the times are called persistence PDFs and the exponent ${\it\gamma}$ is called the persistence exponent (see, e.g. Perlekar et al. Reference Perlekar, Ray, Mitra and Pandit2011).

Typical waiting-times series constructed from intervals between successive bursts in the signal are shown in figure 14, demonstrating intermittency and the scale hierarchy in the wavelet decomposition. Typically, the PDFs of these series are not decayed exponentially. A typical power law exponent of the PDF, $P({\rm\Delta}t)\sim ({\rm\Delta}t)^{{\it\gamma}}$ (see example in figure 15 is of ${\it\gamma}\approx -3$ to $-2$ , Savin et al. Reference Savin, Budaev, Zelenyi, Amata, Lutsenko, Borodkova, Zhang, Angelopoulos, Safrankova and Nemecek2011). For the normal random process with Gaussian statistics it should be ${\sim}1$ . The power law of the waiting-time PDF was observed in experiments on T-10 tokamak (Budaev et al. Reference Budaev, Ohno, Masuzaki, Morisaki, Komori and Takamura2008a ,Reference Budaev, Savin, Zelenyi, Ohno, Takamura and Amata b ), Large Helical Device (Ohno et al. Reference Ohno, Masuzaki, Miyoshi, Takamura, Budaev, Morisaki, Ohyabu and Komori2006), D-IIID tokamak and other devices (Sánchez et al. Reference Sánchez, van Milligen, Newman and Carreras2003) and in the Earth’s magnetosphere (Savin et al. Reference Savin, Budaev, Zelenyi, Amata, Lutsenko, Borodkova, Zhang, Angelopoulos, Safrankova and Nemecek2011).

Figure 14.

Intermittent waiting-time series ${\rm\Delta}t$ and its wavelet decomposition. Hierarchy of scales is observed as an evidence of fractality and long-term correlation. Cross-field flux in the SOL of T-10 tokamak.

Figure 15.

The PDF of waiting times for extreme events (peaks above 2 standard deviations) in log–log plot. 2 – TBL, $B_{z}$ , CLUSTER, 02.02.2003, 3 – TBL, $nV_{z}$ , CLUSTER, 02.02.2003, 4 – TBL, $B_{y}$ , GEOTAIL, 16.04.1996, 18:54-20:04 UT, 6 – TBL, $nV$ , GEOTAIL, 16.04.1996, 18:54-20:04 UT.

If the triggering of the turbulent bursts are not correlated, such a process may follow the Poisson process, and the probability density function of the waiting times, ${\rm\Delta}t$ , should be an exponential law: $P({\rm\Delta}t)={\it\gamma}^{-1}\text{e}^{-{\rm\Delta}t/{\it\gamma}}$ – the law of rare events. This exponential PDF is considered in the frame of self-organized criticality (SOC) hypothesis (Bak, Tang & Wiesenfeld Reference Bak, Tang and Wiesenfeld1988). However, statistical analysis shows a general lack of an exponential decay of the waiting-time PDF observed in the plasma TBLs, suggesting long-range correlation. This has raised criticism about the applicability of the simple SOC concept to the plasma TBLs.

3.2. Generalized scale invariance

A statistical description of intermittent turbulence is available through the structure functions. The structure functions are the various order moments of the PDF of the fluctuating quantity: $S_{q}({\it\tau})=\langle |{\it\delta}_{{\it\tau}}X|^{q}\rangle$ , where ${\it\delta}_{{\it\tau}}X=X(t+{\it\tau})-X(t)$ , $\langle \cdots \rangle$ is the ensemble average of experimental time-series $X(t)$ . Typically, the structure functions $S_{q}({\it\tau})$ exhibit a saturation over some ‘coarse’ time scale ( ${>}1~\text{s}$ in the Earth’s magnetosphere TBL and ${>}10~{\rm\mu}\text{s}$ in fusion devices), figure 16. The power dependence $S_{q}({\it\tau})\sim {\it\tau}^{{\it\zeta}(q)}$ is observed only on limited time scales. Such behaviour is typical for data from plasma TBLs. This was observed in space plasma (INTERBALL, CLUSTER, WIND, SPECTR-R, DOUBLE STAR, GEOTAIL, POLAR spacecrafts) and laboratory plasma devices (tokamaks, stellarators and linear machines), Budaev et al. (Reference Budaev, Savin and Zelenyi2011).

Figure 16.

The high-order structure functions $S_{q}({\it\tau})$ of different orders ( $q=2$ , 3, 4, 5, 6, 7, 8 from bottom up) versus time scale ${\it\tau}$ in log–log plot. (a) SW ion flux, BMSW data SPECTR-R on 28.09.2011; (b) plasma density in edge of T-10 tokamak.

Hydrodynamic turbulence (Benzi et al. Reference Benzi, Ciliberto, Tripiccione, Baudet, Massaioli and Succi1993) in wind tunnel experiments have shown that scaling laws of the velocity increments can be extended up into a dissipative range: $S_{q}(l)$ has a power-law dependence on $S_{3}(l)$ ,

over a range which is substantially longer (for long ranges of scale $l\geqslant 5{\it\eta}$ , ${\it\eta}$ – dissipation scale) than the scaling range obtained by plotting $S_{q}(l)$ as a function of $l$ . This behaviour holds even for intermittent turbulence at a moderate Reynolds number and was named an ESS. This ESS property is observed on the broad range of scales from large to small scales including the dissipation range. The ESS corresponds to the scaling in a turbulent cascade not with respect to the usual distance, but with respect to an effective scale defined by the third-order moment of the velocity field.

This phenomenological observation led to the criterion of the generalized self-similarity in the form $S_{q}(l)\sim S_{p}(l)^{{\it\zeta}(q)/{\it\zeta}(p)}$ , for any pair of structure functions. Such a property is explained in the frameworks of the log-Poisson cascade models (generalization of multifractal models) of intermittent turbulence, see below. In the general case, scaling laws ${\it\zeta}(q)$ represent a nonlinear function of $q$ . So, generalized self-similarity (scale invariance) is considered to be the ESS property and nonlinear scaling laws ${\it\zeta}(q)$ describing a universal property of scale invariance in the framework of the log-Poisson cascade models (see below).

The ESS property is illustrated in the log–log plots: the dependence of $S_{q}({\it\tau})$ on $S_{3}({\it\tau})$ is seen to be linear almost over up to three orders of magnitude in time (figure 17). All data from edge plasma in fusion devices (tokamaks T-10, HYBTOK-II, JT-60U, heliotron LHD, linear NAGDIS-II (Budaev et al. Reference Budaev, Ohno, Masuzaki, Morisaki, Komori and Takamura2008a ,Reference Budaev, Savin, Zelenyi, Ohno, Takamura and Amata b )) and space plasma (SPECTR-R, INTERBALL, CLUSTER, WIND) demonstrate such an ESS property, which confirms the universality of this property, Budaev et al. (Reference Budaev, Savin and Zelenyi2011). Such observations lead to the hypothesis of the universality of the scaling in turbulence boundary layers which is currently widely discussed in the literature (see Budaev et al. Reference Budaev, Savin and Zelenyi2011 and other related papers). With the exact result ${\it\zeta}(3)=1$ and by using the ESS property, with experimental data one can obtain rather accurate values of ${\it\zeta}(q)$ considering ${\it\zeta}(q)/{\it\zeta}(3)$ by plotting $S(q)$ as a function of $S(3)$ .

Figure 17.

The ESS plot of the structure function $S_{q}({\it\tau})$ of different orders ( $q=2,3,4,5,6,7,8$ from bottom up) versus the third-order structure function: (a) the magnetic field $B_{x}$ in the TBL near MP, (b) plasma density in edge of T-10 tokamak, (c) plasma density in NAGDIS-II device, (d) SW ion flux, BMSW data SPECTR-R on 28.09.2011 (for $q=2,3,4,5,6,7,8,9$ from bottom up). A linear behaviour is interpreted that the ESS holds.

From this point, we are going to discuss some critical issues regarding the ESS property and determination of the intermittency exponents.

The property of ESS suggests that scaling properties in turbulence should be investigated as a function of the generalized scale ${\it\Omega}(l)$ , rather than a function of the resolution scale $l$ . This is similar to critical phenomena in a finite size system in which the scaling properties are considered as a function of a correlation length, depending on the system size and diverging at the critical point. The ESS property can become fully compatible with a generalized scale transformation moment hierarchy in the log-Poisson She–Leveque–Dubrulle model (see appendix B) in terms of a generalized scale ${\it\Omega}(l)=\langle \langle {\it\varepsilon}_{l}\rangle ^{-1}{\it\delta}{\it\upsilon}_{l}^{3}\rangle$ , (see Dubrulle Reference Dubrulle1994) i.e. depending on the scale defined by the third-order moment. The She–Leveque–Dubrulle model postulates that the moment hierarchy originates from some hidden symmetry of the NSE, Dubrulle (Reference Dubrulle1994). The plasma turbulence is treated by governing equations which are more complicated than the NSE. At the same time, all these governing equations remain formally invariant (as with the NSE) under any affine contraction of the space–time $x\rightarrow x/{\it\lambda}$ , $t\rightarrow t/{\it\lambda}^{1-h}$ , (of scale ratios ${\it\lambda}$ ), see discussion in Cartes et al. (Reference Cartes, Bustamante, Pouquet and Brachet2009). There are no limitations to suggest that hidden symmetry, similar to NSE, can regulate plasma dynamics: symmetry transforms for MHD equations and other equations used for a description of plasma dynamics are considered in Gridnev (Reference Gridnev1968), Nucci (Reference Nucci1984), Samokhin (Reference Samokhin1985), Gusyatnikova et al. (Reference Gusyatnikova, Samokhin, Titov, Vinogradov and Yamaguzhin1989), Bogoyavlenskij (Reference Bogoyavlenskij2002). Thus, observation of the ESS property for plasma data is an argument to examine the scalings proposed by the log-Poisson models in the same way as in hydrodynamics.

Questions regarding similarity of the energy transfer cascade process in neutral flow and magnetofluids can arise on this way. In laboratory plasma, dissipation occurs on different scales, including large scales, which supposes an inverse cascade process. While in hydrodynamics, theoretical models assume a single direct cascade to small scales in terms of a scalar cascade approach. However in 3-D hydrodyamic flows, the existence of an inverse cascade is still an open question (see discussions in Paret & Tabeling Reference Paret and Tabeling1998; Boffetta et al. Reference Boffetta, Cenedese, Espa and Musacchio2005; Celani, Musacchio & Vincenzi Reference Celani, Musacchio and Vincenzi2010). Strictly speaking, in 3-D, the scalar approach is insufficient for a description of both the hydrodynamics and plasma dynamics, and turbulence should be studied using vector Lie cascades (see discussion in appendix A). However, the main features of cascades can be revealed using the scalar approach in terms of multifractals. The multifractal approach to the intermittency proposes self-similarity analysis independently of the issue of cascade direction (direct or inverse) – multifractal measures are a very general outcome of stochastic cascades.

It is worth noting that the absolute value of the velocity differences are often used to construct structure functions from experimental space data. Such structure functions of absolute values of velocity are not described by any theory. Whereas it is often the first step to make meaningful analysis of statistics from experimental data.

A fundamental open question connected to intermittency is the dependency of velocity statistics at various temporal scales on large scale forcing and boundary conditions: the so-called problem of universality (see Arneodo et al. Reference Arneodo, Benzi, Berg, Biferale, Bodenschatz, Busse, Calzavarini, Castaing, Cencini and Chevillard2008). Universality features are linked to the degree of anisotropy and non-homogeneities of the turbulent statistics. Different forms of anisotropy might have different effects on the small scales of turbulence. In hydrodynamics, the conjecture of local isotropy has received much attention, and has been tested in hydrodynamics flows with various kinds of anisotropy (see e.g. Biferale & Procaccia Reference Biferale and Procaccia2005). Understanding the way turbulence tends towards local isotropy from various states of anisotropy remains an important challenge in the study of turbulence. Different forms of anisotropy might have different effects on the small scales of plasma turbulence. In hydrodynamics, the conjecture of local isotropy has received much attention, and has been tested in flows with various kinds of anisotropy (see e.g. Biferale & Procaccia Reference Biferale and Procaccia2005). Finite Reynolds number effects and anisotropy may lead to a discrepancy of longitudinal and transverse components of the structure functions and scaling exponents (Biferale & Procaccia Reference Biferale and Procaccia2005). These authors have suggested an approach to treat the anisotropy problem in the following way: it is necessary to decompose the structure function of each order into the symmetry group determined by the large-scale flow and intermittency exponents calculated sector-by-sector. In a system where the large-scale isotropy is broken, for example in the plasma TBL, such an approach will help to improve intermittency investigation in plasma turbulence. This is an important consideration in the case of interpreting SW measurements in terms of anisotropic turbulence theories (see, e.g. Chen et al. Reference Chen, Wicks, Horbury and Schekochihin2010).

To summarise, ESS is a procedure that remarkably extends the range of scaling for structure functions and thus allows improved determination of intermittency exponents. In spite of several attempts to explain the success of ESS (see discussion, e.g. in Chakraborty, Frisch & Ray Reference Chakraborty, Frisch and Ray2010), the latter is still not fully understood. It was shown by Chakraborty et al. (Reference Chakraborty, Frisch and Ray2010), that by the process of ESS, the sub-leading terms to the leading scaling exponent are made smaller. This was shown for the Burgers equation and speculated for the NSE. ESS has been applied to Burgers turbulence at high Reynolds numbers and a gain of about three-quarters of a decade was numerically observed: there is a reduction of the subdominant contributions to scaling when going from the standard structure function representation to the ESS one (Chakraborty et al. Reference Chakraborty, Frisch and Ray2010).

4.1. Comparison with She–Leveque–Dubrulle model

The phenomenon of intermittency is explained in the framework of the log-Poisson models. The log-Poisson models were developed by the generalization of ${\it\beta}$ -model incorporating asymmetry and the hypothesis of scale invariance and cascade. The assumption of the existence of a random multiplicative process for the inertial range dynamics is important. Over a finite range of scales, a quantized cascade provides a log-Poisson process containing an infinite number of discrete values for the multipliers. It considerably contrasts to previous discrete cascade models which contain a finite number of multipliers: the ${\it\alpha}$ -model and $p$ -model contain two multipliers, the random ${\it\beta}$ -model contains three multipliers. Comparative study of the velocity and magnetic field in the interplanetary medium regarding scalings proposed by the ${\it\alpha}$ -model, $p$ -model, the random ${\it\beta}$ -model, IK model has been done in a wide range of scales (see Carbone et al. Reference Carbone, Veltry and Bruno1996; Budaev et al. Reference Budaev, Savin and Zelenyi2011). The log-Poisson intermittency model reproduces these experimental and numerical findings in hydrodynamic and MHD (see Carbone et al. Reference Carbone, Veltry and Bruno1996; Müller & Biskamp Reference Müller and Biskamp2003; Budaev et al. Reference Budaev, Savin and Zelenyi2011) turbulence very well.

The She–Leveque–Dubrulle model (Dubrulle Reference Dubrulle1994; She & Leveque Reference She and Leveque1994) is explained in appendix B. It assumes the existence of a limiting value ${\it\varepsilon}_{l}^{\infty }$ associated with the most dissipative structures. Advantages of the log-Poisson models are the consideration of intermittent turbulence properties such as the scale invariance, power law scaling for dissipation energy of the most intermittent dissipative structures, ${\it\varepsilon}_{l}^{\infty }\sim l^{-{\it\Delta}},l\rightarrow 0$ where the parameter ${\it\Delta}$ is associated with the geometry (dimension) of dissipative structures. To treat the plasma turbulence problem, we can suggest a geometry of dissipative structures from experimental observation of theoretical consideration. In this subsection, we review the comparison of experimental scalings with the prediction of the She–Leveque–Dubrulle model:

For 3-D isotropic hydrodynamic turbulence, She & Leveque (Reference She and Leveque1994) proposed that ${\it\Delta}={\it\beta}=2/3$ . The fitting of experimental data to scaling (4.1) is a way to classify the intermittency property.

She & Waymire (Reference She and Waymire1995) considered a stochastic multiplicative cascade in which complicated topological structures, namely, dissipative structures of different dimensions (in particular, those having fractal dimensions), can be formed simultaneously. In probability theory, such a process is described by the Khintchine–Levy approach (She & Waymire Reference She and Waymire1995). This approach is important for interpretation of the experimental data when experimental scalings deviate insignificantly from scaling (4.1) with fixed ${\it\Delta}$ and ${\it\beta}$ values. The deviation can arise from the complicated geometry of the dissipative structures, or from the fact that structures of different dimensions are present in the process simultaneously. This is why a real turbulent process can be characterized by the fitting values of ${\it\Delta}$ and ${\it\beta}$ .

The log-Poisson model reproduces experimental and numerical results in hydrodynamic (see Frish Reference Frish1995) and plasma turbulence rather well (see Budaev et al. Reference Budaev, Savin and Zelenyi2011). In figure 18, the structure function scalings ${\it\zeta}(q)$ of space and fusion devices data are shown in the same plot with the scalings predicted by the K41 and the standard log-Poisson (She–Leveque, SL) models. The scalings are anomalously deviated from the K41 scaling. Many experimental spectra can be described by the log-Poisson model (4.1) with adjusted parameters ${\it\Delta}$ and ${\it\beta}$ : ${\it\Delta}=0.12{-}0.8$ , ${\it\beta}=0.25{-}0.7$ , see table 1. It is worth stressing that different observables (magnetic field, velocity, density, fluxes) have the closed multifractal measures which characterize intermittency (table 1).

Figure 18.

(a) The scaling law of the high-order structure function ${\it\zeta}(q)/{\it\zeta}(3)$ for the Earth’s magnetosphere and edge plasma fluctuations in fusion devices. The dashed line shows the Kolmogorov K41 scaling $q/3$ , the solid line shows the log-Poisson model prediction for ${\it\beta}={\it\Delta}=2/3$ by the She–Leveque model (SL). (b) The deviation of the scaling laws of the structure function from Kolmogorov K41spectrum. The edge plasma in T-10 tokamak: (T-10 n far SOL) is the plasma density, (T-10 ${\it\Gamma}$ far SOL) is the particle flux; plasma density in NAGDIS-II in the attached mode (N-II attach) and detached mode (N-II detach); the ion flux in the TBL of the Earth’s magnetosphere (TBL ion flux), the magnetic field in the TBL of the Earth’s magnetosphere (TBL $B_{x}$ , 19.06.1998), the magnetic field in the MSH outside TBL of the Earth’s magnetosphere (MSH $B_{x}$ ) and in the SW from the Geotail data (SW, $B_{x}$ ).

Table 1.

Indicies ${\it\Delta}$ and ${\it\beta}$ .

Values of ${\it\Delta}$ between ${\sim}1/3$ and $2/3$ are found (table 1); these cases can be interpreted as an evidence of dominant contribution of 1-D (filament-like) dissipative structures to the process. Such dissipative structures appear to have not a trivial geometrical topology but a fractal one. Only in the velocity shear layer in tokamaks (where the destruction of turbulent eddies is assumed) and in MSH far from TBL of the Earth’s magnetosphere, non-intermittent signals with ${\it\beta}\approx 1$ were observed.

To verify the hypotheses regarding power laws in the log-Poisson model, we can analyse the scalings for the relative moments

and examine the dependence

The index ${\it\delta}_{q}$ can be changed between ${\it\delta}_{0}$ , as averaged flow scales ${\it\Pi}_{0}(l)\sim l^{{\it\delta}0}$ , and ${\it\delta}_{\infty }$ , consistent with limit scaling ${\it\Pi}_{\infty }(l)\sim l^{{\it\delta}\infty }$ . A dependence of ${\it\delta}_{q}$ on $q$ for different models:

1. (I) ${\it\delta}_{q}\equiv 0$ – Kolmogorov K41.

2. (II) ${\it\delta}_{q}\equiv \text{const.}$ , isolated vorticities in the turbulence

3. (III) ${\it\delta}_{0}=0$ , ${\it\delta}_{\infty }=2/3$ – She–Leveque model.

4. (IV) ${\it\delta}_{0}=\text{const.}$ , ${\it\delta}_{\infty }$ is limited, She–Leveque–Dubrulle model ${\it\delta}_{q}={\it\delta}_{\infty }+{\it\zeta}_{3}{\rm\Delta}Q(q)$ , where $Q(q)$ -monotonic decay function $Q(0)=1$ , $Q(\infty )=0$ , e.g.  $Q(q)=\exp (-a_{c}q)$ .

Figure 19 illustrates experimental observations of the case (IV). We point out that these results support hypothesis of moments hierarchy and generalized scale invariance of the plasma turbulence in terms of She–Leveque–Dubrulle model (see appendix B).

Figure 19.

Index ${\it\delta}_{q}$ versus $q$ . Magnetic field $B_{x}$ in the Earth’s magnetosphere TBL, plasma density in SOL (T-10 SOL) at $r=31~\text{cm}$ and in the far SOL (T-10 far SOL) at $r=36~\text{cm}$ in the T-10 tokamak.

4.2. Anisotropic turbulence cascade and dimension of dissipative structures

The scaling properties depend on a topology of singular dissipative structures (see Frish Reference Frish1995). The dimension of dissipative structures $\mathsf{D}$ (see figure 20 a) relates to the probability distribution of active turbulent zones in intermittent turbulence. Therefore it is important to know which dissipative structures are dominant (if structures of different topology co-exist) and determine the scaling properties.

Figure 20.

(a) Diagram of singular objects (dissipative structures in magnetic field $B$ ) of different dimension $D$ in a unit cube. The probability of having a given ${\it\delta}{\it\upsilon}_{l}$ in a sphere of radius $l$ by an object with dimension $D$ is $\mathsf{P}(l)\sim l^{3\text{-}D}$ at $l\rightarrow 0$ (see Frish Reference Frish1995). (b) Diagram of the nonlinear interaction of filamentary dissipative structures ( $D=1$ ) in the cross-field direction.

In the She–Leveque model of 3-D isotropic hydrodynamics, dissipative structures are considered to be 1-D filaments. In the Biskamp–Mueller model (BM) of 3-D isotropic MHD turbulence, dissipative structures are assumed to be 2-D current sheets (Müller & Biskamp Reference Müller and Biskamp2003). A diagram of the process when nonlinear interaction across the magnetic field is of Kolmogorov’s type locality, i.e. only modes with close $k\bot$ interact nonlinearly, is presented in figure 20(b). This scheme illustrates the condition in the SOL (edge plasma) of tokamaks and other fusion devices: the magnetic field lines are open and turbulence is characterized by flute-like structures along the magnetic field (which could be a result of interchange instability). Inside the Earth’s magnetosphere, field-aligned perturbations are also observed. In boundary layers with a high ratio of thermal to magnetic pressure, the jets are extended along the direction of the mean flow velocity (see Savin et al. Reference Savin, Skalsky, Zelenyi, Avanov, Borodkova, Klimov, Lutsenko, Panov, Romanov and Smirnov2005 and references therein). In the general case, as was discussed in the pioneering work by Zeldovich et al. (Reference Zeldovich, Molchanov, Ruzmaikin and Sokolov1987), the filamentary structures produce the dominant contribution to intermittent manifestation of magnetized plasma dynamics.

In the framework of the log-Poisson model, to consider a geometry of the most singular dissipative structures we have to treat a scaling of energy dissipation ${\it\varepsilon}_{l}^{\infty }\sim l^{-{\it\Delta}}$ and the scaling of velocity ${\it\upsilon}_{l}\sim l^{1/g}$ (Müller & Biskamp Reference Müller and Biskamp2003) with the parameter $g=3$ for K41, $g=4$ for IK. In a random cascade model, the intermittency index ${\it\beta}$ relates with ${\it\Delta}$ and co-dimension $C_{0}$ of the most singular dissipative structures: ${\it\beta}=1-{\it\Delta}/C_{0}$ (see Politano & Pouquet Reference Politano and Pouquet1995; Boldyrev Reference Boldyrev2002) $C_{0}=3-D$ , where $D$ is the dimension of dissipative structures. In 3-D, $C_{0}=2$ for filaments ( $D=1$ ), $C_{0}=1$ for micro-sheets ( $D=2$ ). The log-Poisson model scaling:

Usually ${\it\Delta}$ and $g$ are related assuming an equal scaling for the time scale $t_{l}^{\infty }$ of the rate ${\it\varepsilon}_{l}^{\infty }\sim E^{\infty }/t_{l}^{\infty }$ ( $E^{\infty }$ is the amount of the energy dissipated in the most singular structures) and for the nonlinear transfer time $t_{l}^{NL}$ of the energy cascade rate ${\it\varepsilon}\sim {\it\delta}{\it\upsilon}_{l}^{2}/t_{l}^{NL}$ . From ${\it\varepsilon}_{l}^{\infty }\sim l^{-{\it\Delta}}$ and ${\it\upsilon}_{l}\sim l^{1/g}$ , we obtain ${\it\Delta}=2/g$ . The hydrodynamic SL formula results from $C_{0}=2$ , $g=3$ , ${\it\Delta}=2/3$ . The IK MHD model assumes $g=4$ , ${\it\Delta}=1/2$ , and $C_{0}=1$ with dissipative structures interpreted as 2-D current sheets. In DNS of 3-D isotropic MHD turbulence (the BM model, BM Müller & Biskamp Reference Müller and Biskamp2003) scalings are reproduced well with the combination $g=3$ , ${\it\Delta}=2/3$ , and $C_{0}=1$ implying hydrodynamic scaling and sheet-like dissipative structures. To consider anisotropy statistics of the cascade strength, we may follow the phenomenological interpretation of Biskamp and Mueller by dropping the scaling equality of $t_{l}^{NL}$ and $t_{l}^{\infty \prime }$ . Instead, $t_{l}^{\infty }$ is fixed to the K41 time scale, $t_{l}^{\infty }\sim l/{\it\delta}{\it\upsilon}_{l}\sim l^{1-1/g}$ , ${\it\Delta}=1-1/g$ , which with $C_{0}=1$ leads to Müller & Biskamp (Reference Müller and Biskamp2003):

As an alternative we can consider the same scaling $t_{l}^{\infty }\sim l/{\it\delta}{\it\upsilon}_{l}\sim l^{1-1/gf}$ with $C_{0}=2$ of 1-D filament-like structures (see figure 20 b). On this basis, the anomalous scaling that captures 1-D filament-like structures was proposed by Budaev (Reference Budaev2009):

The quantity $g_{f}/3$ expresses the cascade strength relative to the isotropic K41 case (here we use the notation $g_{f}$ to distinguish from the case $g$ in (4.5)). The modified transfer time $t_{l}^{NL}\sim (l/l_{0})^{{\it\theta}}~(l/{\it\delta}{\it\upsilon}_{l})$ can be considered detaching $t_{l}^{NL}$ and $t_{l}^{\infty }$ to characterize the strength of the field-perpendicular and field-parallel polarized fluctuations. Here, $l_{0}$ is an arbitrary reference length, and ${\it\theta}$ is a dimensionless efficiency parameter. Assuming a constant energy cascade ${\it\varepsilon}\sim {\it\delta}{\it\upsilon}_{l}^{2}/t_{l}^{NL}=\text{const.}$ , $t_{l}^{NL}\sim l^{(1+{\it\theta})2/3}$ . In a standard phenomenology $t_{l}^{NL}\sim l^{2/g}$ . In a such approach, the cascade efficiency is controlled by the $(l/l_{0})^{{\it\theta}}$ factor (Müller & Biskamp Reference Müller and Biskamp2003):

1. (i) ${\it\theta}=0~(g,g_{f}=3)$ yields the isotropic K41 cascade;

2. (ii) ${\it\theta}<0~(g,g_{f}>3)$ corresponds to a cascade enhancement;

3. (iii) ${\it\theta}>0~(g,g_{f}<3)$ corresponds to a cascade depletion.

Scaling laws (4.5) and (6.5) are helpful to classify experimental data and to prove the hypothesis of a topology and the dimension of dominant dissipative structures.

For the T-10 tokamak scaling (4.6) was tested in Budaev (Reference Budaev2009). In the main edge region of T-10 ( $30~\text{cm}<r<34~\text{cm}$ ), the scaling is described by the model with 1-D dissipative structures (4.6), figure 21(a). Only in the far edge (far SOL) region ( $r>34~\text{cm}$ ), the scaling is close to the scaling (4.5) of the model which captured 2-D sheet-like dissipative structures. This result gives evidence of the dominant presence of 1-D singular dissipative structures in edge plasma of a fusion device. Observation of space plasma (see examples in figure 21 b) demonstrate scaling properties which captured 1-D filamentary and 2-D sheet-like dissipative structures. Table 2 lists values of $g$ and $g_{f}$ in space and laboratory plasmas. In most experiments, the parameter $g_{f}$ is close to 3: filament-like dissipative structures (1-D topology) dominate intermittent turbulence. It appears that the intermittency observed in space and laboratory plasmas displays universal properties.

Figure 21.

(a) Scaling for the structure function (its deviation from the K41 prediction). Edge plasma of the T-10 tokamak. Shown are the plasma density at $r=34~\text{cm}$ in the SOL region (triangles) and the cross-field particle flux ${\it\Gamma}$ at $r=36~\text{cm}$ in the far SOL region (circles). The K41 model (dashed curves), the log-Poisson model with two-dimension 2-D (solid curves – fitting by (4.5) with $g=2.85$ ) and one-dimension 1-D (dotted curves – fitting by (4.6) with $g_{f}=3.03$ ) dissipative structures in three dimensions with anisotropy from magnetic field. (b) Scalings for the structure function. SPECTR-R, 20.03.12, ion flux, TBL, 12-21 UT, 3 Hz (circles), CLUSTER-4, 20.03.12, ion flux, SW/BS/MSH/MP (triangles). Kolmogorov K41 model (dashed line). The log-Poisson model with two-dimension 2-D (dashed-dotted curves – fitting by (4.5) with $g=2.90$ ); and one-dimension 1-D (solid curves – fitting by (4.6) with $g_{f}=3.008$ ) dissipative structures in 3-D with anisotropy from magnetic field.

Table 2.

Indicies $g$ and $g_{f}$ .

The above approach makes it possible to utilize the language of the structure function’s scaling in order to characterize turbulent structures of different spatial scales. The results of the above analysis provide insight into the topology of the structures responsible for dissipation in plasma turbulence. The theoretical prediction of scaling laws depending on dissipative structure dimension and cascade are summarized in table 3. All cascade models of intermittent turbulence have adjustable parameters that are difficult to determine from first principles and physical arguments and that provide enough freedom to account for the experimental data (e.g. determined by the scaling ${\it\zeta}(q)$ of the structure functions). The scaling ${\it\zeta}(q)$ is helpful to validate cascade models of developed turbulence with intermittency.

Table 3.

Scaling laws predicted by different cascade models.

The results of the above analysis in the framework of the log-Poisson model can be interpreted by considering the topology of the structures responsible for a dissipation in the plasma TBL. They can be associated both with the large-scale coherent structures (which are vivid in experiments) and with small-scale structures (which are usually not seen) of intermittent turbulence. We note that structures of different topology can coexist in the framework of the log-Poisson model (She & Waymire Reference She and Waymire1995).

A theoretical analysis (She & Waymire Reference She and Waymire1995) of a stochastic multiplicative cascade (such as the log-Poisson cascade) shows that the process can involve generating dissipative structures of different (in particular, fractal) geometries (as was also discussed by Zeldovich et al. Reference Zeldovich, Molchanov, Ruzmaikin and Sokolov1987). As was discussed above, the properties of the experimental scalings for 1-D dissipative structures are adequately described by the log-Poisson model. The approach used here does not provide information on the particular spatial shape of quasi-1-D dissipative structures. From the properties of MHD flows, it is possible to assume that they have complex shape, stretched predominantly along the magnetic field. Additional experimental observations are needed to clarify this issue.