2 Measurements and methods

The BMSW instrument on board the high apogee Spektr-R spacecraft (astrophysical Russian mission) can provide measurements of the ion flux vector, proton density, bulk and thermal velocity in the solar wind and in the Earth’s magnetosheath with time resolution up to 32 ms. The main axis of the instrument is oriented along the Sun–Earth line within the limits $\pm 5$ – $10^{\circ }$ . The deviation angle of the device axis from the Sun–Earth line can be determined with a special solar sensor (DSS) with an accuracy of $1^{\circ }$ . BMSW measurements are based on six Faraday cups. Three sensors are declined from the main axis of the instrument to determine the ion flux direction. Three other sensors are intended for measurements of the basic plasma moments under a Maxwellian approximation. The ion moments can be determined with time resolution ${\approx}32~\text{ms}$ only part of the time (in adaptive mode) the rest of the time the integral energy distribution function and its moments are determined with a time resolution of ${\approx}3~\text{s}$ (sweeping mode). However, the ion flux vector can be measured with the highest time resolution during the whole time of instrument operation. A more detailed description of the instrument principles can be found in Zastenker et al. (Reference Zastenker2013), Šafránková et al. (Reference Šafránková, Němeček, Přech, Zastenker, Čermák, Chesalin, Komárek, Vaverka, Beránek and Pavlů2013a ). The time resolution of BMSW data is sufficient for the analysis of turbulent properties of the ion plasma fluctuations up to 16 Hz. We select several periods of the continuous ion flux measurements in the solar wind with duration longer than 3 h. We use also Wind data because magnetic field measurements are missed on Spektr-R. We try to select intervals as long as possible, but we are limited by the Spektr-R telemetry because only 10 % of full time resolution measurements are transmitted to the Earth.

The selected periods are divided into ${\approx}17$  min subintervals (note that a number of data points expressed by a degree of two is favourable for a Fourier analysis and we use $2^{15}=32\,768$ points in each interval). The requirement of stationarity limits the usage of longer intervals. The length of the intervals is enough to analyse the spectra in the frequency range of 0.01–16 Hz but we use the frequency range only up to 10 Hz because results of the in-flight calibration and laboratory tests show that the noise level can be significant for frequencies higher than 8–10 Hz (Šafránková et al. Reference Šafránková, Němeček, Přech and Zastenker2013b ; Chen et al. Reference Chen, Sorriso-Valvo, Šafránková and Němeček2014a ). Nevertheless, longer intervals can be used to clarify the shape of the spectrum. Intervals are overlapped by half of the interval length and we analyse approximately 700 of such intervals. Frequency spectra are calculated for each interval by the fast Fourier transform (FFT) method smoothed in the frequency domain (with the use of a Hamming window) and sorted by shape using an automated routine based on spectral slopes. The selection is checked manually.

Our selection is not limited to the pristine solar wind. We further sort our intervals in accordance with the catalogue of Yermolaev ftp://ftp.iki.rssi.ru/pub/omni/catalog/ (Yermolaev et al. Reference Yermolaev, Nikolaeva, Lodkina and Yermolaev2009) and we note that our set consists of nearly equal proportions of three types of solar wind: (i) slow solar wind regions ( ${\sim}40\,\%$ of the time); (ii) regions of interplanetary coronal mass ejections (‘EJECTA’) (or complex phenomena such as compressed regions inside EJECTA-like events) ( ${\sim}30\,\%$ of the time); and (iii) compression regions before high-speed streams, so-called corotating interaction regions (CIR), and before EJECTA or magnetic clouds (MC), so-called SHEATH regions (CIR and SHEATH regions are observed ${\sim}30\,\%$ of the time). MCs are not included in our statistics because they are less common (Yermolaev et al. Reference Yermolaev, Nikolaeva, Lodkina and Yermolaev2009). The instrument limitation of the velocity measurements of up to ${\sim}$ 600– $700~\text{km}~\text{s}^{-1}$ do not allow us to observe the high-speed streams.

The averaged values of plasma parameters are calculated for each interval of the BMSW measurements (the proton density $N_{p}$ , the bulk velocity $V_{p}$ , the proton temperature $T_{p}$ , the alpha density $N_{\unicode[STIX]{x1D6FC}}$ and the relative abundance of helium $N_{\unicode[STIX]{x1D6FC}}/N_{p}$ with 3 s time resolution). As it was noted above, data from a magnetometer are not available on board Spektr-R. However, the magnetic field magnitude $B$ is determined for each interval from Wind measurements shifted to the position of Spektr-R (Wind data are downloadable from the CDAWeb data base). This allows us to determine also other parameters which are often considered as affecting the properties of turbulence: the Alfvén velocity $V_{A}$ , the proton plasma beta $\unicode[STIX]{x1D6FD}_{p}$ , the cyclotron frequency $f_{c}$ , the proton gyroradius $\unicode[STIX]{x1D70C}$ , the inertial length $\unicode[STIX]{x1D706}$ , the inertial length frequency $f_{\unicode[STIX]{x1D706}}$ and the gyrostructure frequency $f_{\unicode[STIX]{x1D70C}}$ (see § 1).

According to the Taylor hypothesis, the observed ion flux variations in the solar wind can be considered as the spatial variations passing by the spacecraft with the bulk velocity. To confirm the possibility of application of the Taylor hypothesis we always test the ratio of $V_{p}/V_{A}$ . The mean value of this ratio is ${\sim}10$ for our statistics. We rejected the cases with $V_{p}/V_{A}<3$ . It is known that the Taylor hypothesis can be violated in the case of quasi-parallel propagating whistler waves (e.g. Klein et al. Reference Klein, Howes and TenBarge2014). We exclude the intervals suspected as foreshocks, where whistler waves are mainly observed. We assume that the possible influence of whistler waves on the solar wind turbulence is rather weak (Alexandrova et al. Reference Alexandrova, Carbone, Veltri and Sorriso-Valvo2008).

The statistical properties of the ion flux fluctuations (such as probability distribution functions, their high-order structure functions, moments and scaling) are also analysed for the same time intervals (details will be explained in § 5).

3 Variability of the shapes of the ion flux fluctuation spectra at the ion scales

Spectra of the plasma fluctuations in the kinetic range can always differ strongly from the spectra at the MHD scale (see § 1). The character of these distinctions can vary for different observation intervals at different plasma conditions (Bruno & Carbone Reference Bruno and Carbone2013). In this section, we present several examples of different shapes of spectra and in the next section we compare statistically the plasma parameters typical for different types of spectra.

Figure 1.

Examples of different spectral shapes of the ion flux fluctuations in the solar wind: spectra with two slopes and one break (a), spectra with nonlinear steepening in the kinetic range (b), spectra with flattening in the vicinity of the break (c), spectra with bump in the vicinity of the break (d), spectra without steepening in the kinetic range (e). Black lines show the linear approximation of different areas of spectra. A power law with slope $-5/3$ (Kolmogorov slope) is shown in all graphs by an oblique dotted line. Vertical dotted lines correspond to the frequencies of breaks or peaks.

Usually a significant number of fluctuation spectra in the solar wind can be well fitted by power laws with different slopes in the MHD and in the kinetic ranges (Smith et al. Reference Smith, Hamilton, Vasquez and Leamon2006; Alexandrova et al. Reference Alexandrova, Saur, Lacombe, Mangeney, Mitchell, Schartz and Robert2009; Perri et al. Reference Perri, Carbone and Veltri2010; Bourouaine et al. Reference Bourouaine, Alexandrova, Marsch and Maksimovic2012; Šafránková et al. Reference Šafránková, Němeček, Přech, Zastenker, Čermák, Chesalin, Komárek, Vaverka, Beránek and Pavlů2013a ,Reference Šafránková, Němeček, Přech and Zastenker b ; Riazantseva et al. Reference Riazantseva, Budaev, Zelenyi, Zastenker, Pavlos, Safrankova, Nemecek, Prech and Nemec2015). In this case, the power-law ranges are divided by a clear break point. This type of spectrum is observed in ${\sim}50\,\%$ of the solar wind intervals analysed in our statistics. Below we will call them spectra with two slopes and one break.

Figure 1(a) presents an example spectrum of the solar wind ion flux fluctuations observed on 2nd August 2012 12:44-13:01 UT that exhibits two slopes and one break. The mean values of plasma parameters during this period are: the ion density $N_{p}=12~\text{cm}^{-3}$ , the plasma bulk velocity $V_{p}=406~\text{km}~\text{s}^{-1}$ , the proton temperature $T_{p}=13$  eV, the interplanetary magnetic field magnitude $B=10~\text{nT}$ , the Alfvén velocity $V_{A}=66~\text{km}~\text{s}^{-1}$ , the proton plasma beta $\unicode[STIX]{x1D6FD}_{p}=0.6$ , the cyclotron frequency $f_{c}=0.15~\text{Hz}$ , the proton gyroradius $\unicode[STIX]{x1D70C}=51~\text{km}$ , the proton inertial length $\unicode[STIX]{x1D706}=68~\text{km}$ , the inertial length frequency, $f_{\unicode[STIX]{x1D706}}=0.95~\text{Hz}$ , the gyrostructure frequency $f_{\unicode[STIX]{x1D70C}}=1.3~\text{Hz}$ . Two clear power-law parts of the spectrum with slopes $P_{1}=-1.62$ at the MHD frequency scales, and $P_{2}=-2.67$ at higher frequencies, are observed. In Riazantseva et al. (Reference Riazantseva, Budaev, Zelenyi, Zastenker, Pavlos, Safrankova, Nemecek, Prech and Nemec2015), we have analysed spectra with two slopes and one break separately and have shown that the average slope in the kinetic range is equal to ${\sim}\langle P_{2}\rangle =-2.9$ , which is rather typical for the fluctuations of other plasma parameters such as density and bulk velocity observed on board Spektr-R (Šafránková et al. Reference Šafránková, Němeček, Přech, Zastenker, Čermák, Chesalin, Komárek, Vaverka, Beránek and Pavlů2013a ,Reference Šafránková, Němeček, Přech and Zastenker b , Reference Šafránková, Němeček, Němec, Přech, Chen and Zastenker2016) as well as for the plasma and the magnetic field fluctuations observed in numerous other experiments (see the review of Alexandrova et al. (Reference Alexandrova, Chen, Sorisso-Valvo, Horbury and Bale2013)). Less steep slopes can be observed if the dissipation is weakened (e.g. Smith et al. Reference Smith, Hamilton, Vasquez and Leamon2006). The break in figure 1(a) is determined as the intersection of the slopes obtained by independent linear fits of the lower (MHD) and upper the (kinetic) frequency range (e.g. Bourouaine et al. Reference Bourouaine, Alexandrova, Marsch and Maksimovic2012). The frequency of the break is equal to $F_{b}=1.02~\text{Hz}$ which is ${\sim}7$ times greater than the cyclotron frequency $f_{c}$ , and is approximately equal to the inertial length frequency $f_{\unicode[STIX]{x1D706}}$ . It seems reasonable and in accordance with the relationship of the break frequency in the spectra of the magnetic fluctuations with the inertial length for low plasma $\unicode[STIX]{x1D6FD}_{p}$ (Chen et al. Reference Chen, Leung, Boldyrev, Maruca and Bale2014b ). In Riazantseva et al. (Reference Riazantseva, Budaev, Zelenyi, Zastenker, Pavlos, Safrankova, Nemecek, Prech and Nemec2015), we have shown that the break frequency for the ion flux fluctuation spectra varies in a wide range and equals on average to $\langle F_{b}\rangle =1.9\pm 0.8~\text{Hz}$ . Taking into account the dispersion, this value agrees well with the break frequency for the ion density fluctuations ${\sim}1.6~\text{Hz}$ (Šafránková et al. Reference Šafránková, Němeček, Přech and Zastenker2013b ) and the electron density fluctuations ( ${\sim}1~\text{Hz}$ , Chen et al. (Reference Chen, Salem, Bonnell, Mozer and Bale2012)). The break frequency of the spectrum of the velocity fluctuations (Šafránková et al. Reference Šafránková, Němeček, Němec, Přech, Chen and Zastenker2016) is smaller ( ${\sim}0.4~\text{Hz}$ ) in comparison with the break of the density spectra and this difference tends to decrease with $\unicode[STIX]{x1D6FD}$ . The experimental observations of the magnetic field fluctuations also show a large dispersion of the break frequency ${\sim}0.1$ –0.7 Hz (see review of Alexandrova et al. (Reference Alexandrova, Chen, Sorisso-Valvo, Horbury and Bale2013)). The experimentally observed spectra of the fluctuations of the plasma or the magnetic field do not allow us to establish an unambiguous correlation of the break frequency of one of the plasma characteristic scales (see discussion in § 1). The selection of the main physical processes controlling the break is a complex problem which also has to take into account the different contributions of coherent structures, waves and non-coherent fluctuations which can influence the spectrum formation for different plasma parameters (Lion et al. Reference Lion, Alexandrova and Zaslavsky2016). These factors can probably explain the formation of spectra shapes differing from those discussed above.

The steepening of the spectrum in the kinetic range does not always follow a power law. The second type of spectrum under consideration is a spectrum with nonlinear steepening in the kinetic range. Figure 1(b) is a typical example of a spectrum of this type that was observed on 27 September 2011 22:00-22:17 UT. The mean plasma parameters during this period were: $N_{p}=18~\text{cm}^{-3}$ , $V_{p}=553~\text{km}~\text{s}^{-1}$ , $T_{p}=5$  eV, $B=7~\text{nT}$ , $V_{A}=34~\text{km}~\text{s}^{-1}$ , $\unicode[STIX]{x1D6FD}_{p}=0.9$ , $f_{c}=0.1~\text{Hz}$ , $\unicode[STIX]{x1D70C}=52~\text{km}$ , $\unicode[STIX]{x1D706}=54~\text{km}$ , $f_{\unicode[STIX]{x1D706}}=1.6~\text{Hz}$ , $f_{\unicode[STIX]{x1D70C}}=1.7~\text{Hz}$ . So the discussed event is observed in a rather fast and dense solar wind. The traditional linear fit shows that a clear break point is not observed. The spectrum is smoothly descending while the slope of the spectrum is growing toward higher frequencies and reaching a very high value of $P_{2}=-4.9$ , whereas the power-law slope at the MHD scale is equal to the typical value $P_{1}=-1.65$ . The spectrum begins to steepen at 1–2 Hz which is close to both the inertial length and the gyrostructure frequencies (they are approximately equal to each other because of $\unicode[STIX]{x1D6FD}_{p}\sim 1$ ). It is necessary to stress that the mean velocity during this period is rather high and we can say that this type of spectrum is typically observed in the fast solar wind. This corresponds well to Pitňa et al. (Reference Pitňa, Šafránková, Němeček, Goncharov, Němec, Přech, Chen and Zastenker2016) who showed that the ion density spectra fall faster downstream of interplanetary shocks, where the value of the velocity is always large. We also see the nonlinear rapid steepening of the spectrum in high-velocity streams not associated with shocks. At the same time, Bruno, Trenchi & Telloni (Reference Bruno, Trenchi and Telloni2014a ) have shown non-power-law smooth transition for the spectrum of the interplanetary magnetic field fluctuations, and also the faster fall of the spectrum slope with growing solar wind velocity. We observe this type of spectrum only for 6 % of the intervals. A small number of the observed spectra are probably connected to the small number of intervals with a high value of the velocity in our statistics. The exponential fall was predicted in the dissipation range for ordinary fluid flows (Frisch Reference Frisch1995) and also for the solar wind plasma (Howes et al. Reference Howes, Cowley, Dorland, Hammett, Quataert and Schekochihin2008). The exponential descent of the spectrum of the interplanetary magnetic field fluctuations is observed often in a transition from ion to electron scales and can usually be associated with acceleration of the dissipation processes (Alexandrova et al. Reference Alexandrova, Saur, Lacombe, Mangeney, Mitchell, Schartz and Robert2009).

Sometimes the transition region can be characterized by a more complex spectrum with a plateau between the MHD and the kinetic ranges instead of clear break. Below we call them spectra with a flattening in the vicinity of the break. Figure 1(c) represents an example of this spectrum observed on 2 June 2012 15:34–15:51 UT. The mean plasma parameters during this period are: $N_{p}=22~\text{cm}^{-3}$ , $V_{p}=354~\text{km}~\text{s}^{-1}$ , $T_{p}=3$  eV, $B=4~\text{nT}$ , $V_{A}=20~\text{km}~\text{s}^{-1}$ , $\unicode[STIX]{x1D6FD}_{p}=1.6$ , $f_{c}=0.06~\text{Hz}$ , $\unicode[STIX]{x1D70C}=62~\text{km}$ , $\unicode[STIX]{x1D706}=50~\text{km}$ , $f_{\unicode[STIX]{x1D706}}=1.1~\text{Hz}$ , $f_{\unicode[STIX]{x1D70C}}=0.9~\text{Hz}$ . The spectrum in the figure is fitted by three power laws, the intersection of them gives two break points. The spectrum flattens at the scale ${\sim}0.1$ –0.6 Hz, and can be approximated by a power law $P_{1}=-1.89$ before the flattening (at the end of the MHD scale) and by a power law $\langle P_{2}\rangle =-2.83$ after the flattening. The slope of the flatter part is equal to $\langle P_{0}\rangle =-0.69$ . The first break point separates the MHD scale from the transition scale with the flattening. It is equal to $F_{br1}=0.09~\text{Hz}$ which is of the same order as the cyclotron frequency $f_{c}$ . The second break is equal to $F_{br2}=0.55~\text{Hz}$ , which is close to the gyrostructure frequency. We need to take into account that the gyrostructure frequency is slightly lower than the inertial length frequency because $\unicode[STIX]{x1D6FD}_{p}$ is only slightly greater than unity. As it was shown in Šafránková et al. (Reference Šafránková, Němeček, Nemec, Pitna, Chen and Zastenker2015) that the gyrostructure frequency is the best scaling parameter for the location of the second break point of a similar density fluctuation spectra. It reflects the strong changes of character of the turbulence at scales comparable to the proton gyroradius. Chen et al. (Reference Chen, Leung, Boldyrev, Maruca and Bale2014b ) have shown that the break in the magnetic field fluctuation spectrum can be associated with the gyrostructure frequency only for high $\unicode[STIX]{x1D6FD}_{p}$ (as for the spectrum in figure 1 c), whereas low $\unicode[STIX]{x1D6FD}_{p}$ determines the relationship to the inertial length (as for the spectrum presented above in figure 1 a). The plasma $\unicode[STIX]{x1D6FD}_{p}$ parameter also influences the width of the flatter region, which tends to decrease with increasing $\unicode[STIX]{x1D6FD}_{p}$ (Šafránková et al. Reference Šafránková, Němeček, Němec, Přech, Chen and Zastenker2016). The significant interrelation of the characteristics of spectra and plasma $\unicode[STIX]{x1D6FD}$ is connected with the effect of $\unicode[STIX]{x1D6FD}$ on the compressibility of the system (Servidio et al. Reference Servidio, Valentini, Perrone, Greco, Califano, Matthaeus and Veltri2015). The spectra with the flattening are observed often for the ion density fluctuations (Šafránková et al. Reference Šafránková, Němeček, Přech, Zastenker, Čermák, Chesalin, Komárek, Vaverka, Beránek and Pavlů2013a ,Reference Šafránková, Němeček, Přech and Zastenker b , Reference Šafránková, Němeček, Nemec, Pitna, Chen and Zastenker2015; Chen et al. Reference Chen, Sorriso-Valvo, Šafránková and Němeček2014a ). The ion flux fluctuations considered in the current paper are mainly density fluctuations and exhibit a similar spectrum (Pitňa et al. Reference Pitňa, Šafránková, Němeček, Goncharov, Němec, Přech, Chen and Zastenker2016). The spectra with flattening are observed rather often but not always, in 32 % of intervals in the solar wind. Similar flattening around ion scales of the spectrum was first observed by Unti et al. (Reference Unti, Neugebauer and Goldstein1973) for the ion flux fluctuations and by Celnikier et al. (Reference Celnikier, Harvey, Jegou, Moricet and Kemp1983), Kellogg & Horbury (Reference Kellogg and Horbury2005) and Chen et al. (Reference Chen, Salem, Bonnell, Mozer and Bale2012) for the electron density fluctuations. Several authors attributed the flattening to the temperature anisotropy instabilities (Neugebauer et al. Reference Neugebauer, Wu and Huba1978) or to the dominance of kinetic Alfvén waves (e.g. Chandran et al. Reference Chandran, Quataert, Howes, Xia and Pongkitiwanichakul2009).

Sometimes the spectra of the ion flux fluctuations (of all above types) can be further distorted by a distinct peak with maximum in the transition region between the MHD and the kinetic ranges. Hereafter we call them spectra with a bump in the vicinity of the break. Figure 1(d) demonstrates the spectra of the ion flux fluctuations for the interval 7 October 2011 09:26–09:43 UT. The mean plasma parameters during this period are: $N_{p}=5~\text{cm}^{-3}$ , $V_{p}=393~\text{km}~\text{s}^{-1}$ , $T_{p}=1$  eV, $B=3~\text{nT}$ , $V_{A}=27~\text{km}~\text{s}^{-1}$ , $\unicode[STIX]{x1D6FD}_{p}=0.4$ , $f_{c}=0.04~\text{Hz}$ , $\unicode[STIX]{x1D70C}=65~\text{km}$ , $\unicode[STIX]{x1D706}=107~\text{km}$ , $f_{\unicode[STIX]{x1D706}}=0.6~\text{Hz}$ , $f_{\unicode[STIX]{x1D70C}}=1~\text{Hz}$ . So this event presents the lowest plasma parameters in comparison with other examples discussed in the paper. The peak is visible at a frequency of $F_{bump}=0.11~\text{Hz}$ between the cyclotron frequency $f_{c}$ and the inertial length frequency $f_{\unicode[STIX]{x1D706}}$ . It seems that the bump is located in the background of a spectrum of the third type (spectrum with flattening in the vicinity of the break). It is difficult to determine break(s) in this case but the second break $F_{br2}$ seems to be equal to ${\sim}0.4~\text{Hz}$ (see figure 1 d). The break is again near the inertial length frequency as was shown by Chen et al. (Reference Chen, Leung, Boldyrev, Maruca and Bale2014b ) for low $\unicode[STIX]{x1D6FD}_{p}$ . The slope of the spectra at frequencies higher than $F_{br2}$ is equal to $P_{2}=-2.65$ that is typical for the kinetic range. The flattening at frequencies exceeding 2 Hz can be explained by a low level of density fluctuations (10 times lower than that in the previous figures) that does not exceed the noise level. The spectra with a bump can be observed only in 3 % of cases in the solar wind (only ${\sim}20$ of the spectra in our statistics), they can be observed more frequently in the magnetosheath (see the paper of Rakhmanova et al. (Reference Rakhmanova, Riazantseva, Zastenker and Yermolaev2017) in the current issue). The similar bumps observed in the area of the spectral break are typical for the magnetic field fluctuations in the magnetosheath and they can be associated with Alfvén vortices (Alexandrova Reference Alexandrova2008). Roberts et al. (Reference Roberts, Li, Alexandrova and Li2016) proved the possibility of the existence a quasi-monopolar Alfvén vortex in the solar wind but the low value of Alfvén velocity in our example and the absence of magnetic field data make it difficult to use the same approach in our investigation.

The last type of spectra is the spectrum without steepening in the kinetic range. Figure 1(e) demonstrates an example of such a spectrum observed on 9 October 2011, 13:48-14:05 UT with one slope and without a visible break in the frequency spectrum. The mean plasma parameters during this period are: $N_{p}=10~\text{cm}^{-3}$ , $V_{p}=321~\text{km}~\text{s}^{-1}$ , $T_{p}=3$  eV, $B=6~\text{nT}$ , $V_{A}=39~\text{km}~\text{s}^{-1}$ , $\unicode[STIX]{x1D6FD}_{p}=0.3$ , $f_{c}=0.09~\text{Hz}$ , $\unicode[STIX]{x1D70C}=42~\text{km}$ , $\unicode[STIX]{x1D706}=73~\text{km}$ , $f_{\unicode[STIX]{x1D706}}=0.7~\text{Hz}$ , $f_{\unicode[STIX]{x1D70C}}=1.2~\text{Hz}$ . We can identify only one slope that is rather steep ( $P=-1.99$ ) in the inertial range but is too gradual in the kinetic range. It should be taken into account that we can see here only the high-frequency end of the inertial range which is not necessarily enough for the slope determination. The spectral slope at the MHD scale can be as high as $-2$ when, for example, the local magnetic field direction is parallel to the flow (Horbury, Forman & Oughton Reference Horbury, Forman and Oughton2008). However, the spectrum should steepen in the kinetic range even in these cases. As was shown (Riazantseva et al. Reference Riazantseva, Budaev, Zelenyi, Zastenker, Pavlos, Safrankova, Nemecek, Prech and Nemec2015; Šafránková et al. Reference Šafránková, Němeček, Nemec, Pitna, Chen and Zastenker2015), the slope of the kinetic part of the spectrum can vary over a wide range. The low slopes can be a sign of reduced dissipation (Smith et al. Reference Smith, Hamilton, Vasquez and Leamon2006). In Klein et al. (Reference Klein, Howes and TenBarge2014) whistler waves are considered as the possible reason of the spectral flattening in the high-frequency range. In that case, we cannot see the difference between the MHD and the kinetic ranges and identify the location of the break point. The latest type of spectrum is observed in only ${\sim}6\,\%$ cases but this number cannot be ignored.

Table 1.

The mean values of plasma parameters for different shapes of spectra.

4 Statistical distribution of plasma parameters associated with different types of spectra

The proportions between different shapes of spectra and the mean plasma parameters for each of them are summarized in table 1. Each row of the table corresponds to one of the types discussed above. The remaining ${\sim}2.6\,\%$ of spectra with a more complicated shape were not included into the table. Distributions of plasma parameters shown in figure 2 for all types of spectra correspond to the typical values encountered in the solar wind but several interesting features can be pointed out for certain types of spectra. The distribution of the proton density does not show any significant difference for the different types of spectra. The mean value of the proton density is a bit higher for spectra of type 1 and type 2, $\langle N_{p}\rangle \approx 14$ – $15~~\text{cm}^{-3}$ , whereas the three other types were found to be at densities of $\langle N_{p}\rangle \approx 10$ – $12~\text{cm}^{-3}$ . This difference is caused by a number of the spectra corresponding to a rather high value of proton density $N_{p}>30~\text{cm}^{-3}$ in the first two groups. A similar situation is observed also for helium density $N_{\unicode[STIX]{x1D6FC}}$ : largest mean values of helium density for type 2 spectra $\langle N_{\unicode[STIX]{x1D6FC}}\rangle =0.8~\text{cm}^{-3}$ are associated with a distribution tail with high values of $N_{\unicode[STIX]{x1D6FC}}>0.6~\text{cm}^{-3}$ . This is clearly seen for the distribution of relative helium abundance $N_{\unicode[STIX]{x1D6FC}}/N_{p}$ (figure 2 d): a significant number of intervals with a high value of $N_{\unicode[STIX]{x1D6FC}}/N_{p}\sim 7{-}10\,\%$ characterize type 2 spectra. The mean value of relative helium abundance for the second group of spectra is equal to $\langle N_{\unicode[STIX]{x1D6FC}}/N_{p}\rangle \approx 6\,\%$ , but the distribution contains two clear separated parts: approximately half of the spectra are characterized by high helium abundance $N_{\unicode[STIX]{x1D6FC}}/N_{p}>7\,\%$ , the other half lie in the limits ${\sim}1$ –5 %. The temperature distribution shows a large dispersion (figure 2 c).

Figure 2.

Distributions of solar wind parameters: the proton density $N_{p}$ (a), the helium ( $\unicode[STIX]{x1D6FC}$ particle) density $N_{\unicode[STIX]{x1D6FC}}$ (b), the proton temperature $T_{p}$ (c), the helium abundance $N_{\unicode[STIX]{x1D6FC}}/N_{p}$ (d), the bulk velocity $V_{p}$ (e), the Alfvén velocity $V_{A}$ (f), the magnetic field magnitude $B$ (g) and the proton plasma parameter $\unicode[STIX]{x1D6FD}_{p}$ (h) for different types of spectra. The arrows on panels (a,f,g and h) show that the last bins combine all values of the largest amplitude of the current parameter (greater than the left border of the bin).

The majority of intervals shows $T_{p}$ within in the limits ${\sim}1$ –4 eV, but the mean value is greater as a result of the contribution of intervals with hotter plasma. The distribution for type 4 and type 5 spectra shows the maximum at a rather low temperature ( $T_{p}\sim 1{-}2$  eV) with the shifted mean value $\langle T_{p}\rangle \approx 3$ –3.5 eV.

The distribution for type 1 and type 3 spectra exhibits a maximum at temperature $T_{p}\sim 1$ –4 eV with shifted mean value $\langle T_{p}\rangle \approx 4.5$  eV. The distribution for type 2 spectra is shifted to higher temperatures, its mean value $\langle T_{p}\rangle \approx 6.5$  eV. In view of wide variations of the temperature distribution, the difference between groups cannot be considered sufficiently reliable. The clearest result is seen for the distribution of the bulk velocity (figure 2 e). The distribution for type 2 spectra shows the significantly higher values of bulk velocity $V_{p}\sim 500$ – $600~\text{km}~\text{s}^{-1}$ than the distribution for the other types of spectra ( $V_{p}\sim 300$ – $400~\text{km}~\text{s}^{-1}$ ). The mean values of $\langle V_{p}\rangle$ are equal to $\langle V_{p}\rangle \approx 530~\text{km}~\text{s}^{-1}$ for type 2 spectra, $\langle V_{p}\rangle \approx 430$ – $450~\text{km}~\text{s}^{-1}$ for type 1 and type 3 spectra, $\langle V_{p}\rangle \approx 370~\text{km}~\text{s}^{-1}$ for type 4 and type 5 spectra. The mean value of the bulk velocity $V_{p}$ for type 1 and type 3 spectra is caused by the elongated tail of the distribution. The distribution of Alfvén velocity varies in a wide range for all types of spectra (figure 2 f). Only type 4 spectra have a clearly distinguished peak at a low value of Alfvén velocity, $V_{A}\sim 20$ – $30~\text{km}~\text{s}^{-1}$ (the mean value of the distribution is $\langle V_{A}\rangle =38~\text{km}~\text{s}^{-1}$ ).

The other types of spectra have a maximum of the distribution in the broader limits $V_{A}\sim 20$ – $50~\text{km}~\text{s}^{-1}$ . The mean value of the Alfvén velocity for type 1 and type 5 spectra is equal to $\langle V_{A}\rangle \approx 40$ – $50~\text{km}~\text{s}^{-1}$ , whereas it’s mean value for type 2 and type 3 spectra is $\langle V_{A}\rangle \approx 60~\text{km}~\text{s}^{-1}$ because of the presence of a rather large amount of spectra with a comparatively large value of $V_{A}({>}50~\text{km}~\text{s}^{-1})$ . The magnetic field distribution (see figure 2 g) for all types of spectra except type 2 has a maximum at $B\sim 2$ –6 nT with mean values $\langle B\rangle \approx 8~\text{nT}$ for type 1 and type 3 spectra and $\langle B\rangle \approx 5~\text{nT}$ for type 4 and type 5 spectra. The distribution for type 2 spectra seems to be wider with a mean value of $\langle B\rangle \approx 10~\text{nT}$ due to a number of spectra with large $B$ ( ${>}16~\text{nT}$ ). The distribution of plasma $\unicode[STIX]{x1D6FD}_{p}$ does not show any significant difference among the groups of spectra (figure 2 h). The majority of the spectra have $\unicode[STIX]{x1D6FD}_{p}<0.9$ with a maximum at $\unicode[STIX]{x1D6FD}_{p}\sim 0.3$ –0.5, except the intervals of type 2 that have two peaks: one at a lower value $\unicode[STIX]{x1D6FD}_{p}\sim 0.1$ –0.3 and the second at $\unicode[STIX]{x1D6FD}_{p}\sim 0.9$ –1.1. Meanwhile, a significant number of spectra with a high value of $\unicode[STIX]{x1D6FD}_{p}>1.9$ is observed for type 1, type 2 and type 5 spectra, which leads to mean values $\langle \unicode[STIX]{x1D6FD}_{p}\rangle \approx 1$ . On the other hand, the mean value of $\unicode[STIX]{x1D6FD}_{p}$ for type 3 and type 4 spectra is equal to $\langle \unicode[STIX]{x1D6FD}_{p}\rangle \approx 0.6$ .

The differences of the solar wind parameters for the various types of spectral shapes bring us to believe that these or other spectral shapes are connected with the individual type of solar wind. Table 2 presents the proportions of spectral shapes referring to the several types of the solar wind observed in the paper (see § 2).

Table 2.

The proportions of spectral shapes referring to different types of the solar wind.

The most numerous types of spectral shapes (type 1 and type 3) are observed in all types of the solar wind in approximately equal proportions. We can only note that in the pristine slow solar wind, the relative number of spectra type 3 is larger and approaches the relative number of spectra type 1. Also we can note that the spectra of type 2 and type 4 are often observed during ‘EJECTA’ or complex phenomena such as compressed regions inside EJECTA-like events, but the statistics for such subclassification is rather poor. More accurate analysis will be carried out in our subsequent works on more extensive statistical material.

5 Statistical properties of turbulence for different shapes of the ion flux fluctuation spectra in the solar wind

The traditional approach describes the behaviour of the fully developed isotropic turbulence (Kolmogorov Reference Kolmogorov1941) and does not take into account a local breaking of turbulence homogeneity (so-called intermittency) considered first by Novikov & Stewart (Reference Novikov and Stewart1964). The statistical properties of intermittent flow strongly deviate from the statistics of the Kolmogorov model (K41). The statistical methods taking into account the probability distribution function (PDF) and its moments can better describe the properties of a turbulent medium. Intermittency is typically observed in space experiments as non-Gaussian, scale-dependent statistics of parameters which lead to the breaking of trivial self-similarity (Marsch & Tu Reference Marsch and Tu1997; Bruno & Carbone Reference Bruno and Carbone2013 and references therein). Fluctuations of parameters in the space plasma correspond usually to the presence of multifractality and self-similarity at all scales (Zelenyi & Milovanov Reference Zelenyi and Milovanov2004). Long-range correlations and memory effects observed in the cosmic plasma are the result of self-similarity and intermittency (Budaev, Savin & Zelenyi Reference Budaev, Savin and Zelenyi2011). Statistical properties of turbulence follow the universal laws despite the various conditions in the plasma in different areas of the near-Earth space and also in laboratory plasma (Budaev et al. Reference Budaev, Savin and Zelenyi2011; Budaev, Zelenyi & Savin Reference Budaev, Zelenyi and Savin2015). We will show below that the solar wind ion flux fluctuations for different shapes of spectra have similar statistical properties. For each analysed interval, we calculate the PDF, high-order moments of the PDF and analyse their scaling. Then we compare the average statistical properties for each type of spectra defined in §§ 3 and 4.

Figure 3.

An average value of the fourth-order moment (flatness) for the PDF of the ion flux fluctuations versus scale parameter $1/\unicode[STIX]{x1D70F}$ for different types of spectra.

Figure 4.

High-order structure function versus time scale $\unicode[STIX]{x1D70F}$ (a,b) and versus the third-order structure function (c,d) for different types of spectra: spectra with nonlinear steepening in the kinetic range, 27 September 2011 21:09–22:34 UT (a,c); spectra with flattening in the vicinity of the break, 2 June 2012 14:25–15:51 UT (b,d).

Intermittency can be quantified by a growing of flatness (the fourth-order moment of the PDF which reflects the deviation from Gaussianity) toward smaller spatial scales (or high-frequency scales in the sense of frequency of variations) (Frisch Reference Frisch1995). Figure 3 presents the dependence of the flatness $F$ of the ion flux fluctuations versus the time scale parameter $1/\unicode[STIX]{x1D70F}$ for the groups of different shapes of spectra (where $F(\unicode[STIX]{x1D70F})=\langle (\text{Flux}_{i}(t+\unicode[STIX]{x1D70F})-\text{Flux}_{i}(t))^{4}\rangle /(\langle (\text{Flux}_{i}(t+\unicode[STIX]{x1D70F})-\text{Flux}_{i}(t))^{2}\rangle )^{2}$ , $\unicode[STIX]{x1D70F}$ ranges from 0.1 to 256 s). The dashed line corresponds to $F=3$ for a Gaussian PDF. One can see a significant growing of flatness toward the kinetic scales that continues in the kinetic range with a smaller slope. The increase of non-Gaussianity toward the kinetic range was earlier observed for the magnetic field and bulk velocity fluctuations (Marsch & Tu Reference Marsch and Tu1997; Sorriso-Valvo et al. Reference Sorriso-Valvo, Carbone, Veltri, Consolini and Bruno1999; Bruno et al. Reference Bruno, Carbone, Sorriso-Valvo and Bavassano2003; Salem et al. Reference Salem, Mangeney, Bale and Veltri2009) and separately for the fluctuations of plasma parameters (Hnat, Chapman & Rowlands Reference Hnat, Chapman and Rowlands2003; Riazantseva & Zastenker Reference Riazantseva and Zastenker2008; Riazantseva, Zastenker & Karavaev Reference Riazantseva, Zastenker and Karavaev2010; Bruno et al. Reference Bruno, Telloni, Primavera, Pietropaolo, D’Amicisi, Sorriso-Valvo, Carbone, Malara and Veltri2014b ). The statistical properties presented in these papers show a similar behaviour for the magnetic field and plasma fluctuations. The flatness continues to grow (Alexandrova et al. Reference Alexandrova, Carbone, Veltri and Sorriso-Valvo2008; Kiyani et al. Reference Kiyani, Chapman, Khotyaintsev, Dunlop and Sahraoui2009; Yordanova et al. Reference Yordanova, Balogh, Noullez and von Steiger2009; Riazantseva et al. Reference Riazantseva, Budaev, Zelenyi, Zastenker, Pavlos, Safrankova, Nemecek, Prech and Nemec2015) or remains almost constant (Chen et al. Reference Chen, Sorriso-Valvo, Šafránková and Němeček2014a ) at kinetic scales. Riazantseva et al. (Reference Riazantseva, Budaev, Rakhmanova, Zastenker, Safrankova, Nemecek and Prech2016) have shown that flatness can vary in a broad range for different time intervals but, on average, it tends to grow up to a scale of ${\sim}$ 0.1 s for the ion flux fluctuations. The authors also show the presence of intermittency in the whole range of discussed scales. The behaviour of the flatness for different types of spectra (shown by different colours and markers in figure 3) is similar but the fastest growing of $F(1/\unicode[STIX]{x1D70F})$ is observed for the group of spectra without steepening in the kinetic range. In general, the differences among the flatness profiles for different types of spectra are comparable with the statistical errors. It is necessary to take into account the qualitative character of the flatness dependence. It indicates the presence of intermittency but does not allow us to establish the precise quantitative difference of the intermittency level between different spectral types. Nevertheless, we can say that the intermittent flow is observed for all types of spectra, which is typical for a multifractal signal (Bruno & Carbone Reference Bruno and Carbone2013). The scale invariance of the intermittent ion flux in the solar wind can be confirmed by analysis of the dependence of the high-order structure functions on the time scale parameter $\unicode[STIX]{x1D70F}:S_{q}(\unicode[STIX]{x1D70F})=\langle |\text{Flux}_{i}(t+\unicode[STIX]{x1D70F})-\text{Flux}_{i}(t)|^{q}\rangle$ . The nonlinear dependence $S_{q}(\unicode[STIX]{x1D70F})$ was shown by Riazantseva et al. (Reference Riazantseva, Budaev, Zelenyi, Zastenker, Pavlos, Safrankova, Nemecek, Prech and Nemec2015) for type 1 spectra (spectra with two slopes and one break). We present the nonlinear dependence of $S_{q}(\unicode[STIX]{x1D70F})$ for a wide range of scales for type 2 spectrum (spectra with nonlinear steepening in the kinetic range) in figure 4(a) and for the type 3 spectrum (spectra with flattening in the vicinity of the break) in figure 4(b). These figures present the structure functions up to order 6 (the analysis of higher orders is not statistically significant, Dudok de Wit et al. Reference Dudok de Wit, Alexandrova, Furno, Sorriso-Valvo and Zimbardo2013). For analysis of the higher orders of the structure functions, we atempt to use longer intervals than we have used for the analysis of the spectra. For calculation of the structure functions in figure 4 we select periods during which when the same type of spectrum is observed over a rather long period of time ( ${\sim}1.5~\text{h}$ ). The selected intervals include the intervals presented in figure 1.

The observed dependences demonstrate strong deviation from a linear scaling $\unicode[STIX]{x1D701}(q)=q/3$ typical for Kolmogorov turbulence (Kolmogorov Reference Kolmogorov1941). The measured structure function power law $S_{q}(\unicode[STIX]{x1D70F})\sim \unicode[STIX]{x1D70F}^{\unicode[STIX]{x1D701}(q)}$ shows the nonlinear scaling $\unicode[STIX]{x1D701}(q)$ and excludes a trivial self-similarity and scale invariance for the corresponding scales. It is a rather typical situation for the solar wind turbulent flow (Burlaga Reference Burlaga1991; Hnat, Chapman & Rowlands Reference Hnat, Chapman and Rowlands2005; Budaev et al. Reference Budaev, Zelenyi and Savin2015). Herewith the extended self-similarity (ESS) can be shown by linear dependence of $\log (S_{q}(\unicode[STIX]{x1D70F}))$ from $\log (S_{3}(\unicode[STIX]{x1D70F}))$ (Benzi et al. Reference Benzi, Ciliberto, Baudet, Ruiz Chavarria and Tripiccione1993). The ESS is a universal property of statistical symmetry in the system of plasma turbulence in space, laboratory and even neutral fluids (Budaev et al. Reference Budaev, Savin and Zelenyi2011) that leads to hidden statistical symmetries of the motion equations and multifractality. It provides the scale invariance of the process over a wide range of scales. This is clearly shown for almost two to three orders of magnitude $S_{q}(l)$ for spectra with a nonlinear descent in the kinetic range and spectra with a flattening in the vicinity of the break (see figure 4 c,d). We do not show here the ESS plots for other types of spectra but they are always similar. In the solar wind plasma the ESS is observed predominately for the plasma fluctuations at inertial scales (Carbone, Veltri & Bruno Reference Carbone, Veltri and Bruno1995; Hnat et al. Reference Hnat, Chapman and Rowlands2005). Recently, it was also shown to be the case for the ion flux fluctuations, including the kinetic range (Budaev et al. Reference Budaev, Zelenyi and Savin2015; Riazantseva et al. Reference Riazantseva, Budaev, Zelenyi, Zastenker, Pavlos, Safrankova, Nemecek, Prech and Nemec2015), but only for spectra with two slopes and one break.

The analysis of the experimental scaling $\unicode[STIX]{x1D701}(q)$ can help to validate a preferred model of the intermittent turbulence (Budaev et al. Reference Budaev, Savin and Zelenyi2011, Reference Budaev, Zelenyi and Savin2015). The description of the observations of intermittent turbulence with the ESS can be implemented to the log-Poisson turbulence models. These models are a generalization of fractal models of turbulence with intermittency (Dubrulle Reference Dubrulle1994; She & Leveque Reference She and Leveque1994).

Figure 5 demonstrates typical examples of comparison of normalized experimental scaling $S_{3}(l)^{\unicode[STIX]{x1D701}(q)/\unicode[STIX]{x1D701}(3)}$ of the PDF in the solar wind flow with a linear scaling $q/3$ for the K41 (Kolmogorov) model (dashed blue line) and with the predictions of the models of turbulence with intermittency: She–Leveque model (solid line) and Biskamp–Mueller model (dotted line). Different colours and marks show different subintervals with several types of spectra. The intervals correspond to the following types of spectra: spectra with two slopes and one break correspond to intervals 1.1, 1.2, 1.3 in figure 5(a) and intervals 2.1, 2.2 in figure 5(b); spectra with nonlinear steepening in the kinetic range correspond to intervals 1.4, 1.5, 1.6, 1.7 in figure 5(a); spectra with flattening in the vicinity of the break correspond to intervals 1.8, 1.9, 1.10 in figure 5(a) and intervals 2.3, 2.4 in figure 5(b); spectra with a bump in the vicinity of the break correspond to intervals 2.5, 2.6, 2.7 in figure 5(b); spectra without steepening in the kinetic range correspond to intervals 2.8, 2.9 in figure 5(b). Thus the scaling can differ from one interval to another, but the clear deviation from the Kolmogorov scaling is observed for all intervals. The degree of distinction between the experimental scaling and the Kolmogorov model scaling can be different for the same type of spectra. For example type 1 spectra demonstrate weak deviation from the Kolmogorov scaling for the 1.2, 1.3 intervals of figure 5(a) and strong deviation for interval 1.1 in figure 5(a) and intervals 2.1, 2.3 in figure 5(b).

Figure 5.

Comparison of the experimental scaling with the scaling of the Kolmogorov (K41) model (dashed line – $q/3$ ), the She–Leveque log-Poisson model (SL) (solid line) and with the Biskamp–Mueller (BM) log-Poisson model (dashed-dotted line). The curves of different colours and signs correspond to the scaling of the different selective subintervals with the different types of fluctuation spectra (the types are shown in the legend) on 26 September, 2011 12:15–15:00 UT (a) and on 7 October 2011 01:00–09:35 UT (b).

The experimental scaling in plasma turbulence can be well described and parameterized in the approach of She–Leveque–Dubrulle log-Poisson model (Budaev et al. Reference Budaev, Savin and Zelenyi2011, Reference Budaev, Zelenyi and Savin2015):

where $\unicode[STIX]{x1D6E5}$ and $\unicode[STIX]{x1D6FD}_{SLD}$ are changeable parameters, determined from analysis of experimental data. $\unicode[STIX]{x1D6FD}_{SLD}$ determines the degree of intermittency: $\unicode[STIX]{x1D6FD}_{SLD}=1$ for homogeneous developed turbulence without intermittency (as in the Kolmogorov model), $\unicode[STIX]{x1D6FD}_{SLD}<1$ indicates an intermittent flow. The parameter $\unicode[STIX]{x1D6E5}$ determines the scaling of energy $\unicode[STIX]{x1D700}_{l}^{\infty }\sim l^{-\unicode[STIX]{x1D6E5}}$ ( $l$ -scale parameter), where the energy rate value $\unicode[STIX]{x1D700}_{l}^{\infty }$ is limited by the presence of singular dissipative structures (Dubrulle Reference Dubrulle1994; She & Leveque Reference She and Leveque1994). Three-dimensional isotropic hydrodynamic turbulence is characterized by $\unicode[STIX]{x1D6E5}=\unicode[STIX]{x1D6FD}_{SLD}=2/3$ (She & Leveque Reference She and Leveque1994).

We calculate the parameterization coefficient for the data set used above. We determine the scaling $\unicode[STIX]{x1D701}(q)$ by the Wavelet transform modulus maxima (WTMM) procedure (Budaev et al. Reference Budaev, Savin and Zelenyi2011, Reference Budaev, Zelenyi and Savin2015), based on wavelet analysis. This method takes into account the property of scale invariance of the turbulence and can significantly improve the accuracy of estimation of the high-order structure functions (up to $q=9$ for the used number of points). We have found that the mean value of $\langle \unicode[STIX]{x1D6FD}_{SLD}\rangle =0.1$ and $\langle \unicode[STIX]{x1D6E5}\rangle =0.2$ , which proves the high level of intermittency for the majority of the analysed intervals. The value of $\unicode[STIX]{x1D6E5}$ can be explained by the presence of dissipative structures with complicated fractal topology (Budaev et al. Reference Budaev, Savin and Zelenyi2011, Reference Budaev, Zelenyi and Savin2015). The $\unicode[STIX]{x1D6FD}_{SLD}\sim 1$ is observed only in 10 % of the events and can be associated with non-intermittent plasma flows for all kinds of fluctuation spectra. Riazantseva et al. (Reference Riazantseva, Budaev, Zelenyi, Zastenker, Pavlos, Safrankova, Nemecek, Prech and Nemec2015) have shown that the absence of intermittency is typically observed in the fast solar wind with a low density level but this statement cannot be reversed, high-velocity flow is not always characterized by a low level of intermittency. For example, spectra with nonlinear steepening in the kinetic range are generally characterized by high velocity (see § 4) but the majority of them are also characterized by a high level of intermittency.

The topology of dissipative structures affecting the scaling properties determines the selection of the model of three-dimensional isotropic turbulence. The two-dimensional current sheets are considered for the Biskamp–Mueller model (Biskamp & Mueller Reference Biskamp and Mueller2003), based on the traditional Iroshnikov–Kraichnan phenomenology for the plasma. The scaling can be written through the parameterization factor $g$ as:

On the other hand, one-dimensional filaments are considered for the She–Leveque model (She & Leveque Reference She and Leveque1994). Parameterization of the scaling in this case was proposed by Budaev (Reference Budaev2009) and can be written through the parameterization factor $g_{f}$ as:

The cascade efficiency in comparison to the Kolmogorov model is reflected in the values of coefficients $g$ and $g_{f}$ . A constant cascade of the energy rate is assumed in the anisotropic case and the scaling is supposed as $\unicode[STIX]{x1D6FF}_{l}\unicode[STIX]{x1D710}\sim l^{1/g}$ (where $\unicode[STIX]{x1D6FF}_{l}\unicode[STIX]{x1D710}$ is a velocity difference and $l$ is a scale parameter, see Budaev et al. (Reference Budaev, Savin and Zelenyi2011)). The experimental scaling of the structure functions $\unicode[STIX]{x1D701}(q)$ (reflected e.g. in figure 5) can be parameterized by $g$ and $g_{f}$ coefficients using the WTMM procedure (Budaev et al. Reference Budaev, Savin and Zelenyi2011). Values of $g$ or $g_{f}$ of approximately 3 reflect the efficiency of the cascade and determines the choice of dominant contribution of the dissipation structures. Coefficient values higher or lower than 3 correspond to enhancement or depletion of the cascade, respectively. The mean values of geometrical parameterization parameters for all analysed intervals are equal to $\langle g\rangle =2.5$ and $\langle g_{f}\rangle =2.9$ . Proximity of $g_{f}$ value to 3 prefers a model with a dominant contribution of filament structures. The dominance of one-dimensional dissipation structures was also shown for the magnetic field fluctuations in the magnetospheric boundary layers (Budaev et al. Reference Budaev, Savin and Zelenyi2011). Earlier Carbone et al. (Reference Carbone, Veltri and Bruno1995) produced an analysis of high-order moments and compared the scaling laws between the low-frequency MHD turbulence in the solar wind, ordinary fluid flow and the She–Leveque model. The observed difference is explained by the contributions of different kinds of singular dissipative structures: two-dimensional planar sheets for the MHD solar wind turbulence and one-dimensional filaments for ordinary fluid flow. However, this analysis was produced only for the low-frequency solar wind turbulence whereas our analysis is concerned with the high-frequency solar wind plasma turbulence.

6 Discussion and conclusions

High time resolution plasma measurements allow us to focus on the features of the turbulence near the ion scales. The analysis of spectral and statistical properties of the high-frequency plasma fluctuations shows a non-stationary character of the turbulent flow. Various deviations of the frequency spectra and the probability distribution functions of fluctuations from the corresponding characteristics of classical stationary turbulence models can be observed.

It is known that the interplanetary magnetic field fluctuation spectra have different shapes in the fast and slow solar wind (Bruno & Carbone Reference Bruno and Carbone2013). Whistler waves can lead to flattening in the high-frequency part of the magnetic field fluctuation spectra (Klein et al. Reference Klein, Howes and TenBarge2014) and the Alfvén ion cyclotron waves can give rise to bumps in the spectra (Alexandrova Reference Alexandrova2008). The flattening at a frequency lower than the break frequency can be due to the contribution of coherent structures (Lion et al. Reference Lion, Alexandrova and Zaslavsky2016), kinetic Alfvén waves (e.g. Chandran et al. Reference Chandran, Quataert, Howes, Xia and Pongkitiwanichakul2009) or thermal instabilities (e.g. Neugebauer et al. Reference Neugebauer, Wu and Huba1978). High variability of the magnetic field fluctuation spectra around the ion scales in the solar wind can be explained by the proportion of different physical processes which are working at these scales (Lion et al. Reference Lion, Alexandrova and Zaslavsky2016). The clear break is typical for a coherent spectrum whereas a spectrum with a smooth decrease without a break is observed for non-coherent fluctuations.

Several earlier density measurements exhibited predominately spectra with a flattening around the ion scales (Unti et al. Reference Unti, Neugebauer and Goldstein1973; Neugebauer et al. Reference Neugebauer, Wu and Huba1978; Celnikier et al. Reference Celnikier, Harvey, Jegou, Moricet and Kemp1983; Kellogg & Horbury Reference Kellogg and Horbury2005; Chen et al. Reference Chen, Salem, Bonnell, Mozer and Bale2012) including recent Spektr-R measurements (Šafránková et al. Reference Šafránková, Němeček, Přech, Zastenker, Čermák, Chesalin, Komárek, Vaverka, Beránek and Pavlů2013a ,Reference Šafránková, Němeček, Přech and Zastenker b , Reference Šafránková, Němeček, Nemec, Pitna, Chen and Zastenker2015) but the spectra of the ion flux fluctuations demonstrate predominately a shape with a clear break (Riazantseva et al. Reference Riazantseva, Budaev, Zelenyi, Zastenker, Pavlos, Safrankova, Nemecek, Prech and Nemec2015). In this paper, we observe a wide variety of spectra in Spektr-R ion flux measurements and determine the relative proportion of different types of spectra. The most common shapes are spectra with two slopes and one break ( ${\sim}50\,\%$ ) and spectra with flattening in the vicinity of the break ( ${\sim}32\,\%$ ) often described in literature both for plasma and for the magnetic field fluctuations (see the review of Alexandrova et al. (Reference Alexandrova, Chen, Sorisso-Valvo, Horbury and Bale2013)). Other shapes of spectra can be observed in a much smaller but not negligible number of cases: ${\sim}6\,\%$ – spectra with nonlinear steepening in the kinetic range, ${\sim}6\,\%$ – spectra without steepening in the kinetic range and ${\sim}3\,\%$ – spectra with a bump in the vicinity of the break.

Using the joint analysis of Spektr-R and Wind measurements, we also determine typical plasma conditions for each type of spectra. The spectra with nonlinear steepening in the kinetic range are observed predominately for the solar wind with a rather high bulk velocity $V_{p}>500~\text{km}~\text{s}^{-1}$ . This fact is consistent with the recently established relationship between bulk velocity and degree of spectra steepening for the magnetic field fluctuations (Bruno et al. Reference Bruno, Trenchi and Telloni2014a ,Reference Bruno, Telloni, Primavera, Pietropaolo, D’Amicisi, Sorriso-Valvo, Carbone, Malara and Veltri b ). A significant part of the spectra with nonlinear steepening in the kinetic range is also connected with a high helium abundance and, to a lesser degree, by a large value of the density, the temperature, the magnetic field magnitude and the Alfvén velocity. The relationship between the turbulent properties and helium abundance seems to be a rather interesting point because the abundance of alpha particles can change the dispersive properties of the plasma and excite an instability which in turn can affect the spectrum of turbulent fluctuations (Hellinger & Travnicek Reference Hellinger and Travnicek2013; Valentini Reference Valentini2016). The lowest values of plasma parameters are typical of spectra with a bump in the vicinity of the break and spectra without steepening in the kinetic range. Solar wind plasma parameters for spectra with two slopes and one break and spectra with a flattening in the vicinity of the break are rather ordinary. The peak of the distribution of Alfvén velocities for spectra with a bump in the vicinity of the break at low values is rather surprising but the statistics for this type of spectrum is poor. The role of the plasma parameter $\unicode[STIX]{x1D6FD}_{p}$ is not clear from our statistics. We can say that low $\unicode[STIX]{x1D6FD}_{p}$ is observed in approximately equal degrees for all types of spectra, whereas $\unicode[STIX]{x1D6FD}_{p}\sim 1$ and $\unicode[STIX]{x1D6FD}_{p}>1$ can be observed predominately for spectra with two slopes and one break, spectra with nonlinear steepening in the kinetic range and spectra without a steepening in the kinetic range. The examples discussed in § 3 demonstrate the organization of the break frequency (where it can be determined) by $\unicode[STIX]{x1D6FD}_{p}$ : the break tends to be controlled by the inertial length frequency for low $\unicode[STIX]{x1D6FD}_{p}$ and by the gyrostructure frequency for high $\unicode[STIX]{x1D6FD}_{p}$ , as it was earlier shown for the interplanetary magnetic field spectra (Chen et al. Reference Chen, Leung, Boldyrev, Maruca and Bale2014b ). The investigation of the relationship between spectrum shape and $\unicode[STIX]{x1D6FD}_{p}$ should continue using a set containing extreme values of $\unicode[STIX]{x1D6FD}_{p}$ .

The probability distribution functions of the ion flux fluctuations and their statistical characteristics also vary considerably in the frequency range ${\sim}0.1$ –10 Hz for different intervals in the solar wind, but they always differ significantly from a Gaussian distribution. The increase of flatness (reflecting the deviation of distribution tails from the Gaussian distribution) at the high-frequency scales is observed for all shapes of fluctuation spectra in the solar wind and indicates the presence of intermittency. This was already shown for the magnetic field and plasma parameters in interplanetary medium up to the dissipation range (Bruno & Carbone Reference Bruno and Carbone2013) but the plasma fluctuations in the kinetic range sometimes exhibit non-intermittent behaviour (Chen et al. Reference Chen, Sorriso-Valvo, Šafránková and Němeček2014a ).

The turbulence in all analysed intervals has a complicated multifractal structure and exhibits extended self-similarity which provide scale invariance to the system. This is true for the statistics presented in the paper and also for different turbulent plasma flows in space and the laboratory (Budaev et al. Reference Budaev, Savin and Zelenyi2011, Reference Budaev, Zelenyi and Savin2015). Herewith statistical and spectral properties of the ion flux fluctuations in the solar wind can be different for various plasma conditions. But this diversity lies inside the class of phenomena described in the sense of the ESS.

In a dominant part of the ion flux fluctuation measurements, the scaling of the structure functions may be well described by the log-Poisson model approach that takes into account the presence of intermittency. Using the log-Poisson parameterization we show the dominant contribution of filament-like structures in the formation of the solar wind plasma intermittency for intervals corresponding to different types of spectra in the solar wind. The geometry of the intermittent structures has been discussed for decades. The idea of the crucial role of filament structures in the solar wind (‘spaghetti model’) was born a long time ago on the basis of earlier solar wind observations (e.g. Bartley et al. Reference Bartley, Bakata, McCracken and Rao1966) and still remains relevant with the new experimental results (Bruno et al. Reference Bruno, Carbone, Veltri, Pietropaolo and Bavassano2001; Borovsky Reference Borovsky2008). The authors have assumed that the solar wind consists of flow tubes of different sizes, with boundaries representing tangential discontinuities. They display the interlacement of tubes containing the distinct plasmas with different axial orientations for each. The intermittent events are located at the boundaries between of two adjacent flux tubes (Bruno et al. Reference Bruno, Carbone, Veltri, Pietropaolo and Bavassano2001). The possible relation of the dissipation mechanisms with the intermittent events (Matthaeus et al. Reference Matthaeus, Wan, Servidio, Greco, Osman, Oughton and Dmitruk2015) has revived the interest in the topic of the geometry of the solar wind structures. A large number of authors have associated the intermittency in the solar wind turbulence with two-dimensional structures such as currents sheets (e.g. Boldyrev & Perez Reference Boldyrev and Perez2012; Greco et al. Reference Greco, Matthaeus, D’Amicis, Servidio and Dmitruk2012; Perri et al. Reference Perri, Goldstein, Dorelli and Sahraoui2012). However, Alexandrova (Reference Alexandrova2008) have shown that the intermittency in the solar wind turbulence is related to the filamentary structures such as Alfvén vortices. The analysis of high-order moments in Carbone et al. (Reference Carbone, Veltri and Bruno1995) has shown that the most intermittent structures can be represented as filaments only for ordinary fluid flow, whereas for the low-frequency MHD solar wind turbulence they can be associated with the planar sheets. Budaev et al. (Reference Budaev, Savin and Zelenyi2011) have produced the log-Poisson parameterization of the high-order scaling for the turbulent space plasma in the boundary layers of the Earth’s magnetosphere and demonstrated the better agreement with the model assuming one-dimensional dissipation structures. Later, it was proven for the solar wind plasma by using the high-resolution plasma measurements of Spektr-R (Budaev et al. Reference Budaev, Zelenyi and Savin2015; Riazantseva et al. Reference Riazantseva, Budaev, Zelenyi, Zastenker, Pavlos, Safrankova, Nemecek, Prech and Nemec2015). In this paper we also show the dominant role of the filamentary intermittent structures in the solar wind for a wide range of plasma parameters.