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Comparison of properties of small-scale ion flux fluctuations in the flank magnetosheath and in the solar wind
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ScienceDirect Advances in Space Research xxx (2016) xxx–xxx www.elsevier.com/locate/asr
Comparison of properties of small-scale ion flux fluctuations in the flank magnetosheath and in the solar wind M.O. Riazantseva a,b,⇑, V.P. Budaev b,c, L.S. Rakhmanova b, G.N. Zastenker b, J. Sˇafra´nkova´ d, Z. Neˇmecˇek d, L. Prˇech d a
Skobeltsyn Institute of Nuclear Physics, Lomonosov Moscow State University (SINP MSU), 1(2), Leninskie gory, Moscow 119991, Russia b Space Research Institute of the Russian Academy of Science (IKI), 84/32 Profsoyuznaya Str., Moscow 117997, Russia c National Research Centre ‘‘Kurchatov Institute”, 1, Akademika Kurchatova pl., Moscow 123182, Russia d Faculty of Mathematics and Physics, Charles University, V Holesovickach 2, Prague 8 18000, Czech Republic Received 25 April 2015; received in revised form 13 December 2015; accepted 15 December 2015
Abstract We present a comparison of spectral and statistical properties of ion flux fluctuations in the turbulent solar wind and in the flank Earth’s magnetosheath. We use the data of the BMSW device operating in frame of the SPECTR-R mission with an extremely hightime resolution (up to 30 ms). Fourier spectra of ion flux fluctuations are systematically analyzed both in the solar wind and in the magnetosheath on the inertial scale and on a transition to the dissipation scale in the range of 0.01–10 Hz. We show that ion flux fluctuation spectra in the flank magnetosheath are similar to those observed in the solar wind and we demonstrate the presence of the break at frequencies of 1–2 Hz. Spectra are slightly steeper in the flank magnetosheath but the break frequency is near twice less in a comparison to the solar wind. The magnetosheath ion flux turbulent flow is intermittent as it was shown earlier for the solar wind. We discuss the level of intermittency of ion flux fluctuations in both regions and we determine the characteristics of structure functions. Finally, we demonstrate extended self-similarity in the magnetosheath. Ó 2015 COSPAR. Published by Elsevier Ltd. All rights reserved.
Keywords: Magnetosheath; Solar wind; Plasma turbulence; Intermittency; Kinetic scales
1. Introduction The Earth’s magnetosheath (MSH) delivers solar wind (SW) plasma into the magnetosphere, thus solarterrestrial investigations need the analysis of variations of SW parameters and their modifications in the MSH. A study of SW and MSH turbulence presents an opportunity to analyze the dynamics of the space plasma for different boundary conditions. SW turbulence freely developes in ⇑ Corresponding author at: Skobeltsyn Institute of Nuclear Physics, Lomonosov Moscow State University (SINP MSU), 1(2), Leninskie gory, Moscow 119991, Russia. Tel.: +7 4959392810. E-mail address: [email protected] (M.O. Riazantseva).
the space during long-time intervals, whereas MSH turbulence developes between two boundaries – the magnetopause and bow shock. A large number of various types of waves originate at these boundaries and inside the MSH (Omidi et al., 1994; Lacombe and Belmont, 1995). A high level of plasma and magnetic field fluctuations are constantly observed in the MSH as a result of these processes (Neˇmecˇek et al., 2001; Zastenker et al., 2002; Shevyrev et al., 2003), thus they cannot be described by traditional models of a laminar plasma flow (Neˇmecˇek et al., 2000; Zastenker et al., 2002). The recent hybrid models (e.g., Karimabadi et al., 2014) demonstrate the presence of a high level of turbulent fluctuations downstream the quasi-parallel bow shock. The
http://dx.doi.org/10.1016/j.asr.2015.12.022 0273-1177/Ó 2015 COSPAR. Published by Elsevier Ltd. All rights reserved.
Please cite this article in press as: Riazantseva, M.O., et al. Comparison of properties of small-scale ion flux fluctuations in the flank magnetosheath and in the solar wind. Adv. Space Res. (2016), http://dx.doi.org/10.1016/j.asr.2015.12.022
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quasi-parallel bow shock modifies and amplifies SW fluctuations, so the level of fluctuations is increased behind it (Shevyrev et al., 2003; Alexandrova, 2008). Shevyrev and Zastenker (2005) show that the amplitudes of high frequency fluctuations are also higher behind the quasiparallel bow shock as well as in the foreshock located in front of it. Fluctuations of different nature are observed in the MSH because all outer regions, the bow shock (e.g., Neˇmecˇek et al., 2002) foreshock (e.g., Blanco-Cano et al., 2006) and magnetopause (e.g., Rezeau et al., 1999; Gutynska et al., 2012) contribute to the formation of MSH turbulence. A high level of turbulent fluctuations in the MSH leads to a low level of correlations between the magnetic field and plasma in the MSH and SW (Gutynska et al., 2012; Rakhmanova et al., 2015a,b). Moreover, the magnetic field near the magnetopause can strongly differ from that in the SW (Sˇafra´nkova´ et al., 2009; Pulinets et al., 2014) and this fact significantly complicates a creation of adequate models. In the MSH, the fluctuations of the SW origin and the fluctuations born on boundaries of the MSH and within it are mixed (Zastenker et al., 2002). Low-frequency fluctuations come mainly from the SW, whereas high-frequency fluctuations have as a rule a local origin (Gutynska et al., 2012; Rakhmanova et al., 2015a,b), however, it is rather difficult to separate these sources in each specific case. For example, the dynamics of some significant structures in the SW (as the interplanetary shocks, or large amplitude discontinuities) can be traced (Koval et al., 2006; Sˇafra´nkova´ et al., 2007; Rakhmanova et al., 2012) also in the MSH. However, the understanding of the dynamics of small SW plasma and magnetic field structures downstream the bow shock requires a joint analysis of the turbulent properties of the SW and the MSH (Savin et al., 2014). The power character of the spectra of turbulent fluctuations are similar in the SW and in the MSH (Shevyrev and Zastenker, 2005; Alexandrova, 2008) with significant differences for so called inertial and dissipation scales and with the break between them (Alexandrova, 2008). Different types of waves can influence the spectra formation in the MSH (Alexandrova et al., 2008), e.g. the presence of Alfven vortices in the MSH often leads to the observation of the spectrum peak near to the spectral break (Alexandrova et al., 2004, 2006). An Alfvenic fluctuations dominate in a wide frequency range in the MSH for small plasma b parameters and the spectra become anisotropic (Anderson et al., 1994), whereas the spectra are practically isotropic for large b plasma. High-resolution plasma measurements of the BMSW instrument open a possibility to analyze and to compare the turbulent properties of plasma fluctuations in the SW and in the MSH up to the frequency 10 Hz. Such comparison is necessary to study the dynamics of small-scale SW fluctuations in the MSH and also for understanding the reasons of frequently observed low correlations of the SW and the MSH simultaneous data series.
2. The experimental data set We use the ion flux measurements by the BMSW (Bright Monitor of Solar Wind) spectrometer of the Plasma-F experiment on board of the SPECTR-R spacecraft (Sˇafra´nkova´ et al., 2008, 2013a; Zastenker et al., 2013; Zelenyi et al., 2013). The regular systematic measurements in the SW and the MSH are available due to the highapogee orbit and due to the long-term employment of the device (it operates since August 2011 until present). The main advantage of the BMSW instrument is a possibility of measurements of the ion flux value and its direction with high-time resolution up to 31 ms. The plasma parameters (as the ion density, bulk velocity and ion temperature) are measured in a part of time with the same high-time resolution (in the adaptive mode). In the rest of time, the same set of parameters and also the density of helium is determined with a 3 s time resolution from the energy distribution of ions (in the sweeping mode). The BMSW is directed to the Sun with a precision of 5–10° (the orientation of the device relative to the Sun direction is determined with the accuracy of 1° by the special solar sensor). The ion flux deflection from the device axis can be determined from the ratio of currents from the three multidirectional sensors. If the deflection is large (as often in the MSH), the currents may be sufficiently small on some of sensors and the noise can become significant in these cases. It can lead to the errors in a determination of plasma parameters. So we consider only the cases with the angle between the SW stream direction and main axis of the BMSW instrument less than 20°. We selected rather long time intervals (more than 3 h) of the SW and the MSH flank measurements. Then we smashed these intervals in pieces with the length of 17 min (i.e., 32768 points because it is better to use the time series with a number of points equal to a power of two for the Fourier transform) with overlapping by a half of their length with respect one to another. As the next step, we analyze the ion flux fluctuation properties for 363 17-min intervals in the SW and for 427 intervals with the same length in the flank MSH. Note that we used the statistics presented earlier in Riazantseva et al. (2015) for the SW analysis. We compute the fast Fourier transform for a spectral analysis and smooth the spectra using a Hanning window in the frequency domain. We focus on the behavior of the power spectra at frequencies less than 10 Hz to exclude a noticeable contribution of the instrumental noise to higher frequencies. This frequency limit is obtained from the in-flight calibration and in laboratory tests (Sˇafra´nkova´ et al., 2013a; Chen et al., 2014). A low-frequency limit is determined by a length of intervals and becomes near 0.01 Hz. A study of features of probability distribution functions and structure functions of ion flux fluctuations is performed for the same intervals and for the same frequency ranges.
Please cite this article in press as: Riazantseva, M.O., et al. Comparison of properties of small-scale ion flux fluctuations in the flank magnetosheath and in the solar wind. Adv. Space Res. (2016), http://dx.doi.org/10.1016/j.asr.2015.12.022
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3. Comparison of ion flux spectra in the solar wind and magnetosheath As the BMSW instrument can measure either in the SW or in the MSH simultaneous measurements are not available, thus we could not directly compare the synchronous spectra of fluctuations in these two areas. Fig. 1 presents typical examples of spectra of the ion flux fluctuations in the MSH (for October 24, 2012, 0841-0858 UT) and in the SW (for September 28, 2011, 0903-0920 UT). Both spectra are similar, i.e., the steepening of a spectrum is observed at high frequencies (above 1 Hz). The power spectral density of the ion flux fluctuations is nearly one order of magnitude higher in the MSH than in the SW for scales below 1 s. This difference slightly disappears at high frequency scales due to a more rapid decay of the MSH spectrum. This qualitatively corresponds to previous Interball-1 results (Shevyrev and Zastenker, 2005). A SW spectrum follows a power law with the slope of P1SW = 1.7 ± 0.1 in the low-frequency range below the ion break, which is near to 5/3 of the Kolmogorov like spectrum (dashed line in Fig. 1). Similarly, in the MSH, this slope is equal to P1MSH = 1.65 ± 0.1, again near to the Kolmogorov like spectrum (Kolmogorov, 1941). In the high-frequency range above the break (1–10 Hz), the slopes are equal to P2SW = 2.9 ± 0.1, P2MSH = 3.2 ± 0.1, respectively; the slope in the MSH being slightly steeper than in the SW. The break frequencies between low- and high-frequency parts (determined automatically as the intersection of a linear approximation of two branches of the spectra) are equal to Fb_SW = 1.3 ± 0.1 and Fb_MSH = 0.75 ± 0.1, respectively, so the break frequency in the SW is almost twice larger than that in the MSH. A flattening of the spectra at the junction of these two scales can be observed sometimes in the SW (Unti et al., 1973; Celnikier et al., 1987; Kellogg and Horbury, 2005;
Fig. 1. Examples of power density spectra of ion flux fluctuations in the MSH (October 24, 2012, 0841-0858 UT – red points (grey in the printed version)) and in the SW (September 28, 2011, 0903-0920 UT blue points (black in the printed version)). The dashed line represents the spectrum with a slope of 5/3. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
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Chen et al., 2012; Sˇafra´nkova´ et al., 2015). Some features around the break frequency can be observed also in the MSH spectra (Alexandrova et al., 2004, 2006). However, in the present study, we consider the spectra with two legible slopes without flattening or some peaks to avoid any uncertainties. A third range of the spectrum can be also sometimes observed at high frequencies (up to 4–5 Hz and more), but the level of a noise at these frequencies does not allow to distinguish this range in all spectra legibly, so we did not included such spectra into this study. We compute the values of spectra indices (two slopes and the break frequency between them) for each spectrum in both the SW and flank MSH and to compare them statistically. Fig. 2 presents the distributions of the spectral indices, P1 (Fig. 2a), P2 (Fig. 2b) and the break frequency, Fb (Fig. 2c). The spectral indices change over a wide range in both regions. In a low-frequency range, the maxima of the slope distributions are P1MSH = 1.7 to 1.8 and P1sw = 1.6 to 1.7 respectively (Fig. 2a) and the average values of the MSH = 1.75 ± 0.2 is slightly larger than those in the SW – = 1.6 ± 0.2. Taking into account the standard deviations of these values, both slopes are again near to 5/3 as the numerous experimental results (see, Bruno and Carbone, 2013; Alexandrova et al., 2013 and references therein) show for the magnetic field and plasma density. In the high-frequency range (Fig. 2b), the maxima of distributions are approximately P2MSH 2.8 to 3.0 and P2SW 2.6 to 2.8, and average values are = 3.0 ± 0.4 and = 2.9 ± 0.5. Similarly to the low-frequency range, the slopes are slightly steeper in the flank MSH, but the difference is practically negligible. In the range of >103 Hz, the SW turbulent spectra typically have a power-law shape with a Kolmogorov like slope of 5/3 on the scale up to 0.1–1 Hz and steepen up to 2.5 to 3 in higher frequencies for both the interplanetary magnetic field fluctuations (this aspect has been widely discussed in many papers (e.g., see the reviews of Bruno and Carbone, 2013; Alexandrova et al., 2013 and references therein)) as well as the plasma fluctuations (e.g., Celnikier et al., 1987; Chen et al., 2012; Sˇafra´nkova´ et al., 2013b, 2015; Riazantseva et al., 2015). According to Alexandrova et al. (2008), Kolmogorov-like spectra of magnetic field fluctuations in the MSH are observed only at the flanks where the plasma transit time is rather long and turbulence has enough time to be developed (Alexandrova, 2008). Low-frequency spectra of the magnetic field fluctuations in the MSH are steeper near to the magnetopause and they are characterized by strong anisotropy (Sahraoui et al., 2006) and oppositely, their slopes in the low-frequency part are nearer to 1 around the bow shock (Czaykowska et al. (2001), while the high-frequency decay generally has the slope between 3 and 2 (Czaykowska et al., 2001; Alexandrova et al., 2008; Huang et al., 2014). It is rather typical downstream quasi-parallel and quasi-perpendicular shocks (Czaykowska et al., 2001; Shevyrev and Zastenker, 2005).
Please cite this article in press as: Riazantseva, M.O., et al. Comparison of properties of small-scale ion flux fluctuations in the flank magnetosheath and in the solar wind. Adv. Space Res. (2016), http://dx.doi.org/10.1016/j.asr.2015.12.022
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Fig. 2. The distributions of the spectral indices P1 (a), P2 (b) and break frequencies Fb (c) for ion flux fluctuation spectra in the SW (grey bins) and in the flank MSH (dark bins).
The slope in high frequency range becomes slightly larger (3.4) near to the magnetopause (Rezeau et al., 1999). So, the spectrum with near the Kolmogorov slope in the inertial scale, and with the noticeable steepening of the spectra at the high-frequency dissipation scale is rather typical in the flank MSH and the SW as it is observed for ion flux fluctuations (current study) and for magnetic field fluctuations (Alexandrova, 2008). Such similarity of the power spectra of fluctuations demonstrates the universal properties of turbulence in the near Earth environment. The numerous papers discussed the possible explanations of the steepening of the turbulent plasma spectrum at frequencies above the break (see the reviews of Zelenyi and Milovanov, 2004; Howes, 2015 and references therein), but there is no full explanation of this behavior. The break frequency for analyzed spectra also varies in a broad range. The distribution of the break frequency is presented in Fig 2c. The distribution of SW ion flux fluctuation is very wide, its maxima correspond to 2–2.5 Hz and the mean value of the break frequency is equal to = 1.9 ± 0.8 Hz. The maxima of the break frequency distribution in the MSH are 0.5–1 Hz with the mean value = 0.9 ± 0.5 Hz. So the break frequency in the flank MSH is almost twice less than that in the SW. An absence of simultaneous magnetic field measurements onboard of SPECTR-R does not allow us to verify possible relations of the break frequency with the cyclotron frequency. In the SW, it is possible to use the measurements of the magnetic field from other spacecrafts shifted by the SW propagation time as it was done in Sˇafra´nkova´ et al. (2013b) and Sˇafra´nkova´ et al. (2015),
but the propagation time is difficult to determine exactly because of a large number of uncertainties always for the MSH, and sometimes even for the SW. The break frequency does not equal to the cyclotron frequency in average, but several papers discuss some connection between the value of the break frequency and the cyclotron frequency both in the SW (Leamon et al., 2000; Sˇafra´nkova´ et al., 2013b, 2015) and in the MSH (Alexandrova, 2008). Typically, the cyclotron frequency in the MSH is larger than in the SW as the magnetic field increases in the MSH, however, the MSH magnetic field varies with a large amplitude which can result in the uncertainties in a determination of the value of the cyclotron frequency and the variations further complicate a determination of the possible relation of the break frequency with the cyclotron frequency. We suggest that the large dispersion of the spectral break values can be a result of the interaction of the background turbulence with different instabilities, coherent structures and nonlinear waves in plasma (Alexandrova et al., 2004, 2006; Gary, 2015). So the different nature and dynamics of such processes in the MSH and the SW can cause the great differences of the spectral break position. 4. A statistical study of ion flux fluctuations in the magnetosheath Developed turbulence is characterized by a large number of coupled modes, a small-scale structure and random fluctuations of velocities and fields. Hence, it can be best described by the statistical methods using a probability
Please cite this article in press as: Riazantseva, M.O., et al. Comparison of properties of small-scale ion flux fluctuations in the flank magnetosheath and in the solar wind. Adv. Space Res. (2016), http://dx.doi.org/10.1016/j.asr.2015.12.022
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distribution function (PDF) or, equivalently, by the moments of the distribution function which can be measured in experiments. In the simplest case, the PDF of the fluctuation amplitudes obeys the Gaussian (so called normal) law. In numerous hydrodynamic and plasma experiments, the non-Gaussian statistics of fluctuations is observed and this phenomenon is called intermittency. Intermittency was considered first by Novikov and Stewart (1964); they considered it as a local breaking of turbulence homogeneity when active regions (with strong localized fluctuations) coexist with passive (with small amplitude fluctuations) ones. Universal properties of intermittency are observed both for hydrodynamic and magnetohydrodynamic turbulent flows (see, e.g., review of Budaev et al., 2011 and the reference therein). Intermittency is observed in the SW flow: Statistics of small-scale fluctuations is non-Gaussian but it is nearly Gaussian for large-scale fluctuations. It was demonstrated for the interplanetary magnetic field fluctuations (Marsch and Tu, 1997; Sorriso-Valvo et al., 1999; Bruno et al., 2003; Alexandrova et al., 2009; Kiyani et al., 2009) and also for the SW plasma fluctuations (Bruno et al., 2003; Salem et al., 2009; Riazantseva et al., 2007, 2010; Riazantseva and Zastenker, 2008). BMSW measurements in the SW also show the intermittent nature up to scale 0.1 s. (Riazantseva et al., 2015). Intermittency of MSH magnetic field fluctuations on the same scale was reported in Kozak et al. (2011). Fig. 3 demonstrates the same tendency for ion flux fluctuations calculated as ds Flux (t) = Flux (t + s) Flux (t) (where s is the time-scale of variations) on different scales. The
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experimental PDF is calculated for the interval of 3 h of ion flux fluctuations measured by the BMSW instrument in the flank MSH. PDFs noticeably deviate from a Gaussian shape (calculated with the standard deviation of the signal under consideration) on the time scale 1 s (Fig. 3a and b), slightly deviate from it on the time scale 1 min (c), and are near Gaussian on about a half-hour time scale (d). This trend can be better described by the dependence of 4th order moment of the PDF (so called flatness) on the time scale. Flatness is calculated as F(s) = hjds X(t)j4i/(hjds Xi(t)j2i)2 where ds X(t) = X(t + s) X(t) is the variation of the analyzed parameter on the length-scale s. F(s) = 3 for a Gaussian distribution, and exceed three for the distribution with fatter tails. A high value of flatness reflects a high probability of large amplitude fluctuations. Fig. 4 presents the dependencies of the flatness coefficient of the ion flux fluctuations on the time scales s in the flank MSH (Fig. 4a) and in the SW (Fig. 4b). A time scale s is changed in the range of 0.03–256 s. Each black line corresponds to one of the intervals from discussed above statistics. Solid lines demonstrate the average values of the flatness coefficient for the MSH (red line on Fig. 4a) and the SW (blue line on Fig. 4b), respectively. The green dashed line corresponds the flatness coefficient equal to 3, and illustrates a case of the Gaussian statistics on all scales. One can see that the flatness coefficients are strongly different for various intervals. So it is difficult to discuss the numerical comparison of flatness values for different intervals or even compare the mean values of the flatness in the MSH and in the SW. In spite of a broad range of the flatness
Fig. 3. Examples of the PDFs of ion flux fluctuations in the MSH (solid line) on scales of 1/8 s (a); 1 s (b); 64 s (c); and 2048 s (d), and corresponding Gaussian distributions (dashed line) for period of October 13, 2011 from 1511 to 1835 UT. A value of the time scale s and the corresponding flatness coefficient are presented in the top right corners of each panel.
Please cite this article in press as: Riazantseva, M.O., et al. Comparison of properties of small-scale ion flux fluctuations in the flank magnetosheath and in the solar wind. Adv. Space Res. (2016), http://dx.doi.org/10.1016/j.asr.2015.12.022
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Fig. 4. The dependences of the flatness coefficient for the ion flux fluctuations from the time scale s in the flank MSH (a) and in the SW (b). The solid lines show the average values of the flatness coefficient for the MSH ((a) – red line) and the SW ((b) – blue line), respectively. The green dashed lines correspond to the flatness coefficient equal to 3 for all time scales. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Fig. 5. An analysis of high order structure functions in the flank MSH on November 22, 2011 from 0915 to 1245 UT: (a) the dependence of structure functions from the time scale s; (b) the ESS plot – the dependence of structure functions of q-order Sq from the structure function of 3-rd order S3 (q-is the order of structure functions); (c) nonlinear scaling of the structure function vs. order q derived from the ESS plot – the deviation of normalized scaling f(q)/f(3) from linear scaling q/3 of the K41 model (f(q)/f(3) q/3) (curves of different colors and symbols for several subintervals are shown). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Please cite this article in press as: Riazantseva, M.O., et al. Comparison of properties of small-scale ion flux fluctuations in the flank magnetosheath and in the solar wind. Adv. Space Res. (2016), http://dx.doi.org/10.1016/j.asr.2015.12.022
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coefficients, we can see the tendency of flatness increasing from large to smaller scales for a majority of intervals for both the SW and the flank MSH. In some of the intervals, the flatness increases up to the smallest scale, in other cases, it increases up to scale 1–10 s and it is descending or stays stable after. Such growing of the flatness coefficient is a typical demonstration of the multifractality of the signal, and according to the approach of Bruno et al. (2003), this indicate intermittency in the observed flow. The time series are more intermittent if flatness grows faster. Nonetheless, the flatness may be considered only as a qualitative characteristics of intermittency. Generally, we can say that the flatness grows faster in average for the SW intervals than for the MSH intervals. It is interesting to note that a lot of intervals with flatness coefficients near 3 at all time scales can be observed in the MSH. We can suggest that the ion flow in the flank MSH is less intermittent than that in the SW. For high-frequency magnetic field fluctuations in the MSH, the flatness also grows with the decreasing of the scales (Kozak et al., 2011). These results are similar to those published earlier for the interplanetary magnetic field fluctuations (Bruno et al., 2003; Alexandrova et al., 2009) and SW the plasma fluctuations (Bruno et al., 2003; Riazantseva et al., 2010, 2015; Chen et al., 2014). The analysis of high-order structure functions Sq(s) = h|ds X(t)|qi (where q-is the order of the moment, ds X(t) was defined above) open the possibility to study the scale invariance of the experimental signal. An example of the dependence of high-order structure functions on the time scale s is presented in Fig. 5a for ion flux fluctuations measured by the BMSW instrument in the flank MSH. We present the structure functions only up to order q = 6 because the larger ones can be strongly affected by statistical errors (see, for example, Dudok de Wit et al., 2013). The dependence demonstrates a typical behavior of the structure functions for intermittent signal. It is expected non-linear scaling f(q) on the q order of structure function power law Sq(s) sf(q). Remind that a linear scaling f(q) = q/3 usually characterizes fully developed isotropic turbulence of the Kolmogorov model (Kolmogorov, 1941). In our previous work (Riazantseva et al., 2015), we show that the nonlinear scaling f(q) is observed for ion flux fluctuations in the SW, which is rather typical for the solar wind as a whole (Burlaga, 1991). In Riazantseva et al. (2015), we demonstrated the extended self-similarity (ESS) of fluctuations of the SW ion flux on the basis of a linear dependence log(Sq(s)) from log(S3(s)) according to the concept of Benzi et al. (1993) for an intermittent turbulent flow. The similar analysis was also provide by Chen et al. (2014). Fig. 5b reflects the same trend of ESS observations for MSH ion flux fluctuations: almost two-three orders of magnitudes Sq(l) has a power-law dependence on S3(l), Sq(l) S3(l)f(q)/f(3). The ESS provides an opportunity to estimate the scaling f(q) which is nonlinear (Fig. 5c), it allows us a comparison with the predictions of intermittent turbulence models (e.g., Budaev et al., 2011). For MSH magnetic field high-frequency fluctuations, the ESS
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properties were shown earlier in Kozak et al. (2011) based on the Interball-1 data. In Budaev et al. (2011), an ESS property is observed in neutral fluids and also in space and laboratory plasmas. 5. Conclusions The BMSW high-resolution measurements allow us to study the main turbulent properties of a plasma flow in the SW and in the flank MSH up to scale 0.03 s. They show the difference between turbulent spectra in these regions on smallest scales which were not reached in the previous experiments. The analysis shows that turbulence of the SW plasma flow substantially retains its properties in the flank MSH. The power density spectra of ion flux fluctuations are similar in their basic features: Spectra are usually separated into two clear branches and form low- and high-frequency ranges that are characterized by different spectral slopes. Alexandrova, 2008 reported also a comparison of the values of the spectral indices in the SW and MSH for magnetic field fluctuations and has reached the similar conclusions at the flank MSH. Sometimes the influence of Alfven vortices (e.g., Alexandrova et al., 2004, 2006) can add some features to the area of the ion break point; however, we excluded such spectra from our statistics. Below, we summarize the results of the analysis of turbulent ion flux fluctuations in the flank MSH (427 intervals) comparatively to those in the SW (363 intervals) by the BMSW measurements. – An inertial range of ion flux fluctuation spectra at the flank MSH is characterized by a spectral slope equal to 1.75 ± 0.2. In the SW, the spectra are less steep than those in the MSH, their slopes are equal to 1.6 ± 0.2 and correspond better to Kolmogorov like spectra. – In a high-frequency range, almost twice steeper slopes were found for both flank MSH and SW ion flux fluctuations spectra. The average slopes are = 3.0 ± 0.4 and = 2.9 ± 0.5 for the MSH and SW, respectively. – The average break frequency is equal 0.9 ± 0.5 Hz in the flank MSH which is two times smaller than the value of 1.9 ± 0.8 Hz observed in the SW. – The PDFs of high-frequency ion flux fluctuations in the flank MSH exhibit flatter tails in a comparison to the Gaussian distribution of non-intermittent flows. The flatness coefficient tends to increase to the small scales which demonstrates the presence of multifractal, intermittent signal. So, MSH turbulence is intermittent, similarly to SW turbulence. – The behavior of high-order structure functions reflects the trend of an extended self-similarity in the observed multifractal signal of the ion flow in the flank MSH as it was early shown for ion flux in the SW (Chen et al., 2014; Riazantseva et al., 2015). As we mentioned in the introduction, the large number of turbulent fluctuations in the MSH are generated by the
Please cite this article in press as: Riazantseva, M.O., et al. Comparison of properties of small-scale ion flux fluctuations in the flank magnetosheath and in the solar wind. Adv. Space Res. (2016), http://dx.doi.org/10.1016/j.asr.2015.12.022
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bow shock. At the same time all plasma parameters change strongly in the MSH. The plasma decelerates and becomes denser, the temperature increases and becomes anisotropic. Also the waves can be amplified as they cross the bow shock. These changes can affect turbulence properties as we demonstrate by the present analysis. From it follows that MSH turbulence is more developed and, as a consequence, it can be less intermittent in some cases. Acknowledgments The work produced with the support of RFBR Grant Nos. 13-02-00819 and 16-02-00669 and the program of Russian Academy of Sciences P-22 in Space Research Institute (Moscow), with the support of RFBR Grants 12-0500984 and 15-05-04965 in Skobeltsyn Institute of Nuclear Physics of Lomonosov Moscow State University, with the support of projects of the Czech Grant Agency, 1419376S and the Ministry of Education, LH14193 in Charles University in Prague. The work of L. Rakhmanova is produce with the support of RFBR grant 16-32-00818. The authors also thank the big team of the PLASMA-F experiment in Space Research Institute and in the space industry enterprise NPOL. References Alexandrova, O., Mangeney, A., Maksimovic, M., et al., 2004. Cluster observations of finite amplitude Alfve´n waves and small-scale magnetic filaments downstream of a quasi-perpendicular shock. J. Geophys. Res. 109, A05207. http://dx.doi.org/10.1029/2003JA010056. Alexandrova, O., Mangeney, A., Maksimovic, M., et al., 2006. Alfve´n vortex filaments observed in magnetosheath downstream of a quasiperpendicular bow shock. J. Geophys. Res. 111, A12208. http://dx.doi.org/10.1029/2006JA011934. Alexandrova, O., 2008. Solar wind vs magnetosheath turbulence and Alfve´n vortices. Nonlin. Processes Geophys. 15 (1), 95–108, www. nonlin-processes-geophys.net/15/95/2008/. Alexandrova, O., Lacombe, C., Mangeney, A., 2008. Spectra and anisotropy of magnetic fluctuations in the Earth’s magnetosheath: cluster observations. Ann. Geophys. 26 (11), 3585–3596. http://dx.doi. org/10.5194/angeo-26-3585-2008. Alexandrova, O., Saur, J., Lacombe, C., et al., 2009. Universality of solarwind turbulent spectrum from MHD to electron scales. Phys. Rev. Lett. 103 (16), 165003. http://dx.doi.org/10.1103/PhysRevLett. 103.165003. Alexandrova, O., Chen, C.H.K., Sorisso-Valvo, L., et al., 2013. Solar wind turbulence and the role of ion instabilities. Space Sci. Rev. 178 (2–4), 101–139. http://dx.doi.org/10.1007/s11214-013-0004-8. Anderson, B.J., Fuselier, S.A., Gary, S.P., Denton, R.E., 1994. Magnetic spectral signatures in the Earth’s magnetosheath and plasma depletion layer. J. Geophys. Res. 99, 5877–5891. http://dx.doi.org/10.1029/ 93JA02827. Benzi, R., Ciliberto, S., Tripiccione, R., et al., 1993. Extended selfsimilarity in turbulent flows. Phys. Rev. E 48, 29–35. http://dx.doi.org/ 10.1209/0295-5075/24/4/007. Blanco-Cano, X., Omidi, N., Russell, C.T., 2006. Macrostructure of collisionless bow shocks: 2. ULF waves in the foreshock and magnetosheath. J. Geophys. Res. 111, A10205. http://dx.doi.org/ 10.1029/2005JA011421. Bruno, R., Carbone, V., Sorriso-Valvo, L., Bavassano, B., 2003. Radial evolution of solar wind intermittency in the inner heliosphere. J. Geophys. Res. 108 (A3), 1130. http://dx.doi.org/10.1029/ 2002JA009615.
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Please cite this article in press as: Riazantseva, M.O., et al. Comparison of properties of small-scale ion flux fluctuations in the flank magnetosheath and in the solar wind. Adv. Space Res. (2016), http://dx.doi.org/10.1016/j.asr.2015.12.022
ScienceDirect Advances in Space Research xxx (2016) xxx–xxx www.elsevier.com/locate/asr
Comparison of properties of small-scale ion flux fluctuations in the flank magnetosheath and in the solar wind M.O. Riazantseva a,b,⇑, V.P. Budaev b,c, L.S. Rakhmanova b, G.N. Zastenker b, J. Sˇafra´nkova´ d, Z. Neˇmecˇek d, L. Prˇech d a
Skobeltsyn Institute of Nuclear Physics, Lomonosov Moscow State University (SINP MSU), 1(2), Leninskie gory, Moscow 119991, Russia b Space Research Institute of the Russian Academy of Science (IKI), 84/32 Profsoyuznaya Str., Moscow 117997, Russia c National Research Centre ‘‘Kurchatov Institute”, 1, Akademika Kurchatova pl., Moscow 123182, Russia d Faculty of Mathematics and Physics, Charles University, V Holesovickach 2, Prague 8 18000, Czech Republic Received 25 April 2015; received in revised form 13 December 2015; accepted 15 December 2015
Abstract We present a comparison of spectral and statistical properties of ion flux fluctuations in the turbulent solar wind and in the flank Earth’s magnetosheath. We use the data of the BMSW device operating in frame of the SPECTR-R mission with an extremely hightime resolution (up to 30 ms). Fourier spectra of ion flux fluctuations are systematically analyzed both in the solar wind and in the magnetosheath on the inertial scale and on a transition to the dissipation scale in the range of 0.01–10 Hz. We show that ion flux fluctuation spectra in the flank magnetosheath are similar to those observed in the solar wind and we demonstrate the presence of the break at frequencies of 1–2 Hz. Spectra are slightly steeper in the flank magnetosheath but the break frequency is near twice less in a comparison to the solar wind. The magnetosheath ion flux turbulent flow is intermittent as it was shown earlier for the solar wind. We discuss the level of intermittency of ion flux fluctuations in both regions and we determine the characteristics of structure functions. Finally, we demonstrate extended self-similarity in the magnetosheath. Ó 2015 COSPAR. Published by Elsevier Ltd. All rights reserved.
Keywords: Magnetosheath; Solar wind; Plasma turbulence; Intermittency; Kinetic scales
1. Introduction The Earth’s magnetosheath (MSH) delivers solar wind (SW) plasma into the magnetosphere, thus solarterrestrial investigations need the analysis of variations of SW parameters and their modifications in the MSH. A study of SW and MSH turbulence presents an opportunity to analyze the dynamics of the space plasma for different boundary conditions. SW turbulence freely developes in ⇑ Corresponding author at: Skobeltsyn Institute of Nuclear Physics, Lomonosov Moscow State University (SINP MSU), 1(2), Leninskie gory, Moscow 119991, Russia. Tel.: +7 4959392810. E-mail address: [email protected] (M.O. Riazantseva).
the space during long-time intervals, whereas MSH turbulence developes between two boundaries – the magnetopause and bow shock. A large number of various types of waves originate at these boundaries and inside the MSH (Omidi et al., 1994; Lacombe and Belmont, 1995). A high level of plasma and magnetic field fluctuations are constantly observed in the MSH as a result of these processes (Neˇmecˇek et al., 2001; Zastenker et al., 2002; Shevyrev et al., 2003), thus they cannot be described by traditional models of a laminar plasma flow (Neˇmecˇek et al., 2000; Zastenker et al., 2002). The recent hybrid models (e.g., Karimabadi et al., 2014) demonstrate the presence of a high level of turbulent fluctuations downstream the quasi-parallel bow shock. The
http://dx.doi.org/10.1016/j.asr.2015.12.022 0273-1177/Ó 2015 COSPAR. Published by Elsevier Ltd. All rights reserved.
Please cite this article in press as: Riazantseva, M.O., et al. Comparison of properties of small-scale ion flux fluctuations in the flank magnetosheath and in the solar wind. Adv. Space Res. (2016), http://dx.doi.org/10.1016/j.asr.2015.12.022
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quasi-parallel bow shock modifies and amplifies SW fluctuations, so the level of fluctuations is increased behind it (Shevyrev et al., 2003; Alexandrova, 2008). Shevyrev and Zastenker (2005) show that the amplitudes of high frequency fluctuations are also higher behind the quasiparallel bow shock as well as in the foreshock located in front of it. Fluctuations of different nature are observed in the MSH because all outer regions, the bow shock (e.g., Neˇmecˇek et al., 2002) foreshock (e.g., Blanco-Cano et al., 2006) and magnetopause (e.g., Rezeau et al., 1999; Gutynska et al., 2012) contribute to the formation of MSH turbulence. A high level of turbulent fluctuations in the MSH leads to a low level of correlations between the magnetic field and plasma in the MSH and SW (Gutynska et al., 2012; Rakhmanova et al., 2015a,b). Moreover, the magnetic field near the magnetopause can strongly differ from that in the SW (Sˇafra´nkova´ et al., 2009; Pulinets et al., 2014) and this fact significantly complicates a creation of adequate models. In the MSH, the fluctuations of the SW origin and the fluctuations born on boundaries of the MSH and within it are mixed (Zastenker et al., 2002). Low-frequency fluctuations come mainly from the SW, whereas high-frequency fluctuations have as a rule a local origin (Gutynska et al., 2012; Rakhmanova et al., 2015a,b), however, it is rather difficult to separate these sources in each specific case. For example, the dynamics of some significant structures in the SW (as the interplanetary shocks, or large amplitude discontinuities) can be traced (Koval et al., 2006; Sˇafra´nkova´ et al., 2007; Rakhmanova et al., 2012) also in the MSH. However, the understanding of the dynamics of small SW plasma and magnetic field structures downstream the bow shock requires a joint analysis of the turbulent properties of the SW and the MSH (Savin et al., 2014). The power character of the spectra of turbulent fluctuations are similar in the SW and in the MSH (Shevyrev and Zastenker, 2005; Alexandrova, 2008) with significant differences for so called inertial and dissipation scales and with the break between them (Alexandrova, 2008). Different types of waves can influence the spectra formation in the MSH (Alexandrova et al., 2008), e.g. the presence of Alfven vortices in the MSH often leads to the observation of the spectrum peak near to the spectral break (Alexandrova et al., 2004, 2006). An Alfvenic fluctuations dominate in a wide frequency range in the MSH for small plasma b parameters and the spectra become anisotropic (Anderson et al., 1994), whereas the spectra are practically isotropic for large b plasma. High-resolution plasma measurements of the BMSW instrument open a possibility to analyze and to compare the turbulent properties of plasma fluctuations in the SW and in the MSH up to the frequency 10 Hz. Such comparison is necessary to study the dynamics of small-scale SW fluctuations in the MSH and also for understanding the reasons of frequently observed low correlations of the SW and the MSH simultaneous data series.
2. The experimental data set We use the ion flux measurements by the BMSW (Bright Monitor of Solar Wind) spectrometer of the Plasma-F experiment on board of the SPECTR-R spacecraft (Sˇafra´nkova´ et al., 2008, 2013a; Zastenker et al., 2013; Zelenyi et al., 2013). The regular systematic measurements in the SW and the MSH are available due to the highapogee orbit and due to the long-term employment of the device (it operates since August 2011 until present). The main advantage of the BMSW instrument is a possibility of measurements of the ion flux value and its direction with high-time resolution up to 31 ms. The plasma parameters (as the ion density, bulk velocity and ion temperature) are measured in a part of time with the same high-time resolution (in the adaptive mode). In the rest of time, the same set of parameters and also the density of helium is determined with a 3 s time resolution from the energy distribution of ions (in the sweeping mode). The BMSW is directed to the Sun with a precision of 5–10° (the orientation of the device relative to the Sun direction is determined with the accuracy of 1° by the special solar sensor). The ion flux deflection from the device axis can be determined from the ratio of currents from the three multidirectional sensors. If the deflection is large (as often in the MSH), the currents may be sufficiently small on some of sensors and the noise can become significant in these cases. It can lead to the errors in a determination of plasma parameters. So we consider only the cases with the angle between the SW stream direction and main axis of the BMSW instrument less than 20°. We selected rather long time intervals (more than 3 h) of the SW and the MSH flank measurements. Then we smashed these intervals in pieces with the length of 17 min (i.e., 32768 points because it is better to use the time series with a number of points equal to a power of two for the Fourier transform) with overlapping by a half of their length with respect one to another. As the next step, we analyze the ion flux fluctuation properties for 363 17-min intervals in the SW and for 427 intervals with the same length in the flank MSH. Note that we used the statistics presented earlier in Riazantseva et al. (2015) for the SW analysis. We compute the fast Fourier transform for a spectral analysis and smooth the spectra using a Hanning window in the frequency domain. We focus on the behavior of the power spectra at frequencies less than 10 Hz to exclude a noticeable contribution of the instrumental noise to higher frequencies. This frequency limit is obtained from the in-flight calibration and in laboratory tests (Sˇafra´nkova´ et al., 2013a; Chen et al., 2014). A low-frequency limit is determined by a length of intervals and becomes near 0.01 Hz. A study of features of probability distribution functions and structure functions of ion flux fluctuations is performed for the same intervals and for the same frequency ranges.
Please cite this article in press as: Riazantseva, M.O., et al. Comparison of properties of small-scale ion flux fluctuations in the flank magnetosheath and in the solar wind. Adv. Space Res. (2016), http://dx.doi.org/10.1016/j.asr.2015.12.022
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3. Comparison of ion flux spectra in the solar wind and magnetosheath As the BMSW instrument can measure either in the SW or in the MSH simultaneous measurements are not available, thus we could not directly compare the synchronous spectra of fluctuations in these two areas. Fig. 1 presents typical examples of spectra of the ion flux fluctuations in the MSH (for October 24, 2012, 0841-0858 UT) and in the SW (for September 28, 2011, 0903-0920 UT). Both spectra are similar, i.e., the steepening of a spectrum is observed at high frequencies (above 1 Hz). The power spectral density of the ion flux fluctuations is nearly one order of magnitude higher in the MSH than in the SW for scales below 1 s. This difference slightly disappears at high frequency scales due to a more rapid decay of the MSH spectrum. This qualitatively corresponds to previous Interball-1 results (Shevyrev and Zastenker, 2005). A SW spectrum follows a power law with the slope of P1SW = 1.7 ± 0.1 in the low-frequency range below the ion break, which is near to 5/3 of the Kolmogorov like spectrum (dashed line in Fig. 1). Similarly, in the MSH, this slope is equal to P1MSH = 1.65 ± 0.1, again near to the Kolmogorov like spectrum (Kolmogorov, 1941). In the high-frequency range above the break (1–10 Hz), the slopes are equal to P2SW = 2.9 ± 0.1, P2MSH = 3.2 ± 0.1, respectively; the slope in the MSH being slightly steeper than in the SW. The break frequencies between low- and high-frequency parts (determined automatically as the intersection of a linear approximation of two branches of the spectra) are equal to Fb_SW = 1.3 ± 0.1 and Fb_MSH = 0.75 ± 0.1, respectively, so the break frequency in the SW is almost twice larger than that in the MSH. A flattening of the spectra at the junction of these two scales can be observed sometimes in the SW (Unti et al., 1973; Celnikier et al., 1987; Kellogg and Horbury, 2005;
Fig. 1. Examples of power density spectra of ion flux fluctuations in the MSH (October 24, 2012, 0841-0858 UT – red points (grey in the printed version)) and in the SW (September 28, 2011, 0903-0920 UT blue points (black in the printed version)). The dashed line represents the spectrum with a slope of 5/3. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
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Chen et al., 2012; Sˇafra´nkova´ et al., 2015). Some features around the break frequency can be observed also in the MSH spectra (Alexandrova et al., 2004, 2006). However, in the present study, we consider the spectra with two legible slopes without flattening or some peaks to avoid any uncertainties. A third range of the spectrum can be also sometimes observed at high frequencies (up to 4–5 Hz and more), but the level of a noise at these frequencies does not allow to distinguish this range in all spectra legibly, so we did not included such spectra into this study. We compute the values of spectra indices (two slopes and the break frequency between them) for each spectrum in both the SW and flank MSH and to compare them statistically. Fig. 2 presents the distributions of the spectral indices, P1 (Fig. 2a), P2 (Fig. 2b) and the break frequency, Fb (Fig. 2c). The spectral indices change over a wide range in both regions. In a low-frequency range, the maxima of the slope distributions are P1MSH = 1.7 to 1.8 and P1sw = 1.6 to 1.7 respectively (Fig. 2a) and the average values of the MSH
Please cite this article in press as: Riazantseva, M.O., et al. Comparison of properties of small-scale ion flux fluctuations in the flank magnetosheath and in the solar wind. Adv. Space Res. (2016), http://dx.doi.org/10.1016/j.asr.2015.12.022
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Fig. 2. The distributions of the spectral indices P1 (a), P2 (b) and break frequencies Fb (c) for ion flux fluctuation spectra in the SW (grey bins) and in the flank MSH (dark bins).
The slope in high frequency range becomes slightly larger (3.4) near to the magnetopause (Rezeau et al., 1999). So, the spectrum with near the Kolmogorov slope in the inertial scale, and with the noticeable steepening of the spectra at the high-frequency dissipation scale is rather typical in the flank MSH and the SW as it is observed for ion flux fluctuations (current study) and for magnetic field fluctuations (Alexandrova, 2008). Such similarity of the power spectra of fluctuations demonstrates the universal properties of turbulence in the near Earth environment. The numerous papers discussed the possible explanations of the steepening of the turbulent plasma spectrum at frequencies above the break (see the reviews of Zelenyi and Milovanov, 2004; Howes, 2015 and references therein), but there is no full explanation of this behavior. The break frequency for analyzed spectra also varies in a broad range. The distribution of the break frequency is presented in Fig 2c. The distribution of SW ion flux fluctuation is very wide, its maxima correspond to 2–2.5 Hz and the mean value of the break frequency is equal to
but the propagation time is difficult to determine exactly because of a large number of uncertainties always for the MSH, and sometimes even for the SW. The break frequency does not equal to the cyclotron frequency in average, but several papers discuss some connection between the value of the break frequency and the cyclotron frequency both in the SW (Leamon et al., 2000; Sˇafra´nkova´ et al., 2013b, 2015) and in the MSH (Alexandrova, 2008). Typically, the cyclotron frequency in the MSH is larger than in the SW as the magnetic field increases in the MSH, however, the MSH magnetic field varies with a large amplitude which can result in the uncertainties in a determination of the value of the cyclotron frequency and the variations further complicate a determination of the possible relation of the break frequency with the cyclotron frequency. We suggest that the large dispersion of the spectral break values can be a result of the interaction of the background turbulence with different instabilities, coherent structures and nonlinear waves in plasma (Alexandrova et al., 2004, 2006; Gary, 2015). So the different nature and dynamics of such processes in the MSH and the SW can cause the great differences of the spectral break position. 4. A statistical study of ion flux fluctuations in the magnetosheath Developed turbulence is characterized by a large number of coupled modes, a small-scale structure and random fluctuations of velocities and fields. Hence, it can be best described by the statistical methods using a probability
Please cite this article in press as: Riazantseva, M.O., et al. Comparison of properties of small-scale ion flux fluctuations in the flank magnetosheath and in the solar wind. Adv. Space Res. (2016), http://dx.doi.org/10.1016/j.asr.2015.12.022
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distribution function (PDF) or, equivalently, by the moments of the distribution function which can be measured in experiments. In the simplest case, the PDF of the fluctuation amplitudes obeys the Gaussian (so called normal) law. In numerous hydrodynamic and plasma experiments, the non-Gaussian statistics of fluctuations is observed and this phenomenon is called intermittency. Intermittency was considered first by Novikov and Stewart (1964); they considered it as a local breaking of turbulence homogeneity when active regions (with strong localized fluctuations) coexist with passive (with small amplitude fluctuations) ones. Universal properties of intermittency are observed both for hydrodynamic and magnetohydrodynamic turbulent flows (see, e.g., review of Budaev et al., 2011 and the reference therein). Intermittency is observed in the SW flow: Statistics of small-scale fluctuations is non-Gaussian but it is nearly Gaussian for large-scale fluctuations. It was demonstrated for the interplanetary magnetic field fluctuations (Marsch and Tu, 1997; Sorriso-Valvo et al., 1999; Bruno et al., 2003; Alexandrova et al., 2009; Kiyani et al., 2009) and also for the SW plasma fluctuations (Bruno et al., 2003; Salem et al., 2009; Riazantseva et al., 2007, 2010; Riazantseva and Zastenker, 2008). BMSW measurements in the SW also show the intermittent nature up to scale 0.1 s. (Riazantseva et al., 2015). Intermittency of MSH magnetic field fluctuations on the same scale was reported in Kozak et al. (2011). Fig. 3 demonstrates the same tendency for ion flux fluctuations calculated as ds Flux (t) = Flux (t + s) Flux (t) (where s is the time-scale of variations) on different scales. The
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experimental PDF is calculated for the interval of 3 h of ion flux fluctuations measured by the BMSW instrument in the flank MSH. PDFs noticeably deviate from a Gaussian shape (calculated with the standard deviation of the signal under consideration) on the time scale 1 s (Fig. 3a and b), slightly deviate from it on the time scale 1 min (c), and are near Gaussian on about a half-hour time scale (d). This trend can be better described by the dependence of 4th order moment of the PDF (so called flatness) on the time scale. Flatness is calculated as F(s) = hjds X(t)j4i/(hjds Xi(t)j2i)2 where ds X(t) = X(t + s) X(t) is the variation of the analyzed parameter on the length-scale s. F(s) = 3 for a Gaussian distribution, and exceed three for the distribution with fatter tails. A high value of flatness reflects a high probability of large amplitude fluctuations. Fig. 4 presents the dependencies of the flatness coefficient of the ion flux fluctuations on the time scales s in the flank MSH (Fig. 4a) and in the SW (Fig. 4b). A time scale s is changed in the range of 0.03–256 s. Each black line corresponds to one of the intervals from discussed above statistics. Solid lines demonstrate the average values of the flatness coefficient for the MSH (red line on Fig. 4a) and the SW (blue line on Fig. 4b), respectively. The green dashed line corresponds the flatness coefficient equal to 3, and illustrates a case of the Gaussian statistics on all scales. One can see that the flatness coefficients are strongly different for various intervals. So it is difficult to discuss the numerical comparison of flatness values for different intervals or even compare the mean values of the flatness in the MSH and in the SW. In spite of a broad range of the flatness
Fig. 3. Examples of the PDFs of ion flux fluctuations in the MSH (solid line) on scales of 1/8 s (a); 1 s (b); 64 s (c); and 2048 s (d), and corresponding Gaussian distributions (dashed line) for period of October 13, 2011 from 1511 to 1835 UT. A value of the time scale s and the corresponding flatness coefficient are presented in the top right corners of each panel.
Please cite this article in press as: Riazantseva, M.O., et al. Comparison of properties of small-scale ion flux fluctuations in the flank magnetosheath and in the solar wind. Adv. Space Res. (2016), http://dx.doi.org/10.1016/j.asr.2015.12.022
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Fig. 4. The dependences of the flatness coefficient for the ion flux fluctuations from the time scale s in the flank MSH (a) and in the SW (b). The solid lines show the average values of the flatness coefficient for the MSH ((a) – red line) and the SW ((b) – blue line), respectively. The green dashed lines correspond to the flatness coefficient equal to 3 for all time scales. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Fig. 5. An analysis of high order structure functions in the flank MSH on November 22, 2011 from 0915 to 1245 UT: (a) the dependence of structure functions from the time scale s; (b) the ESS plot – the dependence of structure functions of q-order Sq from the structure function of 3-rd order S3 (q-is the order of structure functions); (c) nonlinear scaling of the structure function vs. order q derived from the ESS plot – the deviation of normalized scaling f(q)/f(3) from linear scaling q/3 of the K41 model (f(q)/f(3) q/3) (curves of different colors and symbols for several subintervals are shown). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Please cite this article in press as: Riazantseva, M.O., et al. Comparison of properties of small-scale ion flux fluctuations in the flank magnetosheath and in the solar wind. Adv. Space Res. (2016), http://dx.doi.org/10.1016/j.asr.2015.12.022
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coefficients, we can see the tendency of flatness increasing from large to smaller scales for a majority of intervals for both the SW and the flank MSH. In some of the intervals, the flatness increases up to the smallest scale, in other cases, it increases up to scale 1–10 s and it is descending or stays stable after. Such growing of the flatness coefficient is a typical demonstration of the multifractality of the signal, and according to the approach of Bruno et al. (2003), this indicate intermittency in the observed flow. The time series are more intermittent if flatness grows faster. Nonetheless, the flatness may be considered only as a qualitative characteristics of intermittency. Generally, we can say that the flatness grows faster in average for the SW intervals than for the MSH intervals. It is interesting to note that a lot of intervals with flatness coefficients near 3 at all time scales can be observed in the MSH. We can suggest that the ion flow in the flank MSH is less intermittent than that in the SW. For high-frequency magnetic field fluctuations in the MSH, the flatness also grows with the decreasing of the scales (Kozak et al., 2011). These results are similar to those published earlier for the interplanetary magnetic field fluctuations (Bruno et al., 2003; Alexandrova et al., 2009) and SW the plasma fluctuations (Bruno et al., 2003; Riazantseva et al., 2010, 2015; Chen et al., 2014). The analysis of high-order structure functions Sq(s) = h|ds X(t)|qi (where q-is the order of the moment, ds X(t) was defined above) open the possibility to study the scale invariance of the experimental signal. An example of the dependence of high-order structure functions on the time scale s is presented in Fig. 5a for ion flux fluctuations measured by the BMSW instrument in the flank MSH. We present the structure functions only up to order q = 6 because the larger ones can be strongly affected by statistical errors (see, for example, Dudok de Wit et al., 2013). The dependence demonstrates a typical behavior of the structure functions for intermittent signal. It is expected non-linear scaling f(q) on the q order of structure function power law Sq(s) sf(q). Remind that a linear scaling f(q) = q/3 usually characterizes fully developed isotropic turbulence of the Kolmogorov model (Kolmogorov, 1941). In our previous work (Riazantseva et al., 2015), we show that the nonlinear scaling f(q) is observed for ion flux fluctuations in the SW, which is rather typical for the solar wind as a whole (Burlaga, 1991). In Riazantseva et al. (2015), we demonstrated the extended self-similarity (ESS) of fluctuations of the SW ion flux on the basis of a linear dependence log(Sq(s)) from log(S3(s)) according to the concept of Benzi et al. (1993) for an intermittent turbulent flow. The similar analysis was also provide by Chen et al. (2014). Fig. 5b reflects the same trend of ESS observations for MSH ion flux fluctuations: almost two-three orders of magnitudes Sq(l) has a power-law dependence on S3(l), Sq(l) S3(l)f(q)/f(3). The ESS provides an opportunity to estimate the scaling f(q) which is nonlinear (Fig. 5c), it allows us a comparison with the predictions of intermittent turbulence models (e.g., Budaev et al., 2011). For MSH magnetic field high-frequency fluctuations, the ESS
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properties were shown earlier in Kozak et al. (2011) based on the Interball-1 data. In Budaev et al. (2011), an ESS property is observed in neutral fluids and also in space and laboratory plasmas. 5. Conclusions The BMSW high-resolution measurements allow us to study the main turbulent properties of a plasma flow in the SW and in the flank MSH up to scale 0.03 s. They show the difference between turbulent spectra in these regions on smallest scales which were not reached in the previous experiments. The analysis shows that turbulence of the SW plasma flow substantially retains its properties in the flank MSH. The power density spectra of ion flux fluctuations are similar in their basic features: Spectra are usually separated into two clear branches and form low- and high-frequency ranges that are characterized by different spectral slopes. Alexandrova, 2008 reported also a comparison of the values of the spectral indices in the SW and MSH for magnetic field fluctuations and has reached the similar conclusions at the flank MSH. Sometimes the influence of Alfven vortices (e.g., Alexandrova et al., 2004, 2006) can add some features to the area of the ion break point; however, we excluded such spectra from our statistics. Below, we summarize the results of the analysis of turbulent ion flux fluctuations in the flank MSH (427 intervals) comparatively to those in the SW (363 intervals) by the BMSW measurements. – An inertial range of ion flux fluctuation spectra at the flank MSH is characterized by a spectral slope equal to 1.75 ± 0.2. In the SW, the spectra are less steep than those in the MSH, their slopes are equal to 1.6 ± 0.2 and correspond better to Kolmogorov like spectra. – In a high-frequency range, almost twice steeper slopes were found for both flank MSH and SW ion flux fluctuations spectra. The average slopes are
Please cite this article in press as: Riazantseva, M.O., et al. Comparison of properties of small-scale ion flux fluctuations in the flank magnetosheath and in the solar wind. Adv. Space Res. (2016), http://dx.doi.org/10.1016/j.asr.2015.12.022
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bow shock. At the same time all plasma parameters change strongly in the MSH. The plasma decelerates and becomes denser, the temperature increases and becomes anisotropic. Also the waves can be amplified as they cross the bow shock. These changes can affect turbulence properties as we demonstrate by the present analysis. From it follows that MSH turbulence is more developed and, as a consequence, it can be less intermittent in some cases. Acknowledgments The work produced with the support of RFBR Grant Nos. 13-02-00819 and 16-02-00669 and the program of Russian Academy of Sciences P-22 in Space Research Institute (Moscow), with the support of RFBR Grants 12-0500984 and 15-05-04965 in Skobeltsyn Institute of Nuclear Physics of Lomonosov Moscow State University, with the support of projects of the Czech Grant Agency, 1419376S and the Ministry of Education, LH14193 in Charles University in Prague. The work of L. Rakhmanova is produce with the support of RFBR grant 16-32-00818. The authors also thank the big team of the PLASMA-F experiment in Space Research Institute and in the space industry enterprise NPOL. References Alexandrova, O., Mangeney, A., Maksimovic, M., et al., 2004. Cluster observations of finite amplitude Alfve´n waves and small-scale magnetic filaments downstream of a quasi-perpendicular shock. J. Geophys. Res. 109, A05207. http://dx.doi.org/10.1029/2003JA010056. Alexandrova, O., Mangeney, A., Maksimovic, M., et al., 2006. Alfve´n vortex filaments observed in magnetosheath downstream of a quasiperpendicular bow shock. J. Geophys. Res. 111, A12208. http://dx.doi.org/10.1029/2006JA011934. Alexandrova, O., 2008. Solar wind vs magnetosheath turbulence and Alfve´n vortices. Nonlin. Processes Geophys. 15 (1), 95–108, www. nonlin-processes-geophys.net/15/95/2008/. Alexandrova, O., Lacombe, C., Mangeney, A., 2008. Spectra and anisotropy of magnetic fluctuations in the Earth’s magnetosheath: cluster observations. Ann. Geophys. 26 (11), 3585–3596. http://dx.doi. org/10.5194/angeo-26-3585-2008. Alexandrova, O., Saur, J., Lacombe, C., et al., 2009. Universality of solarwind turbulent spectrum from MHD to electron scales. Phys. Rev. Lett. 103 (16), 165003. http://dx.doi.org/10.1103/PhysRevLett. 103.165003. Alexandrova, O., Chen, C.H.K., Sorisso-Valvo, L., et al., 2013. Solar wind turbulence and the role of ion instabilities. Space Sci. Rev. 178 (2–4), 101–139. http://dx.doi.org/10.1007/s11214-013-0004-8. Anderson, B.J., Fuselier, S.A., Gary, S.P., Denton, R.E., 1994. Magnetic spectral signatures in the Earth’s magnetosheath and plasma depletion layer. J. Geophys. Res. 99, 5877–5891. http://dx.doi.org/10.1029/ 93JA02827. Benzi, R., Ciliberto, S., Tripiccione, R., et al., 1993. Extended selfsimilarity in turbulent flows. Phys. Rev. E 48, 29–35. http://dx.doi.org/ 10.1209/0295-5075/24/4/007. Blanco-Cano, X., Omidi, N., Russell, C.T., 2006. Macrostructure of collisionless bow shocks: 2. ULF waves in the foreshock and magnetosheath. J. Geophys. Res. 111, A10205. http://dx.doi.org/ 10.1029/2005JA011421. Bruno, R., Carbone, V., Sorriso-Valvo, L., Bavassano, B., 2003. Radial evolution of solar wind intermittency in the inner heliosphere. J. Geophys. Res. 108 (A3), 1130. http://dx.doi.org/10.1029/ 2002JA009615.
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Please cite this article in press as: Riazantseva, M.O., et al. Comparison of properties of small-scale ion flux fluctuations in the flank magnetosheath and in the solar wind. Adv. Space Res. (2016), http://dx.doi.org/10.1016/j.asr.2015.12.022
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