Anisotropies of the magnetic field fluctuations at kinetic scales in the solar wind : Cluster observations
We present the first statistical study of the anisotropy of the magnetic field turbulence in the solar wind between 1 and 200 Hz, i.e. from proton to sub-electron scales. We consider 93 10-minute intervals of Cluster/STAFF measurements. We find that the fluctuations are not gyrotropic at a given frequency , a property already observed at larger scales (/ mean parallel/perpendicular to the average magnetic ). This non-gyrotropy gives indications on the angular distribution of the wave vectors : at 10 Hz, we find that , mainly in the fast wind; at 10 Hz, fluctuations with a non-negligible are also present. We then consider the anisotropy ratio , which is a measure of the magnetic compressibility of the fluctuations. This ratio, always smaller than 1, increases with . It reaches a value showing that the fluctuations are more or less isotropic at electron scales, for Hz. From 1 to 15-20 Hz, there is a strong correlation between the observed compressibility and the one expected for the kinetic Alfvén waves (KAW), which only depends on the total plasma . For 15-20 Hz, the observed compressibility is larger than expected for KAW; and it is stronger in the slow wind: this could be an indication of the presence of a slow-ion acoustic mode of fluctuations, which is more compressive and is favored by the larger values of the electron to proton temperature ratio generally observed in the slow wind.
For tens of years, the three orthogonal components of the magnetic field fluctuations have been observed in the solar wind, as functions of the frequency in the frame of a single spacecraft. Their frequency spectra display two kinds of anisotropy. A first anisotropy is only related to the direction of the average magnetic field : there is less energy in the compressive fluctuations parallel to than in the transverse fluctuations perpendicular to . The other anisotropy is related to the direction of and to the radial direction , which is close to the direction of the expansion velocity of the solar wind: the fluctuations are non-gyrotropic (non-axisymmetric with respect to ), with less energy in the direction perpendicular to in the plane than in the direction, perpendicular to both and (see e.g., Bruno & Carbone, 2013; Oughton et al., 2015). These two kinds of anisotropy have been observed at large scales, in the inertial range of the magnetic turbulence (Belcher & Davis, 1971; Bieber et al., 1996; Wicks et al., 2012). The fluctuations are still anisotropic and non-gyrotropic at proton scales (around [0.1, 1] Hz) and at sub-proton scales up to 10 Hz, a range sometimes called dissipation range (Leamon et al., 1998; Turner et al., 2011). In the present work, we study the anisotropies of from 1 Hz to about 200 Hz, in a sample of 93 intervals of 10 minutes. Indeed, the nature of the turbulent fluctuations in the solar wind is still under investigation, and a study of their anisotropies can throw some light on this question: previous works have shown that the anisotropy gives indications on the nature of the dominant mode of the fluctuations, while the non-gyrotropy of gives indications on the directions of the wave vectors , i.e. the anisotropy of the -distribution.
The general shape of the -distribution can be deduced from the non-gyrotropy of , using the fact that fluctuations with different contribute to the reduced spectral density observed at a same . If is gyrotropic, a 2D turbulence (i.e. ) will be observed with a non-gyrotropic in the satellite frame; for a slab turbulence (i.e. ), will be gyrotropic in the satellite frame (Bieber et al., 1996). Previous observations in the inertial range have shown that the -distribution is mainly 2D. However, the slab component is also present and its proportion increases when reaches the proton scales [0.1, 1] Hz (Bieber et al., 1996; Leamon et al., 1998; Hamilton et al., 2008).
The anisotropy of the -distribution of the fluctuations has also been estimated by a comparison of the correlation lengths parallel and perpendicular to , with a single spacecraft, assuming a stationary and gyrotropic -distribution in the MHD range, below Hz (Matthaeus et al., 1990) and below Hz (Dasso et al., 2005). Multispacecraft measurements also allow a comparison of the correlation lengths, parallel and perpendicular to , in the inertial range, still assuming a gyrotropic -distribution. Osman & Horbury (2007), with Cluster observations, find an indication that the -distributions in the inertial range are more 2D for the compressive fluctuations than for the transverse fluctuations .
Using data from the four Cluster spacecraft, Chen et al. (2010) find that and have a -distribution with , below 10 Hz. Using the -filtering technique, Sahraoui et al. (2010) find up to 2 Hz. Still using the -filtering technique below 1 Hz, for 52 intervals of Cluster data, Roberts et al. (2015) find ; except in four intervals with a relatively fast wind, where quasi-parallel wave vectors are also present above 0.05 Hz, up to 0.3 Hz where they disappear (Roberts & Li, 2015). These quasi-parallel wave vectors correspond to Alfvén Ion Cyclotron waves (AIC) which are usually observed in this frequency range, in a fast wind (Jian et al., 2014). At proton scales, in the range [0.1,2.5] Hz, Perrone et al. (2016) find that the magnetic fluctuations in a slow wind are mainly compressive coherent structures, with . At the same scales, but in a fast wind, Lion et al. (2016) find Alfvénic coherent structures with , and quasi-monochromatic AIC waves with , superimposed on a non-coherent and non-polarised component of the turbulence. In these analyses, as well as in Perschke et al. (2013), there is no clear mention of a non-gyrotropy of the wave vector energy distribution in the solar wind.
The second kind of anisotropy, the ratio , is a measure of the compressibility of the magnetic fluctuations. In the inertial range, as well as at proton scales, the compressibility increases when the proton factor increases (Smith et al., 2006; Hamilton et al., 2008). () is the ratio between the proton (electron) thermal energy and the magnetic energy. The compressibility also increases when increases, from the inertial range to proton scales (Hamilton et al., 2008; Salem et al., 2012) and from 0.3 to 4 Hz (Podesta & TenBarge, 2012). At sub-proton scales, the compressibility still increases with up to 10 Hz (Alexandrova et al., 2008a) and still increases with up to 100 Hz (Kiyani et al., 2013). We call and the proton and electron inertial lengths, and the proton and electron gyroradii, calculated with the temperatures perpendicular and perpendicular to .
Even if isolated, coherent and non-linear structures are present in the solar wind (Perri et al., 2012; Perrone et al., 2016; Lion et al., 2016), the comparison of observed dimensionless ratios of the fluctuating fields and plasma quantities (transport ratios) with the linear properties of the plasma wave modes can give indications, in a first approximation, on the nature of the dominant type of the fluctuations (Gary, 1992; Krauss-Varban et al., 1994; Denton et al., 1998). This usual modelling of the turbulence as a superposition of linear waves is discussed by Klein et al. (2012) and TenBarge et al. (2012) who consider Alfvén, fast, whistler and slow modes from the MHD to the kinetic range.
The slow mode is frequently neglected because it is strongly damped, except when the electron to proton temperature ratio is larger than 1. However Wind observations at large scales (), in the inertial range, show that the anticorrelation between the density and the compressive magnetic fluctuations is typical of quasi-perpendicular slow modes; kinetic slow waves may then be cascaded as passive fluctuations by Alfvénic fluctuations, and thus exist at proton scales (Howes et al., 2012; Klein et al., 2012). Narita & Marsch (2015) underline how difficult it is to distinguish between kinetic slow modes and kinetic Alfvén waves (KAW), mainly in a high-beta plasma.
Salem et al. (2012) compare the observed compressibility with the compressibility expected for whistler waves or for KAW, and find that the compressibility up to 1 Hz can be explained by KAW-like fluctuations, with nearly perpendicular wave vectors. Quasi-perpendicular KAW are generally found up to proton scales, i.e. up to 1 to 3 Hz (Sahraoui et al., 2010; He et al., 2012; Podesta & TenBarge, 2012; Podesta, 2013; Roberts et al., 2015). Conversely, Smith et al. (2012) find that KAW cannot be the only component below 1 Hz, at least when is larger than 1. Comparing the spectra of the density and of the magnetic field fluctuations measured on the ARTEMIS-P2 spacecraft, Chen et al. (2013) find KAW between 2.5 and 7.5 Hz, i.e. 5 to 14.
Calculations of Podesta et al. (2010) show that the linear collisionless electron Landau damping prevents KAW to cascade to the electron scales, so that the energy cascade to these scales must be supported by wave modes other than the KAW mode. On the other hand, analytical calculations and numerical simulations of the turbulence at sub-proton scales show that a cascade characterized by KAW-like properties can be sustained between proton and electron scales (Howes et al., 2011; TenBarge et al., 2013; Franci et al., 2015; Schreiner & Saur, 2017) and can explain the shape of the spectra observed by Alexandrova et al. (2012), controlled by the electron gyroradius.
With 2D kinetic simulations, Camporeale & Burgess (2011b) analyse the dispersion relation and the electron compressibility (i.e. the ratio between the electron density and the magnetic field modulus of the fluctuations) up to 1 or more, for . They find that the electron compressibility of the simulated fluctuations is too small to be related to slow-ion acoustic waves, and much too large to be related to whistler waves, while KAW are damped. They conclude that the fluctuations are probably a mixture of different modes.
In this paper, we consider the two types of anisotropy discussed above: (i) the non-gyrotropy of and (ii) the compressibility of the magnetic fluctuations . After a presentation of the Cluster/STAFF instrument (section 2) and of the data selection procedure (section 3), we study the non-gyrotropy of . In section 4, we compare our observations, from 1 Hz to 50 Hz or more, in the fast wind and in the slow wind, with the calculations of Saur & Bieber (1999): this gives us indications on the -distribution, assumed to be gyrotropic in the plasma frame. Note that the values of the field-to-flow angle are near in our sample, otherwise Cluster would be magnetically connected to the Earth’s bow shock. In section 5, we study the compressibility of the magnetic fluctuations. We show how the compressibility depends on the plasma parameters in our sample. Then, we compare our observations with the predictions for KAW. We show that the observed magnetic compressibility agrees with the KAW compressibility up to 15-20 Hz, which corresponds to in our sample (where the electron inertial length is about 1 to 3 km). Above 20 Hz, the magnetic fluctuations are more compressive than KAW, and more compressive in the slow wind than in the fast wind. It is well-known that the electron to proton temperature ratio is larger in the slow wind. A larger increases the damping of KAW for several values of and (Schreiner & Saur, 2017) but reduces the damping of the slow-ion acoustic mode; so that we cannot exclude the presence of this last mode for 20 Hz, i.e. for 0.3, a mode which would be less damped and more compressive than the KAW mode.
2. Instruments and data
The present study relies on data sets from different experiments onboard a spacecraft (Cluster 1) of the Cluster fleet. The Spatio-Temporal Analysis of Field Fluctuations (STAFF) experiment on Cluster (Cornilleau-Wehrlin et al., 1997, 2003) measures the three orthogonal components of the magnetic field fluctuations in the frequency range 0.1 Hz - 4 kHz, and comprises two on board analysers, a wave form unit (SC) and a Spectrum Analyser (SA). STAFF-SC provides digitized wave forms, which are projected in the Magnetic Field Aligned (MFA) frame given by the FGM experiment every 4 seconds; Morlet wavelet spectra are then calculated between 1 and 9 Hz. The Spectrum Analyser builds a 3x3 spectral matrix every 4 s, at 27 frequencies between 8 Hz and 4 kHz. This spectral matrix is also projected in the MFA frame: the Propagation Analysis of STAFF-SA Data with Coherency Tests (PRASSADCO program) gives the fluctuation properties, direction of propagation, phase and polarisation with respect to (Santolík et al., 2003). Both experiments, SC and SA, allow to see whether the observed fluctuations are polarised or not. In some intervals, circularly-polarised fluctuations are observed, with a direction of propagation near , at frequencies displaying a small or large spectral bump. These fluctuations are quasi-parallel whistler waves (Lacombe et al., 2014). In the present work we only consider intervals without such quasi-parallel whistler waves.
The WHISPER experiment (Décréau et al., 1997) is used to check that Cluster is in the free solar wind, i.e. that the magnetic field line through Cluster does not intersects the Earth’s bow shock: there is no electrostatic or Langmuir wave, typical of the foreshock, in our sample. Some of the intervals studied by Perri et al. (2009), Narita et al. (2010, 2011a, 2011b), Narita (2014), Comişel et al. (2014), and Perschke et al. (2014) contain incursions of Cluster in the foreshock.
The Cluster Science Data System gives the magnetic field every 4 s (FGM experiment, Balogh et al. (1997)), the proton density , the wind velocity and the proton temperatures and , parallel and perpendicular to of the whole proton distribution (CIS experiment, Rème et al. (1997)). The electron parameters given by the Low Energy Electron Analyser of the PEACE experiment (Johnstone et al., 1997) are taken from the Cluster Science Archive: we use the electron temperatures and of the whole electron distribution.
3. Data set
We first consider the same sample of solar wind intervals as Alexandrova et al. (2012), observed on Cluster 1 from 2001 to 2005. This sample contains 100 intervals of 10 minutes, when magnetic fluctuations (which are not circularly polarised) are observed above 1 Hz, up to 20 to 400 Hz according to the intensity of the fluctuations. These frequencies are above the proton cyclotron frequency [0.08 - 0.3 Hz] in our sample. With the Taylor hypothesis , where is the wavenumber of the fluctuations, the scales corresponding to the observed frequencies decrease from to ( 0.4 to 130).
For each 10-minute interval, average spectra are calculated over 150 points (4 s spectra) in the three directions of the MFA frame. The 4 s FGM magnetic field can be considered as a quasi-local mean field, giving an average frame valid for fluctuations above 1 Hz. It is not really a local mean , which would be computed with waveforms at each scale: such a computation is impossible with the STAFF-SA data, above 1 Hz. A consequence of this use of a quasi-local mean , in place of a local field, will be mentioned in section 4.1.
Alexandrova et al. (2012) studied the spectral shape of the total variance of the fluctuations, summed in the three directions (without subtraction of the background noise). We now want to study the anisotropies of the variance of the magnetic fluctuations. It is well known that the variance is anisotropic, in the inertial range and at kinetic scales, and that the anisotropies are weaker at small scales (see e.g. Turner et al. (2011), and references therein; Kiyani et al. (2013)). At high frequencies, the Power Spectral Density (PSD) is weaker, and tends towards the instrument background noise, which is an additive noise. It is thus necessary to subtract the background noise, not only for the total 3D magnetic variance but also for the 1D variance in each direction. This subtraction has been done as explained in Appendix. As a result, 7 intervals among 100 have been withdrawn from the sample because their signal-to-noise ratio was smaller than 3 at 20 Hz and above. We thus obtain 93 intervals with 1D and 3D spectra intense enough up to 20 Hz or more, after subtraction of 1D and 3D background noise. The importance of the noise subtraction is illustrated in Figure 4 of Howes et al. (2008).
During these intervals in the free solar wind, the Cluster orbit implies that the average magnetic field makes a large angle with the solar wind velocity: the sampling direction is far from the direction. The acute angle between and , which is close to the field-to-flow angle, is always larger than in our sample; its average value is .
Figure 1 gives the ranges of the plasma properties which can be found in our data set. It displays scatter plots of the beta factors and , of the solar wind speed , of the electron to proton temperature ratio , and of the anisotropy ratios of the proton and electron temperature, and . varies between 0.28 and 5.1, and between 0.08 and 3.9 (Fig. 1a). varies between 0.22 and 2.2, and between 300 and 690 km/s (Fig. 1b). varies between 0.12 and 0.97 (Fig. 1c), and between 0.57 and 1 (Fig. 1d). Within these ranges, the plasma is expected to be sufficiently far from the thresholds of proton and electron kinetic instabilities (e.g., Hellinger et al., 2006; Štverák et al., 2008; Matteini et al., 2013; Chen et al., 2016), at least in the majority of the intervals analysed. We then exclude that a possible wave activity associated to these processes may significantly affect the results of this work.
4. Observations, and wave vector distribution
4.1. Spectra and anisotropies
We consider the anisotropic Power Spectral Density (PSD) of the magnetic fluctuations as a function of . The PSD of the transverse fluctuations, perpendicular to , is . is the PSD perpendicular to and to , and is the PSD perpendicular to in the plane . We call compressive the fluctuations parallel to i.e. to , . We call the sum of the fluctuations in the three directions . Two spectra of our sample are shown in Figure 2. For each spectrum, the solid line gives , the dashed line and the dash-dot line . We see that is the most intense spectrum and that is generally the least intense. is close to at the lowest frequencies, and close to at the highest frequencies.
Figure 3a displays the anisotropy of the variance of the fluctuations in the two directions and perpendicular to . The 93 solid lines are the ratios of the 10-minute averages of the PSD in these two directions, anisotropy ratios as functions of (see a discussion about different averaging methods in TenBarge et al. (2012)). The red diamonds are average values of these anisotropy ratios over the 93 events, at each frequency. is always larger than 1. This implies that the transverse fluctuations are not gyrotropic at a given frequency in the spacecraft frame. The fact that the non-gyrotropy of could be compatible with a gyrotropic wave vector energy distribution has been suggested by Alexandrova et al. (2008b). By analytical calculations and numerical simulations, Turner et al. (2011) have shown that this suggestion was valid for the magnetic fluctuations in the solar wind, from the inertial range to 10 Hz, if the frequency is only due to the sampling along (Taylor hypothesis) and if the modulus of the spectral index of the fluctuations is larger than 1.
The compressibility of the 93 events, observed as a function of , is given in Figure 3b: the average value of at each frequency is shown by red diamonds. The compressibility increases when increases. The Figure 10 of TenBarge et al. (2012) illustrates how the compressibility depends on the chosen local field: the compressiblity calculated with a field averaged over 1 hour can be 2 to 3 times larger than the compressibility calculated with a local field. We thus think that our quasi-local field (4 s average) can lead to a slight overestimation of the compressibility at the highest frequencies.
According to Figure 3 (see the horizontal bars in each panel), the frequencies corresponding to the proton gyroscale () are generally found between 1 and 4 Hz, and the frequencies between 0.7 and 2 Hz. For the electron scales, is mainly found between 30 and 200 Hz, and between 30 and 100 Hz.
In the following subsection, we shall look for wave vector distributions consistent with the non-gyrotropy in the frequency domain displayed in Figure 3a. The other anisotropy, the compressibility (Fig. 3b), will be studied in section 5.
4.2. Angular distributions of the wave vectors
Saur & Bieber (1999) consider four models for the wave vector angular distribution in the solar wind, and the resultant PSD anisotropies in the frequency domain, assuming that the fluctuations are stationary and frozen-in (Taylor hypothesis). The four distributions are:
a) a slab symmetry with respect to i.e. strictly parallel to .
b) a gyrotropic 2D distribution, with strictly perpendicular to . The corresponding fluctuations are in planes which contain , and can make any angle with it, with a component parallel to and a component perpendicular to .
c) an isotropic wave vector distribution.
d) a slab symmetry with respect to the radial direction i.e. strictly parallel to .
Equations (33) to (43) of Saur & Bieber (1999) give the values of , and for the four above distributions, as functions of , and . The PSD are power laws . The spectral index of the fluctuations is assumed to be the same in the three directions , and , and constant over the whole frequency range.
The equations of Saur & Bieber (1999) are valid at MHD scales, larger than and . However, at smaller scales, the Hall term starts to show its effects in the induction equation. As clearly shown by Kiyani et al. (2013) (see also the detailed derivation of Schekochihin et al., 2009), the Hall term implies interactions between the compressive and transverse fluctuations, and an enhancement of the magnetic compressibility. Thus, the MHD terms and given by Saur & Bieber (1999) are no more valid at proton and electron scales. The term , which only involves transverse fluctuations, can be considered as less affected by the Hall effect: it does not depend on . It is thus able to give indications on the wave vector distribution at scales smaller than the MHD scales.
The anisotropy can be calculated with the formulas of Saur & Bieber (1999) for the four above distributions. For the slab symmetry with respect to ,
This distribution gives a gyrotropic variance, which is not observed. For the 2D distribution, and depend on in the same way, so that their ratio does not depend on it:
where is the spectral index of and . For the isotropic distribution, and do not depend on :
where is the field-to-flow angle. For the slab symmetry with respect to ,
In Figure 4, the observations of are compared with two models (Eqs. 2 and 3). The calculated anisotropies depend on the observed spectral index . To get an average value of the spectral index at each frequency, we use successive averaging processes: we first calculate the local slope (linear fit over three contiguous frequencies) of the total spectrum of each event of our sample; then, we smooth this local slope over 11 contiguous frequencies for each event; finally, we average the slope over the 93 events, and find .
In Figures 4a and 4b, the observed average anisotropy (red diamonds) shown in Figure 3a is drawn as a function of . As the dispersion of for the 93 events is large (Fig. 3a), the two red lines of Figures 4a and 4b give the error interval (average the standard deviation) at each frequency. Following Eq. (2), for the 2D distribution is equal to the spectral index. Thus, in Figure 4a, the thick black solid line for the 2D -distribution shows the observed average : it varies between 2.7 and 2.9 below 6 Hz; then it increases up to 4 when increases. This regular increase of is due to the exponential shape of the spectra (Alexandrova et al., 2012). The thin black lines give the standard deviation for over the 93 events, i.e. the dispersion of the 2D model itself. Taking into account the standard deviation of the observed ratio (red lines), it is clear that the observations from 1 Hz to 6 Hz are close to the ratio for a 2D -distribution. Above 10 Hz (at sub-proton and electron scales) the observations are further from the 2D model.
In Figure 4b, the observations are compared with the isotropic model (with its standard deviation). Below 10 Hz, the observations are far from the isotropic model. Above 10 Hz, the observations are near the ratio for an isotropic -distribution. At any frequency, the slab model (Eq. 1) is far from the observations. The ratio (Eq. 4) for the model with a slab symmetry with respect to is not drawn in Figure 4 because it is generally larger than 10. We conclude that the 2D model is better below 6 Hz, and the isotropic model better above 10 Hz. However, this conclusion relies on the hypothesis of Saur & Bieber (1999) that the spectral index is constant over the whole frequency range: this is not exact above about 10 Hz (Fig. 4a). Our results are just an indication, given by the non-gyrotropy of the PSD, that the -distribution tends to be 2D below 6 Hz, and more isotropic above 10 Hz.
Several authors assume that the turbulence is in critical balance, i.e. the nonlinear cascade rate is of the order of the linear frequency. Such a model implies relations between and , i.e. privileged directions of the wave vectors, with more and more energy in when the wavenumber increases (see Fig. 1 of Howes et al., 2008). It can be valid from the inertial range to the kinetic range (Goldreich & Sridhar, 1995; Howes et al., 2008; Schekochihin et al., 2009; TenBarge et al., 2013). In our sample, the critical balance model could be valid at low frequency (below 6 Hz) where the wave vectors are quasi-perpendicular; but it is probably less valid at higher frequency, where the -distribution is more isotropic, i.e. with relatively more energy in oblique or parallel wave vectors when the wavenumber increases.
To explain the variations of the observed anisotropies with the field-to-flow angle at a given frequency, different authors consider a linear combination of two of the above distributions (Saur & Bieber (1999) and references therein), slab, slab/R, 2D and isotropic, in different frequency domains in the solar wind. At frequencies [ Hz, 2 Hz], Bieber et al. (1996) find that the turbulence in the inertial range can be described by of the energy in 2D fluctuations, and in slab fluctuations. Leamon et al. (1998) find that the proportion of slab fluctuations increases at proton scales (0.1 to 1 Hz). Hamilton et al. (2008) find an even larger proportion (more than 80%) of slab fluctuations at proton scales. However, the pure 2D + slab combination is oversimplified: an angular lobe of wave vectors along each axis has probably to be considered. This is illustrated by the work of Mangeney et al. (2006) who study the variations, as a function of , of the PSD of the magnetic fluctuations and of the PSD of the electric fluctuations observed by Cluster in the magnetosheath from 8 Hz to 2 kHz. They find that the electromagnetic PSD, up to or more, can be explained by a 2D -distribution, with an aperture angle of about ; the electrostatic fluctuations, above can be explained by a slab distribution, with the same aperture angle for around .
A more or less smooth transition from a 2D to an isotropic -distribution, when increases, could be a model of our observations more realistic than a two-components model with an increasing proportion of slab fluctuations when increases.
4.3. Wavevector distribution in the fast and slow winds
In the inertial range (resolution of about 1-minute), Dasso et al. (2005) and Weygand et al. (2011) find that fast streams are dominated by wave vectors quasi-parallel to , while intermediate and slow streams have quasi-2D -distributions. Osman & Horbury (2007), using 4 s FGM data on Cluster, do not find any relation between the solar wind speed and the ratio of the parallel to perpendicular correlation lengths. Hamilton et al. (2008) do not find a significant difference between the -distribution in the fast and the slow wind, neither in the inertial range nor at proton scales.
In our sample, the wind speed varies from 300 to 700 km/s (Fig. 1b). We consider separately a sample of 16 intervals with km/s, and a sample of 16 intervals with km/s. Figure 5a displays as a function of for the fast wind (red) and for the slow wind (blue). The crosses give the average value of each sample. We see that the average value of is larger for the fast wind, below 10 Hz. Is this difference significant, in view of the large dispersion of in each sample? In Figure 5b, the average values of Figure 5a (crosses) are shown again, as well as the error interval for each sample (average the standard deviation) given by the solid lines of the same color. Between 2 Hz and 10 Hz, the average values given by the red (blue) crosses are at the boundary of the blue (red) error interval: the difference between the averages of in the fast wind and the slow wind is thus significant at these frequencies. Figures 5c and 5d allow a comparison of the averages of observed in the fast and the slow wind (crosses) to calculated for two models of the -distribution given by Saur & Bieber (1999) (Eqs. 2 and 3).
We thus find that the 2D character of the -distribution is stronger in the fast wind than in the slow wind, from 1 to 10 Hz. This is contrary to the observations in the inertial range (quoted above). In the same way, simulations lead Verdini & Grappin (2016) to conjecture that the turbulence is more isotropic in the fast wind than in the slow wind, still in the inertial range. A reason for these differences could be that our frequency range is more than one hundred times higher than the inertial range, so that the operating physical processes can be different.
5. Magnetic compressibility of the fluctuations
The compressibility of the 93 events of our sample (Figure 3b) increases with the frequency. The average value of is about 0.2 at 1 Hz, and reaches 0.5 at 50 Hz, close to the values displayed in Figure 1 of Kiyani et al. (2013). Above 50 Hz, is larger than the value 0.5 corresponding to an isotropic variance, and it still increases up to 0.65 at 200 Hz. To allow comparisons with other works, Figure 6a displays , another definition of the compressibility, as a function of . Salem et al. (2012) find that reaches 0.5 at 2 Hz. This is larger than what we find.
The ranges of values of our sample are given in Figure 1a. The lower red dashed line in Figure 6a is the average of for five events with a small , and the upper red dashed line for five events with a large (see the selected values of in the caption). These red lines show that the compressibility, and the variation of the compressibility with , depend strongly on .
5.1. Compressibility in the wavenumber domain
Converting the observed frequency into a wavenumber (Taylor hypothesis) we calculate the PSD , , and in (in spite of the fact that this conversion is problematic for anisotropic -distributions and reduced spectra; we shall return to this point in the Discussion). In Figure 6b, similar to Figure 6a, we see that the variation of the compressibility with the wavenumber depend strongly on .
We then normalize to the electron inertial length , or to the gyroradii or , and interpolate the normalized PSD at 15 values of , or . Figures 7a and 7b give the total power as functions of and . The slope of the spectra is close to -2.8 at the largest scales. At the smallest scales, the spectra are steeper, with an exponential shape controlled by the electron gyroradius (Alexandrova et al., 2012).
Figure 7c gives the compressibility as a function of . The vertical bars in the abscissae give , 10 and 100. With high-resolution 2D hybrid simulations of a decaying turbulence, for 0.5, Franci et al. (2015) obtain a compressibility increasing from 0.1, at 0.1, to 0.5 at 2, and a nearly constant (plateau) value for 5 to 10. (See also the gyrokinetic simulations in Fig. 9 of TenBarge et al., 2012). After a kind of plateau from 2 to 10, the observed average (Fig. 7c) increases from 10 to 100, but it is always smaller than 0.4 (reached for , ). We conclude that the observed and the simulated compressibility have similar variations up to ; but the observed compressibility is 2 to 3 times weaker: we shall discuss this last point in section 6. For , a domain not reached by these simulations, the observed compressibility still increases.
5.2. Compressibility in the fast and slow winds
In Figure 8a, we compare the compressibility for a sample of 16 intervals in the fast wind (red) and 16 intervals in the slow wind (blue). The crosses give the average values for each sample. In Figure 8b, taking into account the standard deviation of each sample, we see that the compressibility is the same in the fast wind and in the slow wind for 10 Hz. For 15-20 Hz, it is larger in the slow wind. It is well-known that there is a correlation between and , in the solar wind: fast winds have a higher proton temperature than slow winds, the electron temperature having a smaller range of variation (see e.g. Mangeney et al., 1999): is anticorrelated with (Fig. 1b). Thus, the larger compressibility observed in the slow wind above 15-20 Hz could be related to larger values of .
5.3. Role of
Smith et al. (2006) in the inertial range, Hamilton et al. (2008) up to 0.8 Hz, Alexandrova et al. (2008a) from 0.3 Hz to 10 Hz, find that the compressibility increases when increases. This is clear in Figure 9, for the present sample, at 2 Hz and 5 Hz. The slope of as a function of (solid lines in Figure 9) decreases when increases, from 5 to 22 Hz. At 56 Hz, the compressibility is no more correlated with .
5.4. Compressibility versus different plasma parameters
The correlations of the compressibility with different plasma parameters in the solar wind can help to identify the nature of the dominant mode in the magnetic fluctuations. Figures 10a to 10f display the correlation coefficients of with local parameters related to the proton or electron temperature. Scatter plots of these plasma parameters are shown in Figure 1. The correlation of with (Fig. 10a), as a function of , is larger than 0.7 from 3 to 10 Hz. It decreases clearly for 15 Hz. The correlation with (Fig. 10c) is weaker than with , but it extends up to 20-30 Hz. The correlation with (Fig. 10b) is good (around 0.6) from 10 to 50 Hz. The correlation with (not shown) is always weaker than 0.55.
Boldyrev et al. (2013) (see also Schekochihin et al., 2009; TenBarge et al., 2012) give analytical formulas for the magnetic and the electron compressibilities of sub-proton electromagnetic fluctuations, i.e. kinetic Alfvén waves (KAW) and whistler waves (see also Gary & Smith, 2009). The whistler magnetic compressibility does not depend on , but only on the wave vector direction. Conversely, the magnetic compressibility for kinetic Alfvén waves, in the framework of the electron-reduced MHD ( and ), depends on and , not on the wavenumber:
Figure 10d displays the correlation between the observed magnetic compressibility and , as a function of . This correlation is everywhere larger than the correlation with (Fig. 10a). It is larger than the correlation with (Fig. 10c) except in a narrow frequency range around 15 to 25 Hz. The correlation of Figure 10d is larger than 0.7 between 1.5 and 15 Hz.
5.5. Modes of the fluctuations
5.5.1 For 15-20 Hz
At these frequencies (corresponding to ) our observations can be compared with the Vlasov-Maxwell results of TenBarge et al. (2012). These authors calculate for Alfvén, fast and slow modes up to , for different values of , assuming that the turbulence is in critical balance () i.e. with quasi-perpendicular wave vectors.
We have seen (Fig. 9) that increases when increases: this is typical of KAW, but excludes the presence of fast waves for which decreases when increases. The observations of Figure 7d are also consistent with KAW properties up to (Fig. 2 of TenBarge et al., 2012): for the lowest values of , the observed (upper solid red line) decreases strongly when increases, while decreases weakly for the largest values of (lower solid red line). Finally, Figure 10d confirms that fluctuations with a KAW polarisation are generally dominant at sub-proton scales, for 15-20 Hz).
However, the observed profiles (Fig. 7d) have a large dispersion, so that they could be consistent, in some cases, with other modes, oblique whistler waves with a critical balance distribution (, Figure 4 of TenBarge et al., 2012); or with different mixtures of Alfvén, fast and slow wave modes (Figs. 2 and 6 of TenBarge et al., 2012) for which changes strongly from to .
5.5.2 For 15-20 Hz
The weaker correlation between and observed in Figure 10d above 15 Hz (and below 1.5 Hz) can be due to different reasons: i) Eq. 5 for is valid in a limited scale range, ii) the KAW are damped, and/or iii) another mode, with a different compressibility, is present. Let us discuss these three points:
i) validity of Eq. 5? Eq. 5 is valid for and . This is shown in Figure 2a of TenBarge et al. (2012) where the compressibility is indeed equal to for 2 to 10, at least for the observed values . In our sample, for all the events, if 3 Hz, and if 50 Hz. Thus, in Figure 10d, the weaker correlation below 3 Hz can be due to the lack of validity of Eq. 5. Conversely, the weaker correlation from 15-20 Hz to 50 Hz has to be due to other reasons.
ii) KAW damping? Figure 10e shows the correlation between and as a function of . This correlation is zero up to 10 Hz, and still very small up to 50 Hz. Above 80 Hz, the correlation reaches 0.5 among about 20 events (a number given by the dotted line, see the caption) which are the most intense at these frequencies. Figure 10f shows that the correlation between and (Fig. 10e) could be related to the anticorrelation of with , above 20 Hz. According to Figure 1 of Schreiner & Saur (2017) larger values of increase the damping of KAW, at least for the values of our sample, and for small scales (see also Howes et al., 2006). The decrease of the correlation between and above 20 Hz (Fig. 10d) can thus be due to a stronger damping of KAW, itself due to relatively larger values of . Note that Bruno & Telloni (2015) observe that KAW (below 4 Hz) tend to disappear when the wind speed decreases.
iii) another mode? The weaker correlation between and for 15-20 Hz (Fig. 10d) can be due to the presence of another mode, with another compressibility, superimposed or not on the KAW mode. What could be this other mode? At electron scales, neither the electron temperature anisotropy (Fig. 1d) nor the heat flux are unstable: whistler waves are not observed. The values are generally too low to give an electron parallel firehose instability (Matteini et al., 2013). The correlation of with (Fig. 10e) above 80 Hz, which could imply the presence of slow modes, is not strong. But it is consistent with the results of Figure 8, above 20 Hz: the compressibility is larger in the slow wind, where is larger. varies between 0.22 and 2.2 in our sample, with 60% of the intervals for (Fig. 1b). These values seem to be too low to prevent the damping of slow-ion acoustic modes. However, we recall that and are average values for the whole distributions, without distinction between the temperatures of core, halo, strahl or beam. A better description of the distribution functions could show whether slow-ion acoustic modes could be less damped above 20 Hz, mainly in the slow wind: Tong et al. (2015) underline how the KAW damping depends on realistic distribution functions, with electron drifts. Furthermore, the correlation of the compressibility with (Fig. 10b) is larger for 15-20 Hz than in the KAW frequency domain. According to Narita & Marsch (2015), quasi-perpendicular kinetic slow modes could produce a proton heating so that the correlation of Figure 10b could be another indication for the presence of kinetic slow modes, cascading above 15-20 Hz.
In Figure 11, observed at four fixed frequencies is drawn as a function of (Eq. 5). On each panel, gives the correlation coefficient between and . At 2 Hz the observed compressibility is always weaker than , the compressibility expected for KAW (Fig. 11a). At 6.6 Hz (Fig. 11b), where peaks, is still weaker but closer to . At 22 Hz the observed compressibility is, on an average, equal to (Fig. 11c). At 56 Hz, is mainly larger than (Fig. 11d).
Taking into account that the observed magnetic compressibility can be underestimated with respect to simulations or numerical calculations (see section 6), we conclude that is larger than the KAW compressibility for 15-20 Hz, i.e. .
A few simulations of the slow-ion acoustic mode reach the scales or corresponding to our observations. The hybrid Vlasov-Maxwell simulations of Valentini et al. (2008) and Valentini & Veltri (2009) reach and show the generation of ion-acoustic turbulence; but these simulations imply a temperature ratio much larger than the values of our sample (Fig. 1b). Camporeale & Burgess (2011b) reach but the electron compressibility of their simulated fluctuations is weaker than the calculated electron compressibility of slow-ion acoustic waves. An interesting result of Figure 3 of Camporeale & Burgess (2011a) is that the electron compressibility of the slow-ion acoustic mode is larger for a small propagation angle () than for 60 or 80. This is another point in favor of the presence of slow-ion acoustic modes in our data. Indeed, we have concluded (section 4.2) that the -distribution is more isotropic, so that small angles are more frequent, when increases. Thus, a larger electron compressibility should be found when increases ( decreases) if slow-ion acoustic modes are present. We did not measure the electron compressibility, but Figures 6a and 11 show that the magnetic compressibility increases when increases. Calculations similar to those of Gary & Smith (2009) would tell us whether the magnetic compressibility of slow-ion acoustic modes increases when decreases.
In this work, we consider the reduced power spectral densities , and of the magnetic field fluctuations observed at a given frequency in the three directions with respect to the quasi-local magnetic field (4 s average), and . The sampling direction is in the plane (x, z) i.e. perpendicular to the y direction.
In section 5, the magnetic compressibility observed in the frequency domain, is compared with analytical and numerical calculations and simulations in the wave vector domain, which only consider two components and , and give the compressibility . But this comparison is not really valid: we cannot assume that is equal to . Indeed, if there is only one wave vector perpendicular to , along for instance, the PSD in the directions perpendicular to , and , will be observed as and at the same frequency (see Podesta & TenBarge, 2012, section 2.1), and will display a phase relation between and corresponding to their polarisation. If there is also a wave vector perpendicular to , along , with a wavenumber , and will be observed as and at the same frequency , but will be much smaller than because is more or less perpendicular to the sampling direction . Thus, even if two wave vectors perpendicular to have the same , they will not be observed at the same frequency in , and . Inversely, at a given frequency, , and do not correspond to the same wavenumbers and wave vectors: they do not belong to the same fluctuations, there is no clear phase relation between the components , and . Furthermore, a PSD like , or like or , at a given frequency, is a mixture of fluctuations with different wavenumbers and different phases. Thus, analytical and 2D numerical calculations, which involve the same wave vectors for all the PSD, and simulations which do not ”fly through” the simulation box with the solar wind velocity, cannot predict exactly what is observed at a given frequency. This is probably one of the reasons why the observed compressibility is sometimes weaker than the simulated or calculated compressibility (see section 5).
The Taylor hypothesis has been assumed to be valid from 1 to 100 Hz: in Figures 6b and 7, the wavenumber dependency relies on this hypothesis. However, at proton or electron scales, some fluctuations can be present, with a frequency in the solar wind frame non-negligible with respect to . Thus, any detailed comparison of observations with calculations in the wavenumber domain (section 5.1) has to take into account the possibility that is not always proportional to , in the kinetic range, and that wave modes could be present with a non-negligible . What could be these wave modes?
Howes et al. (2014) show that the Taylor hypothesis is valid for all the linear modes (Alfvén, fast, slow) in the near-Earth solar wind, except for quasi-parallel whistler waves. Our frequency range [1 - 200 Hz] is below the electron gyrofrequency range : in our sample, varies from 150 to 550 Hz, with an average value of 300 Hz. Quasi-parallel whistler waves could thus be present below (Lacombe et al., 2014; Stansby et al., 2016; Kajdič et al., 2016); but their characteristic right-handed polarisation with respect to , and the corresponding spectral bump, are not observed, in our sample; so that even if weak underlying quasi-parallel whistlers were present, they could not play an important role.
The observed frequency range is well above the proton gyrofrequency range [0.08 - 0.3 Hz]. Thus, ion Bernstein modes, at low harmonics (1, 2 or 3) of cannot be observed.
Our frequency range is well below the proton plasma frequency ( varies from 400 Hz to 1.5 kHz, average 800 Hz): slow-ion acoustic waves could be present, which propagate below . But the only magnetic compressibility cannot allow the identification of non-Alfvénic wave modes (Narita & Marsch, 2015). This is illustrated by TenBarge et al. (2012). To remove these ambiguities, fluctuations other than the magnetic field fluctuations, or other transport ratios, would be considered, notwithstanding the fact that each wave identifier has its own ambiguity (Krauss-Varban et al., 1994; Denton et al., 1998; Klein et al., 2012). However, at small scales ( 15-20 Hz, see section 5.5.2), the observed compressibility is larger than the KAW compressibility (Fig. 11). It is larger in the slow wind than in the fast wind (Fig. 8) i.e. larger when the electron to proton temperature ratio is larger (Fig. 1b). Is there a mode which has a compressibility larger than the KAW compressibility, which could be more present in the slow wind, where is larger, and which can propagate at oblique or small angles with respect to ? According to Figure 1 of Gary (1992), if and , there is always a slow mode which is only weakly damped for small propagation angles. This is valid for MHD modes, and up to = 0.1 (Gary & Winske, 1992; Krauss-Varban et al., 1994). This is still valid up to (Camporeale & Burgess, 2011a) at least for . The magnetic compressibility of this kinetic mode has to be calculated to be compared to our observations.
The comparison of the observed polarisation ratios with those of linear modes is not always justified. But it could be more valid at the smallest scales ( 20 Hz, or 0.2), when some angles are probably smaller than the values giving the critical balance condition, i.e. smaller than the values at which the role of non-linear effects becomes dominant (Fig. 1 of Howes et al., 2008).
7. Summary and conclusion
We have analyzed 93 10-minute intervals of the solar wind magnetic field fluctuations observed on Cluster 1, from 1 Hz to about 200 Hz. These intervals, between 2001 and 2005, are selected with two conditions: 1) there is no indication of a connection to the Earth’s bow shock; as a consequence, the Cluster orbit implies that the field-to-flow angle is large; 2) there is no indication of the presence of quasi-parallel whistler waves, which are easily detected by their quasi-circular right-handed polarisation (Lacombe et al., 2014).
The power spectral density (PSD) of the magnetic field turbulence is measured in the three directions of the magnetic field aligned (MFA) frame, an average quasi-local frame given by the 4 s magnetic field data. The PSD is anisotropic. It decreases when increases; it goes down to the instrumental background noise at different frequencies in the different directions. As explained in section 3 and Appendix, we only consider the PSD when it is larger than 3 times the noise in any direction; then, we subtract the noise to obtain the corrected signal.
The corrected power spectral density is itself anisotropic. It is maximum in the direction, which is perpendicular to and to the radial direction. is weaker in the direction, perpendicular to and : the ratio implies that the transverse fluctuations are not gyrotropic at a given frequency in the spacecraft frame.
Following Saur & Bieber (1999), the non gyrotropic ratio gives indications on the shape of a gyrotropic distribution of . We find that the -distribution is mainly 2D (i.e. ) up to about 6 Hz, i.e. or . There is also a slight proportion of wave vectors parallel to . By a separate study of intervals of fast wind and of slow wind, we show that is closer to a 2D distribution in the fast wind, and contains a larger proportion of wave vectors parallel to in the slow wind, at 10 Hz. At frequencies 10 Hz, is less 2D, and tends towards isotropy around 50 Hz: wave vectors oblique or parallel to become relatively more important when the frequency increases (section 4.2).
We call compressibility the anisotropy ratio between the compressive and the transverse fluctuations: this polarisation ratio increases from 1 to 200 Hz. From 2 Hz to about 15-20 Hz, the observed compressibility is very well correlated (Fig. 10d) and nearly equal (Fig. 11) to the compressibility (Eq. 5) of linear kinetic Alfvén waves (KAW), which only depends on the beta factor : fluctuations with a KAW-like polarisation are dominant below 20 Hz, i.e. below or .
At smaller scales, 20 Hz, we suggest that the fluctuations could be related to a slow-ion acoustic mode. The identification of the three modes, Alfvénic or compressive, and the estimation of their damping rates and of their polarisation and transport ratios is a difficult task in the kinetic range where the modes cross each other in the dispersion plane (Krauss-Varban et al., 1994; Klein & Howes, 2015). Calculations of the dispersion relation and of the compressibility of the modes, for different propagation directions, different and different , would be necessary to confirm (or infirm) the presence of slow-ion acoustic modes with a large magnetic compressibility and relatively small propagation angles, for .
Appendix A Subtraction of the background noise
For a precise measurement of weak spectral levels on Cluster, the subtraction of the instrument noise has been justified by Alexandrova et al. (2010): the frequency-dependent background instrument noise is a random variable; the solar wind turbulence is a random variable; these random variables are independent. It is well known that the average value of the sum of two independent random variables is the sum of their average values. Thus, the true spectral level at the frequency is the difference , where is the average of the trace of the spectral matrix measured on the triaxial search coils (signal + noise), and is the average of the trace of the background noise (additive noise). The background noise is generally called sensitivity, but this name is misleading because it implies a threshold, which has to be crossed, not an additive noise which has to be subtracted.
The average background noise is easily measured every year in Summer, when Cluster crosses the magnetospheric lobes: in these regions, the in situ fluctuations are so weak that the measured signal is considered as the receiver noise. This instrument noise is close to the values obtained by Earth’s ground measurements (Cornilleau-Wehrlin et al., 2003). Robert et al. (2014) show, in their Figure 9, that the background noise is rather stable from 2001 to 2004, and even until 2011. At the STAFF-SC frequencies (below 8 Hz) the observed solar wind spectra (Figure 2) are intense enough, so that there is no need to subtract the noise. On STAFF-SA (above 8 Hz), from 2001 to 2005, we consider that the 3D-background noise is given by a one-hour average of the 3D power measured in a lobe on 2004/06/03 (14:00-15:00 UT) on Cluster 1. Histograms of during this interval, at different , show that the instantaneous background noise can be as large as 2 to 3 times the average value below 60 Hz, and 1.3 to 1.5 times above 60 Hz. (Above 60 Hz, the power of each STAFF-SA spectrum relies on 8 times more measures than below 60 Hz, so that the relative uncertainty on the power level is weaker). To reach the true spectral level by a subtraction , we have to take into account this uncertainty on , which is at most 3 times . The corrected spectral level is the average value of , but at the only frequencies where is larger than ,
There is no simple reliable way to separate the signal and the instrument noise at frequencies for which is weaker than 3.
For the background noise in every direction , and , we use (as in Figure 2). Even if the three search coils in the spacecraft frame (SR2) have not exactly the same 1D-background noise, it is not useful to distinguish them: indeed, the on-board (SR2) spectral matrix (signal + noise) is projected and analyzed in the Magnetic Field Aligned frame, , and , so that the role played by each search coil is time-dependent, through the angles between the SR2 and the MFA frames. Anyway, the considered uncertainty (3 times the noise) broadly takes into account the differences between the three search coils.
The present work is based upon measurements of the PSD anisotropies. We have to subtract the 1D background noise from , and . As (dashed lines in Fig. 2) is the weakest 1D spectrum (above 8 Hz), we first have to check that the noise subtraction is significant for: indeed, the subtraction from can be valid (if is larger than ), while the subtraction from is not valid, if is equal to the 1D background noise. To prevent errors in the subtraction of the noise from we thus put a second condition:
As is always weaker than and , this second condition implies that the 1D noise can be subtracted without errors in every direction , and . Figure 12 illustrates the two conditions (Eqs. A1 and A2) to get valid measurements of a 3D spectrum, of , and , and thus of the anisotropy ratios of Figure 3. The lower dotted lines in Figure 12 give the background noise for the 3D noise (left panel) and the 1D noise (right panel). The upper dotted lines respectively give and . The dashed lines give (left panel) and (right panel) before any subtraction. The solid lines give the corrected spectra, which obey the two above conditions, and after subtraction respectively of and .
All the spectra used in the present paper are corrected in this way.
- Alexandrova et al. (2008a) Alexandrova, O., Carbone, V., Veltri, P., & Sorriso-Valvo, L. 2008a, ApJ, 674, 1153
- Alexandrova et al. (2008b) Alexandrova, O., Lacombe, C., & Mangeney, A. 2008b, Annales Geophysicae, 26, 3585
- Alexandrova et al. (2012) Alexandrova, O., Lacombe, C., Mangeney, A., Grappin, R., & Maksimovic, M. 2012, ApJ, 760, 121
- Alexandrova et al. (2010) Alexandrova, O., Saur, J., Lacombe, C., Mangeney, A., Schwartz, S. J., Mitchell, J., Grappin, R., & Robert, P. 2010, Twelfth International Solar Wind Conference, 1216, 144
- Balogh et al. (1997) Balogh, A., et al. 1997, Space Sci. Rev., 79, 65
- Belcher & Davis (1971) Belcher, J. W., & Davis, L. 1971, J. Geophys. Res., 76, 3534
- Bieber et al. (1996) Bieber, J. W., Wanner, W., & Matthaeus, W. H. 1996, J. Geophys. Res., 101, 2511
- Boldyrev et al. (2013) Boldyrev, S., Horaites, K., Xia, Q., & Perez, J. C. 2013, ApJ, 777, 41
- Bruno & Carbone (2013) Bruno, R., & Carbone, V. 2013, Living Reviews in Solar Physics, 10, 2
- Bruno & Telloni (2015) Bruno, R., & Telloni, D. 2015, ApJ, 811, L17
- Camporeale & Burgess (2011a) Camporeale, E., & Burgess, D. 2011a, ApJ, 735, 67
- Camporeale & Burgess (2011b) —. 2011b, ApJ, 730, 114
- Chen & Boldyrev (2017) Chen, C. H. K., & Boldyrev, S. 2017, ApJ, 842, 122
- Chen et al. (2013) Chen, C. H. K., Boldyrev, S., Xia, Q., & Perez, J. C. 2013, Phys. Rev. Lett., 110, 225002
- Chen et al. (2010) Chen, C. H. K., Horbury, T. S., Schekochihin, A. A., Wicks, R. T., Alexandrova, O., & Mitchell, J. 2010, Phys. Rev. Lett., 104, 255002
- Chen et al. (2016) Chen, C. H. K., Matteini, L., Schekochihin, A. A., Stevens, M. L., Salem, C. S., Maruca, B. A., Kunz, M. W., & Bale, S. D. 2016, ApJ, 825, L26
- Comişel et al. (2014) Comişel, H., Narita, Y., & Motschmann, U. 2014, Nonlinear Processes in Geophysics, 21, 1075
- Cornilleau-Wehrlin et al. (1997) Cornilleau-Wehrlin, N., et al. 1997, Space Sci. Rev., 79, 107
- Cornilleau-Wehrlin et al. (2003) —. 2003, Annales Geophysicae, 21, 437
- Dasso et al. (2005) Dasso, S., Milano, L. J., Matthaeus, W. H., & Smith, C. W. 2005, ApJ, 635, L181
- Décréau et al. (1997) Décréau, P. M. E., et al. 1997, Space Sci. Rev., 79, 157
- Denton et al. (1998) Denton, R. E., Lessard, M. R., LaBelle, J. W., & Gary, S. P. 1998, J. Geophys. Res., 103, 23661
- Franci et al. (2015) Franci, L., Verdini, A., Matteini, L., Landi, S., & Hellinger, P. 2015, ApJ, 804, L39
- Gary (1992) Gary, S. P. 1992, J. Geophys. Res., 97, 8519
- Gary & Smith (2009) Gary, S. P., & Smith, C. W. 2009, Journal of Geophysical Research (Space Physics), 114, A12105
- Gary & Winske (1992) Gary, S. P., & Winske, D. 1992, J. Geophys. Res., 97, 3103
- Goldreich & Sridhar (1995) Goldreich, P., & Sridhar, S. 1995, ApJ, 438, 763
- Hamilton et al. (2008) Hamilton, K., Smith, C. W., Vasquez, B. J., & Leamon, R. J. 2008, J. Geophys. Res., 113, 1106
- He et al. (2012) He, J., Tu, C., Marsch, E., & Yao, S. 2012, ApJ, 745, L8
- Hellinger et al. (2006) Hellinger, P., Trávníček, P., Kasper, J. C., & Lazarus, A. J. 2006, Geophys. Res. Lett., 33, 9101
- Howes et al. (2012) Howes, G. G., Bale, S. D., Klein, K. G., Chen, C. H. K., Salem, C. S., & TenBarge, J. M. 2012, ApJ, 753, L19
- Howes et al. (2006) Howes, G. G., Cowley, S. C., Dorland, W., Hammett, G. W., Quataert, E., & Schekochihin, A. A. 2006, ApJ, 651, 590
- Howes et al. (2008) —. 2008, J. Geophys. Res., 113, 5103
- Howes et al. (2014) Howes, G. G., Klein, K. G., & TenBarge, J. M. 2014, ApJ, 789, 106
- Howes et al. (2011) Howes, G. G., TenBarge, J. M., Dorland, W., Quataert, E., Schekochihin, A. A., Numata, R., & Tatsuno, T. 2011, Phys. Rev. Lett., 107, 035004
- Jian et al. (2014) Jian, L. K., et al. 2014, ApJ, 786, 123
- Johnstone et al. (1997) Johnstone, A. D., et al. 1997, Space Sci. Rev., 79, 351
- Kajdič et al. (2016) Kajdič, P., Alexandrova, O., Maksimovic, M., Lacombe, C., & Fazakerley, A. N. 2016, ApJ, 833, 172
- Kiyani et al. (2013) Kiyani, K. H., Chapman, S. C., Sahraoui, F., Hnat, B., Fauvarque, O., & Khotyaintsev, Y. V. 2013, ApJ, 763, 10
- Klein & Howes (2015) Klein, K. G., & Howes, G. G. 2015, Physics of Plasmas, 22, 032903
- Klein et al. (2012) Klein, K. G., Howes, G. G., TenBarge, J. M., Bale, S. D., Chen, C. H. K., & Salem, C. S. 2012, ApJ, 755, 159
- Krauss-Varban et al. (1994) Krauss-Varban, D., Omidi, N., & Quest, K. B. 1994, J. Geophys. Res., 99, 5987
- Lacombe et al. (2014) Lacombe, C., Alexandrova, O., Matteini, L., Santolík, O., Cornilleau-Wehrlin, N., Mangeney, A., de Conchy, Y., & Maksimovic, M. 2014, ApJ, 796, 5
- Leamon et al. (1998) Leamon, R. J., Smith, C. W., Ness, N. F., Matthaeus, W. H., & Wong, H. K. 1998, J. Geophys. Res., 103, 4775
- Lion et al. (2016) Lion, S., Alexandrova, O., & Zaslavsky, A. 2016, ApJ, 824, 47
- Mangeney et al. (2006) Mangeney, A., Lacombe, C., Maksimovic, M., Samsonov, A. A., Cornilleau-Wehrlin, N., Harvey, C. C., Bosqued, J.-M., & Trávníček, P. 2006, Annales Geophysicae, 24, 3507
- Mangeney et al. (1999) Mangeney, A., et al. 1999, Annales Geophysicae, 17, 307
- Matteini et al. (2013) Matteini, L., Hellinger, P., Goldstein, B. E., Landi, S., Velli, M., & Neugebauer, M. 2013, J. Geophys. Res., 118, 2771
- Matthaeus et al. (1990) Matthaeus, W. H., Goldstein, M. L., & Roberts, D. A. 1990, J. Geophys. Res., 95, 20673
- Narita (2014) Narita, Y. 2014, Nonlinear Processes in Geophysics, 21, 41
- Narita et al. (2011a) Narita, Y., Gary, S. P., Saito, S., Glassmeier, K.-H., & Motschmann, U. 2011a, Geophys. Res. Lett., 38, 5101
- Narita et al. (2011b) Narita, Y., Glassmeier, K.-H., Goldstein, M. L., Motschmann, U., & Sahraoui, F. 2011b, Annales Geophysicae, 29, 1731
- Narita et al. (2010) Narita, Y., Glassmeier, K.-H., Sahraoui, F., & Goldstein, M. L. 2010, Phys. Rev. Lett., 104, 171101
- Narita & Marsch (2015) Narita, Y., & Marsch, E. 2015, ApJ, 805, 24
- Osman & Horbury (2007) Osman, K. T., & Horbury, T. S. 2007, ApJ, 654, L103
- Oughton et al. (2015) Oughton, S., Matthaeus, W. H., Wan, M., & Osman, K. T. 2015, Philosophical Transactions of the Royal Society A, 373, 20140152
- Passot et al. (2017) Passot, T., Sulem, P. L., & Tassi, E. 2017, J. Plasma Phys., 83, 4.
- Perri et al. (2012) Perri, S., Goldstein, M. L., Dorelli, J. C., & Sahraoui, F. 2012, Physical Review Letters, 109, 191101
- Perri et al. (2009) Perri, S., Yordanova, E., Carbone, V., Veltri, P., Sorriso-Valvo, L., Bruno, R., & André, M. 2009, Journal of Geophysical Research: Space Physics, 114, n/a
- Perrone et al. (2016) Perrone, D., Alexandrova, O., Mangeney, A., Maksimovic, M., Lacombe, C., Rakoto, V., Kasper, J. C., & Jovanovic, D. 2016, ApJ, 826, 196
- Perschke et al. (2013) Perschke, C., Narita, Y., Gary, S. P., Motschmann, U., & Glassmeier, K.-H. 2013, Annales Geophysicae, 31, 1949
- Perschke et al. (2014) Perschke, C., Narita, Y., Motschmann, U., & Glassmeier, K.-H. 2014, ApJ, 793, L25
- Podesta (2013) Podesta, J. J. 2013, Sol. Phys., 286, 529
- Podesta et al. (2010) Podesta, J. J., Borovsky, J. E., & Gary, S. P. 2010, ApJ, 712, 685
- Podesta & TenBarge (2012) Podesta, J. J., & TenBarge, J. M. 2012, Journal of Geophysical Research (Space Physics), 117, A10106
- Rème et al. (1997) Rème, H., et al. 1997, Space Sci. Rev., 79, 303
- Robert et al. (2014) Robert, P., et al. 2014, Geoscientific Instrumentation, Methods and Data Systems, 3, 153
- Roberts & Li (2015) Roberts, O. W., & Li, X. 2015, ApJ, 802, 1
- Roberts et al. (2015) Roberts, O. W., Li, X., & Jeska, L. 2015, ApJ, 802, 2
- Sahraoui et al. (2010) Sahraoui, F., Goldstein, M. L., Belmont, G., Canu, P., & Rezeau, L. 2010, Phys. Rev. Lett., 105, 131101
- Salem et al. (2012) Salem, C. S., Howes, G. G., Sundkvist, D., Bale, S. D., Chaston, C. C., Chen, C. H. K., & Mozer, F. S. 2012, ApJ, 745, L9
- Santolík et al. (2003) Santolík, O., Parrot, M., & Lefeuvre, F. 2003, Radio Science, 38, 1010
- Saur & Bieber (1999) Saur, J., & Bieber, J. W. 1999, J. Geophys. Res., 104, 9975
- Schekochihin et al. (2009) Schekochihin, A. A., Cowley, S. C., Dorland, W., Hammett, G. W., Howes, G. G., Quataert, E., & Tatsuno, T. 2009, ApJS, 182, 310
- Schreiner & Saur (2017) Schreiner, A., & Saur, J. 2017, ApJ, 835, 133
- Smith et al. (2006) Smith, C. W., Vasquez, B. J., & Hamilton, K. 2006, J. Geophys. Res., 111, 9111
- Smith et al. (2012) Smith, C. W., Vasquez, B. J., & Hollweg, J. V. 2012, ApJ, 745, 8
- Stansby et al. (2016) Stansby, D., Horbury, T. S., Chen, C. H. K., & Matteini, L. 2016, ApJ, 829, L16
- TenBarge et al. (2013) TenBarge, J. M., Howes, G. G., & Dorland, W. 2013, ApJ, 774, 139
- TenBarge et al. (2012) TenBarge, J. M., Podesta, J. J., Klein, K. G., & Howes, G. G. 2012, ApJ, 753, 107
- Tong et al. (2015) Tong, Y., Bale, S. D., Chen, C. H. K., Salem, C. S., & Verscharen, D. 2015, ApJ, 804, L36
- Turner et al. (2011) Turner, A. J., Gogoberidze, G., Chapman, S. C., Hnat, B., & Müller, W.-C. 2011, Phys. Rev. Lett., 107, 095002
- Štverák et al. (2008) Štverák, Š., Trávníček, P., Maksimovic, M., Marsch, E., Fazakerley, A. N., & Scime, E. E. 2008, J. Geophys. Res., 113
- Valentini & Veltri (2009) Valentini, F., & Veltri, P. 2009, Physical Review Letters, 102, 225001
- Valentini et al. (2008) Valentini, F., Veltri, P., Califano, F., & Mangeney, A. 2008, Physical Review Letters, 101, 025006
- Verdini & Grappin (2016) Verdini, A., & Grappin, R. 2016, ApJ, 831, 179
- Weygand et al. (2011) Weygand, J. M., Matthaeus, W. H., Dasso, S., & Kivelson, M. G. 2011, Journal of Geophysical Research (Space Physics), 116, A08102
- Wicks et al. (2012) Wicks, R. T., Forman, M. A., Horbury, T. S., & Oughton, S. 2012, ApJ, 746, 103