Anisotropies of the Magnetic Field Fluctuations at Kinetic Scales in the Solar Wind: Cluster Observations

For decades, the three orthogonal components of the magnetic field fluctuations have been observed in the solar wind as functions of the frequency f in the frame of a single spacecraft. Their frequency spectra display two kinds of anisotropy. The first anisotropy is only related to the direction of the average magnetic field ${{\boldsymbol{B}}}_{0}$: there is less energy in the compressive fluctuations $\delta {B}_{\parallel }^{2}(f)$ parallel to ${{\boldsymbol{B}}}_{0}$ than in the transverse fluctuations $\delta {B}_{\perp }^{2}(f)$ perpendicular to ${{\boldsymbol{B}}}_{0}$. The other anisotropy is related to the direction of ${{\boldsymbol{B}}}_{0}$ and to the radial direction ${\boldsymbol{R}}$, which is close to the direction of the expansion velocity of the solar wind: the fluctuations $\delta {B}_{\perp }^{2}(f)$ are non-gyrotropic (non-axisymmetric with respect to ${{\boldsymbol{B}}}_{0}$), with less energy in the ${\boldsymbol{x}}$ direction, which is perpendicular to ${{\boldsymbol{B}}}_{0}$ in the $({\boldsymbol{R}},{{\boldsymbol{B}}}_{0})$ plane, than in the ${\boldsymbol{y}}$ direction, which is perpendicular to both ${{\boldsymbol{B}}}_{0}$ and ${\boldsymbol{R}}$ (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 the dissipation range (Leamon et al. 1998; Turner et al. 2011). In the present work, we study the anisotropies of $\delta {\boldsymbol{B}}$ 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 shed some light into this question: previous works have shown that the anisotropy $\delta {B}_{\parallel }^{2}/\delta {B}_{\perp }^{2}$ gives indications of the nature of the dominant mode of the fluctuations, while the non-gyrotropy of $\delta {B}_{\perp }^{2}$ gives indications of the directions of the wave vectors ${\boldsymbol{k}}$, i.e., the anisotropy of the ${\boldsymbol{k}}$-distribution.

The general shape $P({\boldsymbol{k}})$ of the ${\boldsymbol{k}}$-distribution can be deduced from the non-gyrotropy of $\delta {B}_{\perp }^{2}(f)$, using the fact that fluctuations with different ${\boldsymbol{k}}$ contribute to the reduced spectral density observed at the same f. If $P({\boldsymbol{k}})$ is gyrotropic, 2D turbulence (i.e., ${k}_{\perp }\gg {k}_{\parallel }$) will be observed with a non-gyrotropic $\delta {B}_{\perp }^{2}(f)$ in the satellite frame; for slab turbulence (i.e., ${k}_{\perp }\ll {k}_{\parallel }$), $\delta {B}_{\perp }^{2}(f)$ will be gyrotropic in the satellite frame (Bieber et al. 1996). Previous observations in the inertial range have shown that the ${\boldsymbol{k}}$-distribution is mainly 2D. However, the slab component is also present and its proportion increases when f reaches the proton scales [0.1, 1] Hz (Bieber et al. 1996; Leamon et al. 1998; Hamilton et al. 2008).

The anisotropy of the ${\boldsymbol{k}}$-distribution of the fluctuations has also been estimated from a comparison of the correlation lengths parallel and perpendicular to ${{\boldsymbol{B}}}_{0}$, measured by a single spacecraft, assuming a stationary and gyrotropic ${\boldsymbol{k}}$-distribution in the MHD range below 10⁻³ Hz (Matthaeus et al. 1990) and below 10⁻² Hz (Dasso et al. 2005). Multispacecraft measurements also allow a comparison of the correlation lengths parallel and perpendicular to ${{\boldsymbol{B}}}_{0}$ in the inertial range, still assuming a gyrotropic ${\boldsymbol{k}}$-distribution. Osman & Horbury (2007), with Cluster observations, find an indication that the ${\boldsymbol{k}}$-distributions in the inertial range are more 2D for compressive fluctuations $\delta {B}_{\parallel }^{2}$ than for transverse fluctuations $\delta {B}_{\perp }^{2}$.

Using data from the four Cluster spacecraft, Chen et al. (2010) find that $\delta {B}_{\parallel }^{2}$ and $\delta {B}_{\perp }^{2}$ have a ${\boldsymbol{k}}$-distribution with ${k}_{\perp }\gt {k}_{\parallel }$ below 10 Hz. Using the k-filtering technique, Sahraoui et al. (2010) find ${k}_{\perp }\gg {k}_{\parallel }$ up to 2 Hz. Still using the k-filtering technique below 1 Hz, for 52 intervals of Cluster data, Roberts et al. (2015) find ${k}_{\perp }\gg {k}_{\parallel }$, 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 (AIC) waves, 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 ${k}_{\perp }\gg {k}_{\parallel }$. At the same scales but in a fast wind, Lion et al. (2016) find Alfvénic coherent structures with ${k}_{\perp }\gg {k}_{\parallel }$ and quasi-monochromatic AIC waves with ${k}_{\parallel }\gg {k}_{\perp }$ superimposed on a non-coherent and non-polarized component of the turbulence. In these analyses, as well as in Perschke et al. (2013), there is no clear mention of the non-gyrotropy of the wave vector energy distribution in the solar wind.

The second kind of anisotropy, the ratio $\delta {B}_{\parallel }^{2}/\delta {B}_{\perp }^{2}$, 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 ${\beta }_{p}$ factor increases (Smith et al. 2006; Hamilton et al. 2008). ${\beta }_{p}$ (${\beta }_{e}$) is the ratio between the proton (electron) thermal energy and magnetic energy. The compressibility also increases when f 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 ${\beta }_{p}$ up to 10 Hz (Alexandrova et al. 2008a) and still increases with f up to 100 Hz (Kiyani et al. 2013). We call d_p and d_e the proton and electron inertial lengths, and ${\rho }_{p}$ and ${\rho }_{e}$ the proton and electron gyroradii, calculated with the temperatures T_p⊥ and T_e⊥ perpendicular to ${{\boldsymbol{B}}}_{0}$.

Even if isolated, coherent nonlinear structures are present in the solar wind (Perri et al. 2012; Lion et al. 2016; Perrone et al. 2016, 2017), a comparison of the observed dimensionless ratios of the fluctuating fields and plasma quantities (transport ratios) with the linear properties of the plasma wave modes can give indications, on a first approximation, of the nature of the dominant type of fluctuation (Gary 1992; Krauss-Varban et al. 1994; Denton et al. 1998). This usual modeling 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 ${T}_{e}/{T}_{p}$ is larger than 1. However, Wind observations at large scales ($k{\rho }_{p}\lt 0.05$), 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 (KAWs), especially in a high-beta plasma.

Salem et al. (2012) compare the observed compressibility with the compressibility expected for whistler waves or for KAWs, and find that the compressibility up to 1 Hz can be explained by KAW-like fluctuations with nearly perpendicular wave vectors. Quasi-perpendicular KAWs are generally found up to proton scales, i.e., up to 1–3 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 KAWs cannot be the only component below 1 Hz, at least when ${\beta }_{p}$ 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 KAWs between 2.5 and 7.5 Hz, i.e., $k{\rho }_{p}\simeq 5$–14.

Calculations by Podesta et al. (2010) show that linear collisionless electron Landau damping prevents KAWs from cascading to 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), which is controlled by the electron gyroradius.

Using 2D kinetic simulations, Camporeale & Burgess (2011b) analyzed the dispersion relation and electron compressibility (i.e., the ratio between the electron density and the magnetic field modulus of the fluctuations) up to $k{\rho }_{e}=1$ or more, for ${\beta }_{p}={\beta }_{e}=0.5$. 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 KAWs 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 $\delta {{\boldsymbol{B}}}_{\perp }$ and (ii) the compressibility of the magnetic fluctuations $\delta {B}_{\parallel }^{2}/\delta {B}_{\perp }^{2}$. After a presentation of the Cluster/STAFF instrument (Section 2) and the data selection procedure (Section 3), we study the non-gyrotropy of $\delta {{\boldsymbol{B}}}_{\perp }$. 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 of the ${\boldsymbol{k}}$-distribution, assumed to be gyrotropic in the plasma frame. Note that the values of the field-to-flow angle θ are near 90° 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 KAWs. We show that the observed magnetic compressibility agrees with the KAW compressibility up to 15–20 Hz, which corresponds to ${{kd}}_{e}\simeq 0.3$ in our sample (where the electron inertial length d_e is about 1 to 3 km). Above 20 Hz, the magnetic fluctuations are more compressive than KAWs, and are more compressive in the slow wind than in the fast wind. It is well-known that the electron to proton temperature ratio ${T}_{e}/{T}_{p}$ is larger in the slow wind. A larger ${T}_{e}/{T}_{p}$ increases the damping of KAWs for several values of ${\beta }_{p}$ and k (Schreiner & Saur 2017) but reduces the damping of the slow-ion acoustic mode; hence, we cannot exclude the presence of this last mode for $f\gt 20$ Hz, i.e., for ${{kd}}_{e}\geqslant 0.3$, a mode which would be less damped and more compressive than the KAW mode.

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 onboard analyzers, a wave form unit (SC) and a Spectrum Analyzer (SA). STAFF-SC provides digitized waveforms, which are projected in the Magnetic Field Aligned (MFA) frame given by the FGM experiment every 4 s; Morlet wavelet spectra are then calculated between 1 and 9 Hz. The SA builds a 3 × 3 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 polarization with respect to ${{\boldsymbol{B}}}_{0}$ (Santolík et al. 2003). Both experiments, SC and SA, allow us to see whether or not the observed fluctuations are polarized. In some intervals, circularly polarized fluctuations are observed, with the direction of propagation near ${{\boldsymbol{B}}}_{0}$, 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 intersect 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 into the foreshock.

The Cluster Science Data System gives the magnetic field ${{\boldsymbol{B}}}_{0}$ every 4 s (FGM experiment; Balogh et al. 1997), the proton density N_p, the wind velocity V_sw, and the proton temperatures of the entire proton distribution, T_p∥ and T_p⊥, respectively, parallel and perpendicular to ${{\boldsymbol{B}}}_{0}$ (CIS experiment; Rème et al. 1997). The electron parameters given by the Low Energy Electron Analyzer of the PEACE experiment (Johnstone et al. 1997) are taken from the Cluster Science Archive: we use the electron temperatures T_e∥ and T_e⊥ of the entire electron distribution.

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 polarized) are observed above 1 Hz up to 20–400 Hz, according to the intensity of the fluctuations. These frequencies are above the proton cyclotron frequency ${f}_{{cp}}=$ [0.08–0.3 Hz] in our sample. With the Taylor hypothesis $2\pi f=\omega \simeq {\boldsymbol{k}}\cdot {{\boldsymbol{V}}}_{\mathrm{sw}}$, where k is the wavenumber of the fluctuations, the scales corresponding to the observed frequencies decrease from ${{kd}}_{e}\simeq 0.01$ to ${{kd}}_{e}\simeq 3$ (${{kd}}_{p}\simeq 0.4$–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 ${{\boldsymbol{B}}}_{0}$ 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 ${{\boldsymbol{B}}}_{0}$, 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 ${{\boldsymbol{B}}}_{0}$ 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 toward 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, seven intervals among the 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 the 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 ${{\boldsymbol{B}}}_{0}$ makes a large angle with the solar wind velocity: the sampling direction is far from the ${{\boldsymbol{B}}}_{0}$ direction. The acute angle θ between ${\text{}}{B}_{0}$ and ${\boldsymbol{R}}$, which is close to the field-to-flow angle, is always larger than 52° in our sample; its average value is 78°.

Figure 1 gives the ranges of the plasma properties that can be found in our data set. It displays scatter plots of the beta factors ${\beta }_{e}$ and ${\beta }_{p}$, the solar wind speed V_sw, the electron to proton temperature ratio ${T}_{e}/{T}_{p}$, and the anisotropy ratios of the proton and electron temperature, ${T}_{p\perp }/{T}_{p\parallel }$ and ${T}_{e\perp }/{T}_{e\parallel }$. ${\beta }_{p}$ varies between 0.28 and 5.1, and ${\beta }_{e}$ between 0.08 and 3.9 (Figure 1(a)). ${T}_{e}/{T}_{p}$ varies between 0.22 and 2.2, and V_sw between 300 and 690 km s⁻¹ (Figure 1(b)). ${T}_{p\perp }/{T}_{p\parallel }$ varies between 0.12 and 0.97 (Figure 1(c)), and ${T}_{e\perp }/{T}_{e\parallel }$ between 0.57 and 1 (Figure 1(d)). 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 analyzed. We then exclude that a possible wave activity associated with these processes may significantly affect the results of this work.

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4.1. Spectra and Anisotropies

We consider the anisotropic PSD P(f) of the magnetic fluctuations as a function of f. The PSD of the transverse fluctuations, perpendicular to ${{\boldsymbol{B}}}_{0}$, is ${P}_{\perp }(f)={P}_{{xx}}(f)+{P}_{{yy}}(f)$. P_yy is the PSD perpendicular to ${{\boldsymbol{B}}}_{0}$ and to ${\boldsymbol{R}}$, and P_xx is the PSD perpendicular to ${{\boldsymbol{B}}}_{0}$ in the plane $({\boldsymbol{R}},{{\boldsymbol{B}}}_{0})$. We call the fluctuations parallel to ${{\boldsymbol{B}}}_{0}$, i.e., to ${\boldsymbol{z}}$, ${P}_{\parallel }={P}_{{zz}}$, compressive. We call ${P}_{3{\rm{D}}}(f)$ the sum of the fluctuations in the three directions, ${P}_{3{\rm{D}}}={P}_{\parallel }+{P}_{\perp }$. Two spectra of our sample are shown in Figure 2. For each spectrum, the solid line gives P_yy, the dashed line P_xx, and the dashed–dotted line P_zz. We see that P_yy is the most intense spectrum and that P_xx is generally the least intense. P_zz is close to P_xx at the lowest frequencies and close to P_yy at the highest frequencies.

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Figure 3(a) displays the anisotropy of the variance of the fluctuations in the two directions ${\boldsymbol{x}}$ and ${\boldsymbol{y}}$ perpendicular to ${{\boldsymbol{B}}}_{0}$. The 93 solid lines are the ratios of the 10 minute averages of the PSD in these two directions—the anisotropy ratios as functions of f (see a discussion on different averaging methods in TenBarge et al. 2012). The red diamonds are the average values of these anisotropy ratios over the 93 events at each frequency. ${P}_{{yy}}/{P}_{{xx}}$ 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 P(f) could be compatible with a gyrotropic wave vector energy distribution has been suggested by Alexandrova et al. (2008b). Using 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 f is only due to the sampling along V_sw (Taylor hypothesis) and if the modulus q of the spectral index of the fluctuations is larger than 1.

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The compressibility ${P}_{\parallel }/{P}_{\perp }$ of the 93 events, observed as a function of f, is given in Figure 3(b): the average value of ${P}_{\parallel }/{P}_{\perp }$ at each frequency is shown by red diamonds. The compressibility increases when f increases. Figure 10 of TenBarge et al. (2012) illustrates how the compressibility depends on the chosen local ${{\boldsymbol{B}}}_{0}$ field: the compressiblity calculated with a field averaged over 1 hr can be two to three 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 ${f}_{{\rho }_{p}}$ corresponding to the proton gyroscale ($k{\rho }_{p}=1$) are generally found between 1 and 4 Hz, and the frequencies ${f}_{{d}_{p}}$ between 0.7 and 2 Hz. For the electron scales, ${f}_{{\rho }_{e}}$ is mainly found between 30 and 200 Hz, and ${f}_{{d}_{e}}$ 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 3(a). The other anisotropy, the compressibility ${P}_{\parallel }/{P}_{\perp }$ (Figure 3(b)), 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 ${{\boldsymbol{B}}}_{0}$, i.e., ${\boldsymbol{k}}$ strictly parallel to ${{\boldsymbol{B}}}_{0}$;
• (b)  
  a gyrotropic 2D distribution, with ${\boldsymbol{k}}$ strictly perpendicular to ${{\boldsymbol{B}}}_{0}$. The corresponding fluctuations are in planes that contain ${{\boldsymbol{B}}}_{0}$ and can make any angle with it, with a component P_∥ parallel to ${{\boldsymbol{B}}}_{0}$ and a component P_⊥ perpendicular to ${{\boldsymbol{B}}}_{0}$;
• (c)  
  an isotropic wave vector distribution;
• (d)  
  a slab symmetry with respect to the radial direction, i.e., ${\boldsymbol{k}}$ strictly parallel to ${\boldsymbol{R}}$.

Equations (33) to (43) of Saur & Bieber (1999) give the values of P_xx(f), P_yy(f), and P_zz(f) for the four above distributions as functions of f, θ, and $({P}_{\parallel }/{P}_{\perp })(f)$. The PSD are power laws $\simeq {f}^{-q}$. The spectral index q of the fluctuations is assumed to be the same in the three directions x, y, and z, and constant over the whole frequency range.

The equations of Saur & Bieber (1999) are valid at MHD scales, larger than ${\rho }_{p}$ and d_p. 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 P_zz and transverse P_⊥ fluctuations, and an enhancement of the magnetic compressibility. Thus, the MHD terms ${P}_{{zz}}/{P}_{{yy}}$ and ${P}_{{zz}}/{P}_{{xx}}$ given by Saur & Bieber (1999) are no longer valid at proton and electron scales. The term ${P}_{{yy}}/{P}_{{xx}}$, which only involves transverse fluctuations, can be considered to be less affected by the Hall effect: it does not depend on ${P}_{\parallel }/{P}_{\perp }$. It is thus able to give indications of the wave vector distribution at scales smaller than the MHD scales.

The anisotropy ${P}_{{yy}}/{P}_{{xx}}$ can be calculated with the formulas of Saur & Bieber (1999) for the four above distributions. For the slab symmetry with respect to B₀,

$$\begin{eqnarray}&&({P}_{{yy}}/{P}_{{xx}})(f)=1.\end{eqnarray} \tag{ 1 }$$

This distribution gives a gyrotropic variance, which is not observed. For the 2D distribution, P_yy(f) and P_xx(f) depend on $({P}_{\parallel }/{P}_{\perp })(f)$ in the same way, and so their ratio does not depend on it:

$$\begin{eqnarray}&&({P}_{{yy}}/{P}_{{xx}})(f)=q,\end{eqnarray} \tag{ 2 }$$

where q is the spectral index of P_xx and P_yy. For the isotropic distribution, P_yy(f) and P_xx(f) do not depend on $({P}_{\parallel }/{P}_{\perp })(f)$:

$$\begin{eqnarray}&&({P}_{{yy}}/{P}_{{xx}})(f)=\displaystyle \frac{(1+q)/2}{{\cos }^{2}\theta (1+q)/2+{\sin }^{2}\theta },\end{eqnarray} \tag{ 3 }$$

where θ is the field-to-flow angle. For the slab symmetry with respect to ${\boldsymbol{R}}$,

$$\begin{eqnarray}&&({P}_{{yy}}/{P}_{{xx}})(f)=1/{\cos }^{2}\theta .\end{eqnarray} \tag{ 4 }$$

In Figure 4, the observations of ${P}_{{yy}}/{P}_{{xx}}$ are compared with two models (Equations (2) and (3)). The calculated anisotropies depend on the observed spectral index q. To get the average value $\langle q\rangle$ 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 P_3D 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 $\langle q\rangle$.

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In Figures 4(a) and (b), the observed average anisotropy (red diamonds) ${P}_{{yy}}/{P}_{{xx}}$ shown in Figure 3(a) is drawn as a function of f. As the dispersion of ${P}_{{yy}}/{P}_{{xx}}$ for the 93 events is large (Figure 3(a)), the two red lines in Figures 4(a) and (b) give the error interval (average ± the standard deviation) at each frequency. Following Equation (2), ${P}_{{yy}}/{P}_{{xx}}$ for the 2D distribution is equal to the spectral index. Thus, in Figure 4(a), the thick black solid line for the 2D ${\boldsymbol{k}}$-distribution shows the observed average $\langle q\rangle$: it varies between 2.7 and 2.9 below 6 Hz; then it increases up to 4 when f increases. This regular increase of $\langle q\rangle$ is due to the exponential shape of the spectra (Alexandrova et al. 2012). The thin black lines give the standard deviation for $\langle q\rangle$ over the 93 events, i.e., the dispersion of the 2D model itself. Taking into account the standard deviation of the observed ratio ${P}_{{yy}}/{P}_{{xx}}$ (red lines), it is clear that the observations from 1 to 6 Hz are close to the ratio for a 2D ${\boldsymbol{k}}$-distribution. Above 10 Hz (at sub-proton and electron scales), the observations are further from the 2D model.

In Figure 4(b), 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 ${\boldsymbol{k}}$-distribution. At any frequency, the slab model (Equation (1)) is far from the observations. The ratio (Equation (4)) for the model with a slab symmetry with respect to ${\boldsymbol{R}}$ 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 q is constant over the whole frequency range: this is not exact above about 10 Hz (Figure 4(a)). Our results are just an indication, given by the non-gyrotropy of the PSD, that the k-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 k_∥ and k_⊥, i.e., privileged directions of the wave vectors, with increasing energy in k_⊥ when the wavenumber increases (see Figure 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 ${\boldsymbol{k}}$-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 θ 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 $[{10}^{-3}$ Hz, 2 10⁻² Hz], Bieber et al. (1996) find that the turbulence in the inertial range can be described by $\simeq 85 \%$ of the energy in 2D fluctuations and $\simeq 15 \%$ in slab fluctuations. Leamon et al. (1998) find that the proportion of slab fluctuations increases at proton scales (0.1–1 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 probably has to be considered. This is illustrated by the work of Mangeney et al. (2006), who study the variations, as a function of θ, 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 ${{kd}}_{e}\simeq 30$ or more, can be explained by a 2D ${\boldsymbol{k}}$-distribution, with an aperture angle of about 10°; the electrostatic fluctuations above ${{kd}}_{e}\simeq 160$ can be explained by a slab distribution, with the same aperture angle for ${\boldsymbol{k}}$ around ${{\boldsymbol{B}}}_{0}$.

A more or less smooth transition from a 2D to an isotropic ${\boldsymbol{k}}$-distribution when f increases could be a model of our observations that is more realistic than a two-component model with an increasing proportion of slab fluctuations when f increases.

4.3. Wave Vector 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 ${{\boldsymbol{B}}}_{0}$, while intermediate and slow streams have quasi-2D ${\boldsymbol{k}}$-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 ${\boldsymbol{k}}$-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⁻¹ (Figure 1(b)). We consider separately a sample of 16 intervals with ${V}_{\mathrm{sw}}\gt 630$ km s⁻¹, and a sample of 16 intervals with ${V}_{\mathrm{sw}}\lt 380$ km s⁻¹. Figure 5(a) displays ${P}_{{yy}}/{P}_{{xx}}$ as a function of f 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 ${P}_{{yy}}/{P}_{{xx}}$ is larger for the fast wind, below 10 Hz. Is this difference significant, in view of the large dispersion of ${P}_{{yy}}/{P}_{{xx}}$ in each sample? In Figure 5(b), the average values of Figure 5(a) (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 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 ${P}_{{yy}}/{P}_{{xx}}$ in the fast wind and the slow wind is thus significant at these frequencies. Figures 5(c) and (d) allow a comparison of the averages of ${P}_{{yy}}/{P}_{{xx}}$ observed in the fast and the slow wind (crosses) to the ${P}_{{yy}}/{P}_{{xx}}$ calculated for the two models of the ${\boldsymbol{k}}$-distribution given by Saur & Bieber (1999) (Equations (2) and (3)).

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We thus find that the 2D character of the ${\boldsymbol{k}}$-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 led 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 100 times higher than the inertial range, so that the operating physical processes can be different.

The compressibility ${P}_{\parallel }/{P}_{\perp }$ of the 93 events of our sample (Figure 3(b)) increases with frequency. The average value of ${P}_{\parallel }/{P}_{\perp }$ 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, ${P}_{\parallel }/{P}_{\perp }$ is larger than the value of 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 6(a) displays ${P}_{\parallel }/{P}_{3{\rm{D}}}$, another definition of the compressibility, as a function of f. Salem et al. (2012) find that ${P}_{\parallel }/{P}_{3{\rm{D}}}$ reaches 0.5 at 2 Hz. This is larger than what we find.

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The ranges of the ${\beta }_{p}$ values of our sample are given in Figure 1(a). The lower red dashed line in Figure 6(a) is the average of ${P}_{\parallel }/{P}_{3{\rm{D}}}$ for five events with a small ${\beta }_{p}$, and the upper red dashed line is for five events with a large ${\beta }_{p}$ (see the selected values of ${\beta }_{p}$ in the caption). These red lines show that the compressibility and the variation of the compressibility with f depend strongly on ${\beta }_{p}$.

5.1. Compressibility in the Wavenumber Domain

Converting the observed frequency f into a wavenumber $k=2\pi f/{V}_{\mathrm{sw}}$ (Taylor hypothesis), we calculate the PSDs P_xx(k), P_yy(k), P_zz(k), and ${P}_{3{\rm{D}}}(k)$ in ${\mathrm{nT}}^{2}\ \mathrm{km}$ (despite the fact that this conversion is problematic for anisotropic ${\boldsymbol{k}}$-distributions and reduced spectra; we shall return to this point in Section 6). In Figure 6(b), similar to Figure 6(a), we see that the variation of the compressibility ${P}_{\parallel }/{P}_{3{\rm{D}}}$ with the wavenumber k depends strongly on ${\beta }_{p}$.

We then normalize k to the electron inertial length d_e, or to the gyroradii ${\rho }_{e}$ or ${\rho }_{p}$, and interpolate the normalized PSD at 15 values of kd_e, $k{\rho }_{e}$, or $k{\rho }_{p}$. Figures 7(a) and (b) give the total power P_3D as functions of kd_e and $k{\rho }_{e}$. 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).

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Figure 7(c) gives the compressibility ${P}_{\parallel }/{P}_{3{\rm{D}}}$ as a function of kd_e. The vertical bars in the abscissae give kd_p = 1, 10, and 100. With high-resolution 2D hybrid simulations of a decaying turbulence, for ${\beta }_{p}={\beta }_{e}=0.5$, Franci et al. (2015) obtain a compressibility ${P}_{\parallel }/{P}_{3{\rm{D}}}$ increasing from 0.1, at ${{kd}}_{p}=0.1,$ to 0.5 at ${{kd}}_{p}=2$, and a nearly constant (plateau) value $\simeq 0.6$ for ${{kd}}_{p}=5$–10. (See also the gyrokinetic simulations in Figure 9 of TenBarge et al. 2012). After a kind of plateau from ${{kd}}_{p}=2$ to 10, the observed average ${P}_{\parallel }/{P}_{3{\rm{D}}}$ (Figure 7(c)) increases from ${{kd}}_{p}=10$ to 100, but it is always smaller than 0.4 (reached for ${{kd}}_{e}\simeq 2$, kd_p = 100). We conclude that the observed and the simulated compressibility have similar variations up to ${{kd}}_{p}=10,$ but the observed compressibility is two to three times weaker: we shall discuss this last point in Section 6. For ${{kd}}_{p}\gt 10$, a domain not reached by these simulations, the observed compressibility still increases.

5.2. Compressibility in the Fast and Slow Winds

In Figure 8(a), we compare the compressibility ${P}_{\parallel }/{P}_{\perp }$ 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 8(b), 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 $f\lt$ 10 Hz. For $f\gt 15\mbox{--}20\,\mathrm{Hz}$, it is larger in the slow wind. It is well-known that there is a correlation between V_sw and T_p 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): ${T}_{e}/{T}_{p}$ is anticorrelated with V_sw (Figure 1(b)). Thus, the larger compressibility observed in the slow wind above 15–20 Hz could be related to larger values of ${T}_{e}/{T}_{p}$.

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5.3. Role of ${\beta }_{p}$

Smith et al. (2006) in the inertial range, Hamilton et al. (2008) up to 0.8 Hz, and Alexandrova et al. (2008a) from 0.3 to 10 Hz find that the compressibility increases when ${\beta }_{p}$ increases. This is clear in Figure 9 for the present sample at 2 and 5 Hz. The slope of ${P}_{\parallel }/{P}_{\perp }$ as a function of $\mathrm{log}{\beta }_{p}$ (solid lines in Figure 9) decreases when f increases, from 5 to 22 Hz. At 56 Hz, the compressibility is no longer correlated with ${\beta }_{p}$.

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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 10(a) to (f) display the correlation coefficients of $({P}_{\parallel }/{P}_{\perp })(f)$ with local parameters related to the proton or electron temperature. Scatter plots of these plasma parameters are shown in Figure 1. The correlation of ${P}_{\parallel }/{P}_{\perp }$ with ${\beta }_{p}$ (Figure 10(a)), as a function of f, is larger than 0.7 from 3 to 10 Hz. It decreases clearly for $f\gt 15$ Hz. The correlation with ${\beta }_{e}$ (Figure 10(c)) is weaker than with ${\beta }_{p}$, but it extends up to 20–30 Hz. The correlation with ${T}_{p\perp }/{T}_{p\parallel }$ (Figure 10(b)) is good (around 0.6) from 10 to 50 Hz. The correlation with ${T}_{e\perp }/{T}_{e\parallel }$ (not shown) is always weaker than 0.55.

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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., KAWs and whistler waves ;(see also Gary & Smith 2009). The whistler magnetic compressibility does not depend on β, only on the wave vector direction. Conversely, the magnetic compressibility ${C}_{\parallel /\perp }^{{ka}}$ for KAWs, in the framework of the electron-reduced MHD ($k{\rho }_{p}\gt 1$ and $k{\rho }_{e}\lt 1$), depends on ${\beta }_{p}$ and ${\beta }_{e}$, not on the wavenumber:

$$\begin{eqnarray}&&{C}_{\parallel /\perp }^{{ka}}=\displaystyle \frac{{\beta }_{p}+{\beta }_{e}}{2+{\beta }_{p}+{\beta }_{e}}.\end{eqnarray} \tag{ 5 }$$

Figure 10(d) displays the correlation between the observed magnetic compressibility ${P}_{\parallel }/{P}_{\perp }$ and ${C}_{\parallel /\perp }^{{ka}}$ as a function of f. This correlation is everywhere larger than the correlation with ${\beta }_{p}$ (Figure 10(a)). It is larger than the correlation with ${\beta }_{e}$ (Figure 10(c)) except in a narrow frequency range around 15–25 Hz. The correlation of Figure 10(d) is larger than 0.7 between 1.5 and 15 Hz.

5.5. Modes of the Fluctuations

5.5.1. For $f\lt 15\mbox{--}20\,\mathrm{Hz}$

At these frequencies (corresponding to $k{\rho }_{p}\leqslant 10$), our observations can be compared with the Vlasov–Maxwell results of TenBarge et al. (2012). These authors calculate ${P}_{\perp }/{P}_{\parallel }$ for Alfvén, fast, and slow modes up to $k{\rho }_{p}=10$ for different values of ${\beta }_{p}$, assuming that the turbulence is in critical balance (${k}_{\parallel }\propto {k}_{\perp }^{2/3}$), i.e., with quasi-perpendicular wave vectors.

We have seen (Figure 9) that ${P}_{\parallel }/{P}_{\perp }$ increases when ${\beta }_{p}$ increases: this is typical of KAWs, but excludes the presence of fast waves for which ${P}_{\parallel }/{P}_{\perp }$ decreases when ${\beta }_{p}$ increases. The observations of Figure 7(d) are also consistent with KAW properties up to $k{\rho }_{p}=10$ (Figure 2 of TenBarge et al. 2012): for the lowest values of ${\beta }_{p}$, the observed ${P}_{\perp }/{P}_{\parallel }$ (upper solid red line) decreases strongly when $k{\rho }_{p}$ increases, while ${P}_{\perp }/{P}_{\parallel }$ decreases weakly for the largest values of ${\beta }_{p}$ (lower solid red line). Finally, Figure 10(d) confirms that fluctuations with a KAW polarization are generally dominant at sub-proton scales, for $f\lt$ 15–20 Hz.

However, the observed ${P}_{\perp }/{P}_{\parallel }$ profiles (Figure 7(d)) have a large dispersion, so that they could be consistent, in some cases, with other modes, oblique whistler waves with a critical balance distribution (${k}_{\parallel }\propto {k}_{\perp }^{1/3}$, Figure 4 of TenBarge et al. 2012), or with different mixtures of Alfvén, fast, and slow wave modes (Figures 2 and 6 of TenBarge et al. 2012) for which ${P}_{\perp }/{P}_{\parallel }$ changes strongly from $k{\rho }_{p}\simeq 0.5$ to $k{\rho }_{p}\simeq 10$.

5.5.2. For f > 15–20 Hz

The weaker correlation between ${P}_{\perp }/{P}_{\parallel }$ and ${C}_{\parallel /\perp }^{{ka}}$ observed in Figure 10(d) above 15 Hz (and below 1.5 Hz) can be due to different reasons: (i) Equation (5) for ${C}_{\parallel /\perp }^{{ka}}$ is valid in a limited scale range, (ii) the KAWs are damped, and/or (iii) another mode, with a different compressibility, is present. Let us discuss these three points.

• (i)  
  Validity of Equation (5)? Equation (5) is valid for $k{\rho }_{p}\gt 1$ and $k{\rho }_{e}\lt 1$. This is shown in Figure 2(a) of TenBarge et al. (2012) where the compressibility is indeed equal to ${C}_{\parallel /\perp }^{{ka}}$ for ${k}_{\perp }{\rho }_{p}=2$ to 10, at least for the observed values ${\beta }_{p}\simeq [0.28\mbox{--}5.1]$ . In our sample, for all events, $k{\rho }_{p}\gt 1$ if $f\gt 3$ Hz, and $k{\rho }_{e}\lt 1$ if $f\lt 50$ Hz. Thus, in Figure 10(d), the weaker correlation below 3 Hz can be due to the lack of validity of Equation (5). Conversely, the weaker correlation from 15 to 20 Hz to 50 Hz has to be due to other reasons.
• (ii)  
  KAW damping? Figure 10(e) shows the correlation between ${P}_{\parallel }/{P}_{\perp }$ and ${T}_{e}/{T}_{p}$ as a function of f. 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 10(f) shows that the correlation between ${P}_{\parallel }/{P}_{\perp }$ and ${T}_{e}/{T}_{p}$ (Figure 10(e)) could be related to the anticorrelation of ${P}_{\parallel }/{P}_{\perp }$ with V_sw above 20 Hz. According to Figure 1 of Schreiner & Saur (2017), larger values of ${T}_{e}/{T}_{p}$ increase the damping of KAWs, at least for the ${\beta }_{p}$ values of our sample, and for small scales ${k}_{\perp }{\rho }_{p}\simeq [10\mbox{--}100]$ (see also Howes et al. 2006). The decrease of the correlation between ${P}_{\perp }/{P}_{\parallel }$ and ${C}_{\parallel /\perp }^{{ka}}$ above 20 Hz (Figure 10(d)) can thus be due to a stronger damping of KAWs, itself due to relatively larger values of ${T}_{e}/{T}_{p}$. Note that Bruno & Telloni (2015) observe that KAWs (below 4 Hz) tend to disappear when the wind speed decreases.
• (iii)  
  Another mode? The weaker correlation between ${P}_{\perp }/{P}_{\parallel }$ and ${C}_{\parallel /\perp }^{{ka}}$ for $f\gt 15\mbox{--}20$ Hz (Figure 10(d)) 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 (Figure 1(d)) nor the heat flux is unstable: whistler waves are not observed. The ${\beta }_{e}$ values are generally too low to give an electron parallel firehose instability (Matteini et al. 2013). The correlation of ${P}_{\parallel }/{P}_{\perp }$ with ${T}_{e}/{T}_{p}$ (Figure 10(e)) 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 ${T}_{e}/{T}_{p}$ is larger. ${T}_{e}/{T}_{p}$ varies between 0.22 and 2.2 in our sample, with 60% of the intervals for ${T}_{e}/{T}_{p}\lt 1$ (Figure 1(b)). These values seem to be too low to prevent the damping of slow-ion acoustic modes. However, we recall that T_e and T_p are average values for the entire distributions, without distinction between the temperatures of the 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 ${T}_{p\perp }/{T}_{p\parallel }$ (Figure 10(b)) is larger for $f\gt 15\mbox{--}20$ Hz than in the KAW frequency domain. According to Narita & Marsch (2015), quasi-perpendicular kinetic slow modes could produce proton heating so that the correlation in Figure 10(b) could be another indication for the presence of kinetic slow modes, cascading above 15–20 Hz.

In Figure 11, the ${P}_{\parallel }/{P}_{\perp }$ observed at four fixed frequencies is drawn as a function of ${C}_{\parallel /\perp }^{{ka}}$ (Equation (5)). In each panel, C_c gives the correlation coefficient between ${P}_{\parallel }/{P}_{\perp }$ and ${C}_{\parallel /\perp }^{{ka}}$. At 2 Hz, the observed compressibility is always weaker than ${C}_{\parallel /\perp }^{{ka}}$, the compressibility expected for KAWs (Figure 11(a)). At 6.6 Hz (Figure 11(b)), where C_c peaks, ${P}_{\parallel }/{P}_{\perp }$ is still weaker but closer to ${C}_{\parallel /\perp }^{{ka}}$. At 22 Hz, the observed compressibility is, on average, equal to ${C}_{\parallel /\perp }^{{ka}}$ (Figure 11(c)). At 56 Hz, ${P}_{\parallel }/{P}_{\perp }$ is mainly larger than ${C}_{\parallel /\perp }^{{ka}}$ (Figure 11(d)).

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Taking into account that the observed magnetic compressibility $({P}_{\parallel }/{P}_{\perp })(f)$ can be underestimated with respect to simulations or numerical calculations (see Section 6), we conclude that ${P}_{\parallel }/{P}_{\perp }$ is larger than the KAW compressibility for $f\gt 15\mbox{--}20$ Hz, i.e., $k{\rho }_{p}\geqslant 10$.

A few simulations of the slow-ion acoustic mode reach the scales $k{\rho }_{p}\simeq 100$ or $k{\rho }_{e}\simeq 1$ corresponding to our observations. The hybrid Vlasov–Maxwell simulations of Valentini et al. (2008) and Valentini & Veltri (2009) reach ${{kd}}_{p}\simeq 100$ and show the generation of ion-acoustic turbulence, but these simulations imply a temperature ratio ${T}_{e}/{T}_{p}=10$, which is much larger than the values of our sample (Figure 1(b)). Camporeale & Burgess (2011b) reach $k{\rho }_{e}\simeq 10$ but the electron compressibility of their simulated fluctuations is weaker than the calculated electron compressibility of the slow-ion acoustic waves. An interesting result in Figure 3 of Camporeale & Burgess (2011a) is that the electron compressibility of the slow-ion acoustic mode is larger for a small propagation angle (${\theta }_{{kB}}=45^\circ$) than for ${\theta }_{{kB}}\,=$ 60° or 80°. This is another point in favor of the presence of slow-ion acoustic modes in our data. Indeed, we concluded (Section 4.2) that the ${\boldsymbol{k}}$-distribution is more isotropic so that small ${\theta }_{{kB}}$ angles are more frequent when f increases. Thus, a larger electron compressibility should be found when f increases (${\theta }_{{kB}}$ decreases) if slow-ion acoustic modes are present. We did not measure the electron compressibility, but Figures 6(a) and 11 show that the magnetic compressibility increases when f increases. Calculations similar to those of Gary & Smith (2009) would tell us whether the magnetic compressibility of slow-ion acoustic modes increases when ${\theta }_{{kB}}$ decreases.

In this work, we consider the reduced power spectral densities P_xx(f), P_yy(f), and P_zz(f) of the magnetic field fluctuations observed at a given frequency f in the three directions with respect to the quasi-local magnetic field ${{\boldsymbol{B}}}_{0}$ (4 s average), ${P}_{\parallel }(f)={P}_{{zz}}(f)$ and ${P}_{\perp }(f)={P}_{{xx}}(f)+{P}_{{yy}}(f)$. The sampling direction is in the plane (${\boldsymbol{x}},{\boldsymbol{z}}$), i.e., perpendicular to the ${\boldsymbol{y}}$ direction.

In Section 5, the magnetic compressibility observed in the frequency domain, $({P}_{\parallel }/{P}_{\perp })(f)$, is compared with analytical and numerical calculations and simulations in the wave vector domain, which only consider two components, k_∥ and k_⊥, and give the compressibility $({P}_{\parallel }/{P}_{\perp })({k}_{\perp })$. But this comparison is not really valid: we cannot assume that $({P}_{\parallel }/{P}_{\perp })(f)$ is equal to $({P}_{\parallel }/{P}_{\perp })({k}_{\perp })$. Indeed, if there is only one wave vector ${{\boldsymbol{k}}}_{1}$ perpendicular to ${{\boldsymbol{B}}}_{0}$, along ${\boldsymbol{x}}$ for instance, the PSD in the directions perpendicular to ${{\boldsymbol{k}}}_{1}$, ${P}_{{yy}}({k}_{1})$ and ${P}_{{zz}}({k}_{1})$, will be observed as ${P}_{{yy}}({f}_{1})$ and ${P}_{{zz}}({f}_{1})$ at the same frequency ${f}_{1}={{\boldsymbol{k}}}_{1}\cdot {{\boldsymbol{V}}}_{\mathrm{sw}}/2\pi$ (see Podesta & TenBarge 2012, Section 2.1) and will display a phase relation between $\delta {B}_{y}$ and $\delta {B}_{z}$ corresponding to their polarization. If there is also a wave vector ${{\boldsymbol{k}}}_{2}$ perpendicular to ${{\boldsymbol{B}}}_{0}$, along ${\boldsymbol{y}}$, with wavenumber ${k}_{2}={k}_{1}$, ${P}_{{xx}}({k}_{2})$ and ${P}_{{zz}}({k}_{2})$ will be observed as ${P}_{{xx}}({f}_{2})$ and ${P}_{{zz}}({f}_{2})$ at the same frequency ${f}_{2}={{\boldsymbol{k}}}_{2}\cdot {{\boldsymbol{V}}}_{\mathrm{sw}}/2\pi$, but f₂ will be much smaller than f₁ because ${{\boldsymbol{k}}}_{2}$ is more or less perpendicular to the sampling direction ${{\boldsymbol{V}}}_{\mathrm{sw}}$. Thus, even if two wave vectors perpendicular to ${{\boldsymbol{B}}}_{0}$ have the same k_⊥, they will not be observed at the same frequency in P_xx(f), P_yy(f), and P_zz(f). Inversely, at a given frequency, P_xx(f), P_yy(f), and P_zz(f) do not correspond to the same wavenumbers and wave vectors: they do not belong to the same fluctuations, and there is no clear phase relation between the components x, y, and z. Furthermore, a PSD like P_zz or like ${P}_{\perp }={P}_{{xx}}+{P}_{{yy}}$ or ${P}_{3{\rm{D}}}={P}_{\perp }+{P}_{{zz}}$ 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 of the PSDs, and simulations that 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 6(b) and 7, the wavenumber dependence relies on this hypothesis. However, at proton or electron scales, some fluctuations can be present, with a frequency ${\omega }_{0}$ in the solar wind frame non-negligible with respect to ${\boldsymbol{k}}.{{\boldsymbol{V}}}_{\mathrm{sw}}$. Thus, any detailed comparison of the observations with calculations in the wavenumber domain (Section 5.1) has to take into account the possibility that k is not always proportional to ω in the kinetic range, and that wave modes could be present with a non-negligible ${\omega }_{0}$. What could be these wave modes?

Howes et al. (2014) show that the Taylor hypothesis is valid for all of 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 f_ce: in our sample, f_ce varies from 150 to 550 Hz, with an average value of 300 Hz. Quasi-parallel whistler waves could thus be present below f_ce (Lacombe et al. 2014; Kajdič et al. 2016; Stansby et al. 2016), but their characteristic right-handed polarization with respect to ${{\boldsymbol{B}}}_{0}$ 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 ${f}_{{cp}}=[0.08\mbox{--}0.3\,\mathrm{Hz}]$. Thus, ion Bernstein modes, at low harmonics (1, 2 or 3) of f_cp cannot be observed.

Our frequency range is well below the proton plasma frequency (f_pi varies from 400 Hz to 1.5 kHz, average ≃ 800 Hz): slow-ion acoustic waves could be present, which propagate below f_pi. 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 need to 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 ($f\gt 15\mbox{--}20$ Hz; see Section 5.5.2), the observed compressibility is larger than the KAW compressibility (Figure 11). It is larger in the slow wind than in the fast wind (Figure 8), i.e., larger when the electron to proton temperature ratio ${T}_{e}/{T}_{p}$ is larger (Figure 1(b)). Is there a mode that has a compressibility larger than the KAW compressibility, which could be present in the slow wind, where ${T}_{e}/{T}_{p}$ is larger, and which can propagate at oblique or small angles with respect to B₀? According to Figure 1 of Gary (1992), if ${T}_{e}/{T}_{p}\geqslant 1$ and ${\beta }_{p}\gt 1$, there is always a slow mode that is only weakly damped for small propagation angles. This is valid for MHD modes and up to ${{kd}}_{p}$ = 0.1 (Gary & Winske 1992; Krauss-Varban et al. 1994). This is still valid up to $k{\rho }_{e}\simeq 10$ (Camporeale & Burgess 2011a) at least for ${\beta }_{p}={\beta }_{e}=0.5$. The magnetic compressibility of this kinetic mode has to be calculated to be compared to our observations.

The comparison of the observed polarization ratios with those of linear modes is not always justified. But it could be more valid at the smallest scales ($f\gt 20$ Hz or $k{\rho }_{e}\gt$ 0.2), when some angles ${\theta }_{{kB}}$ are probably smaller than the values giving the critical balance condition, i.e., smaller than the values where the role of nonlinear effects becomes dominant (Figure 1 of Howes et al. 2008).

We analyzed 93 ten-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 Earth's bow shock; as a consequence, the Cluster orbit implies that the field-to-flow angle θ is large, and (2) there is no indication of the presence of quasi-parallel whistler waves, which are easily detected by their quasi-circular right-handed polarization (Lacombe et al. 2014).

The PSD of the magnetic field turbulence is measured in the three directions of the MFA frame, an average quasi-local frame given by the 4 s magnetic field data. The PSD is anisotropic. It decreases when f increases; it goes down to the instrumental background noise at different frequencies in the different directions. As explained in Section 3 and the Appendix, we only consider the PSD when it is larger than three times the noise in any direction; then, we subtract the noise to obtain the corrected signal.

The corrected PSD P(f) is itself anisotropic. It is maximum in the ${\boldsymbol{y}}$ direction, which is perpendicular to ${{\boldsymbol{B}}}_{0}$ and to the radial direction. P(f) is weaker in the ${\boldsymbol{x}}$ direction, which is perpendicular to ${{\boldsymbol{B}}}_{0}$ and ${\boldsymbol{y}}$: the ratio ${P}_{{yy}}(f)/{P}_{{xx}}(f)\gt 1$ implies that the transverse fluctuations are not gyrotropic at a given frequency f in the spacecraft frame.

Following Saur & Bieber (1999), the non-gyrotropic ratio $({P}_{{yy}}/{P}_{{xx}})(f)$ gives indications of the shape of a gyrotropic distribution of ${\boldsymbol{k}}$. We find that the ${\boldsymbol{k}}$-distribution $P({\boldsymbol{k}})$ is mainly 2D (i.e., ${k}_{\perp }\gg {k}_{\parallel }$) up to about 6 Hz, i.e., ${{kd}}_{e}\simeq 0.1$ or $k{\rho }_{p}\simeq 3$. There is also a slight proportion of wave vectors parallel to ${{\boldsymbol{B}}}_{0}$. Through a separate study of intervals of fast wind and slow wind, we show that $P({\boldsymbol{k}})$ is closer to a 2D distribution in the fast wind and contains a larger proportion of wave vectors parallel to ${{\boldsymbol{B}}}_{0}$ in the slow wind, at $f\,\lt$ 10 Hz. At frequencies $f\,\gt$ 10 Hz, $P({\boldsymbol{k}})$ is less than 2D and tends toward isotropy around 50 Hz: wave vectors oblique or parallel to ${{\boldsymbol{B}}}_{0}$ become relatively more important when the frequency increases (Section 4.2).

We call the anisotropy ratio $({P}_{\parallel }/{P}_{\perp })(f)$ between the compressive and transverse fluctuations the compressibility: this polarization ratio increases from 1 to 200 Hz. From 2 Hz to about 15–20 Hz, the observed compressibility is very well correlated (Figure 10(d)) and nearly equal (Figure 11) to the compressibility (Equation (5)) of linear KAWs, which only depends on the beta factor $\beta ={\beta }_{p}+{\beta }_{e}$: fluctuations with a KAW-like polarization are dominant below 20 Hz, i.e., below ${{kd}}_{e}\simeq 0.2$ or $k{\rho }_{p}\simeq 10$.

At smaller scales, $f\gt 20$ 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 polarization 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 ${\beta }_{p}$, and different ${T}_{e}/{T}_{p}$, 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 $k{\rho }_{p}\geqslant 10$.

Note, however, that recent studies (Chen & Boldyrev 2017; Passot et al. 2017) suggest that the compressibility of KAWs can further increase when reaching the electron inertial length scale, and thus be in possible good agreement with the trend observed in this data set for frequencies above 20 Hz.

We thank Nicole Cornilleau-Wehrlin and Patrick Canu, PIs of the STAFF experiment, for the use of the STAFF-SC and STAFF-SA data. We thank Pierrette Décréau for the WHISPER data. The FGM data (PIs: A. Balogh and E. Lucek) and the CIS data (PIs: H. Rème and I. Dandouras) come from the Cluster Science Data System (ESA). The PEACE data (PI: A. Fazakerley) are taken from the Cluster Science Archive (ESA). The French contribution to the Cluster project has been supported by the European Space Agency (ESA) and by the Centre National d'Etudes Spatiales (CNES). L.M. was supported by the UK STFC grant ST/N000692/1.

For a precise measurement of the weak spectral levels on Cluster, the subtraction of the instrument noise was 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 f is the difference ${P}_{3{\rm{D}}}(f)-{{bn}}_{3{\rm{D}}}(f)$, where ${P}_{3{\rm{D}}}(f)$ is the average of the trace of the spectral matrix measured on the triaxial search coils (signal + noise) and ${{bn}}_{3{\rm{D}}}(f)$ 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, and not an additive noise that 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 ${{bn}}_{3{\rm{D}}}(f)$ is given by a one-hour average of the 3D power P_lobe measured in a lobe on 2004 June 03 (14:00–15:00 UT) on Cluster 1. Histograms of $\mathrm{log}{P}_{\mathrm{lobe}}(f)$ during this interval, at different f, show that the instantaneous background noise can be as large as two to three times the average value ${{bn}}_{3{\rm{D}}}(f)$ below 60 Hz, and 1.3 to 1.5 times ${{bn}}_{3{\rm{D}}}(f)$ above 60 Hz. (Above 60 Hz, the power of each STAFF-SA spectrum relies on eight 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, ${P}_{3{\rm{D}}}(f)\mbox{--}{{bn}}_{3{\rm{D}}}$, we have to take into account this uncertainty on bn_3D, which is at most three times bn_3D. The corrected spectral level is the average value of ${P}_{3{\rm{D}}}\mbox{--}{{bn}}_{3{\rm{D}}}$, but only at the frequencies where P_3D is larger than $3{{bn}}_{3{\rm{D}}}$,

$$\begin{eqnarray}&&{P}_{3{\rm{D}}}(f)\geqslant 3{{bn}}_{3{\rm{D}}}(f).\end{eqnarray} \tag{ 6 }$$

There is no simple reliable way to separate the signal and instrument noise at frequencies for which P_3D is weaker than $3{{bn}}_{3{\rm{D}}}$.

For the background noise in each direction x, y, and z, we use ${{bn}}_{1{\rm{D}}}(f)={{bn}}_{3{\rm{D}}}(f)/3$ (as in Figure 2). Even if the three search coils in the spacecraft frame (SR2) do not have exactly the same 1D background noise, it is not useful to distinguish them: indeed, the onboard (SR2) spectral matrix (signal + noise) is projected and analyzed in the MFA frame, x, y, and z, 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 (three 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 P_xx(f), P_yy(f), and P_zz(f). As P_xx(f) (dashed lines in Figure 2) is the weakest 1D spectrum (above 8 Hz), we first have to check that the noise subtraction is significant forP_xx: indeed, the subtraction from ${P}_{3{\rm{D}}}(f)$ can be valid (if ${P}_{3{\rm{D}}}(f)$ is larger than $3{{bn}}_{3{\rm{D}}}(f)$), while the subtraction from P_xx(f) is not valid, if P_xx(f) is equal to the 1D background noise. To prevent errors in the subtraction of the noise from P_xx(f), we thus put a second condition,

$$\begin{eqnarray}&&{P}_{{xx}}(f)\geqslant 3{{bn}}_{1{\rm{D}}}(f).\end{eqnarray} \tag{ 7 }$$

As P_xx(f) is always weaker than P_yy(f) and P_zz(f), this second condition implies that the 1D noise can be subtracted without errors in every direction, x, y, and z. Figure 12 illustrates the two conditions (Equations (6) and (7)) to get valid measurements of a 3D spectrum of P_xx(f), P_yy(f), and P_zz(f), 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 $3{{bn}}_{3{\rm{D}}}$ and $3{{bn}}_{1{\rm{D}}}$. The dashed lines give P_3D (left panel) and P_xx (right panel) before any subtraction. The solid lines give the corrected spectra, which obey the two above conditions, and after subtraction respectively of bn_3D and bn_1D.

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All of the spectra used in the present paper are corrected in this way.