Spectra and anisotropy of magnetic fluctuations in the Earth's magnetosheath: Cluster observations

We investigate the spectral shape, the anisotropy of the wave vector distributions and the anisotropy of the amplitudes of the magnetic fluctuations in the Earth's magnetosheath within a broad range of frequencies. We present the first observations of a Kolmogorov-like inertial range of Alfvenic fluctuations in the magnetosheath flanks, below fci. In the vicinity of fci, a spectral break is observed, like in solar wind turbulence. Above the break, the energy of compressive and Alfvenic fluctuations generally follow a power law with a spectral index between -3 and -2. Concerning the anisotropy of the wave vector distribution, we observe a change in its nature in the vicinity of ion characteristic scales: if at MHD scales there is no evidence for a dominance of a slab (k||>kperp) or 2D (kperp>k||) turbulence, above the spectral break, (f>fci, kc/wpi>1) the 2D turbulence dominates. This 2D turbulence is observed in six selected one-hour intervals among which the average proton beta varies from 0.8 to 9. It is observed for both the transverse and compressive magnetic fluctuations, independently on the presence of linearly unstable modes at low frequencies or Alfven vortices at the spectral break. We then analyse the anisotropy of the magnetic fluctuations in a time dependent reference frame based on the field B and the flow velocity V directions. Within the range of the 2D turbulence, at scales [1,30]kc/wpi, and for any beta we find that the magnetic fluctuations at a given frequency in the plane perpendicular to B have more energy along the BxV direction. This non-gyrotropy of the fluctuations is consistent with gyrotropic fluctuations at a given wave vector, with kperp>k||, which suffer a different Doppler shift along and perpendicular to V in the plane perpendicular to B.


Introduction
In the space plasma turbulence, the presence of a mean magnetic field B gives rise to anisotropies with respect to the field direction ( means parallel, and ⊥ means perpendicular to B).There are anisotropies both in the intensities δB 2 of the magnetic fluctuations (δB 2 ⊥ = δB 2 ) and in the distribution of their wave vectors k (k ⊥ = k ), i.e., the energy distribution of the turbulent fluctuations is anisotropic in kspace.
To study the anisotropy of turbulent fluctuations in space plasma, we chose here the Earth's magnetosheath as a laboratory.Downstream of the bow shock, the solar wind plasma slows down, and the plasma density, temperature and magnetic field increase in comparison with the solar wind plasma.The magnetosheath boundaries (bow shock and magnetopause) introduce an important temperature anisotropy T ⊥ > T , and therefore linearly unstable waves, such as Alfvén Ion Cyclotron (AIC) and mirror modes, are present (see the reviews by Schwartz et al., 1996;Lucek et al., 2005;Alexandrova, 2008).In the vicinity of the bow-shock, an f −1 power law spectrum is observed at frequencies below the ion cyclotron frequency, f < f ci , (Czaykowska et al., 2001).The power law spectra ∼ f −5/3 , typical of the solar wind inertial range at f < f ci , have not been observed in the magnetosheath.However, as in the solar wind, the energy of the magnetic fluctuations follows a power law close to ∼ f −3 at frequencies f > f ci (Rezeau et al., 1999;Czaykowska et al., 2001).
The question of the anisotropy of wave vectors in the magnetosheath has been mostly addressed for dominant frequencies in the turbulent spectrum (spectral peaks), below f ci , where linearly unstable modes are expected (Sahraoui et al. 2004;Alexandrova et al., 2004;Schäfer et al., 2005;Narita et al., 2006;Narita and Glassmeier, 2006;Constantinescu et al., 2007).Instead, we are interested in permanent fluctuations in the magnetosheath (and not in spectral peaks) which cover a very broad range of frequencies (more than 5 decades), from frequencies well below f ci to frequencies much higher than f ci .
These permanent fluctuations within the frequency range [0.35, 12.5] Hz, above f ci , and for one decade of scale lengths around Cluster separations (∼ 100 km), have been studied by Sahraoui et al. (2006) using the k-filtering technique.For a relatively short time interval in the inner magnetosheath (close to the magnetopause) and for a proton beta β p ∼ 4, the authors show that the wave-vectors of the fluctuations are mostly perpendicular to the mean magnetic field B, k ⊥ k , and that their frequency ω 0 in the plasma frame is zero.In the plane perpendicular to B, the k-distribution is non-gyrotropic, more intense and with a well-defined power law k −8/3 in a direction along the flow velocity V which was perpendicular to both B and the normal to the magnetopause for this particular case.The presence of linearly unstable large scale mirror mode during the considered time interval makes the authors conclude that the small scale fluctuations with the observed dispersion properties k ⊥ k and ω 0 = 0 result from a non-linear cascade of mirror modes.
At higher frequencies, ∼ [10, 10 3 ] Hz, between about the lower hybrid frequency f lh and 10 times the electron cyclotron frequencie f ce , the permanent fluctuations observed in the magnetosheath, during four intervals of several hours, have been studied by Mangeney et al. (2006) and Lacombe et al. (2006).The corresponding spatial scales, ∼ [0.1, 10] km [0.3, 30]kc/ω pe (c/ω pe being the electron inertial length), are much smaller than the Cluster separations, and so only the one-spacecraft technique could be used to analyze the anisotropy of wave vector distributions.
Magnetic fluctuations with k k ⊥ , usually called slab turbulence, have rapid variations of the correlation function along the field and weak dependence upon the perpendicular coordinates.For the fluctuations with k ⊥ k , called 2D turbulence, the correlation function varies rapidly in the perpendicular plane, and there is no dependence along the field direction.So, measurements along different directions with respect to the mean field can give the information on the wave vector anisotropy.Under the assumption of convected turbulent fluctuations through the spacecraft (i.e., the phase velocity v φ of the fluctuations is small with respect to the flow velocity), these measurements are possible with one spacecraft thanks to the variation of the mean magnetic field B direction with respect to the bulk flow V.While V B, the spacecraft resolve fluctuations with k B, when V ⊥ B, the fluctuations with k ⊥ B are measured.
This idea was already used in the solar wind for studying the wave vector anisotropies of the Alfvénic fluctuations in the inertial range (Matthaeus et al., 1990;Bieber et al., 1996;Saur & Bieber, 1999).The authors suppose that the observed turbulence is a linear superposition of two uncorrelated components, slab and 2D, and both components have a power law energy distribution with the same spectral index s, δB 2 where A 1 and A 2 are the amplitudes of slab and 2D turbulent components, respectively.Bieber et al. (1996) propose two independent observational tests for distinguishing the slab component from the 2D component.
The first test is based on the anisotropy of the power spectral density (PSD) of the magnetic fluctuations in the plane perpendicular to B, i.e., on the non-gyrotropy of the PSD at a given frequency in the spacecraft frame: in the case of a slab turbulence with k B, all the wave vectors suffer the same Doppler shift depending on the angle between k and V, and if the spectral power is gyrotropic in the plasma frame, it will remain gyrotropic in the spacecraft frame; in the case of a 2D turbulence, with k ⊥ B, if the PSD is gyrotropic in the plasma frame, it will be non-gyrotropic in the spacecraft frame because the Dopler shift will be different if k is perpendicular to V and if k has a component along V.
The second test reveals the dependence of the PSD at a fixed frequency on the angle between the plasma flow and the mean field Θ BV (defined between 0 and 90 degrees): For a PSD decreasing with k (like a power-law, for example), in the case of the slab turbulence, the PSD for a given frequency will be more intense for Θ BV = 0 • , and therefore, the PSD decreases while Θ BV increases; for the 2D turbulence the PSD will be more intense for Θ BV = 90 • , and so it increases with Θ BV .Using these tests, Bieber et al. (1996) have shown that the inertial range of the slow solar wind is dominated by a 2D turbulence; however, a small percentage of a slab component is present.Mangeney et al. (2006) proposed a model of anisotropic wave vector distribution without any assumption on the independence of the two turbulence components.In their gyrotropic model, the wave vector can be oblique with respect to the and ⊥ directions.The authors introduce a cone aperture of the angle θ kB between k and B, as a free parameter of the model.They assume a power law distribution of the total energy of the fluctuations ∼ k −s , with s independent on θ kB .For a given k, the turbulent spectrum is modeled by one of the two typical angular distribution ∼ cos(θ kB ) µ for k nearly parallel to B and ∼ sin(θ kB ) µ for k nearly perpendicular to B. For these two distributions, the cone aperture of θ kB is about 20 • for µ = 10 and 7 • for µ = 100.The angle θ kB can be easily represented through Θ BV and so the model can be tested with one-spacecraft measurements.
An advantage of the magnetosheath with respect to the solar wind in ecliptical plane is that the angle Θ BV covers the range from 0 • to 90 • within rather short time periods (one hour, or so) while other plasma conditions remain roughly the same.A comparison of the model described above with the observations of the total PSD of the magnetic fluctuations within the magnetosheath flanks, at frequencies between f lh and 10f ce , shows that these fluctuations have a strongly anisotropic distribution of k, with θ kB = (90 ± 7) • (Mangeney et al., 2006).Actually, this model (as well as the tests of Bieber et al., 1996) is valid not only for a power law energy distribution in k, but for any monotone dependence where the energy decreases with increasing k.Mangeney et al. (2006) have also shown that the variations of δB 2 with Θ BV for a given frequency was not consistent with the presence of waves with a non-negligible phase velocity v φ .In other words, if the observed turbulent fluctuations are a superposition of waves, their v φ has to be much smaller than the flow velocity for any wave number k.This is consistent with the assumption that the wave frequency ω 0 is vanishing: the fluctuations are due to magnetic structures frozen in the plasma frame.These results have been obtained in the magnetosheath flanks for f > 10 Hz, at electron spatial scales ∼ [0.3, 30]kc/ω pe .
In this paper we extend the study of Mangeney et al. (2006) to frequencies below 10 Hz, for the same time periods in the magnetosheath flanks.As a result, we will cover the largest possible scale range, from electron (∼ 1 km) to MHD scales (∼ 10 5 km).At variance with the previous study, we analyse the spectral shapes and anisotropies for parallel (∼ compressive) δB and for transverse (∼ Alfvénic) δB ⊥ fluctuations independently.For Alfvénic fluctuations δB ⊥ we perform the first test of Bieber et al. (1996), i.e., we analyze the gyrotropy of the PSD of the magnetic fluctuations in the plane perpendicular to B, at a given frequency, as a function of Θ BV .

Data and methods of analysis
For our study we use high resolution (22 vectors per second) magnetic field waveforms measured by the FGM instrument (Balogh et al., 2001).Four seconds averages of the PSD of the magnetic fluctuations at 27 logarithmically spaced frequencies, between 8 Hz and 4 kHz, are measured by the STAFF Spectrum Analyser (SA) (Cornilleau-Wehrlin et al., 1997).Plasma parameters with a time resolution of 4 seconds are determined from HIA/CIS measurements (Rème et al., 2001).

Magnetic spectra and decomposition in δB ⊥ and δB
High resolution FGM measurements allow to resolve turbulent spectra up to ∼ 10 Hz.Similar to Alexandrova et al. (2006), we calculate the spectra of the magnetic fluctuations in the GSE directions X, Y and Z, using the Morlet wavelet transform.The total power spectral density (PSD) is δB 2 (f ) = j=X,Y,Z δB 2 j (f ).The PSD of the compressive fluctuations δB 2 (f ) is approximated by the PSD of the modulus of the magnetic field.This is a good approximation when δB 2 B 2 0 , where B 0 is the modulus of the magnetic field at the largest scale of the analysed data set.The PSD of the transverse fluctuations is therefore This approach, based on wavelet decomposition, allows the separation of δB ⊥ and δB with respect to a local mean field, i.e. to the field averaged on a neighbouring scale larger than the scale of the fluctuations.The lower frequency limit of this approach is a scale where the ordering δB 2 B 2 0 is no longer satisfied.
The STAFF-SA instrument measures the spectral matrix δB i (f )δB j (f ) at higher frequencies.Because of a recently detected error about the axes directions in the spin plane (O.Santolik, 2008, private communication) we cannot separate parallel and perpendicular spectra at the STAFF-SA frequencies; however we present here the total PSD, the trace of the spectral matrix.

Anisotropy of the k distribution
The motion of the plasma with respect to a probe allows a 1D analysis of the wave vector distribution along the direction of V, as was discussed in section 1.The 3D wave vector power spectrum I(k) ≡ I(k, θ kB , ϕ k ) depends on the wave number k, on the angle θ kB between k and B, and on the azimuth ϕ k of k in the plane perpendicular to B. If ω 0 is the frequency of a wave in the plasma rest frame (ω 0 and ω are assumed to be positive), the Doppler shifted frequency f = ω/2π in the spacecraft frame is given by The trace of the power spectral density at this frequency is i.e. the sum of the contributions with different k.A is a normalisation factor and δ the Dirac function.
The angle θ kB can be considered as depending on the angle θ kV between k and V, the angle appearing in the Doppler shift frequency, and on the angle Θ BV between B and V (see equation ( 2) of Mangeney et al., 2006).Thus, the variations of δB 2 with Θ BV for a given ω will give information about I(k).As was discussed in section 1, δB 2 increases with increasing Θ BV when the fluctuations have k ⊥ k (2D turbulence) and it decreases for a slab turbulence with k k ⊥ (see Figure 6 of Mangeney et al., 2006).As a consequence, in the case of 2D turbulence, the spectrum of the fluctuations will be higher for large angles Θ BV than for small ones, and vice-versa for the slab geometry.

δB-anisotropy in the BV-frame
To study the distribution of the PSD δB 2 ⊥ (f ) in the plane perpendicular to B, i.e. the gyrotropy of the magnetic fluctuations, taking into account the direction of the flow velocity V, we shall consider the following reference frame (b, bv, bbv): b is the direction of the B field, bv the direction of B × V and bbv the direction of B × (B × V).
The definition of this frame depends on the considered scale (frequency).A local reference frame (defined on a neighbouring scale larger than the scale of the fluctuations) can be defined only for frequencies below the spacecraft spin frequency f spin = 0.25 Hz which limits the plasma moments time resolution to 4 s.That is why, for any frequency f > f spin we shall use the frame (b, bv, bbv) redefined every 4 s.
In this frame, we only consider the frequencies below 10 Hz, i.e., the FGM data (the STAFF-SA data cannot be used because of the error that has to be corrected in the whole data set).We project the wavelet transform of B X , B Y and B Z on the b, bv, bbv directions and we calculate the squares of these projections δB 2 b (f, t), δB 2 bv (f, t) and δB 2 bbv (f, t) which are the diagonal terms of the spectral matrix in this new frame.
Figure 1 displays the average spectra of the FGM data for the transverse fluctuations (solid lines) and for the compressive fluctuations (dashed lines).The total PSD of the STAFF-SA data is the dotted line above 8 Hz.The total covered frequency range is more than six decades, from 3 • 10 −4 Hz to 300 Hz.The small vertical bars just above the abscissae-axis indicate the scales from λ = 10 4 km to 1 km corresponding to the Doppler shift f = V /λ for θ kV = 0 • and for the average velocity V = (246 ± 25) km/s.The diamonds above the abscissae indicate the scales kc/ω pi kr gi 0.01 to 100, corresponding to the frequency f = kV /2π.Precisely, kc/ω pi = 1 appears in the spectrum at f = (0.44 ± 0.05) Hz and kr gi = 1 appears at f = (0.63 ± 0.11) Hz. ⊥ is shown in the left panel, δB 2 is shown in the right panel.Middle and lower panels have the same format, but here the frequencies are respectively 0.52 Hz and 0.2 Hz.In all panels, the thick lines give the median value for bins 5 • wide.
We see in Figure 1 that, in the FGM frequency range, δB 2 ⊥ (f ) is everywhere larger than δB 2 (f ), except around f ∼ 5 • 10 −2 Hz where δB 2 ⊥ ∼ δB 2 , and where the compressive fluctuations display a spectral break.The spectrum of δB ⊥ displays a bump and a break around 0.2 Hz, that can be a signature of Alfvén vortices (Alexandrova et al., 2006).Below the bump, δB 2 ⊥ (f ) ∼ f −1.8 , a power law with an exponent close to the Kolmogorov's one −5/3 (in section 5 we will analyse spectral shapes in more details).Above the bump, for kc/ω pi > 0.2, δB 2 ⊥ (f ) and δB 2 (f ) follow a similar power law ∼ f −2.5 .This power law extends on the STAFF-SA frequency range up to kc/ω pi 50 (kc/ω pe 1.3).It is quite possible that, above these scales, the dissipation of the electromagnetic turbulence starts.However, around f 100 Hz, there is another spectral bump, which is due to whistler waves, identified by their right-handed polarisation.The question of the turbulence dissipation is out of scope of the present paper and will be analysed in details in the future.Now, we consider the anisotropy of the distribution of the wave vectors.Figure 2 shows the dependence of δB 2 ⊥ (left column) and δB 2 (right column) on the angle Θ BV at different frequencies.The thick solid curves give the median values for bins 5 • wide.The upper panels of Figure 2 correspond to f = 3 Hz (kc/ω pi 7).The observed increase of δB 2 ⊥ and δB 2 with Θ BV can be produced only by fluctuations with k ⊥ k , with phase velocities v φ negligible with respect to the plasma bulk velocity, and with decreasing intensity of the fluctuations with increasing k (as was discussed in sections 1 and 2.2).
At larger scales (lower frequencies) we observe the same tendency for f > 0.3 Hz.The middle panels of Figure 2 display δB 2 ⊥ and δB 2 as functions of Θ BV for f = 0.52 Hz (kc/ω pi 1.2): we still observe here a clear increase of δB 2 ⊥ and δB 2 with Θ BV .The lower panels of Figure 2 display δB 2 ⊥ and δB 2 at 0.2 Hz (kc/ω pi 0.5), just at the spectral bump of δB 2 ⊥ (see Figure 1).δB 2 still increases with Θ BV (in spite of a large dispersion of the data points around the median), while δB 2 ⊥ has a flat distribution with Θ BV .This can be due to several reasons: (i) I(k) is no longer a decreasing function with k, (ii) I(θ kB ) is more isotropic in the spectral bump and/or (iii) the fluctuations are not frozen in plasma at this scale.This spectral bump, as we have already mentioned, can be the signature of Alfvén vortices with k ⊥ k , propagating slowly in the plasma frame.It can be also the signature of propagating AIC waves with k k ⊥ , which are unstable for the observed plasma conditions (Mangeney et al., 2006;Samsonov et al., 2007).However, as explained in sections 1 and 2.2, the energy of fluctuations with k k ⊥ would decrease with increasing Θ BV at a given frequency, while in our case δB 2 ⊥ seems to be independent on Θ BV .
As we have just seen from Figure 2 (and as discussed in sections 1 and 2.2), the comparison of the energy level of the turbulent fluctuations for different Θ BV at a given frequency gives us a good estimate of the wave vector anisotropy.We compare now the PSD of the fluctuations observed for large Θ BV and for small Θ BV in the whole frequency range, to estimate the wave vector anisotropy in a large domain of wave vectors.
The upper panel of Figure 3 displays the spectra of the compressive fluctuations δB 2 la for the 10% of the points of the sample with the largest Θ BV (la = large angles, solid line), and δB 2 sa for the 10% of the points with the smallest Θ BV (sa = small angles, dashed line).At frequencies below 0.06 Hz (kc/ω pi = 0.1, indicated by a vertical solid line) the spectra δB 2 sa δB 2 la .At higher frequencies, f > 0.06 Hz (i.e., at smaller scales, kc/ω pi > 0.1) we observe δB 2 la > δB 2 sa .This indicates that 2D turbulence dominates at such small scales.Close to 10 Hz, i.e. at the vicinity of the FGM Nyquist frequency, we see that δB 2 la δB 2 sa 10 −4 nT 2 /Hz, that is the sensitivity limit of the FGM instrument.Therefore, the observations at f > 5 Hz are not physically reliable.
The lower panel of Figure 3 displays the spectra for the transverse fluctuations for large and small angles Θ BV , δB 2 ⊥la (solid line) and δB 2 ⊥sa (dashed line).We observe that δB 2 ⊥la becomes larger than δB 2 ⊥sa at about the same scale of kc/ω pi 0.1 as for compressive fluctuations.However, here within the spectral bump range, ∼ [0.1, 0.3] Hz, we observe δB 2 ⊥la δB 2 ⊥sa .This is consistent with our previous results that in this short frequency range the 2D turbulence model is not valid (cf. Figure 2).A clear dominance of δB 2 ⊥la over δB 2 ⊥sa is then observed for f > 0.3 Hz (kc/ω pi > 0.8, see a vertical solid line).
These observations allow to conclude that, for β p 1, there is a change in the nature of the wave-vector distribution of the magnetic fluctuations in the magnetosheath, in the vicinity of ion characteristic scale: if at MHD scales there is no clear evidence for a dominance of a slab or 2D geometry of the fluctuations, at ion scales (kc/ω pi > 0.1) the 2D turbulence dominates.This is valid for both the Alfvénic and compressive fluctuations.The large scale limit of the 2D turbulence is, however, different for Alfvénic and compressive fluctuations, and seems to depend on the presence of spectral features, as peaks or bumps.We analyse this point more in details by considering other cases.

Other case studies
The comparison between the spectra for large Θ BV and for small Θ BV has been made during four other one-hour intervals, with different average β p and different average shock angles θ BN .For the same intervals, Samsonov et al. (2007) display the observed proton temperature anisotropy and the corresponding thresholds for AIC and mirror instabilities.
Figure 4a gives the average PSD of transverse (solid line) and compressive fluctuations (dashed line).There is a spectral bump for the transverse fluctuations around 0.2 Hz.Be-low the spectral bump, δB 2 ⊥ (f ) ∼ f −5/3 .For the compressive fluctuations there is a spectral bump around 0.07 Hz, probably made of mirror modes.Below the bump, δB 2 (f ) is close to f −1 .
In the two other panels of Figure 4, we display the spectra for large and small Θ BV for compressive and for transverse fluctuations, respectively.In Figure 4b, at frequencies above the bump of δB 2 (f > 0.1 Hz, kc/ω pi ≥ 0.3) we observe δB 2 la > δB 2 sa .In Figure 4c, we observe δB 2 ⊥la δB 2 ⊥sa at large scales (observed at f < 0.05 Hz, i.e. kc/ω pi < 0.1), but at frequencies above the bump of δB 2 ⊥ (kc/ω pi > 1) we observe δB 2 ⊥la > δB 2 ⊥sa .So, the transverse and compressive fluctuations have k ⊥ k at scales smaller than their respective spectral bumps.This confirms the conclusions of section 3.1.
For an interval with β p 0.8 and θ BN 70 • (day 16/12/2001, 08:00-09:00 UT) the average spectrum δB 2 ⊥ (f ) displays a spectral bump around 0.5 Hz.The comparison between the spectra for large and small Θ BV (not shown) indicates that above the bump, for kc/ω pi > 1, the transverse fluctuations can be described by the 2D-turbulence model.For the compressive fluctuations, this model is valid for a larger range of scales, kc/ω pi > 0.3.This confirms the results obtained for β p 1.6, shown in Figure 4 as well as the conclusions of section 3.1.
In an interval with a larger value of β p (day 17/05/2002, 11:00-12:00 UT, β p 4.5, θ BN 73 • ), the analysis of the spectra for large and small Θ BV (not shown) shows that the 2D turbulence takes place for kc/ω pi > 0.3 for the transverse fluctuations, and for kc/ω pi > 0.2 for the compressive fluctuations.So, the 2D turbulence range of scales for the transverse fluctuations is wider in this case.
Figure 5b shows that at low frequencies (i.e., at large scales, kc/ω pi < 0.1) the spectrum for large angle δB la dominates slightly at every frequencies.At smaller scales, kc/ω pi > 0.1, this dominance is more clear.Figure 5c shows that δB ⊥la > δB ⊥sa as far as kc/ω pi > 0.01 (wavelengths smaller than 10 4 km), i.e., the δB ⊥ fluctuations can be described by the 2D-turbulence model at all the scales smaller than the Earth's radius.In this case, with large value of plasma beta, the 2D turbulence range of scales increases again, but the lower limit of 2D turbulence is not related to any spectral features, as was observed for smaller β p .
We may therefore conclude that, at frequencies above the spectral break in the vicinity of f ci , the δB ⊥ and δB fluctuations in the magnetosheath have k mostly perpendicular to B, and this is independent on β p and on θ BN .In terms of spatial scales, this is valid for kc/ω pi > 0.1 (or > 1 when spectral features appear in the vicinity of kc/ω pi = 1).For high val- ues of β p , the range of scales of the 2D turbulence seems to increase: for β p ∼ 10 the fluctuations have k ⊥ k for kc/ω pi > 0.01.This small scale spectral anisotropy is also independent on the presence of transverse and/or compressive spectral features (peaks) at larger scales.Nevertheless, for the moderate values of beta (β p < 3), these spectral peaks appear as the lower limit of 2D turbulence.

Gyrotropy of the magnetic fluctuations
In this section we analyse the anisotropy of the amplitudes of magnetic fluctuations in the plane perpendicular to B. For this purpose, we use the coordinate frame based on B and V, (b, bv, bbv), as explained in section 2.3.
For the same time interval as Figure 1, Figure 6 displays the ratio R = δB 2 bv /δB 2 bbv , amplitude of the fluctuations along B × V over the amplitude along B × (B × V), in the plane perpendicular to B, at four fixed frequencies, as a function of Θ BV .In Figure 6a (f = 10 Hz, kc/ω pi = 23) and Figure 6b (f 3 Hz, kc/ω pi = 7) the ratio R is larger than 1 for Θ BV 20 • .The median value decreases and reaches 1 or less for Θ BV < 20 • .A similar dependence was observed by Bieber et al. (1996) and Saur & Bieber (1999) at MHD scales in the solar wind, indicating the dominance of the 2D turbulence.
At larger scales (Figure 6c, f = 1 Hz, kc/ω pi = 2), in spite of the strong dispersion of R, the median values are slightly larger than 1 for any Θ BV : the 2D turbulence still dominates.At an even larger scale, the scale of the spectral break (Figure 6d, f = 0.3 Hz, kc/ω pi = 0.7), the ratio R is strongly dispersed.The variation of the median does not correspond to a slab or 2D turbulence.That is in agreement with the results obtained within the spectral break frequency range in section 3.1 (cf. Figure 2, lowest panel for δB ⊥ ).
The anisotropy of the magnetic fluctuations for the [10 −3 , 10] Hz frequency range is shown in Figure 7 with average spectra in the three directions b (dashed line), bv (solid line) and bbv (dotted line).The panels (a) to (f) correspond to increasing values of β p for the six considered intervals.For each interval, the vertical solid bar indicates the scale kc/ω pi = 1 and the dotted bar shows f ci .
In Figures 7a, b and d, for kc/ω pi ≥ 1 we observe that the spectra of the components along b and along bbv are nearly equal, δB 2 b δB 2 bbv .In Figures 7e and f, δB 2 b is larger than δB 2 bv : the fluctuations are more compressive for the largest values of β p .All the panels of Figure 7 show that δB 2 bv > δB 2 bbv for kc/ω pi ≥ 1.So, within the 2D turbulence range the PSD is not gyrotropic at a given frequency.
As we have mentioned in section 1, the observed nongyrotropy in the spacecraft frame can be due to the Doppler Average spectra of the magnetic fluctuations in the (b, bv, bbv)-frame, b is parallel to the B field (dashed line), bv is parallel to B × V (solid line), bbv is parallel to B × (B × V) (dotted line).For each of the 6 considered one-hour intervals a vertical dotted bar gives fci, a vertical solid bar gives the Doppler shifted wavenumber kc/ωpi = 1.In each panel the shapes of the power laws f −5/3 , f −1 are indicated; in the high frequency range we show the f −s spectral shape, with s determined in section 5, see Figure 8. shift.Indeed, we have shown in section 3 that the wave vectors k are mainly perpendicular to B, i.e k lies in plane spanned by bv and bbv.Assuming plane 2D turbulence, the relation k ⊥ • δB = 0 holds and thus, the wave vectors along bbv (i.e., along the direction of the flow in the plane perpendicular to B, we denote such wave vectors k bbv ) contribute to the PSD of δB bv and the wave vectors along bv (k bv ) contribute to the PSD of δB bbv .Even if I(k) is gyrotropic, the fluctuations δB bv with k bbv suffer a Doppler shift stronger than the fluctuations δB bbv with k bv .For 2D turbulence, this Doppler shift effect is more pronounced when Θ BV reaches 90 • .This implies that, for a gyrotropic energy distribution, I(k bv ) I(k bbv ), in the plane perpendicular to B, and if the energy decreases with k, for example as a power law I(k) ∼ k −s , the observed frequency spectrum δB 2 bv (f ) will be more intense than δB 2 bbv (f ).In other words, at the same frequency f in the spacecraft frame, we observe the fluctu-ations with |k bv | > |k bbv |.As the larger wave numbers correspond to a weaker intensity (for a monotone energy decrease with k), δB 2 bbv will be smaller than δB 2 bv .Therefore, the non-gyrotropy of δB 2 (f ), observed here, could be due to the non-gyrotropy of the Doppler shift, and could be compatible with a gyrotropic distribution of I(k).This is confirmed by the upper panels of Figure 6: as far as k is mainly perpendicular to B, the Doppler shift is small and gyrotropic for small Θ BV and we observe R ∼ 1, i.e. the PSD is gyrotropic; but for large Θ BV , R > 1.
On the other hand, the ratio R(f ) > 1, observed in Figures 6a and 6b for Θ BV 90 • , is also compatible with the non-gyrotropic k-distribution observed by Sahraoui et al. (2006) near the magnetopause, for Θ BV 90 • .In this case study, the authors show that the turbulent cascade develops along V, perpendicular to B and n, where n is the normal to the magnetopause.In this geometry, the direction V is close to bbv.Therefore, a non-gyrotropic I(k) distribution with δB 2 bv (k)/δB 2 bbv (k) > 1 is expected in the k-domain.This non-gyrotropy of wave vectors is then reinforced by the Doppler shift, and would give δB 2 bv (f )/δB 2 bbv (f ) > 1 in the f -domain.

Spectral shapes
We have mentionned in section 3 that, at frequencies f < f ci , below the spectral break, the spectra of the transverse fluctuations δB 2 ⊥ (f ) follow a power law close to f −5/3 (see Figure 1 and 4a).For the six intervals of Figure 7, Figure 8 displays compensated plots of the transverse spectra f 5/3 δB 2 ⊥ (f ) (solid lines).On the low frequency side of these plots, we see that the compensated spectra oscillate around a horizontal line, in a frequency range which varies slightly from day to day: a power law f −5/3 is thus a good approximation for the observations in this frequency range.In Figures 8a to 8e, for which β p is between 0.8 and 4.5, the Kolmogoroff f −5/3 power law is observed below 0.06 or 0.1 Hz, corresponding to scales kc/ω pi < 0.1 (see the vertical solid bar).
In Figure 8f, for β p (9±3), the f −5/3 power law is only observed below 0.01 Hz, i.e., below the 2D turbulence range of δB ⊥ (see Figure 5c).Above this frequency, as we see in Figure 7f, the spectra of all the components are close to an f −1 power law and the three spectra have nearly the same intensity, δB 2 b (f ) δB 2 bv (f ) δB 2 bbv (f ).This isotropy of the amplitudes of the turbulent fluctuations is natural to observe in a high beta plasma, where the mean magnetic field does not play any important role.Within this frequency range the spectrum can be also formed by a superposition of AIC and mirror waves.For such high β p , the mirror modes are more unstable than the AIC waves, but they have an important Alfvénic component (Génot et al., 2001), that can also contribute to make fluctuations more isotropic.
Above f ci , the spectra δB 2 (f ) and δB 2 ⊥ (f ) follow similar power laws, see Figure 7.The compensated spectra f s δB 2 ⊥ (f ) with s between 2 and 3 are presented in Fig- Compensated spectra f 5/3 δB 2 ⊥ (f ) (solid lines) and f s δB 2 ⊥ (f ) with s indicated in each panel (dashed lines) for the same time periods as Figure 7. ure 8 by dashed lines.We see that the composed spectra f 2.5 δB 2 ⊥ (f ) oscillate around a horizontal line in a few cases (Figures 7b, 7d and 7e), for different values of β p .In Figure 8a, the power law is steeper, s = 3.Actually, in this case the spectral bump is the most clearly pronounced of the six analyzed intervals.This bump is a signature of the Alfvén vortices, which have their own spectrum ∼ k −4 or k −6 , depending on the vortex topology (Alexandrova, 2008).The superposition of the background turbulence with the coherent structures, like magnetic vortices, can produce the observed steep spectrum.In Figure 8c we do not observe any clear evidence for a power law spectrum in this frequency range.

Summary and Discussion
In the present paper, we have analysed six one-hour intervals in the middle of the terrestrial magnetosheath (at more than one hour from the crossing of the bow shock or of the magnetopause).Precisely, we considered intervals in the magnetosheath flanks: the local times for the three considered days are respectively 8, 17 and 18 h (Lacombe et al., 2006).The proton beta varies from one interval to another, β p ∈ [0.8, 9], that allows us to study the plasma turbulence in a very large range of plasma conditions.

Spectral shape
The spectral shape of the magnetic fluctuations in the magnetosheath has been studied by several authors (see Alexandrova, 2008).Rezeau et al. (1999) find a power law f −3.4 above f ci in an interval close to the magnetopause.For intervals close to the bow shock, Czaykowska et al. (2001) find power laws around f −1 below f ci , and f −2.6 above f ci .But in these studies, intervals of 4 minutes have been analyzed, so the minimal resolved frequency is about 10 −2 Hz.In the present paper, the length of the intervals allows to reach frequencies smaller than 10 −3 Hz.
Here, we present, for the first time, the observations of a Kolmogorov-like inertial range for Alfvénic fluctuations δB 2 ⊥ (f ) ∼ f −5/3 in the frequency range f < f ci .It is clearly observed in five of the six studied intervals, those for which β p < 5 and when Alfvénic fluctuations were dominant.Such a Kolmogorov power law is observed in the Alfvénic inertial range of the solar wind turbulence, below the spectral break in the vicinity of f ci .The presence of such power law in the magnetosheath flanks is consistent with the estimations made by Alexandrova (2008): in the flanks, the transit time of the plasma is longer than in the subsolar regions, and it is much longer than the time of nonlinear interactions; therefore, the turbulence has enough time to become developed.
In the high frequency range, f > f ci , we generally observe δB 2 ⊥ (f ) and δB 2 (f ) following similar power law f −s with a spectral index s between 2 and 3, in agreement with previous studies.

Wave-vector anisotropy
We analysed here the anisotropy of wave-vector distribution of the magnetic fluctuations from 10 −3 to 10 Hz.This frequency range corresponds to the spatial scales going from ∼ 10 to 10 5 km (from electron to MHD scales).For this analysis we used a statistical method, based on the dependence of the observed magnetic energy at a given frequency on the Doppler shift for different wave vectors (Bieber et al., 1996;Horbury et al., 2005;Mangeney et al. , 2006).
Within the inertial range of the magnetosheath turbulence (f < f ci , kc/ω pi < 1, kr gi < 1), we do not observe a clear evidence of wave-vector anisotropy.It can be related to the fact that linearly unstable modes, such as AIC modes with k mainly parallel to B and mirror modes with k mainly perpendicular to B, together with Alfvén vortices with k ⊥ k co-exist in this frequency range.
However, above the spectral break in the vicinity of the ion characteristic scales (f > f ci , kc/ω pi > 1, kr gi > 1 and up to electron scales), we observe a clear evidence of 2D turbulence with k ⊥ k for both δB ⊥ and δB , and independently on β p , on the bow-shock geometry θ BN , and on the wave activity within the inertial range at larger scales.This wave vector anisotropy seems to be a general property of the small scale turbulence in the Earth's magnetosheath.
The range of wavenumbers of this 2D turbulence sometimes goes down to kc/ω pi 0.1 (or even to kc/ω pi 0.01), but usually it is limited by kc/ω pi 1, while at kc/ω pi < 1 spectral features (peaks or bumps) appear.As we can conclude from the work of Mangeney et al. (2006), the largest wavenumbers of the 2D turbulence are observed around kc/ω pi ∼ 100, where the dissipation of electromagnetic turbulence begins.This last conjecture must be verified by a deeper analysis.

Anisotropy of magnetic fluctuations
Analyzing the anisotropy of the amplitudes of turbulent fluctuations, we usually find that δB 2 ⊥ > δB 2 ; but for the largest plasma β p , the fluctuations are more isotropic δB 2 ⊥ ∼ δB 2 .This is valid for both the large scale inertial range and the small scale 2D turbulence.The dominance of δB 2 happens only locally in the turbulent spectrum, indicating the presence of an unstable mirror mode.
Concerning the gyrotropy of the amplitude of the magnetic fluctuations in the plane perpendicular to B, there is no universal behavior at large scales.At smaller scales, within the frequency range [0.3 − 10] Hz and for any β p , the 2D turbulence is observed to be non-gyrotropic: the energy δB 2 bv along the direction perpendicular to V and B is larger than the energy δB 2 bbv along the projection of V in the plane perpendicular to B. This non-gyrotropy might be a consequence of different Doppler shifts for fluctuations with k parallel and perpendicular to V in the plane perpendicular to B. The non-gyrotropy at a fixed f is compatible with gyrotropic fluctuations at a given k.On the other hand such a non-gyrotropy will be also observed if the k-distribution is not gyrotropic, but is aligned with the plasma flow, as was observed by Sahraoui et al (2006) in the vicinity of the magnetopause.

Fig. 1 .
Fig. 1.FGM and STAFF-SA/Cluster data on 19/12/2001, 02:00-04:00 UT.Average spectra of the magnetic fluctuations, calculated using the Morlet wavelet transform of the FGM data (f < 10 Hz).Solid line: for the transverse fluctuations δB ⊥ .Dashed line: for the compressive fluctuations δB .Dotted line: the total power spectral density, STAFF-SA data (f > 8 Hz).The diamonds give the scales kc/ωpi krgi 0.01 to 100.The vertical dotted line gives the average proton cyclotron frequency fci.The shapes of the power laws f −1.8 and f −2.5 are shown.

Fig. 2 .
Fig. 2. FGM/Cluster data on 19/12/2001, 02:00-04:00 UT.Upper panels: Scatter plots of the power spectral density of the magnetic fluctuations at f = 3 Hz as a function of the angle between the local mean field and velocity, ΘBV .The distributions of the energy of Alfvénic fluctuations δB 2⊥ is shown in the left panel, δB 2 is shown in the right panel.Middle and lower panels have the same format, but here the frequencies are respectively 0.52 Hz and 0.2 Hz.In all panels, the thick lines give the median value for bins 5 • wide.

Fig. 3 .
Fig. 3. FGM/Cluster data on 19/12/2001, 02:00-04:00 UT.Average power spectral density of the transverse magnetic fluctuations (upper panel) and of the compressive fluctuations (lower panel).In each panel, the solid line is the average spectrum for large ΘBV angles, and the dashed line for small ΘBV .The vertical dotted line gives the average fci, the diamonds indicate kc/ωpi = 0.01, 0.1 and 1.

Fig
Fig. 7.Average spectra of the magnetic fluctuations in the (b, bv, bbv)-frame, b is parallel to the B field (dashed line), bv is parallel to B × V (solid line), bbv is parallel to B × (B × V) (dotted line).For each of the 6 considered one-hour intervals a vertical dotted bar gives fci, a vertical solid bar gives the Doppler shifted wavenumber kc/ωpi = 1.In each panel the shapes of the power laws f −5/3 , f −1 are indicated; in the high frequency range we show the f −s spectral shape, with s determined in section 5, see Figure8.

Fig
Fig. 8.Compensated spectra f 5/3 δB 2 ⊥ (f ) (solid lines) and f s δB 2 ⊥ (f ) with s indicated in each panel (dashed lines) for the same time periods as Figure7.