Cluster observations in the magnetosheath – Part 2: Intensity of the turbulence at electron scales

. The Cluster STAFF Spectral Analyser measures the magnetic and electric power spectral densities (PSD) δB 2 and δE 2 in the magnetosheath between 8 Hz and 4 kHz, i.e. between about the lower hybrid frequency and 10 times the proton plasma frequency. We study about 23 h of data on four different days. We do not consider the whistler waves and the electrostatic pulses (which are not always observed) but the underlying permanent ﬂuctuations. Paper 1 (Man-geney et al., 2006) shows why the permanent PSD at a given frequency f depends strongly on the angle 2 BV between the magnetic ﬁeld B and the ﬂow velocity V : this is observed for the electromagnetic (e.m.) ﬂuctuations, δB 2 and δE 2 em , below the electron cyclotron frequency f ce , and for the electrostatic (e.s.) ﬂuctuations δE 2 es at and above f ce . This dependence is due to the Doppler shift of ﬂuctuations which have ≃ 0.3 (50 km) to kc/ω pe ≃ 30 (500 m), i.e. at electron scales smaller than the Cluster separation.


Introduction
The STAFF Spectral Analyser on Cluster operates between 8 Hz and 4 kHz, i.e. between about the lower hybrid frequency and 10 times the proton plasma frequency f pi in the magnetosheath plasma. As discussed in the companion paper (Mangeney et al., 2006, Paper 1), two types of waves have to be distinguished in this frequency range: waves in the range 8 Hz ≤f ≤300 Hz, the "whistler" range, which are basically electromagnetic (e.m.), and waves in the range 300 Hz ≤f ≤4 kHz, the "ion acoustic" range, which are basically electrostatic (e.s.). In Paper 1, we have shown that, for both types of waves, most of the variations of the power spectral density (PSD) observed at a given frequency f can be explained by the variations of the Doppler shift (k.V ), where k is the wave vector. In the "whistler" range, we have neglected the coherent whistler waves (which Published by Copernicus GmbH on behalf of the European Geosciences Union. have a right-handed magnetic polarisation and are not always observed) and we have only considered the underlying permanent fluctuations. In this range, the magnetic and electric PSD increase strongly when the angle BV between the local magnetic field B and the plasma flow velocity V increases from 0 • to 90 • . This can be modelled with statistically permanent e.m. fluctuations which have a negligible frequency in the plasma frame and are Doppler shifted up to f ; the intensity of these fluctuations varies like a power law of the wave number I 1D ∝k −ν with ν≃3; the wave vectors of the fluctuations are mostly perpendicular to B, i.e. with an intensity I 3D ∝| sin θ kB | µ k −ν−2 for µ≃100, in the wave number range 0.3<kc/ω pe <30. θ kB is the angle between k and B. Meanwhile, in the "ion acoustic" range, the e.s. PSD around 1 kHz decreases when the angle BV increases up to 90 • : this can be modelled with statistically permanent e.s. fluctuations which have wave vectors mainly parallel to B, i.e. with a 3-D intensity I 3D ∝| cos θ kB | µ k −ν−2 , with µ≃100 and ν≃4, in the wave number range 0.1<kλ De < 1. These e.s. fluctuations may be ion acoustic modes Doppler shifted up to about 1 kHz. In this e.s. range (see Paper 1) we have not considered the electrostatic pulses: according to Pickett et al. (2005) the time duration and the amplitude of these pulses, observed in the time domain, do not depend on BV .
Note that the above exponent ν refers to the power law index of the 1-D spectrum I 1D (k) defined by δB 2 = dk I 1D (k) with I 1D (k) = 2πAk 2 π 0 sin θ kB dθ kB I 3D (k, θ kB ) .
Are there solar wind or magnetosheath parameters, other than BV i.e. the Doppler shift, which contribute to the control of the intensity of the fluctuations in the STAFF-SA frequency range? Indeed, at frequencies f <10 Hz, below the STAFF-SA range, several parameters have been found to have an effect upon the e.m. PSD in the magnetosheath: the solar wind Alfvén Mach number (Fairfield and Ness, 1970), the connection to quasi-parallel bow shocks (Luhmann et al., 1986;Czaykowska et al., 2001), and the magnetosheath proton β p Czaykowska et al., 2001). According to Rezeau et al. (1992) the level of the Alfvénic magnetic fluctuations close to the magnetopause is higher when the magnetopause moves earthward (compression) and lower when it moves outwards (expansion).
As for the e.m. waves above 10 Hz, Rodriguez (1985) shows that their occurrence and their intensity mainly depend on the position in the magnetosheath: while this "whistler" turbulence is observed more frequently in the inner region of the sunward magnetosheath (a region where compressional MHD fluctuations have been shown to dominate), the intensity of the 200 Hz whistler mode waves (broadband whistlers and narrowband lion roars) peaks near the magnetopause. For the e.s. waves around 1 kHz, Rodriguez (1979) finds that the wave intensity tends to maximize at the bow shock and to decrease towards the magnetopause and towards the dawn and dusk regions. In the "ion acoustic" range (0.2 to 4 kHz), the intensity of the waves observed in the dayside magnetosheath, just downstream of the bow shock ramp, does not appear to be correlated to the shock Alfvén Mach number, to the angle θ BN between the upstream B field and the bow shock normal, or to the electron ratio β e (Onsager et al., 1989). In the distant magnetosheath, on ISEE-3, Coroniti et al. (1994) observed that the e.s. waves, between about 200 Hz and 3 kHz ("ion acoustic" range), are less intense where the angle BV between the local magnetic field B and the plasma flow velocity V reaches 90 • . Between 8 Hz and 4 kHz, the wave intensity in the magnetosheath seems thus to be only controlled by the position in the magnetosheath and/or by the angle between B and V .
We have seen in Paper 1 that the permanent PSD above 10 Hz in the magnetosheath was mainly controlled by BV , i.e. by the Doppler shift. In the present paper, we show that the e.m. and the e.s. PSDs observed by STAFF-SA depend on the solar wind dynamic pressure P SW DY N , but only at a given frequency in the spacecraft frame. At a given scale k −1 , normalised to the electron inertial length c/ω pe , the intensity of the e.m. k spectra in the plasma frame does not depend on P SW DY N (nor on θ BN or β p , see Sect. 3). As for the e.s. spectra, their intensity does not depend on P DY N (nor on θ BN , β p or T e /T p , Sect. 4), if the frequency f is normalised to the plasma frequency f pi . The e.m. and e.s. turbulence intensities thus strongly depend on the Doppler shift through the angle BV . Numerical 3-D MHD simulations (Sect. 5) allow us to map BV in the magnetosheath, as a function of the angle between the solar wind magnetic field B sw and X GSE upstream of the bow shock. These maps of BV in the magnetosheath allow us to predict where the most intense "whistler" waves and the least intense "ion acoustic" waves will be found in the spacecraft frame.

Data
Every second, the STAFF-SA experiment (Cornilleau-Wehrlin et al., 1997) measures the diagonal terms of the 5×5 spectral matrix, computed with the three components of the magnetic fluctuations δB, and two components of the electric fluctuations δE (at 27 logarithmically spaced frequencies, between 8 Hz and 4 kHz). The variance of the fluctuations is nearly isotropic, so that δB 2 xx ≃δB 2 yy ≃δB 2 zz and δE 2 xx ≃δE 2 yy . We shall use 4-s averages of the total magnetic PSD δB 2 =δB 2 xx + δB 2 yy +δB 2 zz in nT 2 /Hz, and of the electric PSD δE 2 =δE 2 xx + δE 2 yy in (mV/m) 2 /Hz. The data of Cluster 1 (Rumba), in the magnetosheath, are analysed during four intervals. Figure 1 displays the position of Cluster in the GSE planes (X, Y) and (Y, Z) for each interval. Also shown are a paraboloidal bow shock model without aberration, X=14.6[1−(Y 2 +Z 2 )/25.6 2 ] (Filbert and Kellogg, 1979), and a magnetopause model (dashed line; Sibeck et al., 1991), Y 2 +Z 2 =139.2−0.18X 2 −14.2X. The considered intervals are generally far from the bow shock, except the longest interval, on 12 February 2001, which corresponds to a complete crossing of the magnetosheath. Cluster is in the dawn side on 17 May 2002. The plasma properties, proton density, temperature and velocity (Rème et al., 1997) and the electron temperature (Johnstone et al., 1997) are sampled with a time resolution of 4 s. We use 4-s averages of the magnetic field (Balogh et al., 1997).
The date and the duration of the four intervals, lasting from 4 h to 7 h, are given in the caption of Fig. 2, which displays relations between several plasma parameters. The different colours correspond to the four different days. Figure 2a shows the anticorrelations between β p =2µ 0 N p k B T p /B 2 and T p⊥ /T p −1, which are generally observed in the magnetosheath and which lead to the "bounded anisotropy model" , and references therein). T p and T p⊥ are the proton temperatures, respectively, parallel and perpendicular to the B field. P MS DY N =m p N p V 2 is the local dynamic pressure in the magnetosheath. Figure 2b shows that it is correlated to β p for a given day. Figure 2c shows that when the angle BV increases towards 90 • , P MS DY N generally increases for a given day. We shall see in Sect. 5 that this occurs in the magnetosheath flow when the solar wind (upstream) magnetic field and the X GSE axis are quasiperpendicular. Figure 2d displays an anticorrelation between T e /T p and P MS DY N , in the magnetosheath, which will be discussed in Sect. 6. 3 Intensity of the electromagnetic waves (10 to 100 Hz)

At a given frequency ≃10 Hz
At the frequency f =11 Hz (which is near the lower hybrid frequency), we look for a dependence of δB 2 on two solar wind parameters, the dynamic pressure P SW DY N and the bow shock angle θ BN . The solar wind data are those of ACE, near the Lagrange point L1. For each interval of magnetosheath data, we calculate the average time delay between ACE and Cluster which gives the best correlation between the magnetic field components on ACE and on Cluster. These delays vary between 57 and 74 mn, i.e. about 1 h for the four considered days. We thus know what are the upstream solar wind B field and P SW DY N at the time of the PSD measurements in the magnetosheath. The shock angle θ BN between the upstream magnetic field and the shock normal is estimated with the assumption that the bow shock has the quoted paraboloidal shape and is located just upstream of the Cluster 1 position. We see in Fig. 3a   proportional to (P SW DY N ) 1/2 /B. Figure 3b shows that the four intervals were mainly downstream of the quasi-perpendicular shocks; it seems that the waves are more intense for a few (red) points corresponding to θ BN smaller than 45 • (quasiparallel shocks), but this is not confirmed on 19 December 2001 (blue points). We note in Fig. 3c that δB 2 (11 Hz) is as well correlated to P MS DY N in the magnetosheath as it is correlated to P SW DY N (Fig. 3a). Indeed, there is a strong correlation between P SW DY N and P MS DY N . We recall that the strong variations of δB 2 on a given day are due to the variation of the angle BV in the magnetosheath (see Fig. 3d, and Paper 1). As the intensity of the waves observed below 10 Hz in the magnetosheath is generally correlated to the local β p (see the Introduction), we look for this correlation at 11 Hz. We see in Fig. 3e that there is a slight relation between δB 2 (11 Hz) and β p on a given day, mainly on 16 December 2001 (red points). Finally, Fig. 3f shows how δB 2 (11 Hz) decreases when the electron to proton temperature ratio, T e /T p , increases; this fact is related to the anticorrelation between P MS DY N and T e /T p seen in Fig. 2d.

For a given scale kc/ω pe =1
We have shown in Paper 1 that the observed frequencies f of the e.m. waves only result from the Doppler shift of a spatial turbulence frozen in the plasma rest frame, with k mainly perpendicular to B in the range kc/ω pe ≃0.3 to 30. Let us consider what happens to the scale k −1 =c/ω pe if k is parallel to V : its Doppler shift kV /2π will be f Dop =f pe V /c. As f pe and V vary on each day and from day to day, the frequency f Dop at which the considered scale is observed will vary like V N 1/2 p , which itself varies like the square root of the local P DY N . In Fig. 4, we display δB 2 (f ) interpolated at the time dependent frequency f pe V /c. Figures 4a, 4c and 4f show that δB 2 (f pe V /c) does not depend anymore on P SW DY N , on P MS DY N and on T e /T p . There is strictly no dependence of the PSD on the shock angle θ BN (Fig. 4b) and on β p (Fig. 4e), even on 16 December 2001 (red points). The only variation is with BV (Fig. 4d). This regular variation, also observed at a given frequency (Fig. 3d), is due to the fact that the intense fluctuations are those with a small k and with k mostly perpendicular to B (see Paper 1). Indeed, Ann. Geophys., 24, 3523-3531, 2006 www.ann-geophys.net/24/3523/2006/ a given Doppler shift f Dop =kV cos θ kV /2π will be reached by a small k (with a large intensity), if θ kV =0 • : this happens for BV =90 • ; and f Dop will be reached by a large k (small intensity) for BV =0 • .
Besides c/ω pe , we consider another electron scale, the electron gyroradius r ge : we then have f Dop =f ce V /v the for the scale k −1 =r ge , if k is parallel to V . The scatter plots of δB 2 (f ce V /v the ) as a function of P MS DY N or of BV (not shown) are more dispersed than the scatter plots of Figs. 4c and 4d. The scale of the turbulence in the e.m. range is thus c/ω pe , not r ge .
We conclude that the variations of the e.m. PSD δB 2 at a given frequency are mainly due to the variations of the local Doppler shift ∝V N 1/2 p for wavelengths normalised to the scale c/ω pe . This implies that the 3-D spectrum δB 2 (k)∝A em (kc/ω pe ) −ν−2 | sin θ kB | µ in the plasma frame is broadly invariant on a given day, and from day to day, i.e. with a value of A em constant when P SW DY N and P MS DY N varies.

Intensity of the electrostatic waves (≃1 kHz)
We have shown in Paper 1 that the magnetic fluctuations δB 2 for kc/ω pe ≃0.3 to 30 have wave vectors mainly perpendicular to B; the e.m. electric fluctuations δE 2 em , at the same frequencies, have similar properties. For smaller scales, kc/ω pe ≃15 to 150 (kλ De ≃0.1 to 1), the electrostatic fluctuations δE 2 es have k mainly parallel to B. To account for the variations of δE 2 with f and BV , we have been led to assume that, at kc/ω pe ≃30, δE 2 es ≃300 δE 2 em (Paper 1). We look for a dependence of δE 2 es on the magnetosheath dynamic pressure P MS DY N , itself correlated to the solar wind P SW DY N . We only consider the data when BV is large (70 • to 110 • ), because the dispersion of δE 2 es at a given f is weaker when BV is large (see Fig. 5e of Paper 1). Figures 5a and 5b display δE 2 at a given frequency 1414 Hz as a function of P MS DY N and of T e /T p . The broken black line gives the median values in bins of T e /T p with a width equal to 0.02. δE 2 tends to increase when P MS DY N increases from day to day. As P MS DY N is anticorrelated with T e /T p (Fig. 2d), δE 2 (1414 Hz) decreases when T e /T p increases (Fig. 5b). There is no correlation between δE 2 (1414 Hz) and the bow shock angle θ BN or the magnetosheath β p (not shown).
Then, we interpolate δE 2 (f ) at a time-dependent frequency, 1.5 times the proton plasma frequency f pi . In Figs. 5c and 5d we note that δE 2 (1.5f pi ) does not depend anymore on P MS DY N and T e /T p . It does not depend on β p nor on the shock angle θ BN (not shown). The spectrum of the e.s. fluctuations is thus invariant with respect to an energy input like P SW DY N , if the frequency is normalised to the local plasma frequency f pi .

Wave intensity and position in the magnetosheath
Paper 1, and the Sect. 3 of the present paper led us to conclude that the intensities of the e.m. and e.s. fluctuations at a given frequency in the magnetosheath strongly depend on the local Doppler shift. They will depend on the position in the magnetosheath (as observed by Rodriguez, 1979Rodriguez, , 1985, if the Doppler shift itself, i.e. the flow speed V and the angle BV , depend on the position. To map the regions of strong Doppler shift, 3-D MHD simulations of the magnetosheath plasma have been performed (see Samsonov and Hubert, 2004;. Figure 6 gives the flow and field parameters calculated in the middle of the magnetosheath (i.e. on a surface at equal distance from the shock and the magnetopause) and projected in the (Y B , Z B ) plane, which is the (Y GSE , Z GSE ) plane rotated around the X GSE axis, so that the direction of Y B is parallel to the projection of the upstream solar wind magnetic field B sw in the (Y GSE , Z GSE ) plane. This projection is the red arrow in Fig. 6c, with B Xsw ≤0, B Y sw >0 and B Zsw =0.
The mapped values, density, speed and magnetic field intensity, are normalised to the solar wind values. Figures 6a-d are drawn for a cone angle of the solar wind magnetic field (B sw , X GSE )=45 • . The density distribution in the magnetosheath is nearly axisymmetric (Fig. 6a). The velocity  Figure 6e gives the angle BV when the cone angle of the solar wind B field is (B sw , X GSE )=20 • : the largest values BV >75 • are now at low latitudes, for Y B <0, downstream of a quasi-parallel bow shock, and the smallest BV are also at low latitudes, downstream of a quasiperpendicular bow shock (Y B >0). If the angle (B sw , X GSE ) is 90 • (Fig. 6f), BV is larger at high latitudes in the plane (Y B , Z B ). A comparison of Fig. 6b with Fig. 6f shows that V tends to be larger when BV is large. This is typical of a magnetosheath observed when (B sw , X GSE ) is larger than 45 • , and implies that the dynamic pressure will increase when BV increases (see Fig. 2c). These simulations show that the angle BV will be larger for large |Z B | values if the cone angle of the solar wind magnetic field is between 45 • and 90 • . On 17 May 2002, the observed cone angle (B sw , X GSE ) remains between 45 • and 90 • , while Cluster remains nearly in the same GSE position. Yet, as the magnetic field rotates, the direction Y B rotates, and the position of Cluster follows large arcs of circle in the (Y B , Z B ) plane of Fig. 7. Figure 7a gives the position of Cluster in the (Y B , Z B ) frame when the e.m. PSD δB 2 at 11 Hz is large, precisely for 10% of the most intense values of δB 2 over 4 h. Similarly, Fig. 7b gives the position of Cluster when the e.s. PSD δE 2 at 1414 Hz is large. We see that the most intense e.m. fluctuations are found at large Z B , while the most intense e.s. fluctuations are found at small Z B .
We conclude that the Doppler shift (the angle BV and the flow speed) depends on the position in the magnetosheath for a given direction of the solar wind magnetic field B sw ; and it depends strongly on the direction of B sw .

Discussion
We have shown in Sect. 3 that the correlation between the magnetosheath dynamic pressure, P MS DY N and δB 2 , at a given frequency (11 Hz) is only due to the Doppler shift, itself proportional to P MS DY N . This correlation between δB 2 (11 Hz) and P MS DY N implies some correlations between δB 2 (11 Hz) and β p , because β p is correlated with P MS DY N , at least for a given day ( Fig. 2c; see also Farrugia et al., 2000). Some observations display a correlation between δB 2 below 10 Hz (ULF waves) and β p . This correlation in the spacecraft frame is questionable because it could be partly due to the Doppler shift, not to a role played by β p .
We have seen in Sect. 3 that there is no significant relation between the e.m. wave intensity and the bow shock angle θ BN , neither at a given frequency (Fig. 3b) nor for a given scale (Fig. 4b). In fact, our magnetosheath intervals generally correspond to quasi-perpendicular shocks, so that we cannot check whether the fluctuations are more intense downstream of quasi-parallel shocks. However, the numerical simulations of Figs. 6d, 6e and 6f show that the value of the angle BV at a given place in the (Y B , Z B ) plane depends strongly on the  cone angle (B sw , X GSE ), and thus on the bow shock angle θ BN . Thus, a possible dependence of δB 2 on θ BN could be due to the dependence on the angle BV which is simply on the Doppler shift for anisotropic k spectra.
We mentioned in Sect. 2 that there is an anticorrelation between T e /T p and P MS DY N in the magnetosheath for a given day and from day to day (Fig. 2d): for the four days (21 000 points) the correlation coefficient between P MS DY N and T e /T p is -0.82 (the correlation is 0.77 between P MS DY N and T p , and -0.66 between P MS DY N and T e ). In the solar wind, it is well known that T p is larger than T e in the high-speed wind, and T e larger than T p in the low-speed wind. There is a global anticorrelation between T e /T p and V sw (see Mangeney et al., 1999). This anticorrelation upstream of the bow shock will probably also be observed downstream of the bow shock. Indeed, a subcritical shock tends to heat the electrons, while a supercritical shock, with a higher Mach number (i.e. a higher flow speed), tends to heat the protons more than the electrons. The shock will strengthen the anticorrelation between T e /T p and V , and strengthen the anticorrelation between T e /T p and P MS DY N in the magnetosheath. We thus suggest that this latter anticorrelation comes from a solar wind property amplified by the shock heating.
In this paper, we have not considered the fluctuations like the e.m. whistler or the e.s. solitary waves which are intermittently observed in the magnetosheath, but only the underlying permanent fluctuations, between about the nominal lower hybrid frequency and 10 times the nominal proton plasma frequency.
In the magnetosheath, the permanent electromagnetic power spectral density, observed at a given frequency in the spacecraft frame, increases when the solar wind and the magnetosheath dynamic pressure P SW DY N and P MS DY N increase. This correlation is a consequence of the Doppler shift, and disappears if we consider the fluctuations at a given scale (∝ k −1 ) , not at a given frequency. Indeed, we find that the scale of the electromagnetic fluctuations is related to the electron inertial length c/ω pe : the e.m. PSD has a k spectral intensity I 1D ∝(kc/ω pe ) −ν with ν≃3 to 4, for scales ranging from kc/ω pe ≃0.3 to 30. The directions of the wave vectors k are mostly perpendicular to the B field (see Paper 1), so that the e.m. 3-D k spectrum (assumed axisymmetric with respect to B) varies like A em (kc/ω pe ) −ν−2 | sin θ kB | µ with µ≃100. We find that A em is invariant, it does not depend on P SW DY N or P MS DY N , nor on the magnetosheath β p . A em does not depend anymore on the upstream bow shock angle θ BN (but our observations correspond to a limited range of θ BN , downstream of oblique or quasi-perpendicular bow shocks).
The electrostatic permanent fluctuations are probably Doppler shifted ion acoustic waves for scales ranging from kλ De ≃0.1 to 1. The directions of k are mostly parallel to the B field (see Paper 1).
A consequence of the strong anisotropies of the e.m. and e.s. k spectra is that the e.m. fluctuations are more intense and the e.s. fluctuations less intense, at a given frequency in the spacecraft frame, when the local angle BV between the flow velocity and the B field is close to 90 • . 3-D MHD simulations of the flow in the magnetosheath allowed us to map the regions where BV reaches 90 • . These maps depend strongly on the cone angle (B sw , X GSE ) of the solar wind magnetic field. For (B sw , X GSE ) larger than 45 • , the angle BV reaches 90 • in regions with a high latitude with respect to the (X GSE , B) plane; B is the direction of the magnetosheath magnetic field, which remains in a plane nearly parallel to the solar wind (X GSE , B sw ) plane. Then, the e.m. PSD will be more intense at high latitudes, and the e.s. PSD more intense at low latitudes with respect to the (X GSE , B) plane. For a cone angle of the solar wind magnetic field smaller than 45 • , the angle BV reaches 90 • downstream of the quasi-parallel bow shock, at low latitudes with respect to the (X GSE , B) plane: the e.m. PSD will be more intense there, and the e.s. PSD less intense.
We conclude that the intensity (in the spacecraft frame) of the e.m. and e.s. fluctuations in the magnetosheath strongly depend on the Doppler effect. This is observed on scales ranging from kc/ω pe ≃0.3 (50 km) to kλ De ≃1 (30 m), i.e. at electron scales, smaller than the Cluster separation. The Doppler shift is related to the dynamic pressure P DY N and to the angle BV ; the angle BV is itself related to the shock angle θ BN and to the position in the magnetosheath. Thus, all the parameters (P DY N , θ BN , position in the magnetosheath) which have been shown to influence the wave intensity, from the ULF range to the "whistler" and "ion acoustic" ranges, may partly be manifestations of the Doppler effect.