Cluster observations in the magnetosheath – Part 1: Anisotropies of the wave vector distribution of the turbulence at electron scales
We analyse the power spectral density δ B2 and δ E2 of the magnetic and electric fluctuations measured by Cluster 1 (Rumba) in the magnetosheath during 23 h, on four different days. The frequency range of the STAFF Spectral Analyser ( f=8 Hz to 4 kHz) extends from about the lower hybrid frequency, i.e. the electromagnetic (e.m.) range, up to about 10 times the proton plasma frequency, i.e. the electrostatic (e.s.) range. In the e.m. range, we do not consider the whistler waves, which are not always observed, but rather the underlying, more permanent fluctuations. In this e.m. range, δ B2 (at 10 Hz) increases strongly while the local angle Θ BV between the magnetic field B and the flow velocity V increases from 0° to 90°. This behaviour, also observed in the solar wind at lower frequencies, is due to the Doppler effect. It can be modelled if we assume that, for the scales ranging from kc/ωpe≃0.3 to 30 ( c/ωpe is the electron inertial length), the intensity of the e.m. fluctuations for a wave number k (i) varies like k−ν with ν>≃3, (ii) peaks for wave vectors k perpendicular to B like |sinθ kB| µ with µ>≃100. The shape of the observed variations of δ B2 with f and with Θ BV implies that the permanent fluctuations, at these scales, statistically do not obey the dispersion relation for fast/whistler waves or for kinetic Alfvén waves: the fluctuations have a vanishing frequency in the plasma frame, i.e. their phase velocity is negligible with respect to V (Taylor hypothesis). The electrostatic waves around 1 kHz behave differently: δ E2 is minimum for Θ BV>≃90°. This can be modelled, still with the Doppler effect, if we assume that, for the scales ranging from k λDe>≃0.1 to 1 (λ De is the Debye length), the intensity of the e.s. fluctuations (i) varies like k−ν with ν>≃4, (ii) peaks for k parallel to B like |cosθ kB| µ with µ>≃100. These e.s. fluctuations may have a vanishing frequency in the plasma frame, or may be ion acoustic waves. Our observations imply that the e.m. frequencies observed in the magnetosheath result from the Doppler shift of a spatial turbulence frozen in the plasma, and that the intensity of the turbulent k spectrum is strongly anisotropic, for both e.m. and e.s. fluctuations. We conclude that the turbulence has strongly anisotropic k distributions, 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.