Evidence of a Cascade and Dissipation of Solar-Wind Turbulence at the Electron Gyroscale

We report the first direct determination of the dissipation range of magnetofluid turbulence in the solar wind at the electron scales. Combining high resolution magnetic and electric field data of the Cluster spacecraft, we computed the spectrum of turbulence and found two distinct breakpoints in the magnetic spectrum at 0.4 and 35 Hz, which correspond, respectively, to the Doppler-shifted proton and electron gyroscales, $f_{{\rho }_p}$ and $f_{{\rho }_e}$. Below $f_{{\rho }_p}$, the spectrum follows a Kolmogorov scaling $f^{-1.62}$, typical of spectra observed at 1 AU. Above $f_{{\rho }_p}$, a second inertial range is formed with a scaling $f^{-2.3}$ down to $f_{{\rho }_e}$. Above $f_{{\rho }_e}$, the spectrum has a steeper power law $\sim f^{-4.1}$ down to the noise level of the instrument. We interpret this as the dissipation range and show a remarkable agreement with theoretical predictions of a quasi-two-dimensional cascade into Kinetic Alfvén Waves (KAW).

Figure 1

FGM magnetic field data measured by Cluster 2 in the solar wind plotted in the Geocentric Solar Ecliptic (GSE) reference frame. The vertical dotted lines delimit two subintervals of time discussed in the text.

Figure 2

The parallel (black) and perpendicular (red) magnetic spectra of FGM data ($f<33\mathrm{Hz}$) and STAFF-SC data (respectively, light line; green online and dark line; blue online); $1.5<f<225\mathrm{Hz}$). The STAFF-SC noise level as measured in the laboratory and in-flight are plotted as dashed and dotted lines, respectively. The straight black lines are power law fits to the spectra. The arrows indicate characteristic frequencies defined in the text.

Figure 3

High-pass filtered power spectra of the parallel (green line) and perpendicular (blue line) of the magnetic fluctuations measured by STAFF-SC during the time subintervals $\Delta t_1$ (a) and $\Delta t_2$ (b) shown on Fig. 1. Dotted line is the STAFF-SC noise level. The straight black lines are direct power law fits of the spectra. Vertical arrows are defined in the text.

Figure 4

Spectra of data from spacecraft 4 in the Despun System of reference (DS): $E_y$ measured by EFW (bold black curve) and $B_z$ measured by FGM and STAFF-SC merged at 1.5 Hz (light line; green online). The straight black lines are direct power law fits of the spectra. Vertical arrows are the Doppler-shifted proton and electron gyroradius and inertial lengths.

Figure 5

Linear solutions of the Maxwell-Vlasov equations for $\theta =\arccos (\mathbf{k},{\mathbf{B}}_0)=89.99°$, obtained using the plasma parameters given in the text. The real part of the frequency (black line) and the damping rate (light line; red online) are consistent with the kinetic Alfvén wave dispersion relation and damping [6, 13]. The dashed line is the asymptote $\omega /{\omega }_{\mathrm{cp}}=k_{\parallel }{V}_A/{\omega }_{\mathrm{cp}}$. The plots stop at scales where $|\gamma |\sim {\omega }_r$.

Figure 6

The ratio $E_y/B_z$ measured by Cluster 4 (black line) compared to a similar ratio calculated from the linear Maxwell-Vlasov theory (light line; red online). The electric field data have been transformed into the plasma reference frame using the Lorentz transform ${\mathbf{E}}_{\mathrm{plas}}={\mathbf{E}}_{\mathrm{sat}}-\mathbf{V}\times \mathbf{B}$ [7]. The horizontal dashed line is the Alfvén speed. The straight black line is a direct fit of $E_y/B_z$ from $k{\rho }_p=1.5$ to $k{\rho }_p=20$. We used $\theta =\arccos (\mathbf{k},\mathbf{v})=50°$ to transform frequency to $k{\rho }_p$.