Ideal MHD turbulence: the inertial range spectrum with collisionless dissipation

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 approximately 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).