Three Dimensional Anisotropic k Spectra of Turbulence at Subproton Scales in the Solar Wind

We show the first three dimensional (3D) dispersion relations and k spectra of magnetic turbulence in the solar wind at subproton scales. We used the Cluster data with short separations and applied the k-filtering technique to the frequency range where the transition to subproton scales occurs. We show that the cascade is carried by highly oblique kinetic Alfvén waves with ω_plas≤0.1ω_ci down to k_⊥ρ_i∼2. Each k spectrum in the direction perpendicular to B₀ shows two scaling ranges separated by a breakpoint (in the interval [0.4,1]k_⊥ρ_i): a Kolmogorov scaling k_⊥^-1.7 followed by a steeper scaling ∼k_⊥^-4.5. We conjecture that the turbulence undergoes a transition range, where part of the energy is dissipated into proton heating via Landau damping and the remaining energy cascades down to electron scales where electron Landau damping may predominate.