High-resolution hybrid simulations of kinetic plasma turbulence at proton scales

We investigate properties of plasma turbulence from magnetohydrodynamic (MHD) to sub-ion scales by means of two-dimensional, high-resolution hybrid particle-in-cell simulations. We impose an initial ambient magnetic field perpendicular to the simulation box, and we add a spectrum of large-scale magnetic and kinetic fluctuations with energy equipartition and vanishing correlation. Once the turbulence is fully developed, we observe an MHD inertial range, where the spectra of the perpendicular magnetic field and the perpendicular proton bulk velocity fluctuations exhibit power-law scaling with spectral indices of $-5/3$ and $-3/2,$ respectively. This behavior is extended over a full decade in wavevectors and is very stable in time. A transition is observed around proton scales. At sub-ion scales, both spectra steepen, with the former still following a power law with a spectral index of $\sim -3.$ A $-2.8$ slope is observed in the density and parallel magnetic fluctuations, highlighting the presence of compressive effects at kinetic scales. The spectrum of the perpendicular electric fluctuations follows that of the proton bulk velocity at MHD scales, and flattens at small scales. All these features, which we carefully tested against variations of many parameters, are in good agreement with solar wind observations. The turbulent cascade leads to on overall proton energization with similar heating rates in the parallel and perpendicular directions. While the parallel proton heating is found to be independent on the resistivity, the number of particles per cell, and the resolution employed, the perpendicular proton temperature strongly depends on these parameters.