Three-dimensional turbulence is characterized by its capacity to transfer energy from large to small scales where it is finally dissipated. When it occurs in a non-collisional plasma like the solar wind, a kinetic modelisation is necessary a priori. The complexity of such an approach however limits the theoretical developments and forces numerical experiments to be restricted to low Reynolds numbers. To what extent does a single-fluid model such as MHD Hall account for the phenomena observed in the solar wind at ion sub-scales ? It is to this question that this thesis tries to answer. The main idea of this work is to take advantage of the relative simplicity of fluid models and of the high precision of pseudo spectral methods to tackle the problem of turbulence in solar wind by direct numerical simulations massively parallelized at high Reynolds numbers. These simulations have helped to highlight the existence of a spontaneous breaking of chiral symmetry in incompressible Hall MHD turbulence, as well as the existence of a new regime called ion MHD (HDMI). A phenomenological model has been proposed to account for these results and new predictions were made and confirmed numerically. The study of the effect of a strong uniform magnetic field on the turbulent dynamics confirmed an ancient conjecture for the first time. The inertia of the electrons was then taken into account in a still fluid model. By a classical hydrodynamic approach, a universal law has been obtained for the third order structure functions. All these results are in qualitative agreement with in situ measurements of the solar wind and challenge the paradigm according to which the successive steepening of the magnetic fluctuations spectrum is necessarily caused by phenomenon of kinetic origin. More generally, this thesis raises fundamental questions about the non-collisional dissipation process in turbulent plasmas.
