The vertical profiles of the wind speed and direction in atmospheric boundary layers are strongly controlled by turbulent friction. Some global weather forecast and climate models parameterize the turbulent momentum fluxes by means of a downgradient eddy diffusion approach, in which the same stability-dependent eddy viscosity profile is applied to both horizontal wind components. In the present study we diagnose eddy viscosity profiles from large-eddy simulations of a stable, a neutral and six convective boundary layers. Each simulation was forced by the same geostrophic wind of 15 ms$^{-1}$, but with a different surface heat flux. The stably stratified boundary layer sustains the largest friction and largest ageostrophic wind turning, due to its shallow depth, which leads to a steep slope (large vertical divergence) of the momentum fluxes. For convective cases we find that the eddy viscosity profiles for the two horizontal wind components are very different, in particular, we find negative eddy viscosities for the cross-isobaric wind component, indicating that its turbulent transport is counter the mean gradient. This implies that a purely downgradient diffusion approach for turbulent momentum fluxes is inadequate. To assess the consequence of applying an anisotropic diffusion approach, a modified solution of the Ekman spiral is presented. It is found that an anisotropic diffusion approach allows for a different vertical profile of the wind in terms of the height of maximum wind speed and the turning of the wind.