Large-eddy simulation (LES) is used to model turbulent winds in a nominally neutral atmospheric boundary layer at varying mesh resolutions. The boundary layer is driven by wind shear with zero surface heat flux and is capped by a stable inversion. Because of entrainment the boundary layer is in a weakly stably stratified regime. The simulations use meshes varying from 1282×64 to 10242×512 gridpoints in a fixed computational domain of size (2560;2560;896) m. The subgrid-scale (SGS) parameterizations used in the LES vary with the mesh spacing. Low-order statistics, spectra, and structure functions are compared on the different meshes and are used to assess grid convergence in the simulations. As expected, grid convergence is primarily achieved in the middle of the boundary layer where there is scale separation between the energy containing and dissipative eddies. Near the surface second-order statistics do not converge on the meshes studied. The analysis also highlights differences between one-dimensional and two-dimensional velocity spectra; differences are attributed to sampling errors associated with aligning the horizontal coordinates with the vertically veering mean wind direction. Higher-order structure functions reveal non-Gaussian statistics on all scales, but are highly dependent on the mesh resolution. A generalized logarithmic law and a k−1 spectral scaling regime are identified with mesh dependent parameters in agreement with previously published results.