Although spectral methods have been in use for decades, there is still room for innovation, refinement and improvement of the methods in terms of efficiency and accuracy, for generalized homogeneous turbulent flows, and especially for specialized applications like the computation of atmospheric flows and numerical weather prediction. In this thesis, two such innovations are presented. First, inspired by the adaptive mesh refinement (AMR) technique, which was developed for the computation of fluid flows in physical space, an algorithm is presented for accelerating direct numerical simulation (DNS) of isotropic homogeneous turbulence in spectral space. In the adaptive spectral resolution (ASR) technique developed here the spectral resolution in spectral space is dynamically refined based on refinement criteria suited to the special features of isotropic homogeneous turbulence in two, and three dimensions. Applying ASR to computations of two- and three-dimensional turbulence allows significant savings in the computational time with little to no compromise in the accuracy of the solutions. In the second part of this thesis the effect of explicit filtering on large eddy simulation (LES) of atmospheric flows in spectral space is studied. Apply an explicit filter in addition to the implicit filter due to the computational grid and discretization schemes in LES of turbulent flows allows for better control of the numerical error and improvement in the accuracy of the results. Explicit filtering has been extensively applied in LES of turbulent flows in physical space while few studies have been done on explicitly filtered LES of turbulent flows in spectral space because of perceived limitations of the approach, which are shown here to be incorrect. Here, explicit filtering in LES of the turbulent barotropic vorticity equation (BVE) as a first model of the Earth's atmosphere in spectral space is studied. It is shown that explicit filtering increases the accuracy of the results over implicit filtering, particularly where the location of coherent structures is concerned.