Turbulent boundary layers are present in many aspects of life, from the weather and wind currents to transportation, production of energy or mixing processes. Understanding turbulent motions can allow for an improvement and development of technical devices, techniques or diagnosis of phenomena where a fluid flow is in the turbulent regime. From the economical and environmental perspectives, knowledge of turbulent boundary layers may help to reduce the drag on aerodynamic surfaces in transportation, thus leading to a reduction in fuel consumption and emissions. It is also possible to enhance the production of energy from wind sources or the harvest of tidal energy. Otherturbulent motions that have recently impacted society are those related to diffusion such as the transport of aerial diseases, or the motions of air in the respiratory system. In all those examples, external pressure gradients or those produced by the curvature of wall surfaces affect the turbulent structures and thus, the outputs that we study such as drag, transport of substances, energetic output, etc. A relevant case is that of adverse pressure gradient, which enhances the wall-normal convection and redistributes the turbulent energy across the turbulent boundary later. In this work, we study a canonical case of an adverse-pressure-gradient turbulent boundary layer, which is the flow over a flat plate, under near-equilibrium adverse-pressure-gradient conditions. We have extended the previous datasets on flat-plate boundary layers under adverse pressure gradients which were obtained at low Reynolds numbers, with a new numerical simulation reaching high Reynolds numbers, comparable to those of experimental campaigns. This new data set allowed us to study both adverse-pressure-gradient and Reynolds-number effects, where the thicker boundary layer exhibits a clear separation of turbulent scales. The influence of the size of the domain and the wider turbulent scales are analyzed through a set of turbulent channel-flow simulations and the spectral analysis of the Reynolds stresses in high Reynolds numbers turbulent boundary layers. The impact of the wider scales was analyzed and scaling factors were found for different regions of the spectra of the Reynolds stresses. In particular, we propose a new scaling for the energy of the small scales that have been advected to the outer region of the boundary layer.