Atmospheric flows display phenomena on a wide range of spatial scales that interact with each other. Strongly localized features, such as complex orography, can be modelled correctly only if a very high spatial resolution is employed, while larger scale features can be adequately resolved on much coarser meshes. The insufficient resolution of orographic features is compensated in numerical weather predictions (NWP) and climate models by subgrid-scale orographic drag parameterizations. However, the interplay between resolved and parameterized orographic effects is critical and global simulations without drag parameterization have shown that the increase in forecast skill was mainly due to the improved representation of the orography. NWP is therefore an apparently ideal framework for adaptive numerical approaches. However, mesh adaptation strategies have only slowly found their way into the NWP literature, due to limitations of earlier numerical methods and concerns about the accuracy of the representation of atmospheric wave phenomena. We present a quantitative assessment of the mesh refinement capabilities of a recently proposed IMEX-DG method to a number of benchmarks for atmospheric flows. The method is based on a high-order h-adaptive Discontinuous Galerkin (DG) spatial discretization and on a IMplicit-EXplicit Runge Kutta (IMEX-RK) time discretization scheme. The scheme has been implemented in the framework of the deal.II library and the local refining procedure has no significant impact on the parallel performance. We show that simulations with adaptive meshes can increase the accuracy of the local flow description without affecting the larger scales on both idealised benchmarks and test cases over real orographic profiles, thereby reducing the number of degrees of freedom. We also discuss the impact of the use of curved elements for flows over orography, which allow the description of small-scale features and strongly affect the development of lee waves .