Extreme atmospheric events such as tropical cyclones are inherently complex multi-scale phenomena. Such extremes are a challenge to simulate in conventional atmosphere models which typically use rather coarse uniform-grid resolutions. Adaptive Mesh Refinement (AMR) techniques seek to mitigate these challenges. They dynamically place high-resolution grid patches over user-defined features of interest, thus providing sufficient local resolution over e.g. a developing cyclone while limiting the total computational burden. Studying such techniques in idealized simulations enables the assessment of the AMR approach in a controlled environment and can assist in identifying the effective refinement choices for more complex, realistic simulations. In this paper, we present a non-hydrostatic, finite-volume dynamical core, which implements refinement in both space and time on a cubed-sphere grid. Several test cases are discussed including an idealized tropical cyclone simulation. These simulations test the effectiveness of both static and dynamic grid refinements as well as the sensitivity of the model results to various adaptation criteria. Finally, we asses AMR as a possible technique to study scale discrepancies in atmospheric models.