Unstructured triangular-hexagonal computational meshes provide enormous versatility for modeling geophysical flows. They are easily configured with resolution that is spatially variable, in principle by orders of magnitude, and the transition can be very gradual or moderately abrupt. Atmospheric numerical models using these grids can, for example, resolve convective scale processes over a selected geographic area and cover the remainder of the globe with a tractable number of coarser mesh cells. This places the entire range of resolution within the framework of a single model and single seamless grid, and provides an excellent test bed for developing process representations that smoothly transition with grid resolution. We will discuss a specific implementation of this approach, the Ocean-Land-Atmosphere Model (OLAM), whose atmospheric dynamic core uses a finite volume, compressible, nonhydrostatic formulation of the governing equations for applicability to the smallest resolved scales. We will focus on the following topics: (1) coupling between the atmospheric and surface (land and water) grids of OLAM, both of which are unstructured and may be configured independently with different resolution and refined mesh structure, (2) use of cut cells to represent topography with horizontal grid surfaces, and (3) exploring methods to smoothly transition between resolved and unresolved convection.