Many geophysical applications feature a large span of spatial scales, which interact and contribute to the general behavior of the system. Examples are small scale mixing and entrainment processes at cloud boundaries or in atmospheric boundary layers as well as wave interaction in barotropic (e.g. tsunami) waves. In order to support these physical phenomena, dynamically adaptive numerical methods have proved their success. However, there are still many challenges to master before these methods can be used in operational simulation environments.
In this presentation we will focus on the efficiency of adaptive mesh refinement methods. Triangular (or tetrahedral) meshes offer advantages when complex domains are to be represented, as in ocean modeling. On the other hand, triangular meshes are hard to handle efficiently on modern hierarchical computer architectures. In order to support numerical methods on triangular grids, hierarchical refinement tree approaches lead to space filling curve linearization of the mesh structure. This yield highly efficient schemes, which can be combined with Galerkin-type numerical methods for discretizing the underlying partial differential equations. Beyond direct efficiency of numerical schemes, we will also shed light on criteria for refinement, ratios of coarse and fine areas, overhead of mesh control, etc.