Exploring a multi-resolution modeling approach within the shallow-water equations

The ability to solve the global shallow-water equations with a conforming, variable-resolution mesh is evaluated using standard shallow-water test cases. While our long-term motivation is the creation of a global climate modeling framework capable of resolving different spatial and temporal scales in different regions, we begin with an analysis of the shallow-water system in order to better understand the strengths and weaknesses of our approach. The multi-resolution meshes are spherical centroidal Voronoi tessellations where a single, user-supplied density function determines the region(s) of fine- and coarse-mesh resolution. We explore the shallow-water system with a suite of meshes ranging from quasi-uniform resolution meshes, where grid spacing is globally uniform, to highly-variable resolution meshes, where grid spacing varies by a factor of 16 between the fine and coarse regions. We find that potential vorticity is conserved to within machine precision and total available energy is conserved to within time-truncation error. This finding holds for the full suite of meshes, ranging from quasi-uniform resolution and highly-variable resolution meshes. Using shallow-water test cases 2 and 5, we find that solution error is controlled primarily by the grid resolution in the coarsest part of the model domain. This finding is consistent with results obtained by others. When these variable resolution meshes are used for the simulation of an unstable zonal jet, we find that the core features of the growing instability are largely unchanged as the variation in mesh resolution increases. The main differences between the simulations occur outside the region of mesh refinement and these differences are attributed to the additional truncation error that accompanies increases in grid spacing. Overall, the results demonstrate support for this approach as a path toward multi-resolution climate system modeling.