On the Resolution of Triangular Meshes

It is generally agreed that the resolution of a regular quadrilateral mesh is the side length of quadrilateral cells. There is less agreement on the resolution of triangular meshes, exacerbated by the fact that the numbers of edges or cells on triangular meshes are approximately three or two times larger than that of vertices. However, the geometrical resolution of triangular meshes, that is, maximum wavenumbers or smallest wavelengths that can be represented on such meshes, is a well defined quantity, known from solid state physics. These wavenumbers are related to a smallest common mesh cell (primitive unit cell), and the set of mesh translations that map it into itself. They do not depend on whether discrete degrees of freedom are placed on vertices, cells or edges. For equilateral triangles the smallest wavelength equals twice the triangle height. Resolutions of quadrilateral and triangular meshes approximately agree if they have the same numbers of vertices. Plain Language Summary Some models used in climate studies are formulated on triangular computational meshes. We discuss how to determine the smallest scales that are resolved on such meshes. They are referred to as a mesh resolution. The notion of mesh resolution is commonly used to relate climate model results simulated on different meshes. Key Points Geometrical resolution of an equilateral triangular mesh is defined by the height of its triangles Quadrilateral and triangular meshes with the same number of vertices have approximately the same resolution