The well-known inefficiencies and numerical problems in numerical weather prediction associated with the convergence of meridians and the polar singularities of a latitude-longitude-based grid system have spurred the development of polyhedron-based alternative grids, such as the cubed sphere and the (triangular-gridded) icosahedron. Except at the vertices, the continuous mappings for these conﬁgurations can be made perfectly conformal (angle preserving), which substantially simpliﬁes the adaptation of existing grid-based regional models to these global geometries. However, the unavoidable vertex singularities on continuous polyhedral grids still remain too strong to avoid severe numerical difficulties for any model based on spatial ﬁnite differencing.
While no completely continuous gridding of the sphere can be completely free of mathematical singularities, the introduction of small and isolated regions of overlap bounded by pairs of weakly-singular branch points near the original vertices, combined with a mechanism for the frequent reconciliation of the overlapped solutions there, provides a method by which all strong singularities are avoided, while an adequate degree of smooth continuity of the entire computational grid domain is preserved. We have formulated a new method, based on the concept of a Riemann surface in the theory of complex analytic functions, that enables such smooth global grids to be constructed corresponding to any simply-connected polyhedron that exhibits a griddable surface. Moreover, these grids are guaranteed to possess the desirable conformal property which facilitates their application to almost any existing regional dynamical formulations that are based on either square or triangular-hexagonal grids.
The technique is immediately applicable to smooth mappings no longer constrained to correspond to merely convex polyhedra; rather, we are now granted the opportunity to formulate coherent dynamically-adapting global grids with multiple regions of enhanced resolution wherever and whenever there is a need. We describe the method of construction and discuss the opportunities for the potential uniﬁcation of global and regional forecasts and climate simulations that these new gridding methods seem to offer.