Geophysical Modeling on the Sphere with Radial Basis Functions

Modeling data on the sphere is fundamental to many problems in the geosciences. Classical approaches to these problems are based on expansions of spherical harmonics and/or approximations on latitude/longitude based grids. The former are quite algorithmically complex, while the latter suffer from the notorious pole problem. Additionally, neither of the methods can be easily generalized to other manifolds. Radial Basis functions (RBFs), on the other hand, are algorithmically simple, suffer from no coordinate singularities, and generalize to arbitrary geometries. Since RBFs do not depend on any grid and require no meshing, they can be naturally used in concert with so called optimal node configurations. We discuss three recent and non-trivial geophysical applications of RBFs on spherical domains with optimal node sets. The first is on the approximation and decomposition of tangent vector fields on the sphere (e.g. horizontal winds in the atmosphere). The second is the simulation of unsteady nonlinear flows on the sphere described by the shallow water equations. The third and final application is the simulation of thermal convection in a 3-D spherical shell, a situation of interest in modeling the Earth’s mantle.