Motivation
The Dedalus Collaboration has developed a sparse discretization of vector fields in spherical geometry. Key to this process is the understanding that vector components, as opposed to scalars, do not have the same regularities as the standard spherical harmonics. However, by using a basis that does have the right regularity, we can construct a sparse matrix solution to differential equations in curvilinear geometries. This basis is the spin-weighted spherical harmonics, constructed from Jacobi polynomials.
As a part of his senior thesis, Matt Goldberg constructed a test problem of barotropic jet formation on the 2-Sphere, demonstrating the fidelity of this method.