Characteristics of a Spectral Inverse of the Laplacian Using Spherical Harmonic Functions on a Cubed-Sphere Grid for Background Error Covariance Modeling

Abstract

n this study, a spectral inverse method using spherical harmonic functions (SHFs) represented on a cubed-sphere grid (SHF inverse) is proposed. The purpose of the spectral inverse method studied is to help with data assimilation. The grid studied is the one that results from a spectral finite element decomposition of the six faces of the cubed sphere on Gauss?Legendre?Lobatto (GLL) points with equiangular gnomonic projection. For a given discretization of the cube in this form, as the total wavenumber of the test functions increases, there comes a point at which the cube?s eigenstructure fails to be able to replicate the spherical harmonic functions. The authors call this point a limit wavenumber in using the SHF inverse. In common with the authors? previous research, the allowable total wavenumber of the SHF inverse increases more effectively with an enhanced polynomial order. The use of the eigenvectors and eigenvalues of the Laplacian, discretized on the grid spacing used in this study, to the Poisson equation is compared with the benchmark set by using the spherical harmonics solution to the problem. In terms of accuracy, the SHF inverse is superior to a direct inverse of the Laplacian using eigendecomposition. The feasibility of SHF inverse in operational implementation is examined under a massive computational environment.