Eigensolutions of the spherical Laplacian for the cubed‐sphere and icosahedral‐hexagonal grids

Abstract

Eigensolutions of the spherical Laplacian operator on the cubed‐sphere and the icosahedral (‐hexagonal) grid were investigated. For the cubed‐sphere grid, the Laplacian operator was discretized with the Spectral Element Method (SEM) using the Lagrange–Legendre Polynomials (LLP), while, for the icosahedral grid, the Finite Difference Method (FDM) was employed. The discrete Laplacian operator was formulated into matrix equations with respect to the unknowns of the global‐domain grid‐point values. The eigenvalues of the matrix exhibited a step‐function‐like behaviour: there are 2n + 1 identical (degenerate) eigenvalues for the degree n, the total wave‐number‐like index. Unlike the spherical harmonic functions, the eigenvectors were found to be not separable into zonal and meridional structure: a part of them reflected the basic geometry of grids, i.e. the cube (icosahedron) for the cubed‐sphere (icosahedral) grid. These eigenvectors were shown to consist of only a couple of spherical harmonics. The eigensolutions indicated that the Rossby–Haurwitz waves, the normal modes of absolute‐vorticity conserving motion, are affected significantly by the underlying grid system in both horizontal structure (resulting in geodesic Rossby–Haurwitz waves) and phase speed (being larger than theory for the cubed‐sphere SEM, but smaller for the icosahedral FDM). It was implied by the eigensolutions that when a prognostic equation incorporates Laplacian‐type hyperviscosity, the maximum viscosity coefficient for the cubed‐sphere SEM should be given significantly smaller than for the icosahedral FDM, particularly in the case of high resolution with the high‐order LLP. Selective filtering or diffusing small‐scales was revealed to be practiced more effectively for the cubed‐sphere SEM than the icosahedral FDM. The eigenvectors are also discussed in relation to the grid imprinting of the numerical models built on those geodesic grids.