Structure of Eigenvalues in the Advection-Diffusion Equation by the Spectral Element Method on a Cubed-Sphere Grid

To fundamentally understand discretized equations and differential operators, this study investigates the eigenvalues of the spherical gradient matrix and spherical Laplacian matrix discretized by the spectral element method (SEM) on the cubed-sphere grid (CS) for solid-body rotation. The gradient matrix with prescribed wind for the solid-body rotation has small positive real numbers in the eigenvalues, which implies that the solutions of the advection equation for solid-body rotation can be unstable and distorted. However, the unstable model can be effectively controlled by the Laplacian matrix, and all eigenvalues of the Laplacian matrix are aligned along the negative part of the real axis. The maximum eigenvalues of the gradient matrix and Laplacian matrix exhibit the characteristic of a linear function of the resolution, which is important to determine the upper limit of the time-step size for stable time integration. This study suggests suitable e-folding time scale of the diffusion using the 6th order for applications of the advection-diffusion equation. Regarding the characteristics of the eigenvalues, the setting of the stable time step is discussed. A method to implement the diffusion operators in the third-order Runge-Kutta time integration scheme in the advection-diffusion model is also discussed.