Vertical finite-element scheme for the hydrostatic primitive equations on a cubed-sphere
Abstract:A vertical finite-element (VFE) scheme of three-dimensional hydrostatic primitive equations is adopted for the numerical weather prediction system, which is horizontally discretized with spectral elements on a cubed-sphere. The hybrid pressure-based vertical coordinate is employed to discretize a vertical grid, in which only the full levels of the coordinate are used in the VFE. Vertical integrals and derivatives in the hydrostatic equations are derived based on Galerkin-based finite elements with b-spline functions. These basis functions and their first-order derivatives are constructed using the Cox-de Boor algorithm. The computation of vertical integrals, derivatives and advections in the hydrostatic equations are easily done in physical space by matrix multiplication with the corresponding vertical operators.
The VFE discretization scheme implemented into the global three-dimensional hydrostatic model on the cubed-sphere is evaluated by performing ideal test cases including the steady-state, baroclinic wave, 3D Rossby-Haurwitz wave, and mountain-induced Rossby wave train test cases. The two types of the VFE scheme are compared to the vertical finite difference scheme.