Formulation of a Three-Dimensional Spectral Model for the Primitive Equations Using Laguerre Functions

This study proposes a novel formulation of a three-dimensional spectral model for the primitive equations, where the spectral method is applied in both the horizontal and vertical directions. We utilize scaled Laguerre functions as the vertical basis and introduce a scaling parameter that enables flexible control over the distribution of vertical grid points. We demonstrate that the optimal setting of this parameter allows the model top to be placed at significantly higher altitudes while maintaining adequate grid spacing in the upper atmosphere, thereby addressing a practical limitation of previous three-dimensional spectral models. The proposed formulation is implemented as a numerical model and validated through several standard atmospheric benchmark experiments. Comparative experiments reveal that the numerical error of the three-dimensional spectral models converges significantly faster than that of a conventional vertical finite-difference model, exhibiting the rapid error reduction characteristic of spectral methods when the vertical degrees of freedom are increased. Furthermore, we investigate and compare the properties of gravity wave propagation using the proposed discretization and the finite-difference method using linearized two-dimensional primitive equations. This confirms that the proposed discretization method can accurately represent upward-propagating waves.