Development and improvement of a nonhydrostatic spectral model using non-constant coefficient semi-implicit and vertically conservative semi-Lagrangian schemes

A two-dimensional x – z nonhydrostatic spectral model has been developed as a simpler framework for testing and improving numerical methods before they are applied to the corresponding global model. The model uses non-constant coefficient semi-implicit and vertically conservative semi-Lagrangian schemes and is computationally efficient by allowing for long timesteps in simulations. The model incorporates several improvements related to free-slip surface boundary conditions, prognostic variables in spectral space, and the semi-Lagrangian and semi-implicit schemes. These improvements enhance numerical stability, especially in cases with steep orography, and improve model results. The model was tested with various test cases (e.g., mountain wave test cases), confirming that a single recalculation of the non-constant coefficient linear terms is usually sufficient to stably solve the equations associated with the non-constant coefficient semi-implicit scheme, even in the cases with a steep mountain with an average slope of 45°. The model also ran stably even in the case of an extremely steep mountain with an average slope of 63.4° by performing several iterations using the preconditioned general conjugate residual method. In all these cases, good results were obtained with long timesteps.