Spline Model: A Hydrostatic/Non-Hydrostatic Dynamic Core with Space-Time Second-Order Precision and Its Exact Tests

We present a new explicit quasi-Lagrangian integration scheme with the three-dimensional cubic spline function transform (transform = fitting + interpolation, referred to as the “spline format”) on a spherical quasi-uniform longitude–latitude grid. It is a consistent longitude–latitude grid, and to verify the feasibility, accuracy, convergence, and stability of the spline format interpolation scheme for the upstream point on the longitude–latitude grid, which may map a quasi-uniform longitude–latitude grid, a set of ideal, exact test schemes is adopted, which are recognized and proven to be effective internationally. The equilibrium flow test, cross-polar flow test, and Rossby–Haurwitz wave test are used to illustrate the spline scheme uniformity to the linear scheme and to overcome the over-dense grid in the polar region and the non-singularity of the poles. The cross-polar flow test demonstrates that the geostrophic wind crosses the polar area correctly, including the South Pole and North Pole. A non-hydrostatic, fully compressible dynamic core is used to complete the density flow test, demonstrating the existence of a time-varying reference atmosphere and that the spline format can simulate highly nonlinear fine-scale transient flows. It can be compared for the two results of the density flow test between the solution with the spline format and the benchmark reference solution with the linear format. Based on the findings, the non-hydrostatic dynamic core with the spline format is recommended for adoption. When simulated for the flow over an ideal mountain, through the “topographic gravity wave test”, the bicubic surface terrain and terrain-following height coordinates, time-split integration, and vector discrete decomposition can be derived successfully. These may serve as the foundations for a global, unified spline-format numerical model in the future.