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 “spline format”) on a spherical quasi-uniform longitude-latitude grid. It is a consistent longitude-latitude grid, and to verify its 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, which are recognized and effective internationally, are adopted. 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 correctly polar area, including the South Pole and North Pole. A non-hydrostatic fully compressible dynamical 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 of with spline format and the benchmark reference solution of with linear format. The non-hydrostatic dynamical core in the spline format is adopted: it can be successfully simulated for the flow over an ideal mountain, called “topographic gravity wave test”, which demonstrating the bicubic surface terrain and terrain-following height coordinates, time-split integration, and vector discrete decomposition method. These can serve as the foundation for the global, unified spline format, numerical model in future.