A quasi-Lagrangian integration scheme with unify finite-difference scheme of cubic spline function transformation, partⅡ:non-hydrostatic and fully compressible dynamic core with spline scheme and density current test

In this numerical analysis, the cubic spline function consists of cubic spline, bi-cubic surface and tri-cubic (3-D cubic) cube.All three are based on the spline algorithm (spline scheme), which posses the second-order differentiable"convergence"and "optimality" of mathematical laws.With the spline scheme, a set of explicit forecast equations of second-order space-time differential remainder is derived, which brings to a quasi-Lagrangian integration scheme as well as a fully compressible dynamic core of a new nonlinear numerical model. With fitting the cubic spline functions to pressure, temperature and wind fields, and to the field of the generalized Newtonian force acting to unit air mass on rotating earth in the wind equations, which made all of them second-order differentiable, the Lagrangian particle's 3D tracks follow the second Newton's law of motion and its forecast values of the variables can be refined by an explicit, iterative procedure.Meanwhile, we may get the averaged wind divergence in one time step of the 3D paths, and it is with the implicit 3D divergence as adiabatic variability that pressure and temperature fields will be changed (forecasted).With separating time-dependent, static pressure field, which satisfies the hydrostatic equilibrium, from non-static pressure field in the model's atmosphere, and replacing the reference atmosphere profile by the static pressure field of space-time function, the non-static pressure gradient force in vertical, as well as its acting path, is rightly determined.The density current test shows that its extra nonlinear flows can be simulated in model atmosphere with such a non-hydrostatic, fully compressible dynamic core, which primarily identifies its feasibility, accuracy, scientific evaluation and our programs correctness and verifies its consistency, convergence and stability with the cubic spline functions.We further analysed the differences of density current test between the Benchmark reference solution and ours.