A quasi- Lagrangian integration scheme with unify finite-difference scheme of cubic spline function transformation, partⅠ: quasi- Lagrangian advection with spline scheme on quasi-uniform latitude-longitude grid and its exact tests

In this study, with transformation of cubic spline function (spline scheme) on latitude-longitude grid mapping a quasi-uniform latitude-longitude one, a new explicit quasi-Lagrangian integration scheme is introduced.Adopting the original atmospheric equations of motion, which includes atmospheres in the North Pole and the Sorth Pole, a general forecast equation of spline scheme of space-time second-order differential remainder is derived, that can be used for forecast variables of pressure, temperature, humidity, wind and acceleration (general Newtonian force acting to unit air mass on the rotating Earth).Their bicubic surfaces are fitted on latitude-longitude grid, and every"bicubic surface"field is second-order differentiable.So, the track of a Lagrangian air parcel and its interpolated values can be produced by an explic? it, iterative process, but it goes along a "spline scheme" path with fitted slopes, curvatures and torsions of the variable fields.In order to inspect its feasibility of the dynamic core of spline scheme on latitude-longitude grid, a full set of international current exact tests, i.e.balance flow, cross-polar flow and Rossby-Haurwitz wave flow, are experimented to try its different dynamical formulation and program correctness, and to verify accuracy of the spline scheme and uniformity to other scheme.The test of the balance flow verifies that "cubic motion" compatible with "linear motion" (no Gibbs wave) and the Coriolis force in the atmospheric equation of motion (or in quasi-geostrophic wind field) does no work.The test of the cross-polar flow shows that the geostrophic wind passes correctly polar area, including the South Pole and the North Pole, with transformation of fitting bicubic surfaces to those scalar and vector variables on the spherical quasi-uniform latitude-longitude grid.The test of Rossby-Haurwitz wave flow demonstrates that the measured wind-pressure field could keep shape and phase propagation correct in integration but that the disturbed pressure's amplitude changes slowly in the non-divergent wind field, with the stream function field fitted a bicubic surface, too.All of the above exact tests indicate that prediction error must be derived from two aspects:one is the error of second-order space differential remainder in fitting the cubic spline functions, and another is the error between upstream air parcel's path and its exact locus (truncation error), but the predictive error of the spline scheme always assumes certain form.Variation of the amplitude of the disturbed pressure field becomes rounding (flattening), which testifies to the predictive error being convergence, spherical symmetry and bounded monotonic deformation, but no phase-shifting error.It is proved that the spline scheme gets over the classical problem of overcrowding in latitude-longitude meshes in polar area and of traveling wave singularity point at the South Pole and the North Pole.