Semi-analytical solution of the dry baroclinic Lagrange primitive equation and numerical experiment of a non-linear density current

To solve atmospheric primitive equations, the finite difference approach would result in numerous problems, compared to the differential equations. Taking the semi-Lagrange model as an example, there exist two difficult problems-the particle trajectory computation and the solutions of the Helmholtz equations. In this study, based on the substitution of atmosphere pressure, the atmospheric primitive equations are linearized within an integral time step, which are broadly seen as ordinary differential equations and can be derived as semi-analytical solutions (SASs). The variables of SASs are continuous functions of time and discretized in a special direction, so the gradient and divergence terms are solved by the difference method. Since the numerical solution of the SASs can be calculated via a highly precise numerical computational method of exponential matrix-the precise integration method, the numerical solution of SASs at any time in the future can be obtained via step-by-step integration procedure. For the SAS methodology, the pressure, as well as the wind vector and displacement, can be obtained without solving the Helmholtz formulations. Compared to the extrapolated method, the SAS is more reasonable as the displacements of the particle are solved via time integration. In order to test the validity of the algorithms, the SAS model is constructed and the same experiment of a non-linear density current as reported by Straka in 1993 is implemented, which contains non-linear dynamics, transient features and fine-scale structures of the fluid flow. The results of the experiment with 50 m spatial resolution show that the SAS model can capture the characters of generation and development process of the Kelvin-Helmholtz shear instability vortex; the structures of the perturbation potential temperature field are very close to the benchmark solutions given by Straka, as well as the structures of the simulated atmosphere pressure and wind field. To further test the convergence of the numerical solution of the SAS model, the 100 m spatial resolution experiment of the non-linear density current is also implemented for comparison. Although the results from both experiments are similar, the former one is better and the property of mass-energy conservation is comparatively reasonable, and furthermore, the SAS model has a convergent property in the numerical solutions. Therefore, the SAS method is a new tool with efficiency for solving the atmospheric primitive equations.