中文摘要：

正压大气是最简单的大气模型,它可以讨论高空槽脊等天气系统的许多动力学过程。在非均匀基流的情况下,即使对小扰动的线性方程,由于其方程系数非常数,一般情况下也难求出其解析解,只能依靠数值计算得到数值解。但数值解的正确性是需要验证的。在特殊情况下,将球面正压大气方程组化为连带Leg-endre方程,解析求出该方程的频散关系和特征函数,其频率方程为一对重力惯性波的离散谱,其特征函数为连带Legendre函数;对非常系数的正压大气方程设计了离散化方案,在同样特殊情况下求出了数值解;将数值解与解析解进行比较,证明了数值计算方案的正确性和计算精度。这为复杂背景中,球面正压原始方程组特征值问题的数值求解的正确性提供了一个很好的验证方法和途径。

英文摘要：

The barotropic atmosphere is a simplified model which can be applied to discussing the dynamic process associated with high trough and ridge.In the case of nonuniform basic flow,the analytical solution of linearized equations with small perturbation is difficult to resolve,which is mainly due to the variable coefficients in the equations.This calls for the calculation and the verification of the numerical solution.Based on a specific case,the equations of the spherical barotropic atmosphere were transformed into the associated Legendre function and the analytical solution was obtained.The frequency equation was found to be a discrete spectrum of a couple of inertia-gravity waves,while the eigenfunction was the associated Legendre function.The numerical solution was then calculated by a difference scheme.The comparison between the numerical and the analytical solution validates the correctness and the precision of the numerical solution.The results are expected to offer a good method to calculate the numerical solution and check its correctness in solving the eigenvalue problem of the primitive equations in spherical barotropic atmosphere under complex background.