基于质量坐标的非静力谱模式高精度垂直离散格式研究

In a mass-based non-hydrostatic spectral kernel, the vercital operators are complicated and a set of constraints exists between the perators. In practical, a carefully designed finite-differential(FD) formulation of vertical descretization with only 2 order accuracy is applied for the non-hydrostatic kernel. Recent studies have shown that, these vertical constraints are not satisfied in finite-element(FE) approach. To address the lack of high-order vercital discretization scheme for mass-based non-hydrostatic spectral kernel, the research based on hybrid vertical discretization approach will be conducted especially for constraints-fulfilled and high-order discretization schemes for linear part of a SI formulation. Specifically, first, a FD scheme which fulfills the constraints with enlarged set of vertical level will be studied. The enlarged level FD scheme with the help of spline interpolation can get higher order accuracy than usual finite-differential scheme. Second, the analysis solution for the linear system based on green function will be explored. Theoreticaly, the analysis solution which satifies the constraints automatically can get the same accuracy order with the input spline functions . Third, the stabilaty of the hybrid descritization.scheme whose non-linear part uses FD scheme and linear part uses other schemes. The effect of the difference between the non-linear and linear discretization schemes will be investigated. All the aforementioned schemes will be verified and validated using the our NWP models and the super computer resourse of NUDT(National University of Defense Technology).

基于质量坐标的非静力谱模式垂直离散形式复杂且存在一系列约束条件，在应用中一般采用精巧配置二阶精度的有限差分格式。研究表明，由于约束条件的限制，直接应用有限元离散是极其困难的。针对基于质量坐标非静力谱模式缺乏高精度垂直离散方案的问题，本项目基于混合有限元垂直离散方案，对半隐式格式线性部分开发约束条件满足的高精度离散格式进行研究。一 对线性部分研究新的高效扩展分层有限差分格式，在样条插值基础上研究最优扩展分层配置，使得满足约束条件下提高计算精度和计算效率。二 对隐式线性部分利用格林函数研究解析求解方案，在解析的情况下，垂直操作算子的约束条件是自动满足的，理论而言解析解的精度能达到输入样条函数的精度。三 针对混合有限元离散线非线性部分和线性部分离散格式和离散精度的不一致问题，研究格式的稳定性情况。依托我团队现有的全球数值预报模式，和国防科大的超级计算机资源将对上述研究方案进行全面的测试。

本项目主要针对质量坐标的全球谱模式垂直方向高精度离散格式、深厚大气垂直坐标选择、初始场垂直加密等方面开展研究。研究了基于高斯积分的混合有限元高精度垂直离散格式，将原混合离散格式线性计算时均匀增加分层，变为依照高斯积分增加分层，使得垂直积分变为高斯积分，进一步提升了格式的计算精度和稳定性。研究了基于静力气压的深厚大气模式动力框架，并开发了相应的线性方程组和谱空间高效求解算法。传统的基于质量坐标深厚大气模式均采用深厚气压作为其垂直坐标，该深厚气压存在一定的缺陷，而本研究提出的静力气压坐标能恰当的弥补这些缺陷。针对模式预报在平流层偏冷问题，研究了基于气压A,B值的模式层垂直加密算法，对模式层初始场加密后保持垂直分层的均匀和平滑，并使得加密后的初始场能驱动模式稳定积分，结果验证了对初始场顶部局部区间加密能够一定程度修改平流层偏冷问题，提高模式层顶的可预报性。