陡峭地形附近自适应平流追踪算法和气压梯度力的高阶数值方案研究

The advection errors and pressure gradient force (PGF) errors in the vicinity of steep terrain have effects on the performance of numerical weather prediction model. With the increasing of the model resolutions, the discrete representation of the terrains becomes steeper and steeper which intensifies the severe effects on predictions by these two types of errors. The existing methods based on fixed mesh are proposed to deal with either advection errors or PGF errors, but not both of them. The anisotropic adaptive mesh, different from the fixed mesh, uses the dynamical adaptive technique enabling the next mesh adaption to be determined to change the spatial distribution and size of mesh. In this study, focus on the anomalous track changes of the airflow near the steep terrain, first, we design a metric-advection algorithm using the current flow velocity to advect the metric forward in time over the period until the next mesh adapts such that the mesh of the area where the airflow will approach could be denser to reduce the advection errors. Then, to reduce PGF errors, we will construct a third-order polynomial representation of PGF based on finite element method replacing the traditional PGF schemes. Finally, we combine these two methods with the adaptive techniques increasing the local mesh resolution by two error metrics, the concentration of airflow and PGF, to a three-dimensional idealized test case. For further improving the accuracy of simulating the advection and PGF near the steep terrain, it is a potential of the dynamical adaptive mesh with a metric-advection algorithm and a third-order polynomial representation for PGF which lays the foundation to improve the accuracy of predictions of velocity and precipitation.

陡峭地形附近的平流误差和气压梯度力（PGF）误差是影响模式模拟性能的重要因素。近年来，模式分辨率的提高导致模式可分辨的地形坡度增大，进一步凸显了上述两类误差的影响，但是目前基于固定网格的计算方案几乎无法同时减小这两类误差。区别于固定网格，各向异性的自适应网格可根据模拟需要实时改变网格的空间分布和大小。本研究将以陡峭地形附近流场的异常突变为研究对象，从动态计算网格的角度出发研究平流追踪算法，以流场的速度预估数值网格度量张量的实时变率，对下一时刻平流经过的区域提前加密，减小平流计算误差；基于有限元方法构造PGF的高阶多项式逼近方案，提高PGF的计算精度，同时结合自适应网格提高陡峭地形附近网格的分辨率进一步减小这两类误差，为改善陡峭地形附近数值模拟精度提供高精度的PGF计算方案和平流过程的动态网格研究方法，以期为进一步提高陡峭地形附近流场和降水场的定量预报精度奠定基础。

如何提高陡峭地形附近平流过程和气压梯度力的模拟精度和计算效率是提升大气模式模拟性能的重要因素。本研究采用伽辽金有限元方法构造了平流追踪算法和高阶气压梯度力（PGF）逼近方案，结合各向异性动态自适应网格和深度学习技术进一步提升新方案的模拟精度和计算效率，探究了陡峭地形附近影响自适应网格模拟精度的因素。首先运用间断有限元和半隐式时间积分构造平流计算方案，采用LSTM神经网络和后验误差估计理论实现自适应平流追踪算法。相比于固定网格，自适应网格能够进一步减小平流误差的40%；引入LSTM人工智能混合度量张量预测模型，减少网格自适应调整时间至少20%，自适应网格的平流误差主要受网格最大最小边长、后验误差水平和自适应关键判据的影响，而不受平流风向和地形复杂度的影响。其次，在谱元单元内部采用均匀网格剖分和稀疏网格策略建立了高阶局部伽辽金稀疏网格数值方案，由于超级收敛的存在，新方案能够以低阶的多项式逼近方案获得高阶模拟精度，扩大了新方案时间步长的选取以减小计算量，消除了高阶谱元方法的谱间断误差。最后，采用高阶多项式控制体积和两步预估校正方法构建PGF的高阶计算方案，引入大气状态方程，构造基于三维各向异性自适应网格的模式动力框架，采用过山波数值试验证明模式动力框架能够表征二维和三维过山波向下游和向上稳定传播的过程，有效地解决跨尺度自适应网格模拟所带来的计算效率问题，为研究陡峭地形附近的PGF和平流计算提供一种全新的方法。现有的大气模式数值计算网格大多基于固定规则网格/嵌套网格生成，受到现有算力水平的限制，网格的分辨率无法达到捕获小尺度天气系统的要求，随着模式分辨率的提高这种矛盾日益凸显。高精度和高效率兼备的自适应有限元大气模式动力框架有可能成为未来提高模式精准定位中小尺度天气系统能力的关键因素，可用于实现强对流系统内部特征的精准刻画，具有极其可观的应用前景。