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# A 3D Nonhydrostatic Compressible Atmospheric Dynamic Core by Multi-moment Constrained Finite Volume Method

Fund Project:

National Key Research and Development Program of China(Grant Nos. 2017YFC1501901 and 2017YFA0603901) and the Beijing Natural Science Foundation (Grant No. JQ18001)

• A 3D compressible nonhydrostatic dynamic core based on a three-point multi-moment constrained finite-volume (MCV) method is developed by extending the previous 2D nonhydrostatic atmospheric dynamics to 3D on a terrain-following grid. The MCV algorithm defines two types of moments: the point-wise value (PV) and the volume-integrated average (VIA). The unknowns (PV values) are defined at the solution points within each cell and are updated through the time evolution formulations derived from the governing equations. Rigorous numerical conservation is ensured by a constraint on the VIA moment through the flux form formulation. The 3D atmospheric dynamic core reported in this paper is based on a three-point MCV method and has some advantages in comparison with other existing methods, such as uniform third-order accuracy, a compact stencil, and algorithmic simplicity. To check the performance of the 3D nonhydrostatic dynamic core, various benchmark test cases are performed. All the numerical results show that the present dynamic core is very competitive when compared to other existing advanced models, and thus lays the foundation for further developing global atmospheric models in the near future.
摘要: 在原来二维非静力大气模式框架基础上，本文采用3点多矩约束有限体积格式发展了一个含地形的三维完全可压缩非静力有限体积大气模式动力框架。多矩约束有限体积方法定义了两类矩：（1）点值（PV矩），（2）体积积分平均值（VIA矩）。通过大气控制方程，单元网格内未知变量（即PV矩）的时间演变得以更新；而积分平均值（VIA矩）通过有限体积通量形式方法更新其时间演变，进而保证了数值的严格守恒。与现有的其他方法相比，本文发展的三维大气模式框架所采用的3点多矩约束有限体积算法，具有一致的3阶精度、模板紧致和算法简洁等优势。为了检验发展的三维大气模式框架性能，本文进行了各种标准数值测试包括陡峭地形数值测试，数值模拟结果表明新发展的三维大气模式框架可与现有的先进大气模式相媲美。本研究为进一步发展全球多矩约束有限体积模式奠定基础。
• Figure 1.  The locations (black circles) of (a) solution points in the 1D case and (b) solution points over cell Ci,j,k on a 3D Cartesian grid.

Figure 2.  Numerical results of a 3D rising thermal bubble (xz slices at y = 1600 m). The contour lines of (a) potential temperature perturbation (units: K) at T = 0 s are from 0.0 to 2.0 with an interval of 0.2, (b) potential temperature perturbation (units: K) at T = 480 s are from 0.05 to 1.0 with an interval of 0.05, (c) the u wind (units: m s−1) at T = 480 s are from −2.4 to 2.4 with an interval of 0.4, (d) the vertical wind (units: m s−1) at T = 480 s are from −2 to 9 with an interval of 1. Positive contours are presented with solid lines and negative contours with dashed lines.

Figure 3.  Numerical results of a 3D hydrostatic mountain after one hour (xz slices at y = 120 km). The contour lines of (a) the u velocity perturbation (units: m s−1) are from −0.008 to 0.01 with an interval of 0.002, and (b) the vertical velocity (m s−1) are from −0.002 to 0.0015 with an interval of 0.0005. Positive contours are presented with solid lines and negative contours with dashed lines. Zero contour lines are not shown.

Figure 4.  Density perturbations (units: kg m−3) for the linear hydrostatic mountain after five hours. (a) xz cross section in the plane y = 120 km. The contour lines are from −2.5 × 10−5 to 5.0 × 10−5 with an interval of 5 × 10−6. Positive values are displayed by solid lines and negative values by dashed lines. (b) Vertical profile at x = y = 120 km.

Figure 5.  The steep 3D hill profile. Units: m, in both the horizontal and vertical direction.

Figure 6.  Vertical velocity (units: m s−1) along the lower surface of z = h(x, y). The contour lines are from −5.0 to 1.0 with an interval of 0.5. Positive values are denoted by solid lines and negative values by dashed lines. Zero contour lines are not shown. Vector arrows are overlaid.

Figure 7.  (a) Vertical velocity (units: m s−1) in the central xz cross section at y = 0. The contour lines are from −4.0 to 5.0 with an interval of 0.5. (b) Vertical velocity (units: m s−1) in the xy cross section at z = 500 m. The contour lines are from −4.0 to 3.0 with an interval of 0.5. Positive values are denoted by solid lines and negative values by dashed lines. Zero contour lines are not displayed. Vector arrows are overlaid.

Figure 8.  Vertical velocity (units: m s−1) in the xy cross section at z = 2500 m. The contour lines are from −2.0 to 1.5 with an interval of 0.5. Positive values are denoted by solid lines and negative values by dashed lines. Zero contour lines are not displayed. Vector arrows are overlaid.

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## Manuscript History

Manuscript revised: 19 April 2019
Manuscript accepted: 13 May 2019
###### 通讯作者: 陈斌, bchen63@163.com
• 1.

沈阳化工大学材料科学与工程学院 沈阳 110142

## A 3D Nonhydrostatic Compressible Atmospheric Dynamic Core by Multi-moment Constrained Finite Volume Method

###### Corresponding author: Xingliang LI, lixliang@cma.cn
• 1. College of Global Change and Earth System Science, Beijing Normal University, Beijing 100875, China
• 2. Center of Numerical Weather Predication, China Meteorological Administration, Beijing 100081, China
• 3. State Key Laboratory for Strength and Vibration of Mechanical Structures, Xi’an Jiaotong University, Xi’an 710049, China
• 4. Department of Mechanical Engineering, Tokyo Institute of Technology, Tokyo 226-8502, Japan
• 5. School of Atmospheric Sciences, Sun Yat-Sen University, Guangzhou 510275, China
• 6. Beijing Meteorological Observatory, Beijing 100089, China

Abstract: A 3D compressible nonhydrostatic dynamic core based on a three-point multi-moment constrained finite-volume (MCV) method is developed by extending the previous 2D nonhydrostatic atmospheric dynamics to 3D on a terrain-following grid. The MCV algorithm defines two types of moments: the point-wise value (PV) and the volume-integrated average (VIA). The unknowns (PV values) are defined at the solution points within each cell and are updated through the time evolution formulations derived from the governing equations. Rigorous numerical conservation is ensured by a constraint on the VIA moment through the flux form formulation. The 3D atmospheric dynamic core reported in this paper is based on a three-point MCV method and has some advantages in comparison with other existing methods, such as uniform third-order accuracy, a compact stencil, and algorithmic simplicity. To check the performance of the 3D nonhydrostatic dynamic core, various benchmark test cases are performed. All the numerical results show that the present dynamic core is very competitive when compared to other existing advanced models, and thus lays the foundation for further developing global atmospheric models in the near future.

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