# Challenges in Developing Finite-Volume Global Weather and Climate Models with Focus on Numerical Accuracy

## 构建具有数值计算精度的全球有限体积天气气候模式面临的挑战

• Corresponding author: Yuanfu XIE, xieyf@cma.gov.cn
• Funds:

Supported by the National Key Research and Development Program of China (2017YFC1502201) and Basic Scientific Research and Operation Fund of Chinese Academy of Meteorological Sciences (2017Z017)

• doi: 10.1007/s13351-021-0202-3
• High-resolution global non-hydrostatic gridded dynamic models have drawn significant attention in recent years in conjunction with the rising demand for improving weather forecasting and climate predictions. By far it is still challenging to build a high-resolution gridded global model, which is required to meet numerical accuracy, dispersion relation, conservation, and computation requirements. Among these requirements, this review focuses on one significant topic—the numerical accuracy over the entire non-uniform spherical grids. The paper discusses all the topic-related challenges by comparing the schemes adopted in well-known finite-volume-based operational or research dynamical cores. It provides an overview of how these challenges are met in a summary table. The analysis and validation in this review are based on the shallow-water equation system. The conclusions can be applied to more complicated models. These challenges should be critical research topics in the future development of finite-volume global models.
近年来，随着对更精细化天气预报和更精准气候预测的需求的不断增加，高分辨率全球网格非静力平衡预报模式的构建一直是大气科学领域研究与应用的热点。目前构建满足各项数学、物理及计算要求的网格模式仍然遇到严峻的挑战。本综述深入讨论了在构造基于有限体积法的全球数值模式动力框架中面临的主要挑战，即如何保证球面上各种算子的离散数值精度、模式的频散关系、预测变量的守恒特性以及计算效率问题。其中，如何在非均匀球面网格上评估以及改进数值精度是本文讨论的重点。文章回顾了目前已经广泛应用的基于有限体积法的全球数值模式动力框架，逐个分析了其数值精度、频散以及守恒问题，并对各个框架在各项指标上的优缺点进行评估，详尽展示基于有限体积法的全球动力模式的发展状况。尽管文中的分析和讨论主要基于浅水方程系统，但所涉及的大多数概念仍可广泛地用于评估其他更复杂的动力框架模式。优化并解决这些挑战与难点，是未来高精度全球动力模式研究的主要方向。
• Fig. 1.  Two quasi-uniform grids on a sphere, with the icosahedral grid on the left and the cubed sphere on the right.

Fig. 2.  Finite-volume scheme definitions for the A-, B-, C-, C–D, D-, and Z-grid schemes, where ${h}$ is the fluid thickness, ${\boldsymbol{V}}$ is the velocity vector, ${{u}}_{\rm{n}}$ is the normal velocity, ${{u}}_{{\tau }}$ is the tangential velocity component, and $\mathrm{\zeta }$ and $\mathrm{\delta }$ are the vorticity and divergence, respectively.

Fig. 3.  Grids of centroidal center (top left), Voronoi center (top right), icosahedral center (bottom left), circumcenter (bottom middle), and midcenter (bottom right).

Fig. 4.  Illustration of center-arc and edge. The center-arc may not bisect the edge.

Fig. 5.  Definition of ${\lambda }^{\perp }$ of the distance between an arc-center and its edge intersection.

Fig. 6.  Challenge of defining tangential derivatives on irregular grid because the normal and tangential vectors (black and red vectors) do not match and interpolation to vertex location (blue dots) causes accuracy loss.

Fig. A1.  (a) The distribution of cell distortion $\epsilon \left(x\right)$ in the x-direction with two different values of ${\sigma }^{2}$ for scheme B. (b) Time integrations, Heun second-order, Runge–Kutta, Wick–Skamarock (WS) third-order Runge–Kutta, and leapfrog all yield the same solution at a resolution of 161 grid points. A second-order scheme with different time integration schemes yields identical results with the maximum value of 1.0 while the zeroth-order ones have much reduced values. (c) A comparison of the second-order scheme with the zeroth-order scheme A. The conserving schemes with zeroth-order schemes have significant phase errors. (d) A different number of distorted cells in schemes B (blue) and C (yellow and red) result in deviate solutions. Smaller distortion portions yield solutions that are closer to the true solution. (e) Comparison of scheme C with different signs of $\epsilon$, and the true solution. (f) The numerical solutions for the second-order accurate and zeroth-order schemes under different resolutions. Inaccurate schemes do not converge to the true solution with increases in resolution.

•  [1] Adcroft, A. J., C. N. Hill, and J. C. Marshall, 1999: A new treatment of the Coriolis terms in C-grid models at both high and low resolutions. Mon. Wea. Rev., 127, 1928–1936.. [2] Bao, L., R. D. Nair, and H. M. Tufo, 2014: A mass and momentum flux-form high-order discontinuous Galerkin shallow water model on the cubed-sphere. J. Comput. Phys., 271, 224–243.. [3] Du, Q., M. D. Gunzburger, and L. L. Ju, 2003: Constrained centroidal Voronoi tessellations for surfaces. SIAM J. Sci. Comput., 24, 1488–1506.. [4] Eldred, C., and D. Randall, 2017: Total energy and potential enstrophy conserving schemes for the shallow water equations using Hamiltonian methods—Part 1: Derivation and properties. Geosci. Model Dev., 10, 791–810.. [5] Gill, A. E., 1982: Atmosphere–Ocean Dynamics. Academic Press, San Diego, 662 pp. [6] Giraldo, F. X., J. S. Hesthaven, and T. Warburton, 2002: Nodal high-order discontinuous Galerkin methods for the spherical shallow water equations. J. Comput. Phys., 181, 499–525.. [7] Heikes, R., and D. A. Randall, 1995a: Numerical integration of the shallow-water equations on a twisted icosahedral grid. Part I: Basic design and results of tests. Mon. Wea. Rev., 123, 1862–1880.. [8] Heikes, R., and D. A. Randall, 1995b: Numerical integration of the shallow-water equations on a twisted icosahedral grid. Part II: A detailed description of the grid and an analysis of numerical accuracy. Mon. Wea. Rev., 123, 1881–1887.. [9] Heikes, R. P., D. A. Randall, and C. S. Konor, 2013: Optimized icosahedral grids: Performance of finite-difference operators and multigrid solver. Mon. Wea. Rev., 141, 4450–4469.. [10] Kanamitsu, M., J. C. Alpert, K. A. Campana, et al., 1991: Recent changes implemented into the global forecast system at NMC. Wea. Forecasting, 6, 425–435.. [11] Konor, C. S., and D. A. Randall, 2018: Impacts of the horizontal and vertical grids on the numerical solutions of the dynamical equations—Part 1: Nonhydrostatic inertia–gravity modes. Geosci. Model Dev., 11, 1753–1784.. [12] Lax, P. D., and R. D. Richtmyer, 1956: Survey of the stability of linear finite difference equations. Commun. Pure Appl. Math., 9, 267–293.. [13] Lin, S.-J., 2004: A “vertically Lagrangian” finite-volume dynamical core for global models. Mon. Wea. Rev., 132, 2293–2307.. [14] Lin, S.-J., and R. B. Rood, 1996: Multidimensional flux-form semi-Lagrangian transport schemes. Mon. Wea. Rev., 124, 2046–2070.. [15] Putman, W. M., and S.-J. Lin, 2007: Finite-volume transport on various cubed-sphere grids. J. Comput. Phys., 227, 55–78.. [16] Rajpoot, M. K., S. Bhaumik, and T. K. Sengupta, 2012: Solution of linearized rotating shallow water equations by compact schemes with different grid-staggering strategies. J. Comput. Phys., 231, 2300–2327.. [17] Randall, D. A., 1994: Geostrophic adjustment and the finite-difference shallow-water equations. Mon. Wea. Rev., 122, 1371–1377.. [18] Reinecke, P. A., and D. Durran, 2009: The overamplification of gravity waves in numerical solutions to flow over topography. Mon. Wea. Rev., 137, 1533–1549.. [19] Ringler, T. D., J. Thuburn, J. B. Klemp, et al., 2010: A unified approach to energy conservation and potential vorticity dynamics for arbitrarily-structured C-grids. J. Comput. Phys., 229, 3065–3090.. [20] Sadourny, R., A. Arakawa, and Y. Mintz, 1968: Integration of the nondivergent barotropic vorticity equation with an icosahedral-hexagonal grid for the sphere. Mon. Wea. Rev., 96, 351–356.. [21] Saito, K., J.-I. Ishida, K. Aranami, et al., 2007: Nonhydrostatic atmospheric models and operational development at JMA. J. Meteor. Soc. Japan, 85B, 271–304.. [22] Salmon, R., 1998: Lectures on Geophysical Fluid Dynamics. Oxford University Press, New York, 378 pp. [23] Salmon, R., 2007: A general method for conserving energy and potential enstrophy in shallow-water models. J. Atmos. Sci., 64, 515–531.. [24] Satoh, M., T. Matsuno, H. Tomita, et al., 2008: Nonhydrostatic icosahedral atmospheric model (NICAM) for global cloud resolving simulations. J. Comput. Phys., 227, 3486–3514.. [25] Simmons, A. J., D. M. Burridge, M. Jarraud, et al., 1989: The ECMWF medium-range prediction models development of the numerical formulations and the impact of increased resolution. Meteor. Atmos. Phys., 40, 28–60.. [26] Skamarock, W. C., J. B. Klemp, M. G. Duda, et al., 2012: A multiscale nonhydrostatic atmospheric model using centroidal Voronoi tesselations and C-grid staggering. Mon. Wea. Rev., 140, 3090–3105.. [27] Süli, E., and D. F. Mayers, 2003: An Introduction to Numerical Analysis. Cambridge University Press, Cambridge, 444 pp. [28] Thuburn, J., T. D. Ringler, W. C. Skamarock, et al., 2009: Numerical representation of geostrophic modes on arbitrarily structured C-grids. J. Comput. Phys., 228, 8321–8335.. [29] Tomita, H., M. Tsugawa, M. Satoh, et al., 2001: Shallow water model on a modified icosahedral geodesic grid by using spring dynamics. J. Comput. Phys., 174, 579–613.. [30] Tomita, H., M. Satoh, and K. Goto, 2002: An optimization of the icosahedral grid modified by spring dynamics. J. Comput. Phys., 183, 307–331.. [31] Ullrich, P. A., C. Jablonowski, J. Kent, et al., 2017: DCMIP2016: a review of non-hydrostatic dynamical core design and intercomparison of participating models. Geosci. Model Dev., 10, 4477–4509.. [32] Wan, H., M. A. Giorgetta, G. Zängl, et al., 2013: The ICON-1.2 hydrostatic atmospheric dynamical core on triangular grids—Part 1: Formulation and performance of the baseline version. Geosci. Model Dev., 6, 735–763.. [33] Wicker, L. J., and W. C. Skamarock, 2002: Time-splitting methods for elastic models using forward time schemes. Mon. Wea. Rev., 130, 2088–2097.. [34] Williamson, D. L., and P. J. Rasch, 1994: Water vapor transport in the NCAR CCM2. Tellus A Dyn. Meteor. Oceanogr., 46, 34–51.. [35] Xie, Y. F., 2019: Generalized Z-grid model for numerical weather prediction. Atmosphere, 10, 179.. [36] Yu, Y. G., N. Wang, J. Middlecoff, et al., 2020: Comparing numerical accuracy of icosahedral A-grid and C-grid schemes in solving the shallow-water model. Mon. Wea. Rev., 148, 4009–4033..
###### 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

## Challenges in Developing Finite-Volume Global Weather and Climate Models with Focus on Numerical Accuracy

###### Corresponding author: Yuanfu XIE, xieyf@cma.gov.cn;
• 1. State Key Laboratory of Severe Weather, Chinese Academy of Meteorological Sciences, China Meteorological Administration, Beijing 100081
• 2. Guangdong–Hong Kong–Macao Greater Bay Area Weather Research Center for Monitoring Warning and Forecasting (Shenzhen Institute of Meteorological Innovation), Shenzhen 518048
Funds: Supported by the National Key Research and Development Program of China (2017YFC1502201) and Basic Scientific Research and Operation Fund of Chinese Academy of Meteorological Sciences (2017Z017)

Abstract: High-resolution global non-hydrostatic gridded dynamic models have drawn significant attention in recent years in conjunction with the rising demand for improving weather forecasting and climate predictions. By far it is still challenging to build a high-resolution gridded global model, which is required to meet numerical accuracy, dispersion relation, conservation, and computation requirements. Among these requirements, this review focuses on one significant topic—the numerical accuracy over the entire non-uniform spherical grids. The paper discusses all the topic-related challenges by comparing the schemes adopted in well-known finite-volume-based operational or research dynamical cores. It provides an overview of how these challenges are met in a summary table. The analysis and validation in this review are based on the shallow-water equation system. The conclusions can be applied to more complicated models. These challenges should be critical research topics in the future development of finite-volume global models.

Reference (36)

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