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.
Similar content being viewed by others
References
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, doi: https://doi.org/10.1175/1520-0493(1999)127<1928:ANTOTC>2.0.CO;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, doi: https://doi.org/10.1016/j.jcp.2013.11.033.
Du, Q., M. D. Gunzburger, and L. L. Ju, 2003: Constrained centroidal Voronoi tessellations for surfaces. SIAM J. Sci. Comput., 24, 1488–1506, doi: https://doi.org/10.1137/S1064827501391576.
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, doi: https://doi.org/10.5194/gmd-10-791-2017.
Gill, A. E., 1982: Atmosphere-Ocean Dynamics. Academic Press, San Diego, 662 pp.
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, doi: https://doi.org/10.1006/jcph.2002.7139.
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, doi: https://doi.org/10.1175/1520-0493(1995)123<1862:NIOTSW>2.0.CO;2.
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, doi: https://doi.org/10.1175/1520-0493(1995)123<1881:NIOTSW>2.0.CO;2.
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, doi: https://doi.org/10.1175/MWR-D-12-00236.1.
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, doi: https://doi.org/10.1175/1520-0434(1991)006<0425:RCIITG>2.0.CO;2.
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, doi: https://doi.org/10.5194/gmd-11-1753-2018.
Lax, P. D., and R. D. Richtmyer, 1956: Survey of the stability of linear finite difference equations. Commun. Pure Appl. Math., 9, 267–293, doi: https://doi.org/10.1002/cpa.3160090206.
Lin, S.-J., 2004: A “vertically Lagrangian” finite-volume dynamical core for global models. Mon. Wea. Rev., 132, 2293–2307, doi: https://doi.org/10.1175/51520-0493(2004)132<2293:AVLFDC>2.0.CO;2.
Lin, S.-J., and R. B. Rood, 1996: Multidimensional flux-form semi-Lagrangian transport schemes. Mon. Wea. Rev., 124, 2046–2070, doi: https://doi.org/10.1175/1520-4933(1966)124<2046:MFF-SLT>2.0.CO;2.
Putman, W. M., and S.-J. Lin, 2007: Finite-volume transport on various cubed-sphere grids. J. Comput. Phys., 227, 55–78, doi: https://doi.org/10.1016/j.jcp.2007.07.022.
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, doi: https://doi.org/10.1016/j.jcp.2011.11.025.
Randall, D. A., 1994: Geostrophic adjustment and the finite-difference shallow-water equations. Mon. Wea. Rev., 122, 1371–1377, doi: https://doi.org/10.1175/1520-0493(1994)122<1371:GAATFD>2.0.CO;2.
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, doi: https://doi.org/10.1175/2008MWR2630.1.
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, doi: https://doi.org/10.1016/j.jcp.2009.12.007.
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, doi: https://doi.org/10.1175/1520-0493(1968)096<0351:IOTNBV>2.0.CO;2.
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, doi: https://doi.org/10.2151/jmsj.85B.271.
Salmon, R., 1998: Lectures on Geophysical Fluid Dynamics. Oxford University Press, New York, 378 pp.
Salmon, R., 2007: A general method for conserving energy and potential enstrophy in shallow-water models. J. Atmos. Sci., 64, 515–531, doi: https://doi.org/10.1175/JAS3837.1.
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, doi: https://doi.org/10.1016/j.jcp.2007.02.006.
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, doi: https://doi.org/10.1007/BF01027467.
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, doi: https://doi.org/10.1175/MWR-D-11-00215.1.
Süli, E., and D. F. Mayers, 2003: An Introduction to Numerical Analysis. Cambridge University Press, Cambridge, 444 pp.
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, doi: https://doi.org/10.1016/j.jcp.2009.08.006.
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, doi: https://doi.org/10.1006/jcph.2001.6897.
Tomita, H., M. Satoh, and K. Goto, 2002: An optimization of the icosahedral grid modified by spring dynamics. J. Comput. Phys., 183, 307–331, doi: https://doi.org/10.1006/jcph.2002.7193.
Ullrich, P. A., C. Jablonowski, J. Kent, et al., 2017: DCMIP2016: a review of non-hydrostatic dynamical core design and inter-comparison of participating models. Geosci. Model Dev., 10, 4477–4509, doi: https://doi.org/10.5194/gmd-10-4477-2017.
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, doi: https://doi.org/10.5194/gmd-6-735-2013.
Wicker, L. J., and W. C. Skamarock, 2002: Time-splitting methods for elastic models using forward time schemes. Mon. Wea. Rev., 130, 2088–2097, doi: https://doi.org/10.1175/1520-0493(2002)130<2088:TSMFEM>2.0.CO;2.
Williamson, D. L., and P. J. Rasch, 1994: Water vapor transport in the NCAR CCM2. Tellus A Dyn. Meteor. Oceanogr., 46, 34–51, doi: https://doi.org/10.3402/tellusa.v46i1.15426.
Xie, Y. F., 2019: Generalized Z-grid model for numerical weather prediction. Atmosphere, 10, 179, doi: https://doi.org/10.3390/atmos10040179.
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, doi: https://doi.org/10.1175/MWR-D-20-0024.1.
Author information
Authors and Affiliations
Corresponding author
Additional information
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).
Rights and permissions
About this article
Cite this article
Xie, Y., Qin, Z. Challenges in Developing Finite-Volume Global Weather and Climate Models with Focus on Numerical Accuracy. J Meteorol Res 35, 775–788 (2021). https://doi.org/10.1007/s13351-021-0202-3
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s13351-021-0202-3