Journal of the Korean earth science society (한국지구과학회지)

The Korean Earth Science Society (한국지구과학회)
권호 표지
DOI QR코드

DOI QR Code

Effect of Nonuniform Vertical Grid on the Accuracy of Two-Dimensional Transport Model

  • Lee, Chung-Hui (Department of Environmental Atmospheric Sciences, Pukyong National University) ;
  • Cheong, Hyeong-Bin (Department of Environmental Atmospheric Sciences, Pukyong National University) ;
  • Kim, Hyun-Ju (Department of Environmental Atmospheric Sciences, Pukyong National University) ;
  • Kang, Hyun-Gyu (Department of Environmental Atmospheric Sciences, Pukyong National University)
  • Received : 2018.07.25
  • Accepted : 2018.08.23
  • Published : 2018.08.31
Download PDF
⟨ Previous Next ⟩

Abstract

Effect of the nonuniform grid on the two-dimensional transport equation was investigated in terms of theoretical analysis and finite difference method (FDM). The nonuniform grid having a typical structure of the numerical weather forecast model was incorporated in the vertical direction, while the uniform grid was used in the zonal direction. The staggered and non-staggered grid were placed in the vertical and zonal direction, respectively. Time stepping was performed with the third-order Runge Kutta scheme. An error analysis of the spatial discretization on the nonuniform grid was carried out, which indicated that the combined effect of the nonuniform grid and advection velocity produced either numerical diffusion or numerical adverse-diffusion. An analytic function is used for the quantitative evaluation of the errors associated with the discretized transport equation. Numerical experiments with the non-uniformity of vertical grid were found to support the analysis.

Keywords

References

  1. Arakawa, A., 1972, Design of the UCLA general circulation model. UCLA Meteorology Department Tech. Rep. 7, 116 pp.
  2. Arakawa, A., and Lamb, V.R., 1977: Computational design of the basic dynamical processes of the UCLA general circulation model. Methods in Computational Physics, Vol. 17, J. Chang, Ed., Academic Press, 173-265.
  3. Browning, G.L., Hack, J.J., and Swarztrauber, P.N., 1989, A comparison of three numerical methods for solving differential equations on the sphere. Monthly Weather Review, 117, 1058-1075. https://doi.org/10.1175/1520-0493(1989)117<1058:ACOTNM>2.0.CO;2
  4. Cheong, H.B., 2006: A dynamical core with double Fourier series: Comparison with the spherical harmonics method. Monthly Weather Review, 134, 1299-1315. https://doi.org/10.1175/MWR3121.1
  5. Cheong, H.B., Kong, H.J., Kang, H.G., and Lee, J.D., 2015, Fourier finite-element method with linear basis functions on a sphere: Application to elliptic and transport equations. Monthly Weather Review, 143, 1275-1294. https://doi.org/10.1175/MWR-D-14-00093.1
  6. Cheong, H.B. and Park, J.R., 2007, Geopotential field in nonlinear balance with the sectoral mode of Rossby-Haurwitz wave on the inclined rotation axis. Journal of the Korean Earth Science Society, 28, 936-946. https://doi.org/10.5467/JKESS.2007.28.7.936
  7. Choi, S.J, and Hong, S.Y., 2016, A global non-hydrostatic dynamical core using the spectral element method on a cubed-sphere grid. Asia Pacific Journal of Atmospheric Sciences, 52, 291-307. https://doi.org/10.1007/s13143-016-0005-0
  8. Durran, D.R., 1999, Numerical methods for wave equations in geophysical fluid dynamics. Springer, 465pp.
  9. Flyer, N., and Wright, G.B., 2007, Transport schemes on a sphere using radial basis functions. J. Computational Physics, 226, 1059-1084. https://doi.org/10.1016/j.jcp.2007.05.009
  10. Gal-Chen, T., and Somerville, R., 1975: On the use of a coordinate transformation for the solution of the Navier-Stokes equations. Journal of Computational Physics, 17, 209-228. https://doi.org/10.1016/0021-9991(75)90037-6
  11. Gates, W.L., 1992, AMIP: The Atmospheric Model Intercomparison Project. Bulletin of American Meteorological Society, 73, 1962-1970. https://doi.org/10.1175/1520-0477(1992)073<1962:ATAMIP>2.0.CO;2
  12. Haiden, T., Janousek, M., Bidlot, J.R., Ferranti, L., Prates, F., Vitart, F., Bauer, P., and Richardson, D., 2017, Evaluation of ECMWF forecasts, including 2016-2017 upgrades. Technical Memorandum 817, ECMWF, 58 pp.
  13. Haltiner, G.J., and Williams, R.T., 1980, Numerical weather prediction and dynamic meteorology. John Wiley and Sons, 477 pp.
  14. Hoskins, B.J., and Simmons, A.J., 1975, A multilayer spectral model and the semiimplicit method. Quarterly Journal of Royal Meteorological Society, 101, 637-655. https://doi.org/10.1002/qj.49710142918
  15. Kang, H.G., and Cheong, H.B., 2017, An efficient implementation of a high-order filter for a cubed-sphere spectral element model. J. Computational Physics, 332, 66-82. https://doi.org/10.1016/j.jcp.2016.12.001
  16. Kang, H.G., and Cheong, H.B., 2018, Effect of high-order filter on a cubed-sphere spectral element dynamical core. Monthly Weather Review, 146, 2047-2064. https://doi.org/10.1175/MWR-D-17-0226.1
  17. Klemp, J.B., 2011, A Terrain-Following Coordinate with Smoothed Coordinate Surfaces. Monthly Weather Review, 139, 2163-2169. https://doi.org/10.1175/MWR-D-10-05046.1
  18. Leuenberger, D., Koller, M., Fuhrer, O., and Schr, C., 2010, A generalization of the SLEVE vertical coordinate. Monthly Weather Review, 138, 3683-3689. https://doi.org/10.1175/2010MWR3307.1
  19. Marras, S.M., Kopera, A., and Giraldo, F.X., 2015, Simulation of shallow-water jets with a unified elementbased continuous/discontinuous Galerkin model with grid flexibility on the sphere. Quarterly Journal of Royal Meteorological Society, 141, 1727-1739. https://doi.org/10.1002/qj.2474
  20. Nair, R.D., Thomas, S.J., and Loft, R.D., 2005, A discontinuous Galerkin transport scheme on the cubedsphere. Monthly Weather Review, 133, 814-828. https://doi.org/10.1175/MWR2890.1
  21. Rivier, L., Loft, R., and Polvani, L.M., 2002, An efficient spectral dynamical core for distributed memory computers. Monthly Weather Review, 130, 1384-1396. https://doi.org/10.1175/1520-0493(2002)130<1384:AESDCF>2.0.CO;2
  22. Saito, K., Fujita, T., Yamada, Y., Ishida, J.I., Kumagai, Y., Aranami, K., Ohmori, S., Nagasawa, R.,. Kumagai, S., Muroi, C., Kato, T., Eito, H., and Yamazaki, Y., 2006, The operational JMA nonhydrostatic mesoscale model. Monthly Weather Review, 134, 1266-1298. https://doi.org/10.1175/MWR3120.1
  23. Simmons, A.J., and Burridge, D.M., 1981, An energy and angular-momentum conserving vertical finite-difference scheme and hybrid vertical coordinates. Monthly Weather Review, 109, 758-766. https://doi.org/10.1175/1520-0493(1981)109<0758:AEAAMC>2.0.CO;2
  24. Skamarock, W.C., and Klemp, J.B., 2008, A time-split nonhydrostatic atmospheric model for weather research and forecasting applications. J. Computational Physics, 227, 3465-3485. https://doi.org/10.1016/j.jcp.2007.01.037
  25. Smagorinsky, J., 1963, General circulation experiments with the primitive equations: I. The basic experiment. Monthly Weather Review, 91, 99-164. https://doi.org/10.1175/1520-0493(1963)091<0099:GCEWTP>2.3.CO;2
  26. Tomita, H., and Satoh, M., 2004, A new dynamical framework of nonhydrostatic global model using the icosahedral grid. Fluid Dynamics Research, 34, 357-400. https://doi.org/10.1016/j.fluiddyn.2004.03.003
  27. Williamson, D.L., Drake, J.B., Hack, J.J., Jakob, R., and Swarztrauber, P.N., 1992, A standard test set for numerical approximations to the shallow water equations in spherical geometry. J. Computational Physics, 102, 211-224. https://doi.org/10.1016/S0021-9991(05)80016-6
  28. Zhang, H., and Rancic, M., 2007, A global eta model on quasi-uniform grids. Quarterly Journal of Royal Meteorological Society, 133, 517-528. https://doi.org/10.1002/qj.17