Editorial Type:
Article
Article Type:
Research Article

Scale-Dependent Accuracy in Regional Spectral Methods

Hung-Chi Kuo Department of Atmospheric Science, National Taiwan University, Taipei, Taiwan, Republic of China

Search for other papers by Hung-Chi Kuo in
Current site
Google Scholar
PubMed Close
and
R. T. Williams Department of Meteorology, Naval Postgraduate School, Monterey, California

Search for other papers by R. T. Williams in
Current site
Google Scholar
PubMed Close
Print Publication:
01 Oct 1998
DOI:
https://doi.org/10.1175/1520-0493(1998)126<2640:SDAIRS>2.0.CO;2
Page(s):
2640–2647
Restricted access

Abstract

The accuracy of a numerical model is often scale dependent. Large spatial-scale phenomena are expected to be numerically solved with better accuracy, regardless of whether the discretization is spectral, finite difference, or finite element. The purpose of this article is to discuss the scale-dependent accuracy associated with the regional spectral model variables expanded by sine–cosine series. In particular, the scale-dependent accuracy in the Chebyshev-tau, finite difference, and sinusoidal- or polynomial-subtracted sine–cosine expansion methods is considered. With the simplest examples, it is demonstrated that regional spectral models may possess an unusual scale-dependent accuracy. Namely, the numerical accuracy associated with large-spatial-scale phenomena may be worse than the numerical accuracy associated with small-spatial-scale phenomena. This unusual scale-dependent accuracy stems from the higher derivatives of basic-state subtraction functions, which are not periodic. The discontinuity is felt mostly by phenomena with large spatial scale. The derivative discontinuity not only causes the slow convergence of the expanded Fourier series (Gibbs phenomenon) but also results in the unusual scale-dependent numerical accuracy. The unusual scale-dependent accuracy allows large-spatial-scale phenomena in the model perturbation fields to be solved less accurately.

Corresponding author address: Dr. R. T. Williams, Department of Meteorology, Naval Postgraduate School, Monterey, CA 93943-5000.

Abstract

The accuracy of a numerical model is often scale dependent. Large spatial-scale phenomena are expected to be numerically solved with better accuracy, regardless of whether the discretization is spectral, finite difference, or finite element. The purpose of this article is to discuss the scale-dependent accuracy associated with the regional spectral model variables expanded by sine–cosine series. In particular, the scale-dependent accuracy in the Chebyshev-tau, finite difference, and sinusoidal- or polynomial-subtracted sine–cosine expansion methods is considered. With the simplest examples, it is demonstrated that regional spectral models may possess an unusual scale-dependent accuracy. Namely, the numerical accuracy associated with large-spatial-scale phenomena may be worse than the numerical accuracy associated with small-spatial-scale phenomena. This unusual scale-dependent accuracy stems from the higher derivatives of basic-state subtraction functions, which are not periodic. The discontinuity is felt mostly by phenomena with large spatial scale. The derivative discontinuity not only causes the slow convergence of the expanded Fourier series (Gibbs phenomenon) but also results in the unusual scale-dependent numerical accuracy. The unusual scale-dependent accuracy allows large-spatial-scale phenomena in the model perturbation fields to be solved less accurately.

Corresponding author address: Dr. R. T. Williams, Department of Meteorology, Naval Postgraduate School, Monterey, CA 93943-5000.

Save
Cover Monthly Weather Review
Monthly Weather Review

  • Chen, Q.-S., and Y.-H. Kuo, 1992: A harmonic-sine series expansion and its application to the partition and reconstruction problem in a limited area. Mon. Wea. Rev.,120, 91–112.

  • ——, L.-S. Bai, and D. H. Bromwich, 1997: A harmonic spectral limited-area model with an external wind lateral boundary condition. Mon. Wea. Rev.,125, 143–167.

  • Fulton, S. R., and W. H. Schubert, 1987a: Chebyshev spectral methods for limited-area models. Part I: Model problem analysis. Mon. Wea. Rev.,115, 1940–1953.

  • ——, and ——, 1987b: Chebyshev spectral methods for limited-area models. Part II: Shallow water model. Mon. Wea. Rev.,115, 1954–1965.

  • ——, P. E. Ciesielski, and W. H. Schubert, 1986: Multigrid methods for elliptic problems: A review. Mon. Wea. Rev.,114, 943–959.

  • Gottlieb, D., and S. A. Orszag, 1977: Numerical Analysis of Spectral Methods.National Science Foundation-Conference Board of Mathematical Sciences Monogr., No. 26, Society of Industrial and Applied Mathematics. [NTIS AD-A056922.].

  • Haugen, J. E., and B. Machenhauer, 1993: A spectral limited-area model formulation with time-dependent boundary conditions applied to the shallow-water equations. Mon. Wea. Rev.,121, 2618–2630.

  • Hoyer, J. M., 1987: The ECMWF spectral limited-area model. Proc. ECMWF Workshop on Techniques for Horizontal Discretization in Numerical Weather Prediction Models, Berkshire, Reading, United Kingdom, ECMWF, 343–359.

  • Juang, H.-M. H., and M. Kanamitsu, 1994: The NMC nested regional spectral model. Mon. Wea. Rev.,122, 3–26.

  • Kuo, H.-C., and W. H. Schubert, 1988: Stability of cloud-topped boundary layers. Quart. J. Roy. Meteor. Soc.,114, 887–916.

  • ——, and R. T. Williams, 1992: Boundary effects in regional spectral models. Mon. Wea. Rev.,120, 2986–2992.

  • Lanczos, C., 1956: Applied Analysis. Prentice-Hall, 539 pp.

  • Lander, J., and B. J. Hoskins, 1997: Believable scales and parameterizations in a spectral transform model. Mon. Wea. Rev.,125, 292–303.

  • Machenhauer, B., 1986: Limited-area modeling by the HIRLAM project group. HIRLAM Tech. Rep. 1, 13pp. [Available from Danish Meteorlogical Institute, Lyngbyne; 100, 2100 Copenhagen, Denmark.].

  • Orszag, S. A., 1970: Transform method for calculation of vector-coupled sums: Application to the spectral form of the vorticity equation. J. Atmos. Sci.,27, 890–895.

  • Segami, A., K. Kurihara, H. Nakamura, M. Ueno, I. Takano, and Y. Tatsumi, 1989: Operational mesoscale weather prediction with Japan spectral model. J. Meteor. Soc. Japan,67, 907–923.

  • Tatsumi, Y., 1986: A spectral limited-area model with time-dependent lateral boundary conditions and its application to a multi-level primitive equation model. J. Meteor. Soc. Japan,64, 637–664.

All Time Past Year Past 30 Days
Abstract Views 0 0 0
Full Text Views 158 69 3
PDF Downloads 41 24 2