Preprints
https://doi.org/10.5194/gmd-2021-168
https://doi.org/10.5194/gmd-2021-168

Submitted as: development and technical paper 08 Jul 2021

Submitted as: development and technical paper | 08 Jul 2021

Review status: this preprint is currently under review for the journal GMD.

Improved double Fourier series on a sphere and its application to a semi-implicit semi-Lagrangian shallow water model

Hiromasa Yoshimura Hiromasa Yoshimura
  • Meteorological Research Institute, Japan Meteorological Agency, 1-1 Nagamine, Tsukuba, Ibaraki 305-0052, Japan

Abstract. The computational cost of a spectral model using spherical harmonics (SH) increases significantly at high resolution because the transform method with SH requires O(N3) operations, where N is the truncation wavenumber. One way to solve this problem is to use double Fourier series (DFS) instead of SH, which requires O(N2 log N) operations. This paper proposes a new DFS method that improves the numerical stability of the model compared with the conventional DFS methods by adopting the following two improvements: a new expansion method that employs the least-squares method (or the Galerkin method) to calculate the expansion coefficients in order to minimize the error caused by wavenumber truncation, and new basis functions that satisfy the continuity of both scalar and vector variables at the poles. In the semi-implicit semi-Lagrangian shallow water model using the new DFS method, the Williamson test cases 2 and 5 and the Galewsky test case give stable results without the appearance of high-wavenumber noise near the poles, even without using horizontal diffusion. The new DFS model is faster than the SH model, especially at high resolutions, and gives almost the same results.

Hiromasa Yoshimura

Status: final response (author comments only)

Comment types: AC – author | RC – referee | CC – community | EC – editor | CEC – chief editor | : Report abuse
  • RC1: 'Comment on gmd-2021-168', Anonymous Referee #1, 30 Jul 2021
  • RC2: 'Comment on gmd-2021-168', Anonymous Referee #2, 02 Aug 2021
  • RC3: 'Comment on gmd-2021-168', Anonymous Referee #3, 10 Sep 2021
    • AC3: 'Reply on RC3', Hiromasa Yoshimura, 18 Sep 2021

Hiromasa Yoshimura

Data sets

Yoshimura_DFS_SW_Testcase_2021_pre Hiromasa Yoshimura https://climate.mri-jma.go.jp/pre/Yoshimura_DFS_SW_Testcase_2021/

Hiromasa Yoshimura

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Short summary
This paper proposes a new double Fourier series (DFS) method on a sphere that improves the numerical stability of a model compared with conventional DFS methods. The shallow water model using the new DFS method gives stable results without the appearance of high-wavenumber noise near the poles. The new DFS model is faster than the model using spherical harmonics, especially at high resolutions, and gives almost the same results.