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.

Highlights

  • Global spectral atmospheric models using the spectral transform method with spherical harmonics (SH) as basis functions are widely used

  • This paper proposes a new double Fourier series (DFS) method that improves the numerical stability of the model compared with the conventional DFS methods by 10 adopting the following two improvements: a new expansion method that employs the least-squares 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

  • We have confirmed that numerical instability occurs in some test cases in the old DFS model without the zonal Fourier filter, but stable integration is possible in all test cases shown here in the new DFS model, 20 even without the zonal Fourier filter

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Summary

Introduction

Global spectral atmospheric models using the spectral transform method with spherical harmonics (SH) as basis functions are widely used. 20 Yoshimura and Matsumura (2005) and Yoshimura (2012) stably ran the two-time-level semi-implicit semi-Lagrangian hydrostatic and nonhydrostatic atmospheric models using the DFS basis functions of Cheong in Eq 6 These models used meridional truncation with N ≅ J, and U u sin θ and V v sin θ (instead of u⁄sin θ and v⁄sin θ) were transformed from grid space to spectral space, where u is the zonal wind and v is the meridional wind. A new expansion method to calculate DFS expansion coefficients of scalar and vector variables, which adopts the leastsquares method (or the Galerkin method) to minimize the error due to the meridional wavenumber truncation.

Gradient of a scalar variable
New method to calculate expansion coefficients for a scalar variable
Relation between the least-squares method and Galerkin method for a scalar variable
Comparison of new DFS with SH
Application of the new basis functions to a wind vector
New method to calculate expansion coefficients for a wind vector
Relation between the least-squares method and the Galerkin method for the wind vector
Arrangement of equally spaced latitudinal grid points
2.10 Discrete Fourier cosine and sine transforms in latitude
2.11 Zonal Fourier filter
2.13 The Helmholtz equation
2.14 Horizontal diffusion
Equations
Time integration method
R and R are calculated by
Models
Williamson test case 2
Galewsky test case
Conclusions and perspectives
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