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Double Fourier sphere method

## Summary

In mathematics, the double Fourier sphere (DFS) method is a simple technique that transforms a function defined on the surface of the sphere to a function defined on a rectangular domain while preserving periodicity in both the longitude and latitude directions.

## Introduction

First, a function ${\displaystyle f(x,y,z)}$  on the sphere is written as ${\displaystyle f(\lambda ,\theta )}$  using spherical coordinates, i.e.,

${\displaystyle f(\lambda ,\theta )=f(\cos \lambda \sin \theta ,\sin \lambda \sin \theta ,\cos \theta ),(\lambda ,\theta )\in [-\pi ,\pi ]\times [0,\pi ].}$

The function ${\displaystyle f(\lambda ,\theta )}$  is ${\displaystyle 2\pi }$ -periodic in ${\displaystyle \lambda }$ , but not periodic in ${\displaystyle \theta }$ . The periodicity in the latitude direction has been lost. To recover it, the function is "doubled up” and a related function on ${\displaystyle [-\pi ,\pi ]\times [-\pi ,\pi ]}$  is defined as

${\displaystyle {\tilde {f}}(\lambda ,\theta )={\begin{cases}g(\lambda +\pi ,\theta ),&(\lambda ,\theta )\in [-\pi ,0]\times [0,\pi ],\\h(\lambda ,\theta ),&(\lambda ,\theta )\in [0,\pi ]\times [0,\pi ],\\g(\lambda ,-\theta ),&(\lambda ,\theta )\in [0,\pi ]\times [-\pi ,0],\\h(\lambda +\pi ,-\theta ),&(\lambda ,\theta )\in [-\pi ,0]\times [-\pi ,0],\\\end{cases}}}$

where ${\displaystyle g(\lambda ,\theta )=f(\lambda -\pi ,\theta )}$  and ${\displaystyle h(\lambda ,\theta )=f(\lambda ,\theta )}$  for ${\displaystyle (\lambda ,\theta )\in [0,\pi ]\times [0,\pi ]}$ . The new function ${\displaystyle {\tilde {f}}}$  is ${\displaystyle 2\pi }$ -periodic in ${\displaystyle \lambda }$  and ${\displaystyle \theta }$ , and is constant along the lines ${\displaystyle \theta =0}$  and ${\displaystyle \theta =\pm \pi }$ , corresponding to the poles.

The function ${\displaystyle {\tilde {f}}}$  can be expanded into a double Fourier series

${\displaystyle {\tilde {f}}\approx \sum _{j=-n}^{n}\sum _{k=-n}^{n}a_{jk}e^{ij\theta }e^{ik\lambda }}$

## History

The DFS method was proposed by Merilees[1] and developed further by Steven Orszag.[2] The DFS method has been the subject of relatively few investigations since (a notable exception is Fornberg's work),[3] perhaps due to the dominance of spherical harmonics expansions. Over the last fifteen years it has begun to be used for the computation of gravitational fields near black holes[4] and to novel space-time spectral analysis.[5]

## References

1. ^ P. E. Merilees, The pseudospectral approximation applied to the shallow water equations on a sphere, Atmosphere, 11 (1973), pp. 13–20
2. ^ S. A. Orszag, Fourier series on spheres, Mon. Wea. Rev., 102 (1974), pp. 56–75.
3. ^ B. Fornberg, A pseudospectral approach for polar and spherical geometries, SIAM J. Sci. Comp, 16 (1995), pp. 1071–1081
4. ^ R. Bartnik and A. Norton, Numerical methods for the Einstein equations in null quasispherical coordinates, SIAM J. Sci. Comp, 22 (2000), pp. 917–950
5. ^ C. Sun, J. Li, F.-F. Jin, and F. Xie, Contrasting meridional structures of stratospheric and tropospheric planetary wave variability in the northern hemisphere, Tellus A, 66 (2014)