Double Fourier sphere method
This article may be too technical for most readers to understand.(August 2021) |
In mathematics, the double Fourier sphere (DFS) method is a 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
[edit]First, a function on the sphere is written as using spherical coordinates, i.e.,
The function is -periodic in , but not periodic in . The periodicity in the latitude direction has been lost. To recover it, the function is "doubled up” and a related function on is defined as
where and for . The new function is -periodic in and , and is constant along the lines and , corresponding to the poles.
The function can be expanded into a double Fourier series
History
[edit]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
[edit]- ^ P. E. Merilees, The pseudospectral approximation applied to the shallow water equations on a sphere, Atmosphere, 11 (1973), pp. 13–20
- ^ S. A. Orszag, Fourier series on spheres, Mon. Wea. Rev., 102 (1974), pp. 56–75.
- ^ B. Fornberg, A pseudospectral approach for polar and spherical geometries, SIAM J. Sci. Comp, 16 (1995), pp. 1071–1081
- ^ R. Bartnik and A. Norton, Numerical methods for the Einstein equations in null quasispherical coordinates, SIAM J. Sci. Comp, 22 (2000), pp. 917–950
- ^ 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)