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Double Fourier sphere method

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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

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

f ( λ , θ ) = f ( cos ⁡ λ sin ⁡ θ , sin ⁡ λ sin ⁡ θ , cos ⁡ θ ) , ( λ , θ ) ∈ [ − π , π ] × [ 0 , π ] . {\displaystyle f(\lambda ,\theta )=f(\cos \lambda \sin \theta ,\sin \lambda \sin \theta ,\cos \theta ),(\lambda ,\theta )\in \times .}

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

f ~ ( λ , θ ) = { g ( λ + π , θ ) , ( λ , θ ) ∈ [ − π , 0 ] × [ 0 , π ] , h ( λ , θ ) , ( λ , θ ) ∈ [ 0 , π ] × [ 0 , π ] , g ( λ , − θ ) , ( λ , θ ) ∈ [ 0 , π ] × [ − π , 0 ] , h ( λ + π , − θ ) , ( λ , θ ) ∈ [ − π , 0 ] × [ − π , 0 ] , {\displaystyle {\tilde {f}}(\lambda ,\theta )={\begin{cases}g(\lambda +\pi ,\theta ),&(\lambda ,\theta )\in \times ,\\h(\lambda ,\theta ),&(\lambda ,\theta )\in \times ,\\g(\lambda ,-\theta ),&(\lambda ,\theta )\in \times ,\\h(\lambda +\pi ,-\theta ),&(\lambda ,\theta )\in \times ,\\\end{cases}}}

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

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

f ~ ≈ ∑ j = − n n ∑ k = − n n a j k e i j θ e i k λ {\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 and developed further by Steven Orszag. The DFS method has been the subject of relatively few investigations since (a notable exception is Fornberg's work), 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 and to novel space-time spectral analysis.

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)


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