Journal of the Korean earth science society (한국지구과학회지)

The Korean Earth Science Society (한국지구과학회)
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Spherical Harmonics Power-spectrum of Global Geopotential Field of Gaussian-bell Type

  • Cheong, Hyeong-Bin (Department of Environmental Atmospheric Sciences, Pukyong National University) ;
  • Kong, Hae-Jin (Department of Environmental Atmospheric Sciences, Pukyong National University)
  • Received : 2013.07.10
  • Accepted : 2013.08.23
  • Published : 2013.09.30
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Abstract

Spherical harmonics power spectrum of the geopotential field of Gaussian-bell type on the sphere was investigated using integral formula that is associated with Legendre polynomials. The geopotential field of Gaussian-bell type is defined as a function of sine of angular distance from the bell's center in order to guarantee the continuity on the global domain. Since the integral-formula associated with the Legendre polynomials was represented with infinite series of polynomial, an estimation method was developed to make the procedure computationally efficient while preserving the accuracy. The spherical harmonics power spectrum was shown to vary significantly depending on the scale parameter of the Gaussian bell. Due to the accurate procedure of the new method, the power (degree variance) spanning over orders that were far higher than machine roundoff was well explored. When the scale parameter (or width) of the Gaussian bell is large, the spectrum drops sharply with the total wavenumber. On the other hand, in case of small scale parameter the spectrum tends to be flat, showing very slow decaying with the total wavenumber. The accuracy of the new method was compared with theoretical values for various scale parameters. The new method was found advantageous over discrete numerical methods, such as Gaussian quadrature and Fourier method, in that it can produce the power spectrum with accuracy and computational efficiency for all range of total wavenumber. The results of present study help to determine the allowable maximum scale parameter of the geopotential field when a Gaussian-bell type is adopted as a localized function.

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References

  1. Cheong, H.B., Double Fourier Series on a Sphere, 2000, Applications to Elliptic and Vorticity Equations. Journal of Computational Physics, 157, 327-349. https://doi.org/10.1006/jcph.1999.6385
  2. Cheong, H.B. and Park J.R., 2007, Geopotential field in nonlinear balance with the sectoral mode of Rossby-Haurwitz wave on the inclined rotation axis. Journal of the Korean Earth Science Society, 28, 936-946. https://doi.org/10.5467/JKESS.2007.28.7.936
  3. Cheong, H.B. and Park J.R., 2008, On the interpolativeness of discrete Legendre functions. Proceedings of the 2008 International Conference on Scientific Computing, Las Vegas, CSREA press, 252-258.
  4. Cheong, H.B., Park J.R., and Kang H.G., 2012, Fourierseries representation and projection of spherical harmonic functions. Journal of Geodesy, 86, 975-990. https://doi.org/10.1007/s00190-012-0558-3
  5. Dilts G.A., 1985, Computation of Spherical harmonic Expansion Coefficients via FFT's. Journal of Computational Physics, 57, 439-453. https://doi.org/10.1016/0021-9991(85)90189-5
  6. Enomoto, T., Fuchigami H., and Shingu S., 2004, Accurate and robust Legendre transforms at large truncation wavenumbers with the Fourier method. Proceedings of the 2004 Workshop on the Solution of Partial Differential Equations on the Sphere, Yokohama, Japan, 17-19.
  7. Hofsommer, D.J. and Potters M.L., 1960, Table of Fourier Coefficients of Associated Legendre Functions. Proceedings of the KNAW, Series A-Mathematical Sciences, Vol 63, Amsterdam, 460-466.
  8. Hopkins, J., 1973, Computation of Normalized Associated Legendre Functions Using Recursive Relations. Journal of Geophysical Research, 78, 476-477. https://doi.org/10.1029/JB078i002p00476
  9. Jekeli, C., Lee J.K., and Kwon J.H., 2007, On the computation and approximation of ultra-high-degree spherical harmonic series. Journal of Geodesy, 81, 603-615. https://doi.org/10.1007/s00190-006-0123-z
  10. Moriguchi S.I., Udakawa K.H., and Shin H.M., 1990, Formulas for mathematical functions III. The 5thed., Iwanami Shoten, Tokyo, 310 p.
  11. Nehrkorn T., 1990, On the computation of Legendre functions in spectral models. Monthly Weather Review, 118, 2248-2251. https://doi.org/10.1175/1520-0493(1990)118<2248:OTCOLF>2.0.CO;2
  12. Ricardi L.J. and Burrows M.L., 1972, A recurrence technique for expanding a function in spherical harmonics. IEEE Transactions on Computers 21, 583-535.
  13. Risbo T., 1996, Fourier transform summation of Legendre series and D-functions. Journal of Geodesy, 70, 383-396. https://doi.org/10.1007/BF01090814
  14. Rod Blais, J.A., 2008, Discrete spherical harmonic transforms: numerical preconditioning and optimization. Proc. ICCS 2008 Part II (M Bubak et al. Eds.), Springer-Verlag Berlin Heidelberg, LNCS 5102, 683-645.
  15. Sardeshmukh P.D. and Hoskins B.J., 1984, Spatial Smoothing on the sphere. Monthly Weather Review, 121, 2524-2529.
  16. Sneeuw, N. and Bun R., 1996, Global spherical harmonic computation by two-dimensional Fourier method. Journal of Geodesy, 70, 224-232. https://doi.org/10.1007/BF00873703
  17. Swarztrauber, P.N., 1993, The vector harmonic transform method for solving partial differential equations in spherical geometry. Monthly Weather Review, 121, 3415-3437. https://doi.org/10.1175/1520-0493(1993)121<3415:TVHTMF>2.0.CO;2
  18. Wittwer, T., Klees R., and Seitz K., 2008, Ultra-high degree spherical harmonic analysis and synthesis using extended-range arithmetic. Journal of Geodesy, 82, 223-229. https://doi.org/10.1007/s00190-007-0172-y

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