DCL:MATH2:SHTLIB: Spherical Harmonic Functions

4.1 Summary

This is a package of subroutines that performs spectral (spherical harmonic function) transformations, and converts a spherical harmonic function into grid data by expansion, or the vice versa by inverse transformation. The package design is especially suited for data analysis, and a special feature of the package is that it can handle equal-interval grid data. Furthermore, to enhance its capability in spectral data analysis, it is equipped with a wide variety of inverse transformation routines. The FFTLIB subroutine is used within this package.

A spectral inversion with cut-off wavenumber of M (triangular truncation) can be expressed as follows:

(4.1)

Or, by using the inverse Legendre transformation:

(4.2)

(4.1) can be expressed as a product of an inverse Legendre transformation and an inverse Fourier transformation.

(4.3)

Here, &lambda and &phi are latitude and longitude, respectively.

Furthermore, P^m_n(&mu) is an associated Legendre function normalized to 2, and is defined as follows:

(4.4)

(4.5)

The inverse spectral transformation can also be expressed as follows:

(4.6)

As in the case of the inverse transformation, by using the forward Fourier transformation:

(4.7)

(4.6) can be expressed as a product of a forward Fourier transformation and a forward Legendre transformation:

(4.8)

If we assume that G(&lambda, &phi) is a floating-point number, then S^m_n and W^m_(&phi) must satisfy the relationship below.

(4.9)

S^-m_n = {S^m_n}^*

(4.10)

Here, {}^* represents a complex conjugate. Therefore, W^m_(sin_&phi) and ??? needs only to be determined for m≧0. Furthermore, from the above restrictions, W^m_(sin_&phi) and S⁰_n will be floating-point numbers.

This library consists of a group of routines that performs inverse transformation from spectral data (S^m_n) into wave data in an equal-interval meridional plane (W^m(&phij)) and into an equal-interval grid data (G(&phij)) based on Eq. (1)-(3); a group of routines that performs forward transformation from an equal-interval grid data (G(&phij)) into wave data in an equal-interval meridional plane (W^m(&phij)) and into spectral data (S^m_n) based on Eq. (6)-(8); and a group of other auxiliary routines.

Here, it is assumed that the the latitude (&lambdai) and longitude (&phij) of the grid points can be represented as follows using the partition numbers I and J:

(4.11)

(4.12)

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DCL:MATH2:SHTLIB: Spherical Harmonic Functions