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4. Series-Expansion Methods
Suppose that one wishes to compute the spectral coefficients ofthe binary product 1/1X, where 1/1 and X are given by the truncated spherical harmonic expansions
M
N(p)
1/I(J... , J-t) = L L ap,qYp,q(J..., J-t),
p= -Mq=lpl
M N(r)
X(J..., J-t) = L L br,sYr,s(J..., J-t) .
r=-M s=lrl
(4.63)
(4.64)
Let Cm ,n be the coefficient of Ym,n in the spherical harmonic expansion of 1/1X.
Without loss of generality consider the case m 2: 0, since the coefficients for
which m < 0 can be obtained using (4.45). Then
Cm,n = 1
(4.65)
1
Gm (J-t)Pm ,n(J-t) du ,
- I
1 1
7C
Gm(J-t) = -
2n -7C
1/I(J..., J-t)X(J..., J-t)e-rm . J. . . d):
(4.66)
where
The last of the preceding integrals is a Fourier transform, and as discussed in
Section 4.2.2, the discrete Fourier transform of binary products of Fourier series
truncated at wave number M can be evaluated without aliasing error if the transforms are computed using a minimum of (3M - 1)/2 wave numbers. In order to
maximize the efficiency of the fast Fourier transforms in practical applications,
the actual cutoff wave number may be chosen as the smallest product of prime
factors no larger than five that exceeds (3M - 1)/2. This criterion for the cutoff
wave number can be alternatively expressed as a requirement that the physical
mesh include a minimum of 3M + 1 grid points around each latitude circle.
Now consider the evaluation of (4.65). The associated Legendre functions have
the form
where Qm,n is a polynomial in u.of degree n - m . Since Pm,n is not a polynomial
when m is odd, it is not obvious that Gaussian quadrature can be used to integrate (4.65) without error. Neverthelcss, it turns out that the complete integrand
Gm(J-t)Pm,n(J-t) is a polynomial in J-t whose maximum degree can be determined
as follows . Substituting the finite series expansions for 1/1 and X into (4.66) and
using the orthogonality of the Fourier modes,
where the notation below the first summation indicates that the sum should bc
performed for all indices p and r such that Ipl M, Irl M, and p + r = m.
4. Series-Expansion Methods
Suppose that one wishes to compute the spectral coefficients ofthe binary product 1/1X, where 1/1 and X are given by the truncated spherical harmonic expansions
M
N(p)
1/I(J... , J-t) = L L ap,qYp,q(J..., J-t),
p= -Mq=lpl
M N(r)
X(J..., J-t) = L L br,sYr,s(J..., J-t) .
r=-M s=lrl
(4.63)
(4.64)
Let Cm ,n be the coefficient of Ym,n in the spherical harmonic expansion of 1/1X.
Without loss of generality consider the case m 2: 0, since the coefficients for
which m < 0 can be obtained using (4.45). Then
Cm,n = 1
(4.65)
1
Gm (J-t)Pm ,n(J-t) du ,
- I
1 1
7C
Gm(J-t) = -
2n -7C
1/I(J..., J-t)X(J..., J-t)e-rm . J. . . d):
(4.66)
where
The last of the preceding integrals is a Fourier transform, and as discussed in
Section 4.2.2, the discrete Fourier transform of binary products of Fourier series
truncated at wave number M can be evaluated without aliasing error if the transforms are computed using a minimum of (3M - 1)/2 wave numbers. In order to
maximize the efficiency of the fast Fourier transforms in practical applications,
the actual cutoff wave number may be chosen as the smallest product of prime
factors no larger than five that exceeds (3M - 1)/2. This criterion for the cutoff
wave number can be alternatively expressed as a requirement that the physical
mesh include a minimum of 3M + 1 grid points around each latitude circle.
Now consider the evaluation of (4.65). The associated Legendre functions have
the form
where Qm,n is a polynomial in u.of degree n - m . Since Pm,n is not a polynomial
when m is odd, it is not obvious that Gaussian quadrature can be used to integrate (4.65) without error. Neverthelcss, it turns out that the complete integrand
Gm(J-t)Pm,n(J-t) is a polynomial in J-t whose maximum degree can be determined
as follows . Substituting the finite series expansions for 1/1 and X into (4.66) and
using the orthogonality of the Fourier modes,
where the notation below the first summation indicates that the sum should bc
performed for all indices p and r such that Ipl M, Irl M, and p + r = m.