202
4. Series-Expansion Methods
Let M be the highest zonal wave number (in radi ans) resolved on the numerical
mesh; then M /),.). = n , and the preceding stability condition may be expressed as
cM/),.t
- - - S 1.
att cosO
A comparison of this condition with (4.54) shows that the maximum stable time
step that can be used with the finite-difference method on the portion of the mesh
where 0
±rr/2 is far smaller than that which can be used in a spectral model
employing spherical harmonie expansion functions with the same cutoff wave
number.
The restrlctions on the maximum stable time step can be removed altogether
by using trapezoidal time-differencing instead of the forward-backward scheme
in (4.51) and (4.52). This is not a partieularly efficient approach when the Laplacian is approximated using finite differences, since the trapezoidal approximation generates a large system of implicit algebraic equations that must be solved
at every time step. Trapezoidal time-differencing can, however, be implemented
very efficiently in spectral approximations that use spherical harmonie expansion
functions, because the spherical harmonics are eigenfunctions of the horizontal
Laplacian operator on the sphere. As a consequence, the expansion coefficient
for each Yr,s can be computed independently of the other müdes, and the implicit
coupling introduced by trapezoidal time-differencing only generates a trivial twovariable system involving the amplitudes of the divergence and the free-surface
elevation of each mode . The ease with which trapezoidal approximations to (4.51)
and (4.52) can be integrated using spherical harmonics can be used to great advan -
tage in formulating semi-implicit time-differencing approximations to the nonlinear equations goveming fluid flow on a sphere (see Sections 7.2.3 and 7.6 .5) .
4.4.3 Gaussian Quadrature and the Transform Method
In most practical applications some of the forcing terms in the goveming equations contain products of two or more spatially varying functions. Unless the total
number of modes retained in the series expansion is very smalI, a variant of the
transform method described in Section 4.2.2 must be used in order to efficiently
apply spectral methods to such problems. The transform between grid-point values and the spectral coefficients of the spherical harmonie functions is, however,
more cumbersome and computationally less efficient than the fast Fourier transform. The lack of highly efficient transforms is one of the few drawbacks associated with the use of spherical harmonic expansion functions in global spectral
models. Even so, it is far more efficient to use the transform method than the alternative "interaction coefficient" method, in which the forcing is computed from a
summation of products of pairs of the spectral coefficients (Orszag 1970; Eliasen
et al. 1970) .
If y,()., J.t) is approximated by a truncated series of spherical harmonics of the
form (4.50), the transformation from the set of spectral coefficients to points on a
4. Series-Expansion Methods
Let M be the highest zonal wave number (in radi ans) resolved on the numerical
mesh; then M /),.). = n , and the preceding stability condition may be expressed as
cM/),.t
- - - S 1.
att cosO
A comparison of this condition with (4.54) shows that the maximum stable time
step that can be used with the finite-difference method on the portion of the mesh
where 0
±rr/2 is far smaller than that which can be used in a spectral model
employing spherical harmonie expansion functions with the same cutoff wave
number.
The restrlctions on the maximum stable time step can be removed altogether
by using trapezoidal time-differencing instead of the forward-backward scheme
in (4.51) and (4.52). This is not a partieularly efficient approach when the Laplacian is approximated using finite differences, since the trapezoidal approximation generates a large system of implicit algebraic equations that must be solved
at every time step. Trapezoidal time-differencing can, however, be implemented
very efficiently in spectral approximations that use spherical harmonie expansion
functions, because the spherical harmonics are eigenfunctions of the horizontal
Laplacian operator on the sphere. As a consequence, the expansion coefficient
for each Yr,s can be computed independently of the other müdes, and the implicit
coupling introduced by trapezoidal time-differencing only generates a trivial twovariable system involving the amplitudes of the divergence and the free-surface
elevation of each mode . The ease with which trapezoidal approximations to (4.51)
and (4.52) can be integrated using spherical harmonics can be used to great advan -
tage in formulating semi-implicit time-differencing approximations to the nonlinear equations goveming fluid flow on a sphere (see Sections 7.2.3 and 7.6 .5) .
4.4.3 Gaussian Quadrature and the Transform Method
In most practical applications some of the forcing terms in the goveming equations contain products of two or more spatially varying functions. Unless the total
number of modes retained in the series expansion is very smalI, a variant of the
transform method described in Section 4.2.2 must be used in order to efficiently
apply spectral methods to such problems. The transform between grid-point values and the spectral coefficients of the spherical harmonie functions is, however,
more cumbersome and computationally less efficient than the fast Fourier transform. The lack of highly efficient transforms is one of the few drawbacks associated with the use of spherical harmonic expansion functions in global spectral
models. Even so, it is far more efficient to use the transform method than the alternative "interaction coefficient" method, in which the forcing is computed from a
summation of products of pairs of the spectral coefficients (Orszag 1970; Eliasen
et al. 1970) .
If y,()., J.t) is approximated by a truncated series of spherical harmonics of the
form (4.50), the transformation from the set of spectral coefficients to points on a